(ems series of lectures in mathematics) kathrin bringmann, yann bugeaud, titus hilberdink, jurgen...

7/25/2019 (Ems Series of Lectures in Mathematics) Kathrin Bringmann, Yann Bugeaud, Titus Hilberdink, Jurgen Sander-Four F…

http://slidepdf.com/reader/full/ems-series-of-lectures-in-mathematics-kathrin-bringmann-yann-bugeaud-titus 1/200

Series of Lectures in Mathematics

Kathrin Bringmann Yann BugeaudTitus Hilberdink

Jürgen Sander

Four Faces ofNumber Theory



EMS Series of Lectures in Mathematics

Edited by Andrew Ranicki (University of Edinburgh, U.K.)

EMS Series of Lectures in Mathematics is a book series aimed at students, professional

mathematicians and scientists. It publishes polished notes arising from seminars or lecture series

in all fields of pure and applied mathematics, including the reissue of classic texts of continuing

interest. The individual volumes are intended to give a rapid and accessible introduction into

their particular subject, guiding the audience to topics of current research and the more

advanced and specialized literature.

Previously published in this series:Katrin Wehrheim, Uhlenbeck Compactness

Torsten Ekedahl, One Semester of Elliptic Curves

Sergey V. Matveev, Lectures on Algebraic Topology

Joseph C. Várilly, An Introduction to Noncommutative Geometry

Reto Müller, Differential Harnack Inequalities and the Ricci Flow

Eustasio del Barrio, Paul Deheuvels and Sara van de Geer, Lectures on Empirical Processes

Iskander A. Taimanov, Lectures on Differential Geometry

Martin J. Mohlenkamp and María Cristina Pereyra, Wavelets, Their Friends, and What They

Can Do for You

Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles Masoud Khalkhali, Basic Noncommutative Geometry

Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie and Nils Henrik Risebro, Splitting Methods

for Partial Differential Equations with Rough Solutions

Koichiro Harada, “Moonshine” of Finite Groups

Yurii A. Neretin, Lectures on Gaussian Integral Operators and Classical Groups

Damien Calaque and Carlo A. Rossi, Lectures on Duflo Isomorphisms in Lie Algebra and

Complex Geometry

Claudio Carmeli, Lauren Caston and Rita Fioresi, Mathematical Foundations of Supersymmetry

Hans Triebel, Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration

Koen Thas, A Course on Elation Quadrangles Benoît Grébert and Thomas Kappeler, The Defocusing NLS Equation and Its Normal Form

Armen Sergeev, Lectures on Universal Teichmüller Space

Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, 3-Manifold Groups

Hans Triebel, Tempered Homogeneous Function Spaces



Four Faces ofNumber Theory

Kathrin Bringmann Yann BugeaudTitus HilberdinkJürgen Sander



Yann Bugeaud

Institut de Recherche Mathématique Avancée, UMR 7501

Université de Strasbourg et CNRS

7, rue René Descartes

67084 Strasbourg

France

E-mail: [email protected]

Jürgen Sander

Institut für Mathematik und Angewandte Informatik

Universität Hildesheim

Samelsonplatz 1

31141 Hildesheim

Germany


Authors:

Kathrin Bringmann

Mathematisches Institut

Universität zu Köln

Gyrhofstr. 8b

50931 Köln

Germany


Titus Hilberdink

Department of Mathematics and Statistics

University of Reading

Whiteknights P.O. Box 220

Reading RG6 6AX

UK


2010 Mathematics Subject Classification: Primary: 11-02, 11J81; Secondary 11A63, 11J04, 11J13, 11J68, 11J70,

11J87, 68R15

Key words: Transcendence, algebraic number, Schmidt Subspace Theorem, continued fraction, digital

expansion, Diophantine approximation

ISBN 978-3-03719-142-2

The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the

detailed bibliographic data are available on the Internet at http://www.helveticat.ch.

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is

concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting,

reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission

of the copyright owner must be obtained.

© European Mathematical Society 2015

Contact address:

European Mathematical Society Publishing House

Seminar for Applied Mathematics

ETH-Zentrum SEW A27

CH-8092 Zürich

Switzerland

Phone: +41 (0)44 632 34 36

Email: [email protected]

Homepage: www.ems-ph.org

Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany

Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany

∞ Printed on acid free paper

9 8 7 6 5 4 3 2 1



Foreword

Number theory is a fascinating branch of mathematics. There are many examples of number theoretical problems that are easy to understand but difficult to solve, often

enough only by use of advanced methods from other mathematical disciplines. Thismight be one of the reasons why number theory is considered to be such an attractive

field with many connections to other areas of mathematical research.

In August 2012 the number theory group of the Department of Mathematics at

Wurzburg University, under the aegis of Jorn Steuding, organized an international

summer school entitled Four Faces of Number Theory. In the frame of this event

about fifty participants, mostly PhD students from all over the world, but also a few lo-cal participants, even undergraduate students, learned in four courses about different

aspects of modern number theory. These courses highlight a strong interplay betweennumber theory and other fields like combinatorics, functional analysis and graph the-

ory. They will be of interest to (under)graduate students aiming to discover various

aspects of number theory and their relationship with other areas of mathematics.

Kathrin Bringmann from Cologne gave an introduction to the theory of modu-

lar forms and, in particular, so-called Mock theta-functions, a topic which had been

untouched for decades but has obtained much attention during the last five years.Yann Bugeaud from Strasbourg lectured about expansions of algebraic numbers.

Despite some recent progress, presented in his essay, questions like ‘does the digit 7occur infinitely often in the decimal expansion of square root of two?’ remain very

far from being answered. Here combinatorics on words and transcendence theory

are combined to derive new information on the sequence of decimals of algebraic

numbers and on their continued fraction expansions.

Titus Hilberdink from Reading lectured about a recent and rather unexpected ap-

proach to extreme values of the Riemann zeta-function by use of (multiplicative)Toeplitz matrices and functional analysis.

Finally, Jurgen Sander from Hildesheim gave an introduction to algebraic graphtheory and the impact of number theoretical methods on fundamental questions about

the spectra of graphs and the analogue of the Riemann hypothesis.

In this volume the reader can find the course notes from this summer school andin some places further additional material. Each of these courses is essentially self-

contained (although a background in number theory and analysis might be useful).

In all four courses recent research results are included indicating how easily one can

approach frontiers of current research in number theory by elementary and basic an-

alytic methods.The picture on the front page shows the poster created by Nicola Oswald from

Wurzburg University for the summer school. The editing of the course notes had been



vi

done by Rasa Steuding from Wurzburg University. The authors are most grateful to

both of them for their help. Last but not least, we sincerely thank Jorn Steuding from

Wurzburg University for initiating the summer school and the publication of these

notes as well as for his editorial work.

The authors, March 2014



Contents

1 Asymptotic formulas for modular forms and related functions . . . . . 1

Kathrin Bringmann

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Classical modular forms . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Weakly holomorphic modular forms . . . . . . . . . . . . . . . . . . 114 Mock modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Mixed mock modular forms . . . . . . . . . . . . . . . . . . . . . . . 25

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Expansions of algebraic numbers . . . . . . . . . . . . . . . . . . . . . . 31

Yann Bugeaud

1 Representation of real numbers . . . . . . . . . . . . . . . . . . . . . 31

2 Combinatorics on words and complexity . . . . . . . . . . . . . . . . 34

3 Complexity of algebraic numbers . . . . . . . . . . . . . . . . . . . . 374 Combinatorial transcendence criteria . . . . . . . . . . . . . . . . . . 38

5 Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 Diophantine approximation . . . . . . . . . . . . . . . . . . . . . . . 45

7 Sketch of proof and historical comments . . . . . . . . . . . . . . . . 478 A combinatorial lemma . . . . . . . . . . . . . . . . . . . . . . . . . 50

9 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 55

11 A transcendence criterion for quasi-palindromic continued fractions . 6312 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

13 Further notions of complexity . . . . . . . . . . . . . . . . . . . . . . 69Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3 Multiplicative Toeplitz matrices and the Riemann zeta function . . . . . 77

Titus Hilberdink

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2 Toeplitz matrices and operators – a brief overview . . . . . . . . . . . 80

3 Bounded multiplicative Toeplitz matrices and operators . . . . . . . . 854 Unbounded multiplicative Toeplitz operators and matrices . . . . . . . 99

5 Connections to matrices of the form .f.ij=.i;j/2//i;j N . . . . . . . 1 1 8

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120



viii Contents

4 Arithmetical topics in algebraic graph theory . . . . . . . . . . . . . . . 123

J urgen Sander

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2 Some (algebraic) graph theory . . . . . . . . . . . . . . . . . . . . . 1243 The prime number theorem for graphs . . . . . . . . . . . . . . . . . 136

4 Spectral properties of integral circulant graphs . . . . . . . . . . . . . 159

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187



Chapter 1

Asymptotic formulas for modular forms and

related functions

Kathrin Bringmann

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Classical modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Weakly holomorphic modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Mock modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Mixed mock modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1 Introduction

In this paper1, we aim to describe how “modularity” can be useful for studying the

asymptotic behavior of arithmetically interesting functions. We do not attempt to

present all that is known, but rather work with examples to give an idea about basicconcepts.

Let us recall the objects of interest. In the words of Barry Mazur,

“Modular forms are functions on the complex plane that are inordinately symmetric.

They satisfy so many symmetries that their mere existence seem like accidents. Butthey do exist.”

The modular forms alluded to in this quote are meromorphic functions on the complex

upper half-plane H

WD f

2C

IIm./ > 0

g that satisfy (if f is modular of weight

k 2 Z for SL2.Z/)

f

a C b

c C d

D .c C d /k f./

8

a bc d

2 SL2.Z/ WD

a bc d

2 Mat2.Z/I ad bc D 1

: (1.1)

Moreover, as a technical growth condition, one requires the function to be “meromor-

phic at the cusps”. Note that the transformation law can be generalized to subgroups,to include multipliers or half-integral weight. We do not state the specific form of

1The research of the author was supported by the Alfried Krupp Prize for Young University Teachers of the

Krupp Foundation.



2 Kathrin Bringmann

these transformations here, but we will later treat special cases, for example in Theo-

rem 3.1.

An important property of modular forms is that they have Fourier expansions of

the form f ./ D Pn2Z a.n/e2in . This follows from the transformation law (1.1),the fact that

1 10 1

2 SL2.Z/, and the meromorphicity on H. Meromorphicity at the

cusps now says that a.n/ ¤ 0 for only finitely many n < 0. The coefficients a.n/ of-ten encode interesting arithmetic information, such as the number of representations

of n by a (positive definite) quadratic form, just to give one of the numerous examples.Modular forms play an important role in many areas like physics, representation

theory, the theory of elliptic curves (in particular the proof of Fermat’s Last Theo-

rem), quadratic forms, and partitions, just to mention a few. Establishing modularityis of importance because it provides powerful machineries which can be employed

to prove important results. For example, identities may be reduced to a finite cal-

culation of Fourier coefficients (Sturm’s Theorem), asymptotic formulas for Fouriercoefficients can be obtained by Tauberian Theorems or the Circle Method, and con-

gruences may be proven by employing Serre’s theory of p-adic modular forms. Inthis note we are particularly interested in asymptotic and exact formulas for Fourier

coefficients of various modular q-series. In Section 2 we consider holomorphic mod-

ular forms, then in Section 3 allow growth in the cusps, in Section 4 turn to mock modular forms, and finally treat mixed mock modular forms in Section 5.

2 Classical modular forms

In this section we restrict, for simplicity, to forms of even integral weight k for

SL2.Z/. For basic facts on modular forms and most of the details skipped in this

section, we refer the reader to [31].

We call a modular form a holomorphic modular form if it is holomorphic on the

upper half-plane and bounded at 1. The space of holomorphic modular forms of weight k is denoted by M k . If a holomorphic modular form exponentially decays

towards 1 it is called a cusp form. The associated space is denoted by S k. Spe-

cial holomorphic modular forms that are not cusp forms are given by the classical

Eisenstein series and they have very simple explicit Fourier coefficients.

Definition. Formally define for k 2 N the Eisenstein series

Gk . / WDX0

m;n2Z.m C n/k;

where the sum runs through all .m; n/ 2 Z2 n f.0; 0/g. We note that

G2kC1. / 0:

Theorem 2.1. For k 4 even, we have that Gk 2 M k .



1 Asymptotic formulas for modular forms and related functions 3

Proof. (sketch)

Step 1: Prove compact convergence.

Step 2: Apply modular transformations and reorder.

Remark. We also require a normalized version of the Eisenstein series. For this, set

1 WD ˚1 n0 1

I n 2 Z

and for M D a bc d

2 SL2.R/; k 2 Z, and f W H ! C we

define the Petersson slash operator by

f jk M./ WD .c C d /kf

a C b

c C d

:

Define for k 4 an even integer

Ek . /

WD

1

2 XM 21nSL2.Z/

1

jk M./;

where the sum runs through a complete set of representatives of right cosets of 1 in

SL2.Z/. We have that

Ek . / D 1

2.k/Gk ./;

where .s/ WD Pn11

ns (Re.s/ > 1) denotes the Riemann zeta function. Indeed, it

is not hard to see that a set of representatives of 1 n SL2.Z/ may be given by

? ?

c d 2SL2.Z/

I.c;d/

D1 :

Thus

Ek . / D 1

2

X.c;d/D1

.c C d /k D 1

2.k/

Xr1

rkX

.c;d/D1

.c C d /k

D 1

2.k/

X.c;d/D1

r2N

.cr C dr/k :

Now .cr;dr/ runs through Z2 n f.0; 0/g if r runs through N and .c;d / through Z2

under the restriction that .c;d /

D1. This gives the claim.

We next turn to computing the Fourier expansion of the Eisenstein series.

Theorem 2.2. We have the Fourier expansion for k 4 even

Ek . / D 1 2k

Bk

Xn1

k1.n/e2in ;

where .n/ WDPd jn d ` and Bk is the kth Bernoulli number, given by the generating

functionx

ex 1 D Xn0

Bn

xn

nŠ :



4 Kathrin Bringmann

Proof. We instead compute the Fourier expansion of Gk and then use that for k even

.k/ D .1/

k2

C1 .2/kBk

2kŠ :

Due to the absolute convergence of the Eisenstein series we may reorder

Gk . / DX0

m;n2Z.m C n/k D

Xn¤0

nk CX

m¤0

Xn2Z

.m C n/k

D 2.k/ C 2Xm>0

Xn2Z

.m C n/k : (1.2)

Now it is well-known by the Lipschitz summation formula (cf. pp. 65–72 of [30])

that Xn2Z

. C n/k D .2i/k

.k 1/Š

Xn1

nk1e2in :

This gives that the second term in (1.2) equals

2.2i/k

.k 1/Š

Xm1

Xd 1

d k1e2imd D 2.2i/k

.k 1/Š

Xm1

Xd jm

d k1e2im ;

which gives the claim.

Examples. We have the following special cases:

E4. / D 1 C 240Xn1

3.n/e2in ;

E6. / D 1 504Xn1

5.n/e2in ;

E8. / D 1 C 480

Xn1

7.n/e2in :

We next turn to bounding Fourier coefficients of holomorphic modular forms. Forthis we split the space M k into an Eisenstein series part and a cuspidal part.

Theorem 2.3. Assume that k 4 is even. Then

M k D CEk ˚ S k:

Proof. Assume f ./ D

Pn0 a.n/e2in 2 M k . Then

f a.0/Ek 2 S k:




Since the coefficients of the Eisenstein series part were given explicitly in Theo-

rem 2.2, we next turn to bounding coefficients of cups forms.

Theorem 2.4 (Hecke bound). For f ./ D Pn1 a.n/e2in 2 S k with k > 0 , wehave

a.n/ D O

nk2

:

Proof. It is not hard to see that the function

ef ./ WD Im. /k2 jf./j

is bounded on H. Indeed, one can show that

ef is invariant under SL2.Z/ and one

may then bound ef for sufficiently large imaginary part. Using Cauchy’s Theorem,

we can write for n 1

a.n/ D e2ny

Z 1

0

f .x C iy/e2inxdx;

where y > 0 can be chosen arbitrary. This yields that

ja.n/j y k2 e2ny

Z 1

0

ef .x C iy/dx cyk

2 e2ny

for some constant c > 0 (independent of y). Picking y D 1n gives

ja.n/j cnk2 e2 D O

n

k2

:

Remark. The Ramanujan-Petersson Conjecture predicts that for f 2 S k we have forany " > 0 the (optimal) estimate

a.n/ ";f nk1

2 C":

For integral weight k

2 this conjecture was proven by Deligne using highly ad-

vanced techniques from algebraic geometry. His method begins with the fact that the

vector space of cusp forms has a basis of common eigenfunctions of the so-calledHecke algebra and that the Fourier coefficients of these forms coincide with the

eigenvalues.

Corollary 2.5. Assume k 4 is even and f ./ D Pn0 a.n/e2in 2 M k . Then

a.n/ D a.0/2k

Bk

k1.n/ C O

nk2

:

Proof. The claim follows directly by combining Theorem 2.3 and Theorem 2.4.



6 Kathrin Bringmann

Remark. We have that

nr r .n/ D Xd jnd r D nr

Xd jnd r nr

Xd 1

d r D .r/nr :

In particular, we have for k 4 that for f 2 M k

a.n/ D O

nk1

and if f 62 S k, this bound cannot be improved.

We next turn to defining explicit cusp forms whose construction generalizes thatof Eisenstein series.

Definition. For k 2 N and n 2 N0, formally define the Poincare series

P k;n. / WDX

M 21nSL2.Z/

e2in k

M:

Note that

P k;0. / D 2Ek./:

A calculation similar to the proof of Theorem 2.1 establishes the modularity of thesefunctions.

Theorem 2.6. For k 3 the Poincar e series P k;n converges uniformly on every

vertical strip in H. In particular, P k;n 2 M k , and for n > 0 we have P k;n 2 S k.

Poincare series turn out to be useful, as they have explicitly computable Fourier

expansions and integrating cusp forms against them recovers the Fourier coefficients

of these forms.To precisely state this result, we need the Petersson inner product. To define this,

note that for f; g 2 M k

d

WD dxdy

y2

;

f./g./yk

are invariant under SL2.Z/:

Definition. A subset F H is called a fundamental domain of SL2.Z/ if the fol-

lowing hold:

(i) F is closed.

(ii) For every 2 H there exists M 2 SL2.Z/ such that M 2 F .(iii) If and M (M

2SL2.Z/) are in the interior of F , then M

D ˙ 1 00 1 .

We also require the following explicit fundamental domain.




Theorem 2.7. The following is a fundamental domain of SL2.Z/:

F

WD

2H

I j

j 1;

jRe. /

j 1

2 :

Remark. Note that if D x C iy 2 F , then y p

32

.

Definition. For f 2 M k and g 2 S k , formally define the Petersson inner product by

hf; gi WDZ F

f./g./yk d: (1.3)

Later we also require a regularized version of this inner product. To introduce it,denote for T > 0

F T WD 2 HI jxj 12

; j j 1; y T :

Following [7], we define the regularized inner product hf; gireg of forms f; g which

transform like modular forms of weight k , but may grow at the cusps. To be moreprecise, we let hf; gireg be the constant term in the Laurent expansion at s D 0 of the

meromorphic continuation in s of

limT !1

Z F T

g./f./y ksd;

if it exists. Note that if f 2 M k and g has vanishing constant term, then we have

hf; gireg D limT !1

Z F T

g./f./y k d: (1.4)

The following theorem may be easily verified.

Theorem 2.8. The integral (1.3) is absolutely convergent and the following condi-

tions hold:

(i)

hf; g

i D hg; f

i;

(ii) hf; gi is C-linear in f ;

(iii) hf; f i 0 and moreover hf; f i D 0 , f 0.

In particular, integrating against Poincare series yields the Fourier coefficients of

cusp forms.

Theorem 2.9 (Petersson coefficient formula). For f./ D Pm1 a.m/e2im 2 S k ,

we have

˝f; P k;n˛ D (0 for n D 0;

.k2/Š

.4n/k1 a.n/ for n

1:

In particular, Ek?S k with respect to the Petersson scalar product.



8 Kathrin Bringmann

Proof. We only argue formally and ignore questions of convergence. We have

˝f; P k;n˛ D Z F f./ XM 21nSL2.Z/

e2in

jkM ykd

DX

M 21nSL2.Z/

Z F

f ˇ

kM./e2in jk M yk d

DZ S

M 21nSL2.Z/ M F

f./e2in yk d: (1.5)

Note thatS

M 21nSL2.Z/ M F is a fundamental domain for the action of 1 on H

and that f./e2in yk is invariant under the action of 1. Thus we may changeSM 21n M F into

f D x C iyI 0 x 1; y > 0g :

This gives that (1.5) equalsZ 10

Z 1

0

f./e2in yk2dxdy

D

Xm1

a.m/

Z 1

0

e2.nCm/y yk2dy

Z 1

0

e2i.mn/xdx:

Now the claim follows by using thatZ 1

0

e2i`xdx D

1 if ` D 0,

0 otherwise,Z 10

et t k2dt D .k 2/Š:

To show that the Poincare series constitute a basis of S k , we first require the

Valence formula (see [31] for a proof).

Theorem 2.10. If f 6 0 is a meromorphic modular form of weight k 2 Z , we have

with WD e2i

3

ord.f I1/ C 1

2ord.f I i / C 1

3ord.f I / C

Xz2nH

z6i;.mod /

ord.f I z/ D k

12;

where ord.f I1/ D n0 if f ./ D

P1nDn0

a.n/qn with a .n0/ ¤ 0.

One can conclude from Theorem 2.10 the following dimension formulas, theproof of which is omitted here.




Corollary 2.11. For k 2 N0 ,

dim M 2k D 8<:jk6kC

1 if k

61 .mod 6/;jk

6

k if k 1 .mod 6/;

dim S k D

8<:j

k6

k if k 6 1 .mod 6/;j

k6

k 1 if k 1 .mod 6/;k 7;

0 if k D 1:

This easily gives the basis property of the Poincare series.

Corollary 2.12 (Completeness theorem). Let k 4 even and d k WD dim.S k/. Then

a basis of S k is given by ˚P k;nI n D 1 ; : : : ; d k

:

In particular, M k has a basis consisting of Eisenstein series and Poincar e series.

Proof. Set S WD span˚

P k;1; : : : ; P k;d k

S k and let f 2 S k be such that f ?S with respect to the Petersson inner product. Then f has a Fourier expansion of the

form f./ D

Pmd kC1 a.m/e2im . From Corollary 2.11 we know the precise

values of d k , yielding a contradiction to Theorem 2.10. To be more precise, we have

for k 6 2 .mod 12/

d k C 1 D

k

12

C 1 >

k

12 D ord.f I1/ C

Xz2nH

ord.f I z/ d k C 1:

For k 2 .mod 12/, we getk

12 D d k C 1 C 1

6:

Thus ord.f

I1/

Dd k

C1 and X

z2nHord.f I z/ D 1

6;

which is impossible.

We next compute the Fourier coefficients an.m/ of the Poincare series P k;n . Forthis, we require some special functions. Define the Kloosterman sums by

S .m; nI c/ WD

Xa .mod c /?

e2 i a mCanc ;

where the sum runs over all a .mod c/ that are coprime to c and a denotes the



10 Kathrin Bringmann

multiplicative inverse of a .mod c/. Moreover, we let J r be the J -Bessel function

of order r , defined by

J r .x/ WD X0

.

1/`

`Š .` C 1 C r/ x

2rC2`

;

where .x/ denotes the usual gamma-function.

Theorem 2.13. We have for n 2 N

an.m/ Dm

n

k12

0

@ım;n C 2 ik

Xc1

c1S .n;mI c/ J k1

4

p mn

c

1

A; (1.6)

where

ım;n WD

1 if m D n,0 otherwise.

Proof. We again use that a set of representatives of 1nSL2.Z/ is given by? ?c d

2 SL2.Z/I .c;d/ D 1

:

The contribution for c

D0 is easily seen to give the first summand in (1.6). For c

¤0

we use the identitya C b

c C d D a

c 1

c2

C d c

and change d 7! d C mc; where d runs .mod c/? and m 2 Z. This gives

P k;n. / D e2in C 2Xc1

ckX

d .mod c/?

e2ina

c F

C d

c

;

where a is defined by ad

1 .mod c/ and

F . / WDXm2Z

e 2in

c2.Cm/ . C m/k :

Now the classical Poisson summation formula yields

F . / DXm2Z

a.m/e2im ;

where

a.m/ D Z Im. /DC

k e 2inc2 2im d




with C > 0 arbitrary. For m 0 we can deform the path of integration up to infinity

yielding that a.m/ D 0 in this case. For m > 0 we make the substitution Di c1.n=m/

12 w to get

a.m/ D ik1ck1m

n

k2 1

2

Z CCi1

Ci1wke

2c

p mn.ww1/dw:

The claim follows using the fact that for , > 0 the functions

t 7! .t=/1

2 J 1

2p

t

; .t > 0/;

and

w

7!we

w ; .Re.w/ > 0/;

are inverses of each other with respect to the usual Laplace transform (8.412.2 of

[23]).

3 Weakly holomorphic modular forms

We next turn to weakly holomorphic modular forms, which are still holomorphicon H, but admit poles at the “cusps”. The Fourier coefficients of such forms grow

much faster than those of holomorphic forms. Let us in particular describe this in thesituation of the partition function.

Recall that a partition of a positive integer n is a nondecreasing sequence of posi-

tive integers (the parts of the partition) whose sum is n. Let p.n/ denote the numberof partitions of n. For example, the partitions of 4 are

4; 3 C 1; 2 C 2; 2 C 1 C 1; 1 C 1 C 1 C 1;

so that p.4/ D 5. The partition function is very rapidly increasing. For example,

p.3/ D 3;p.4/ D 5;

p.10/ D 42;

p.20/ D 627;

p.100/ D 190569292:

A key observation by Euler is the product identity

P.q/ WD 1 CXn1

p.n/qn D Yn1

1

1

qn:




One can show that for jqj < 1, the function P is holomorphic. Using Euler’s identity

one can embed the partition function into the modular world using the Dedekind

-function .q

De2i /

./ WD q 124 Y

n1

.1 qn/ :

This function is a modular form of weight 1=2. To be more precise, we have thefollowing transformation laws (see e.g. [31]).

Theorem 3.1. We have

. C 1/ D ei12 ./;

1

D

p i./:

Theorem 3.1 in particular gives the asymptotic behavior of as ! 0. To be

more precise, it implies that

P.q/ p

i e i12 . ! 0/:

We moreover note that the coefficients p.n/ are easily seen to be positive and mono-tonic. From this, we may conclude the growth behavior of p.n/ using a Tauberian

Theorem due to Ingham [27].

Theorem 3.2. Assume that f ./

WDqn0 Pn

0 a.n/qn is a holomorphic function on

H , satisfying the following conditions:

(i) For all n 2 N0 , we have

0 < a.n/ a.n C 1/:

(ii) There exist c 2 C , d 2 R , and N > 0 such that

f./ c.i /d e2iN

. ! 0/:

Then

a.n/ c

p 2N 12 .d 1

2 / n

12 .d

32 /

e4

p N n

.n ! 1/:

From this we immediately conclude the growth behavior of the partition function.

Theorem 3.3. We have

p.n/ 1

4np

3 e

q 2n

3 .n ! 1/: (1.7)

We note that the partition function also has the q-hypergeometric series represen-

tationP.q/ D X

n0

qn2

.qI q/2n

; (1.8)




where .aI q/n WD Qn1

j D0.1 aqj /. Showing modularity by just using this represen-tation is still an open problem [3].

Rademacher [36], building on work of Hardy and Ramanujan [22], used the Circle

Method to obtain an exact formula for p.n/. To state his result, we let

I s.x/ WD isJ s.ix/ DXm0

1

mŠ.m C ˛ C 1/

x

2

2mC˛

(1.9)

be the I -Bessel function of order s. Moreover, with 12.x/ WD . 12x

/, we define theKloosterman sum

Ak .n/ WDp

k

4p

3

Xx .mod 24k/

x2124n .mod 24k/

12.x/e2ix

12k :

Note that for k > 0, .h; k/ D 1, and Re.z/ > 0 we may rewrite the transformation

law of the partition generating function as

P

exp

2 i

k .h C iz/

D !h;k

p ze

12k .z1z/P

exp

2 i

k

h0 C i

z

;

(1.10)where hh0 1 .mod k/. Here

!h;k WD exp .is .h; k// ; (1.11)

with

s .h; k/ WD X .mod k/

kh

k

and

..x// WD

x bxc 12

if x 2 RnZ;0 if x 2 Z:

Sometimes it is also useful to rewrite !h;k as [33]

!h;k

D

8<:

kh

e i. 1

4 .2hkh/C 112 .kk1/.2hh0Ch2h0// if h is odd;

h

k e i. 1

4.k

1/

C 112 .k

k1/.2h

h0

Ch2 h0//

if k is odd:

Here

ab

denotes the Jacobi symbol.

Note that we may also write

Ak .n/ DX

h .mod k?/

!h;ke2inh

k : (1.12)

Theorem 3.4. For n 1 , we have

p.n/ D 2

.24n 1/34 X

k1

Ak .n/

k I 32 p

24n

1

6k ! :




We note that this is a very astonishing identity expressing the integer p.n/ as

an infinite sum of transcendental numbers. Recently Bruinier and Ono [16] found

a formula expressing p.n/ as a finite sum of algebraic numbers.

Proof. Here we only give some details of the proof, for more see [4]. By Cauchy’s

Theorem,

p.n/ D 1

2 i

Z C

P.q/

qnC1dq;

where C is any path inside the unit circle surrounding 0 counterclockwise. We may

choose for C the circle centered at 0 with radius D exp. 2N 2

/, with N > 0 fixed(later we let N ! 1). Then

p.n/

Dn

Z 1

0

P . exp .2it// exp .

2int/dt:

Define

# 0h;k WD

1

k .k1 C k/ and # 00

h;k WD 1

k .k2 C k/;

where h1

k1< h

k < h2

k2are adjacent Farey fractions in the Farey sequence of order N

(for example, see p. 72 of [4]). From the theory of Farey fractions it is known that

(j D 1; 2)1

k

Ckj

1

N

C1

:

We decompose the path of integration in paths along the Farey arcs # 0h;k # 00

h;k, where D t h

k. Setting z D k.N 2 i / then yields

p.n/ D exp

2 n

N 2

X1kN 0h<k

.h;k/D1

exp

2inh

k

Z # 00

h;k

# 0h;k

P

exp

2 i

k .h C iz/

exp .2in/d:

For N ! 1 we have that z ! 0, i.e., exp. 2 ik

.h C iz// ! exp2ih

k

. We thus

need to know the behavior of P .q/ as q ! exp

2ihk

. To find it, we apply (1.10)

to obtain

p.n/ D exp

2 n

N 2

X1kN 0h<k.h;k/D1

exp

2inh

k

!h;k

Z #00h;k

#0h;k

z12 exp

12k.z1 z/P exp2 i

kh0C i

z exp .2in/d:




Now exp. 2 ik

.h0 C iz

// ! 0 as z ! 0C. Thus all the terms in P.q/ 1 are

“small”. We therefore write

p.n/

D X1

CX2;

withX1

WD exp

2 n

N 2

X1kN 0h<k.h;k/D1

exp

2inh

k

!h;k

Z # 00

h;k

#0h;k

z12 exp

12k

z1 z

exp .2in/d;

X2

WD exp2 nN 2

X1kN 0h<k.h;k/D1

exp2inhk

!h;kZ #

00

h;k

#0h;k

z12 exp z1

z12k

!

P

exp

2 i

k

h0 C i

z

1

exp .2in/d:

Here we only focus on the main termP

1 and only note thatP

2 contributes to theerror terms, giving

X2 D O N 12

exp2 n

N 2 ! 0

for N ! 1. The proof is given in [4].

Turning to the main term, setting w WD N 2 i yieldsX1

D exp

2 n

N 2

X1kN 0h<k.h;k/D1

exp

2inh

k

!h;kI h;k ;

withI h;k WD i k

12 exp

2nN 2 Z N 2Ci #0

h;k

N 2i # 00h;k

g.w/dw:

Here

g.w/ WD w12 exp

2

n 1

24

w C

12k2w

:

Using the Residue Theorem, we can write

exp2 n

N 2 I h;k

D k

12 i .Lk

I 1

I 2

I 3

I 4

I 5

I 6/ ;




with

Lk

WD Z L z12 exp2 n

1

24 z

C

12k2z dz;

I j WDZ

I j

z12 exp

2

n 1

24

z C

12k2z

dz;

where L and I j denote the paths of integration in the picture

We note without a proof that I 2, I 3, I 4, and I 5 contribute to the error and vanish

as N ! 1. Moreover, as " ! 0

I 1 C I 6 D 2i

Z 10

t 12 exp

2

n 1

24

t

12k2t

dt DW 2iL?

k

This gives that

p.n/ DXk1

Ak.n/k .n/;

where Ak .n/ is defined as in (1.12) and

k.n/ WD ip kLk C 2p kL?k:




To finish the proof, we have to show that

k .n/ D 2

.24n 1/ 34

1

k I 32 p

24n

1

6k ! :

Inserting the power series expansion for the exponential function, interchanging sum-

mation and integration, and then making a change of variables gives that

iLk D 2Xs0

12k2

s

sŠ

2

n 1

24

s 32 1

2 i

Z L

ez zsC12 dz;

where the loop L is as in the above picture.

Using the Hankel loop integral formula

1

s 12

D 1

2 i

Z L

ez zsC12 dz

then yields that

iLk D 1p 2

1

n 1

2432

Xs0

2.n 1

24 /6k2

s

sŠ

s 1

2

:

Treating L?k

similarly, the claim follows using the series representation of the Bessel

function (1.9).

Corollary 3.5. The asymptotic estimate (1.7) is true.

Proof. The claim follows immediately from Theorem 3.4 using that

I .x/ ex

p 2x

.x ! 1/:

4 Mock modular forms

Next we aim to study coefficients of mock modular forms, which are holomorphic

parts of harmonic .weak / Maass forms. These are a generalization of modular formswhich satisfy a transformation law like (1.1) and (weak) growth conditions at the

cusps, but, instead of being meromorphic, they are annihilated by the weight-k hy-

perbolic Laplacian . D x C iy/

k WD y2 @2

@x2 C @2

@y2C i ky @

@x C i @

@y :




We let H k denote the space of harmonic Maass forms of weight k. A theory of har-

monic Maass forms was built by Bruinier and Funke [15]. Previously, Niebur [34, 35]

and Hejhal [24] constructed certain non-holomorphic Poincare series which turn out

to be examples of harmonic weak Maass forms. Since functions that are holomorphicon H are annihilated by k , weakly holomorphic modular forms are harmonic Maass

forms. The simplest (non-weakly holomorphic) harmonic weak Maass form is givenby the non-holomorphic Eisenstein series of weight 2. To be more precise, define

E2. / WD 1 24Xn1

1.n/qn:

Then E2 is translation invariant, but introduces an error term under modular inver-

sion:

E2 1 D 2E2. / C 6

i:

This gives that bE2. / WD E2. / 3

y

is a harmonic weak Maass form of weight 2.

A further example of a mock modular form of weight 32

is given by the generatingfunction of Hurwitz class numbers [25]. To be more precise, the function

bh./ WD Xn0

n0;3 .mod 4/

H.n/qn C i8p

2Z i1

‚.w/

.i . C w//32

dw (1.13)

is a harmonic Maass forms of weight 32

on 0.4/. Here H.n/ is the nth Hur-

witz class number , i.e., the number of equivalence classes of quadratic forms of discriminant n, where each class C is counted with multiplicity 1=Aut.C / and

‚.w/ WD Pn2Z e2in2w is the usual weight- 12

theta function. We note that these

class numbers may also be related to so-called over partitions [9].

In this paper, we are in particular interested in those harmonic Maass forms F for

which there exist a polynomial P F and ` > 0 such that as y ! 1

F . / P F

q1 D O

e`y

:

The function P F .q1/ is called the principal part of F . We denote the associated

space of Maass forms of weight k by H k

.Using the differential equation satisfied by harmonic Maass forms, it is not hard

to see that forms in H k

naturally split into a holomorphic and a non-holomorphic

part:

F . / D F C. / C F ./;




where

F C. / WD

Xn1aC.n/qn;

F . / WD Xn<0

a.n/.1 kI 4jnjy/qn:

Here

.˛I x/ WDZ 1

x

et t ˛1dt

is the incomplete gamma function. The function F C is called a mock modular form

and its coefficients often encode interesting arithmetic information. Each mock mod-

ular form has a hidden companion, its shadow, which is necessary to fully understandthe mock modular form. The shadow, which is a classical cusp form of weight 2

k,

may be obtained from the associated harmonic weak Maass form of weight k by

applying the differential operator k WD 2iyk @@

. A direct calculation gives

k .F / D .4/1kXn1

a.n/n1kqn:

A space “orthogonal” to H k

can be charaterized in terms of the holomorphic

differential operator

Dk1

WD 1

2 i

@

@zk1

:

Bruinier, Ono, and Rhoades [17] showed the following

Theorem 4.1. If k 2 N; k 2 , then the image of the map

Dk1 W H 2k ! M Šk

consists of those h 2 M Šk

which are orthogonal to cusp forms with respect to the

regularized inner product (1.4) , and which also have constant term 0.

Further examples of mock modular forms are given by the so-called mock theta

functions, a collection of 22 q-series including

f.q/ WDXn0

qn2

.qI q/2n

DXn0

˛.n/qn;

which were defined by the visionary Ramanujan in his last letter to Hardy [37]. Note

that this function agrees with the representation (1.8) of P .q/ as a q-hypergeometricfunction up to a simple change of sign. Ramanujan’s last letter to Hardy includes the

claim that

˛.n/ .

1/nC1

2p n e

p

n6

(as n ! 1):




As typical for his writing, Ramanujan left no proof of this claim. Dragonette, a Ph.D.

student of Rademacher, then solved this claim in her thesis [19]. Andrews [1] im-

proved upon her work and together they made the following conjecture.

Conjecture (Andrews, Dragonette). For n 2 N , we have

˛.n/ D

.24n 1/14

Xk1

.1/

jkC1

2

kA2k

n k.1C.1/k/

4

k

I 12

p 24n 1

12n

!:

Note that not even convergence of this exact formula is obvious. In joint work

with K. Ono, we proved this conjecture [12].

Theorem 4.2. The Andrews-Dragonette Conjecture is true.

We note that S. Garthwaite [21] showed in her thesis a similar result for the co-

efficients of Ramanujan’s mock theta function !.q/ defined in (1.14). Later we [13]

proved exact formulas for coefficients of a generic harmonic weak Maass form.The coefficients ˛.n/ also have some combinatorial interpretations. To describe

this, recall that the rank of a partition is its largest part minus the number of its parts.

Dyson [20] introduced this statistic to explain the famous Ramanujan congruences

p.5n

C4/

0 .mod 5/;

p.7n C 5/ 0 .mod 7/;

p.11n C 6/ 0 .mod 11/:

More precisely, Dyson conjectured that the partitions of 5n C 4 (resp. 7n C 5) form

5 (resp. 7) groups of equal size when sorted by their ranks modulo 5 (resp. 7). As anexample, the following table gives the ranks for the partitions of 4.

partition rank rank .mod 5/

4 4 1 = 3 33 C 1 3 2 = 1 12 C 2 2 2 = 0 0

2 C 1 C 1 2 3 = 1 41 C 1 C 1 C 1 1 4 = 3 2

Since a direct calculation confirms that such a splitting in residue classes cannot

hold modulo 11, Dyson postulated the existence of another statistic, which should

be called the crank and which should explain all of the Ramanujan congruences.Dyson’s rank conjecture was solved by Atkin and Swinnerton-Dyer [6]; the crank was

found by Andrews and Garvan [5]. To study ranks it is natural to consider a generating




function. Denoting by N.m;n/ the number of partitions of n of rank m, it is well-

known that

1 CXn1

Xm2Z

N .m;n/zmqn D 1 CXn1

qn2

.zqI q/n .z1qI q/n

:

In particular, letting z D 1, we obtain

1 CXn1

.N e.n/ N o.n//qn D 1 CXn1

qn2

.qI q/2n

D f.q/;

where N e.n/ (resp. N o.n/) denotes the number of partitions of n of even (resp. odd)

rank. This yields the combinatorial interpretation of the coefficients ˛.n/ of the mock theta function f .q/.

Proof of Theorem 4.2. The idea in [12] is to realize f .q/ as the holomorphic part of a Maass-Poincare series. Such Poincare series generate the space of harmonic Maass

forms and have the form XM D

a bc d

21n

.c C d /k

a C b

c C d

;

where is an appropriate subgroup of SL2.Z/ and is a translation-invariant func-tion that satisfies the differential equation of a Maass form. Fourier coefficients of

such Maass-Poincare series are then easily computed using Poisson summation.To be more precise, we need to recall work of Zwegers [39] which completes

f.q/ to a (vector-valued) harmonic weak Maass form. Watson [38] had previously

obtained (mock) transformations for f .q/ and related functions. To state Zwegers’s

result we require a further mock theta function

!.q/ WDXn0

q2n2C2n

.q

Iq2/

2nC

1

(1.14)

to build the vector-valued function

F./ WD .F 0./;F 1./;F 2.//T D

q 124 f.q/;2q

13 !

q12

; 2q

13 !q

12

T

:

We moreover require its non-holomorphic companion, a certain “period integral”,

G./ WD 2ip

3

Z i1

.g1.z/; g0.z/; g2.z//T

p i.z

C /

dz;




where the following cuspidal theta functions of weight 32

are defined as

g0.z/WD Xn2Z.

1/nn

C 1

3 e3 i.nC13 /

2z ;

g1.z/ WD Xn2Z

n C 1

6

e3 i.nC1

6 /2

z ;

g2.z/ WDXn2Z

n C 1

3

e3 i.nC1

3 /2

z:

Then define the completion of F by

bF. /WD

F./

G./:

Theorem 4.3. We have the following transformation laws for the generators of

SL2.Z/:

bF . C 1/ D0@ 1

24 0 00 0 30 3 0

1AbF./;

bF

1

D

p i

0

@0 1 01 0 00 0 1

1

AbF./;

where n WD e2i

n . Moreover,

12

bF

D 0:

From Theorem 4.3 it is then not hard to conclude thatbF 0. / WD F 0.24/ G0.24/

is a harmonic weak Maass form of weight 12

on 0.144/ with Nebentypus character

12.n/

WD 12n . Here 0.N /

WD ˚ a bc d 2

SL2.Z/

Ic

0 .mod N /.

Let us next define the precise Maass-Poincare series. For this, we require furthernotation. For matrices

a bc d

2 0.2/ with c 0, define the multiplier

a bc d

WD(

b24 if c D 0,

i 12 .1/

12

.cCad C1/eaCd

24c a

4 C 3dc

8

!1d;c

if c > 0,

where !h;k was given in (1.11). Here e.x/ WD e2ix . Note that this multiplier is

defined to agree with the automorphy factor of

bF 0 when restricted to 0.2/.

A key function for building the Maass-Poincare series is the special function (s 2C, u 2 Rnf0g)

Ms .u/ WD juj 14 M 1

4 sgn.u/;s 12

.juj/ :




Here M ; is the standard M -Whittaker function, which satisfies the differential

equation

@2

@u2F C1

4 C

u C14

2

u2 !F D 0:

Then set

's. / WD Ms

y

6

e x

24

:

A direct calculation shows that

12

.'s / D

s 1

4

3

4 s

's : (1.15)

From the function 's

we then build the Poincare series

P 12

.sI / WD 2p

XM 21n0.2/

.M/1.c C d /12 's .M / ;

where we always choose representations with non-negative lower-left entry. For

Re.s/ > 1, this series converges absolutely and satisfies for M D a bc d

2 0.2/the correct transformation law

P 12 sI a C b

c C

d D .M/.c C d /12 P 1

2.sI / :

Specializing to s D 34

then formally yields, by (1.15), a function that is annihilated

by 12

. Convergence is however not guaranteed. To overcome this problem, we

analytically continue the function via its Fourier expansion.

Proposition 4.4. We have

P 12

3

4I

D

1 1p

1

2I y

6

q 1

24

CXn1

ˇ.n/qn 124 CXn0

.n/ 12I j24n 1j y

6 qn 124 :

Here

ˇ.n/ D

.24n 1/14

Xk1

.1/bkC1

2 cA2k

n k.1C.1/k/

4

k

I 12

p

24n 1

12k

!;

.n/ D

p

j24n 1j 14 X

k1

.1/bkC1

2 cA2k

n k.1C.1/k/

4 k J 12p j24n

1

j12k !:




Proof. Using Poisson summation, the proof proceeds similarly to the proof of Theo-

rem 2.13. In particular we require the following integral evaluations (see page 357 of

[24]):Z 11

1 i u

1 C i u

k2

M k2 ;s 1

2

4B

1 C u2

exp

2iBu

1 C u2 C 2iAu

du

D .2s/

8ˆ<

ˆ:

2

q jB

A j.sk

2 /W k

2 ;s 12

.4jAj/ J 2s1

4p jABj

if A > 0;

41Cs

.2s1/.sCk2 /.sk

2 /Bs if A D 0;

2

q jB

A j.sC

k2 /

W k2 ;s

1

2

.4jA

j/ I 2s

14p jAB

j if A < 0:

Here W ; is the usual W -Whittaker function. The claim then follows from the spe-cial values of Whittaker functions .y > 0/

W 34

.y/ D ey2 ;

W 34

.y/ D ey2

1

2I y

;

M34

.

y/

D

1

2 p

1

2 Iy e

y2

once convergence is established. Here for s 2 C and u 2 Rnf0g

W s.u/ WD juj 14 W 1

4 sgn.u/;s 1

2.juj/ :

Proving convergence is quite involved and requires modifying a classical argu-

ment of Hooley [26]. To use his method, we rewrite the Kloosterman sums as Salie

sums, which are sums over quadratic congruences. To be more precise, one can show

that

A2k n k.1C.1/k

/4 k

D8<:

1.nI k/p 24k

if k is odd;

i1.nI k/p 24k

2i e

n2k

2.nI k/p

24kif k is even;

where

1.nI k/ WDX

x .mod 48k/

x2124n .mod 48k/

12.x/e x

24k

;

2.n

Ik/

WD Xx .mod 12k/

x2124n .mod 12k/

12.x/ix21

12k e x

24k :




Using the fact that

I 12 p

24n

1

12k ! J 12 p j24n

1

j12k ! j24n

1

j

14

p 6k .k ! 1/

it is not hard to complete the proof once we establish the following estimates ..j; `/ 2f.0;1/;.0;2/;.1;1/g/: ˇ

ˇˇ

Xk1

kj .mod 2/

`.nI k/

k

ˇˇˇ j24n 1j 1

2 :

The proof of this estimate follows similarly as in Hooley. As the Poincare series was constructed such that ˇ.n/ agrees with the formula

conjectured, we are left to show that

˛.n/ D ˇ.n/:

For this we prove that the associated Maass forms coincide. The argument is some-

what lenghty and has been later simplified [13].

5 Mixed mock modular forms

We next turn to the Fourier coefficients of mixed mock modular forms, which are

linear combinations of mock modular forms multiplied by modular forms. We note

that there is no theory of such functions, the reason being that the space of harmonicMaass forms is not closed under multiplication (the eigenfunction property is vio-

lated). In particular, there are no corresponding Poincare series known, and those

were key ingredients for proving an exact formula for the Fourier coefficients of f.q/. To overcome these problems, we [10] developed an amplied version of the

Hardy-Ramanujan Circle Method. Before describing the main modifications neces-sary, we want to give some examples.

The first example comes from combinatorics.

Definition. A partition without sequences [2, 32] is a partition without two consecu-

tive numbers. We denote by s.n/ the number of partitions of n without sequences.

For example the partitions without sequences of 5 are given by

5; 4

C1; 3

C1

C1; 1

C1

C1

C1

C1:

Thus s.5/ D 4.




We have the generating function [2]

G.q/WD Xn0

s.n/qn

D q3I q3

1.q2I q2/1 .q/;

where the product in front is a quotient of -functions (and thus modular) and

.q/ WDXn0

.qI q/n

.q3I q3/n

qn2

is one of Ramanujan’s third-order mock theta functions.

A further example comes from the generating function for Euler numbers of cer-tain moduli spaces [11]. The functions that are of relevance here are (j

2 f0; 1

g)

f j . / WD 1

6. /

Xn0

H.4n C 3j /qnC3j4 DW

Xn0

˛j .n/qn jC14 : (1.16)

The modularity of and (1.13) yields that we have a mixed mock modular form.

A third example comes from Lie superalgebras. Recently, Kac and Wakimoto

[28] found a character formula for certain s`.mj1/^-modules. In terms of q-seriesthis can be written in the following way (s 2 Z):

2qs

2 q2

Iq2

2

1.qI q/mC21 XkD.k1;k2;:::;km1/2Zm1

q12 P

m1iD1 ki .kiC1/

1 C qPm1

iD1 kis :

From this representation it is not clear at all that one has mixed mock modular forms,

but the author proved this in joint work with Ken Ono [14].

One can also consider specialized character formulas for irreducible highest

weight s`.mjn/^ modules. Here an even more complicated class of functions playsa role [8], the so-called almost harmonic Maass forms. Loosely speaking, these are

sums of harmonic Maass forms under iterates of the raising operator Rk WD 2i @@

C ky

(thus themselves non-harmonic weak Maass forms) multiplied by almost holomor-

phic modular forms [29], which are functions that transform like modular forms andare polynomials in 1

y with (weakly) holomorphic coefficients. We call the associated

holomorphic parts almost mock modular forms.

There are many more interesting mixed mock modular forms, for example coming

from certain generating function arising in the theory of black holes [18] or Joyceinvariants.

As a special case, we treat the coefficients ˛j .n/ of f j defined in (1.16). For this,

we require some more notation. We let for k 2 N; g 2 Z, and u 2 R

f k;g .u/ WD 8<:2

sinh2uk

ig

2k if g 6 0 .mod 2k/;

2

sinh2. uk /

k2

u2 if g 0 .mod 2k/:




Furthermore, we define the Kloosterman sums

K j;`.n;m

Ik/

WD X0h<k.h;k/D1

j `.h;h0;k/e 2i

k hnCh0m

4

;

where h0 is defined by hh0 1 .mod k/ and where the j ` are certain explicitmultipliers. Finally, we let

I k;g .n/ WDZ 1

1

f k;g

u

2

I 7

2

k

p .4n .j C 1// .1 u2/

1 u2

74 du:

Theorem 5.1. For n 1 , the Fourier coefficients ˛j .n/ of f j are given by the

following exact formula:

˛j .n/ D

6 .4n .j C 1//

54

Xk1

K j;0.n;0I k/

k I 5

2

k

p 4n .j C 1/

C 1p

2.4n .j C 1//

32

Xk1

K j;0.n;0I k/p k

I 3

k

p 4n .j C 1/

1

8 .4n .j C 1//

74

Xk1

X`2f0;1g

k<g

k

g` .mod 2/

K j;`.n;g2I k/

k2 I k;g .n/:

The above formula is useful as it allows one to determine asymptotic expansions.To be more precise, one can show (see [10, 11]) the following.

Corollary 5.2. The leading asymptotic terms of ˛j .n/ for n ! 1 are

˛j .n/ D

1

96n 3

2 1

32n 7

4 C O

n2

e2p

n:

We note that in contrast to mock modular forms, the shadows of the mixed mock modular forms do contribute to the leading asymptotic terms. One could determinefurther polynomial lower order main terms.

Proof of Theorem 5.1. We only give a sketch of a proof. Details can be found in [11].

From (1.13) and the transformation law of the theta function it follows that

f j

1

k.h C iz/

D z

32

X`2f0;1g

j `.h;h0;k/ei h0

2k i.jC1/h

2k

f 1

k h0 C i

zC 1

4p 2 6 1

k h0 C i

z I 1

kz ;




where

I j .x/

WD Z 1

0

‚j

iw h0

k

.w C x/

3

2

dw:

Using partial fraction decomposition, we obtain

I j .x/ DX

g .mod 2k/gj .mod 2/

e

g2h0

4k

2ı0;gp

x 1p

2k2x

Z 11

e2xu2

f k;g .u/du

;

where ı0;g D 0 unless g 0 .mod 2k/ in which case it equals 1.

Now the main difference compared to the use of the classical Circle Method is thecontribution of integrals of the form

I k;g;b.z/ WD e2b

kz z52

Z 11

e2u2

kz f k;g .u/du:

Similarly to the case of Fourier expansions, we are now interested in their “principal

parts”. To be more precise, we let for b > 0 and g 2 Z

J k;g;b.z/ WD e2b

kz z52

Z p b

p b

e2u2

kz f k;g .u/du:

One may show the following asymptotic bounds for k < g k:

Lemma 5.3.

(i) If b 0 , then ˇ I k;g;b.z/

ˇ jzj 52

(k2

g2 if g ¤ 0;

1 if g D 0:

(ii) If b > 0 , then

I k;g;b.z/ D J k;g;b.z/ C E k;g;b.z/

with E k;g;b satisfying the bound in (i).

With this data one may then use the classical Circle Method.

Acknowledgements

The author thanks Ben Kane and Rene Olivetto for useful comments on an earlier

version.



Bibliography 29

Bibliography

[1] G. Andrews, On the theorems of Watson and Dragonette for Ramanujan’s mock theta func-

tions. Am. J. Math. 88 (1966), 454–490.[2] G. Andrews, Partitions with short sequences and mock theta functions. Proc. Natl. Acad. Sci.

USA 102 (2005), 4666–4671.

[3] G. Andrews, Partitions: At the interface of q-series and modular forms. Ramanujan J. 7

(2003), 385–400.

[4] G. Andrews, The theory of partitions. The Encyclopedia of Mathematics and its Applications

series, Cambridge University Press (1998).

[5] G. Andrews and F. Garvan, Dyson’s crank of a partition. Bull. Am. Math. Soc. 18 (1988),

167–171.

[6] A. Atkin and H. Swinnerton-Dyer, Some properties of partitions. Proc. Lond. Math. Soc. 4(1954), 84–106.

[7] R. Borcherds, Automorphic forms with singularities on Grassmannians. Invent. Math. 132

(1998), 491–562.

[8] K. Bringmann and A. Folsom, Almost harmonic Maass forms and Kac–Wakimoto characters.

J. Reine Angew. Math. (2014), 179–202.

[9] K. Bringmann and J. Lovejoy, Overpartitions and class numbers of binary quadratic forms.

Proc. Natl. Acad. Sci. USA 106 (2009), 5513–5516.

[10] K. Bringmann and K. Mahlburg, An extension of the Hardy-Ramanujan Circle Method and

applications to partitions without sequences. Am. J. Math. 133 (2011), 1151–1178.

[11] K. Bringmann and J. Manschot, From sheaves on P2 to generalisations of the Rademacher

expansion. Am. J. Math. 135 (2013), 1039–1065.

[12] K. Bringmann and K. Ono, The f.q/ mock theta function conjecture and partition ranks.

Invent. Math. 165 (2006), 243–266.

[13] K. Bringmann and K. Ono, Coefficients of harmonic weak Maass forms. Proc. Conf. Part.

Q-ser. Mod. Forms (Univ. Florida), 2008.

[14] K. Bringmann and K. Ono, Some characters of Kac and Wakimoto and nonholomorphic mod-

ular functions. Math. Ann. 345 (2009), 547–558.

[15] J. Bruinier and J. Funke, On two geometric theta lifts. Duke Math. J. 125 (2004), 45–90.

[16] J. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight har-

monic weak Maass forms. Advances in Mathematics 246 (2013), 198–219.

[17] J. Bruinier, K. Ono and R. Rhoades, Differential operators and harmonic weak Maass forms.

Math. Ann. 342 (2008), 673–693.

[18] A. Dabholkar, S. Murthy and D. Zagier, Quantum black holes, wall crossing, and mock mod-

ular forms. Preprint.

[19] L. Dragonette, Some asymptotic formulae for the mock theta series of Ramanujan. Trans. Am.

Math. Soc. 72 (1952), 474–500.

[20] F. Dyson, Some guesses in the theory of partitions. Eureka (Cambridge) 8 (1944), 10–15.

[21] S. Garthwaite, Vector-valued Maass Poincare series. Proc. Am. Math. Soc. 136 (2008), 427–436.




[22] G. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis. Proc. Lond. Math.

Soc., 2, 17 (1918), 75–115.

[23] I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products. Elsevier, Amsterdam

2007.

[24] D. Hejhal, The Selberg trace formula for PSL.2;R/, part 2. Springer Lect. Notes in Math.

1001, Springer-Verlag, Berlin 1983.

[25] F. Hirzebruch and D. Zagier, Intersection numbers on curves on Hilbert modular surfaces and

modular forms of Nebentypus. Invent. Math. 36 (1976), 57–113.

[26] C. Hooley, On the number of divisors of a quadratic polynomial. Acta Math. 110 (1963),

97–114.

[27] A. Ingham, A Tauberian theorem for partitions. Ann. of Math. 42 (1941), 1075–1090.

[28] V. Kac and M. Wakimoto, Integrable highest weight modules over affine superalgebras and

Appell’s function. Comm. Math. Phys. 215 (2011), 631–682.

[29] M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms. The

moduli space of curves, Progr. Math. 129, Birkhauser Boston, Massachusetts 1995, 165–172.

[30] M. Knopp, Modular Functions in Analytic Number Theory. Lectures in Advanced Mathemat-

ics, Markham Publishing Company, Chicago 1970.

[31] M. Kocher and A. Krieg, Elliptische Funktionen und Modulformen. Springer-Verlag, Berlin,

2007.

[32] P. MacMahon, Combinatory Analysis, Cambridge Univ. Press, Cambridge, U.K., Vol II, 49–

58, 1915.

[33] M. Newman, Construction and application of a class of modular functions (II). Proc. Lond. Math. Soc. 9 (1959), 373–397.

[34] D. Niebur, A class of nonanalytic automorphic functions. Nagoya Math. J. 52 (1973), 133–

145.

[35] D. Niebur, Construction of automorphic forms and integrals. Trans. Am. Math. Soc. 191

(1974), 373–385.

[36] H. Rademacher, On the expansion of the partition function in a series. Ann. of Math. 44 (1943),

416–422.

[37] S. Ramanujan, The lost notebook and other unpublished papers. Narosa, New Delhi 1988.

[38] G. Watson, The final problem: An account of the mock theta functions. J. Lond. Math. Soc.11 (1936), 55–88.

[39] S. Zwegers, Mock #-functions and real analytic modular forms. In q-Ser. Appl. Comb. Number

Theory Phys. (ed. B. Berndt and K. Ono), Contemp. Math. 291, AMS (2001), 269–277.



Chapter 2

Expansions of algebraic numbers

Yann Bugeaud

Contents

1 Representation of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 Combinatorics on words and complexity . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Complexity of algebraic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Combinatorial transcendence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Diophantine approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7 Sketch of proof and historical comments . . . . . . . . . . . . . . . . . . . . . . . . . 47

8 A combinatorial lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

9 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

11 A transcendence criterion for quasi-palindromic continued fractions . . . . . . . . . . 63

12 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

13 Further notions of complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

1 Representation of real numbers

The most classical ways to represent real numbers are by means of their continuedfraction expansion or their expansion in some integer base, in particular in base two

or ten. In this text, we consider only these expansions and deliberately ignore ˇ-

expansions, Luroth expansions, Q-Cantor series, etc., as well as the many variations

of the continued fraction algorithm.

The first example of a transcendental number (recall that a real number is alge-

braic if it is a root of a nonzero polynomial with integer coefficients and it is tran-

scendental otherwise) was given by Liouville [51, 52] in 1844. He showed that if the

sequence of partial quotients of an irrational real number grows sufficiently rapidly,then this number is transcendental. He mentioned only at the very end of his note the

now classical example of the series (keeping his notation)

1

a C 1

a1

2 C 1

a1

2

3 C C 1

a1

2

3

m C ;

where a 2 is an integer.



32 Yann Bugeaud

Let b denote an integer at least equal to 2. Any real number has a unique b-ary

expansion, that is, it can be uniquely written as

D b c C X1

a`

b` D b c C 0 a1a2 : : : ; (2.1)

where bc denotes the integer part function, the digits a1; a2; : : : are integers from the

set f0 ; 1 ; : : : ; b 1g and a` differs from b 1 for infinitely many indices `. This

notation will be kept throughout this text.In a seminal paper published in 1909, Emile Borel [21] introduced the notion of

normal number .

Definition 1.1. Let b 2 be an integer. Let be a real number whose b-ary expan-

sion is given by (2.1). We say that is normal to base b if, for every k 1, everyfinite block of k digits in f0 ; 1 ; : : : ; b 1g occurs with the same frequency 1=bk, that

is, if for every k 1 and every d 1; : : : ; d k 2 f0 ; 1 ; : : : ; b 1g we have

limN !C1

#f` W 0 ` < N; a`C1 D d 1; : : : ; a`Ck D d kgN

D 1

bk:

The above definition differs from that given by Borel, but is equivalent to it; see

Chapter 4 of [27] for a proof and further equivalent definitions.We reproduce the fundamental theorem proved by Borel in [21]. Throughout

this text, ‘almost all’ always refers to the Lebesgue measure, unless otherwise speci-fied.

Theorem 1.2. Almost all real numbers are normal to every integer base b 2.

Despite the fact that normality is a property shared by almost all numbers, wedo not know a single explicit example of a number normal to every integer base, let

alone of a number normal to base 2 and to base 3. However, Martin [54] gave in 2001

a nice and simple explicit construction of a real number normal to no integer base.The first explicit example of a real number normal to a given base was given by

Champernowne [34] in 1933.

Theorem 1.3. The real number

0 12345678910111213 : : : ; (2.2)

whose sequence of decimals is the increasing sequence of all positive integers, is

normal to base ten.

Further examples also obtained by concatenation of sequences of integers havebeen given subsequently in [35, 37]. In particular, the real number

0 235711131719232931 : : : ; (2.3)



2 Expansions of algebraic numbers 33

whose sequence of decimals is the increasing sequence of all prime numbers, is nor-

mal to base ten. This is due to the fact that the sequence of prime numbers does not

increase too rapidly. However, we still do not know whether the real numbers (2.2)

and (2.3) are normal to base two.Constructions of a completely different type were found by Stoneham [66] and

Korobov [49]; see Bailey and Crandall [18] for a more general statement which in-cludes the next theorem.

Theorem 1.4. Let b and c be coprime integers, both at least equal to 2. Let d 2be an integer. Then, the real numbersX

j 1

1

cj bcj and

Xj 1

1

cd j bcd j

are normal to base b.

Regarding continued fraction expansions, we can as well define a notion of nor-

mal continued fraction expansion using the Gauss measure (see Section 5) and provethat the continued fraction expansion

˛ D b˛c C Œ0I a1; a2; : : : D b˛c C 1

a1 C 1

a2

C

1

: : :of almost every real number ˛ is a normal continued fraction expansion. Here, the

positive integers a1; a2; : : : are called the partial quotients of ˛ (throughout this text,

a` denotes either a b-ary digit or a partial quotient, but this should be clear from

the context; furthermore, we write for a real number when we consider its b-ary

expansion and ˛ when we study its continued fraction expansion). In 1981, Adler,Keane, and Smorodinsky [13] have constructed a normal

continued fraction in a similar way as Champernowne did for a normal number.

Theorem 1.5. Let 1=2; 1=3; 2=3;1=4; 2=4; 3=4; : : : be the infinite sequence obtained

in writing the rational numbers in .0; 1/ with denominator 2 , then with denominator 3 , denominator 4 , etc., ordered with numerators increasing. Let x1x2x3 : : : be the

sequence of positive integers constructed by concatenating the partial quotients .we

choose the continued fraction expansion which does not end with the digit 1/ of this

sequence of rational numbers. Then, the real number

Œ0I x1; x2; : : : D Œ0I 2; 3; 1; 2; 4; 2; 1; 3; 5; : : :

has a normal continued fraction expansion.

All this shows that the b-ary expansion and the continued fraction expansion of

a real number taken at random are well understood. But what can be said for a specificnumber, like

3p 2; log 2; , etc.?



34 Yann Bugeaud

Actually, not much! We focus our attention on algebraic numbers. Clearly, a real

number is rational if, and only if, its b -ary expansion is ultimately periodic. Analo-

gously, a real number is quadratic if, and only if, its continued fraction expansion is

ultimately periodic; see Section 5. The purpose of the present text is to gather what isknown on the b-ary expansion of an irrational algebraic number and on the continued

fraction expansion of an algebraic number of degree at least three. It is generallybelieved that all these expansions are normal, and some numerical computation tend

to support this guess, but we are very, very far from proving such a strong assertion.We still do not know whether there is an integer b 3 such that at least three dif-

ferent digits occur infinitely often in the b-ary expansion of p

2. And whether there

exist algebraic numbers of degree at least three whose sequence of partial quotientsis bounded. Actually, it is widely believed that algebraic numbers should share most

of the properties of almost all real numbers. This is indeed the case from the point

of view of rational approximation, since Roth’s theorem (see Section 6) asserts thatalgebraic irrational numbers do behave like almost all numbers, in the sense that they

cannot be approximated by rational numbers at an order greater than 2.The present text is organized as follows. Section 2 contains basic results from

combinatorics on words. The main results on the complexity of algebraic numbers

are stated in Section 3. They are proved by combining combinatorial transcendencecriteria given in Section 4 and established in Sections 9 and 10 with a combinatorial

lemma proved in Section 8. In Sections 5 and 6 we present various auxiliary results

from the theory of continued fractions and from Diophantine approximation, respec-tively. Section 7 is devoted to a sketch of the proof of Theorem 4.1 and to a short

historical discussion. We present in Section 11 another combinatorial transcendencecriterion for continued fraction expansions, along with its proof. Section 12 briefly

surveys some refined results which complement Theorem 3.1. Finally, in Section 13,

we discuss other points of view for measuring the complexity of the b-ary expansionof a number.

A proof of Theorem 4.1 can already be found in the surveys [20, 8] and in the

monograph [27]. Here, we provide two different proofs. Historical remarks anddiscussion on the various results which have ultimately led to Theorem 4.2 are given

in [30].

2 Combinatorics on words and complexity

In the sequel, we often identify a real number with the infinite sequence of its b-

ary digits or of its partial quotients. It appears to be convenient to use the point of view from combinatorics on words. Throughout, we denote by A a finite or infinite

set. A finite word over the alphabet A is either the empty word, or a finite string

(or block) of elements from A. An infinite word over A is an infinite sequence of elements from A.




For an infinite word w D w1w2 : : : over the alphabet A and for any positive

integer n, we let

p.n; w;A/

WD#

fwj

C1 : : : wj

Cn

Wj

0

gdenote the number of distinct strings (or blocks) of length n occurring in w. Obvi-ously, putting #A D C1 if A is infinite, we have

1 p.n; w;A/ .#A/n;

and both inequalities are sharp. Furthermore, the function n 7! p.n; w;A/ is non-

decreasing.

Definition 2.1. An infinite word w D w1w2 : : : is ultimately periodic if there exist

positive integers n0 and T such that

wnCT D wn; for every n n0:

The word wn0wn0C1 : : : wn0CT 1 is a period of w. If n0 can be chosen equal to 1,

then w is (purely) periodic, otherwise, w1 : : : wn01 is a preperiod of w.

We establish a seminal result from Morse and Hedlund [55, 56].

Theorem 2.2. Let w be an infinite word over a finite or infinite alphabet A. If w is

ultimately periodic, then there exists a positive constant C such that p.n; w;A/ C for every positive integer n. Otherwise, we have

p.n C 1; w;A/ p.n; w;A/ C 1 for every n 1;

thus,

p.n; w;A/ n C 1 for every n 1:

Proof. Throughout the proof, we write p.; w/ instead of p.; w;A/.Let w be an ultimately periodic infinite word, and assume that it has a preperiod

of length r and a period of length s . Fix h D 1; : : : ; s and let n be a positive integer.For every j 1, the block of length n starting at wrCjsCh is the same as the one

starting at wr

Ch. Consequently, there cannot be more than r

Cs distinct blocks of

length n, thus, p.n; w/ r C s.Write w D w1w2 : : : and assume that there is a positive integer n0 such

that p.n0; w/ D p.n0 C 1; w/. This means that every block of length n0 extends

uniquely to a block of length n0 C 1. It implies that p.n0; w/ D p.n0 C j; w/ holds

for every positive integer j . By the definition of p.n0; w/, two among the words

wj : : : wn0Cj 1, j D 1 ; : : : ; p . n0; w/ C 1, are the same. Consequently, there areintegers k and ` with 0 k < ` p.n0; w/ and wkCm D w`Cm for m D 1 ; : : : ; n0.

Since every block of length n0 extends uniquely to a block of length n0 C 1, this

gives wkCm D w`Cm for every positive integer m. This proves that the word w isultimately periodic.



36 Yann Bugeaud

Consequently, if w is not ultimately periodic, then p.n C 1; w/ p.n; w/ C 1holds for every positive integer n. Then, p.1; w/ 2 and an immediate induction

show that p.n; w/

n

C1 for every n. The proof of the theorem is complete.

We complement Theorem 2.2 by pointing out that there exist uncountably manyinfinite words w over A D f0; 1g such that

p.n; w;A/ D n C 1; for n 1.

These words are called Sturmian words; see e.g. [16].

To prove that a real number is normal to some given integer base, or has a normalcontinued fraction expansion, is in most cases a far too difficult problem. So we are

led to consider weaker questions on the sequence of digits (resp. partial quotients),including the following ones:

Does every digit occur infinitely many times in the b-ary expansion of ?

Are there many non-zero digits in the b-ary expansion of ?

Is the sequence of partial quotients of ˛ bounded from above?

Does the sequence of partial quotients of ˛ tend to infinity?

We may even consider weaker questions, namely, and this is the point of viewwe adopt until the very last sections, we wish to bound from below the number of

different blocks in the infinite word composed of the digits of (resp. partial quotientsof ˛).

Let b 2 be an integer. A natural way to measure the complexity of a real number whose b-ary expansion is given by (2.1) is to count the number of distinct blocks of given length in the infinite word a D a1a2a3 : : : We set p.n; ; b/ D p.n; a; b/, with

a as above. Clearly, we have

p.n;;b/ D #faj C1aj C2 : : : aj Cn W j 0g D p.n; a; f0 ; 1 ; : : : ; b 1g/

and1 p.n;;b/ bn;

where both inequalities are sharp.

Since the b-ary expansion of a real number is ultimately periodic if, and only if,this number is rational, Theorem 2.2 can be restated as follows.

Theorem 2.3. Let b 2 be an integer. If the real number is irrational, then

p.n;;b/ n C 1; for n 1:

Otherwise, the sequence .p.n; ; b//n1 is bounded.

Let ˛ be an irrational real number and write

˛ D b˛c C Œ0I a1; a2; : : : :




Let a denote the infinite word a1a2 : : : over the alphabet Z1. A natural way to

measure the intrinsic complexity of ˛ is to count the number p.n;˛/ WD p.n; a;Z1/of distinct blocks of given length n in the word a.

Since the continued fraction expansion of a real number is ultimately periodic if,and only if, this number is quadratic (see Theorem 5.7), Theorem 2.2 can be restated

as follows.

Theorem 2.4. Let b 2 be an integer. If the real number ˛ is irrational and not

quadratic, then

p.n;˛/ n C 1; for n 1:

If the real number ˛ is quadratic, then the sequence .p.n; ˛//n1 is bounded.

We show in the next section that Theorem 2.3 (resp. 2.4) can be improved when

(resp. ˛) is assumed to be algebraic.

3 Complexity of algebraic numbers

As already mentioned, we focus on the digital expansions and on the continued frac-

tion expansion of algebraic numbers. Until the end of the 20th century, it was onlyknown that the sequence of partial quotients of an algebraic number cannot grow too

rapidly; see Section 12. Regarding b-ary expansions, Ferenczi and Mauduit [43] were

the first to improve the (trivial) lower bound given by Theorem 2.3 for the complexityfunction of the b-ary expansion of an irrational

algebraic number . They showed in 1997 that p.n;; b/ strictly exceeds n C 1for every sufficiently large integer n. Actually, as pointed out a few years later by

Allouche [14], their approach combined with a combinatorial result of Cassaigne

[32] yields a slightly stronger result, namely that

limn!C1

p.n;;b/ n

D C1; (2.4)

for any algebraic irrational number .

The estimate (2.4) follows from a good understanding of the combinatorial struc-ture of Sturmian sequences combined with a combinatorial translation of Ridout’s

theorem 6.6. The transcendence criterion given in Theorem 4.1, established in [10, 3],

yields an improvement of (2.4).

Theorem 3.1. For any irrational algebraic number and any integer b 2 , we have

limn!C1

p.n;;b/

n D C1: (2.5)

Although (2.5) considerably strengthens (2.4), it is still very far from what is

commonly expected, that is, from confirming that p.n;; b/ D bn holds for everypositive n when is algebraic irrational.



38 Yann Bugeaud

Regarding continued fraction expansions, it was proved in [15] that

limn

!C1 p.n;/ n D C1;

for any algebraic number of degree at least three. This is the continued fractionanalogue of (2.4).

Using ideas from [1], the continued fraction analogue of Theorem 3.1 was estab-

lished in [29].

Theorem 3.2. For any algebraic number of degree at least three, we have

limn!C1

p.n;/

n D C1: (2.6)

The main purpose of the present text is to give complete (if one admits Theo-

rem 6.7, whose proof is much too long and involved to be included here) proofs of

Theorems 3.1 and 3.2. They are established by combining combinatorial transcen-

dence criteria (Theorems 4.1 and 4.2) and a combinatorial lemma (Lemma 8.1). This

is explained in details at the end of Section 8.

4 Combinatorial transcendence criteria

In this section, we state the combinatorial transcendence criteria which, combined

with the combinatorial lemma from Section 8, yield Theorems 3.1 and 3.2.

Throughout, the length of a finite word W over the alphabetA, that is, the number

of letters composing W , is denoted by jW j.Let a D .a`/`1 be a sequence of elements from A. We say that a satisfies

Condition ./ if a is notultimately periodic and if there exist three sequences of finite words, .U n/n1,

.V n/n1, and .W n/n1, such that:

(i) For every n

1, the word W nU nV nU n is a prefix of the word a.

(ii) The sequence .jV nj=jU nj/n1 is bounded from above.

(iii) The sequence .jW nj=jU nj/n1 is bounded from above.

(iv) The sequence .jU nj/n1 is increasing.

Theorem 4.1. Let b 2 be an integer. Let a D .a`/`1 be a sequence of elements

from f0 ; 1 ; : : : ; b 1g. If a satisfies Condition ./ , then the real number

WDC1

X`D1

a`

b`

is transcendental.




Theorem 4.2. Let a D .a`/`1 be a sequence of positive integers. Let .p`=q`/`1

denote the sequence of convergents to the real number

˛ WD Œ0I a1; a2; : : : ; a`; : : : :

Assume that the sequence .q1=`

` /`1 is bounded. If a satisfies Condition ./ , then ˛

is transcendental.

The common tool for the proofs of Theorems 4.1 and 4.2 is a powerful theorem

from Diophantine approximation theory, the Subspace Theorem; see Section 6.Let us comment on Condition ./ when, for simplicity, the alphabet A is finite

and has b 2 elements. Take an arbitrary infinite word a1a2 : : : on f0 ; 1 ; : : : ; b 1g.Then, by the Schubfachprinzip, for every positive integer m, there exists (at least) one

finite word U m of length m having (at least) two (possibly overlapping) occurrences inthe prefix a1a2 : : : abmCm. If, for simplicity, we suppose that these two occurrencesdo not overlap, then there exist finite (or empty) words V m; W m; X m such that

a1a2 : : : abmCm D W mU mV mU mX m:

This simple argument gives no additional information on the lengths of W m and V m,which a priori can be as large as bm m. In particular, they can be larger than some

constant greater than 1 raised to the power the length of U m.

We demand much more for a sequence a to satisfy Condition ./, namely we

impose that there exists an integer C such that, for infinitely many m, the lengths of

V m and W m do not exceed C times the length of U m. Such a condition occurs quiterarely.

We end this section with a few comments on de Bruijn words.

Definition 4.3. Let b 2 and n 1 be integers. A de Bruijn word of order n overan alphabet of cardinality b is a word of length b n C n 1 in which every block of

length n occurs exactly once.

A recent result of Becher and Heiber [19] shows that we can extend de Bruijnwords.

Theorem 4.4. Every de Bruijn word of order n over an alphabet with at least three

letters can be extended to a de Bruijn word of order n C 1. Every de Bruijn word

of order n over an alphabet with two letters can be extended to a de Bruijn word of

order n C 2.

Theorem 4.4 shows that there exist infinite de Bruijn words obtained as the induc-

tive limit of extended de Bruijn sequences of order n, for each n (when the alphabet

has at least three letters; for each even n, otherwise). Let b 2 be an integer. Byconstruction, for every m

1, the shortest prefix of an infinite de Bruijn word having

two occurrences of a same word of length m has at least bm C m letters if b 3 andat least 2m1 C m 1 letters if b D 2.



40 Yann Bugeaud

5 Continued fractions

In this section, we briefly present classical results on continued fractions which will

be used in the proofs of Theorems 4.2 and 11.1. We omit most of the proofs andrefer the reader to a text of van der Poorten [58] and to the books of Bugeaud [22],

Cassels [33], Hardy and Wright [44], Khintchine [47], Perron [57], and Schmidt [64],

among many others.

Let x0; x1; : : : be real numbers with x1; x2; : : : positive. A finite continued frac-

tion denotes any expression of the form

Œx0I x1; x2; : : : ; xn D x0 C 1

x1 C 1

x2 C 1

C 1xn

:

We call any expression of the above form or of the form

Œx0I x1; x2; : : : D x0 C 1

x1 C 1

x2 C 1

: : :

D limn!C1 Œx0I x1; x2; : : : ; xn

a continued fraction, provided that the limit exists.Any rational number r has exactly two different continued fraction expansions.

These are Œr and Œr 1I 1 if r is an integer and, otherwise, one of them readsŒa0I a1; : : : ; an1; an with an 2, and the other one is Œa0I a1; : : : ; an1; an 1;1.

Any irrational number has a unique expansion in continued fraction.

Theorem 5.1. Let ˛ D Œa0I a1; a2; : : : be an irrational number. For ` 1 , set

p`=q` WD Œa0I a1; a2; : : : ; a`. Let n be a positive integer. Putting

p1 D 1; q1 D 0; p0 D a0; and q0 D 1;

we havepn D anpn1 C pn2; qn D anqn1 C qn2; (2.7)

and

pn1qn pnqn1 D .1/n: (2.8)

Furthermore, setting ˛nC1 D ŒanC1I anC2; anC3; : : : , we have

˛ D Œa0I a1; : : : ; an; ˛nC1 D pn˛nC1 C pn1

qn˛nC1 C qn1

; (2.9)

thus

qn˛ pn D .1/n

qn˛nC1 C qn1

;




and

1

.anC1 C 2/q2n

< 1

qn.qn C qnC1/

< ˇ˛ pn

qn ˇ < 1

qnqnC1

< 1

anC1q2n 1

q2n

: (2.10)

It follows from (2.9) that any real number whose first partial quotients are

a0; a1; : : : ; an belongs to the interval with endpoints .pn C pn1/=.qn C qn1/ and

pn=qn. Consequently, we get from (2.8) an upper bound for the distance between two

real numbers having the same first partial quotients.

Corollary 5.2. Let ˛ D Œa0I a1; a2; : : : be an irrational number. For ` 0 , let q` be

the denominator of the rational number Œa0I a1; a2; : : : ; a`. Let n be a positive inte-

ger and be a real number such that the first partial quotients of are a0; a1; : : : ; an.

Then,j˛ ˇj 1

qn.qn C qn1/ <

1

q2n

:

Under the assumption of Theorem 5.1, the rational number p`=q` is called the`-th convergent to ˛. It follows from (2.7) that the sequence of denominators of

convergents grows at least exponentially fast.

Theorem 5.3. Let ˛ D Œa0I a1; a2; : : : be an irrational number. For ` 0 , let q` be

the denominator of the rational number Œa0I a1; a2; : : : ; a`. For any positive integers

, h , we have q`Ch q`.p 2/h1

and

q` .1 C maxfa1; : : : ; a`g/`:

Proof. The first assertion follows via induction on h, since qnC2 qnC1 Cqn 2qn

for every n 0. The second assertion is an immediate consequence of (2.7).

The next result is sometimes called the mirror formula.

Theorem 5.4. Let n 2 be an integer and a1; : : : ; an be positive integers. For ` D 1 ; : : : ; n , set p`=q` D Œ0I a1; : : : ; a`. Then,

qn1

qn

D Œ0I an; an1; : : : ; a1:

Proof. We get from (2.7) that

qn

qn1

D an C qn2

qn1

;

for n 1. The theorem then follows by induction.



42 Yann Bugeaud

An alternative proof of Theorem 5.4 goes as follows. Observe that, if a0 D 0 and

n 1, then, by (2.7),

M n WD pn1 pnqn1 qn

D 0 11 a1

0 11 a2

0 11 an

:

Taking the transpose, we immediately get that

tM n Dt

0 11 a1

0 11 a2

0 11 an

D t

0 11 an

t

0 11 an1

t

0 11 a1

D 0 1

1 an0 1

1 an1 0 1

1 a1 D pn

1 qn

1

pn qn ;

which gives Theorem 5.4.

Theorem 5.4 is a particular case of a more general result, which we state belowafter introducing the notion of continuant .

Definition 5.5. Let m 1 and a1; : : : ; am be positive integers. The denominatorof the rational number Œ0I a1; : : : ; am is called the continuant of a1; : : : ; am and is

usually denoted by K m.a1; : : : ; am/.

Theorem 5.6. For any positive integers a1; : : : ; am and any integer k with 1 k m 1 , we have

K m.a1; : : : ; am/ D K m.am; : : : ; a1/; (2.11)

and

K k.a1; : : : ; ak / K mk .akC1; : : : ; am/

K m.a1; : : : ; am/ (2.12)

2 K k.a1; : : : ; ak/ K mk.akC1; : : : ; am/:

Proof. The first statement is an immediate consequence of Theorem 5.4. Combining

K m.a1; : : : ; am/ D amK m1.a1; : : : ; am1/ C K m2.a1; : : : ; am2/

with (2.11), we get

K m.a1; : : : ; am/ D a1K m1.a2; : : : ; am/ C K m2.a3; : : : ; am/;

which implies (2.12) for k D 1. Let k 2 f1 ; 2 ; : : : ; m 2g be such that

K m WD K m.a1; : : : ; am/

DK k .a1; : : : ; ak/

K m

k.ak

C1; : : : ; am/ (2.13)

C K k1.a1; : : : ; ak1/ K mk1.akC2; : : : ; am/;




where we have set K 0 D 1. We then have

K m D K k .a1; : : : ; ak/ akC1K mk1.akC2; : : : ; am/ C K mk2.akC3; : : : ; am/C K k1.a1; : : : ; ak1/ K mk1.akC2; : : : ; am/

D akC1K k.a1; : : : ; ak/ C K k1.a1; : : : ; ak1/

K mk1.akC2; : : : ; am/

C K k.a1; : : : ; ak / K mk2.akC3; : : : ; am/;

giving (2.13) for the index k C 1. This shows that (2.13) and, a fortiori, (2.12) holdfor k D 1 ; : : : ; m 1.

The ‘only if’ part of the next theorem is due to Euler [39], and the ‘if’ part wasestablished by Lagrange [50] in 1770.

Theorem 5.7. The real irrational number ˛ D Œa0I a1; a2; : : : has a periodic con-

tinued fraction expansion .that is, there exist integers r 0 and s 1 such that

anCs D an for all integers n r C 1/ if, and only if, ˛ is a quadratic irrationality.

We display an elementary result on ultimately periodic continued fraction expan-

sions.

Lemma 5.8. Let be a quadratic real number with ultimately periodic continued

fraction expansion

DŒ0

Ia1; : : : ; ar ; ar

C1; : : : ; ar

Cs;

and denote by .p`=q`/`1 the sequence of its convergents. Then, is a root of the

polynomial

.qr1qrCs qr qrCs1/X 2 .qr1prCs qr prCs1 C pr1qrCs pr qrCs1/X

C .pr1prCs pr prCs1/:(2.14)

Proof. It follows from (2.9) that

D Œ0I a1; : : : ; ar ; rC1 D pr 0 C pr1qr 0 C qr1

D prCs 0 C prCs1qrCs 0 C qrCs1

;

where 0 D ŒarC1I arC2; : : : ; arCs; arC1. Consequently, we get

0 D pr1 qr1

qr pr

D prCs1 qrCs1

qrCs prCs

;

from which we obtain

.pr

1

qr

1/.qr

Cs

pr

Cs /

D.pr

Cs

1

qr

Cs

1/.qr

pr /:

This shows that is a root of (2.14).



44 Yann Bugeaud

We do not claim that the polynomial in (2.14) is the minimal polynomial of over

the integers. This is indeed not always true, since its coefficients may have common

prime factors.

The sequence of partial quotients of an irrational real number ˛ in .0; 1/ can beobtained by iterations of the Gauss map T G defined by T G.0/ D 0 and T G.x/ D f1=xgfor x 2 .0; 1/. Namely, if Œ0I a1; a2; : : : denotes the continued fraction expansion of ˛, then T nG .˛/ D Œ0I anC1; anC2; : : : and an D b1=T n1

G .˛/c for n 1.

In the sequel, it is understood that ˛ is a real number in .0; 1/, whose partial quo-tients a1.˛/;a2. ˛/ ; : : : and convergents p1.˛/=q1.˛/;p2.˛/=q2. ˛/ ; : : : are written

a1; a2; : : : and p1=q1; p2=q2; : : :, respectively, when there is no danger of confusion.

The map T G possesses an invariant ergodic probability measure, namely the Gaussmeasure G, which is absolutely continuous with respect to the Lebesgue measure,

with density

G.dx/ D dx.1 C x/ log 2

:

For every function f in L1.G/ and almost every ˛ in .0; 1/, we have (Theorem 3.5.1

in [36])

limn!C1

1

n

n1XkD0

f .T kG ˛/ D 1

log 2

Z 1

0

f.x/

1 C x dx: (2.15)

For subsequent results in the metric theory of continued fractions, we refer the reader

to [47] and to [36].

Definition 5.9. We say that Œ0I a1; a2; : : : is a normal continued fraction, if, for every

integer k 1 and every positive integers d 1; : : : ; d k, we have

limN !C1

#fj W 0 j N k; aj C1 D d 1; : : : ; aj Ck D d kgN

DZ r 0=s0

r=s

G.dx/ D G.d 1;:::;d k /;

(2.16)

where r=s and r 0=s0 denote the rational numbers Œ0I d 1; : : : ; d k1; d k and Œ0I d 1; : : : ;d k

1; d k

C1, ordered so that r=s < r 0=s0, and d 1;:::;d k

DŒr=s;r 0=s0.

Let d 1; : : : ; d k be positive integers. It follows from Theorem 5.1 that the set

d 1;:::;d k of real numbers ˛ in .0; 1/ whose first k partial quotients are d 1; : : : ; d kis an interval of length 1=.qk.qk C qk1//. Applying (2.15) to the function f D1d1;:::;dk

, we get that for almost every ˛ D Œ0I a1; a2; : : : in .0;1/ the limit defined

in (2.16) exists and is equal to G.d 1;:::;d k /. Thus, we have established the followingstatement.

Theorem 5.10. Almost every ˛ in .0; 1/ has a normal continued fraction expansion.

The construction of [13], reproduced in [27], is flexible enough to produce manyexamples of real numbers with a normal continued fraction expansion.




6 Diophantine approximation

In this section, we survey classical results on approximation of real (algebraic) num-

bers by rational numbers.We emphasize one of the consequences of Theorem 5.1.

Theorem 6.1. For every real irrational number , there exist infinitely many rational

numbers p=q with q 1 and ˇ p

q

ˇ <

1

q2:

Theorem 6.1 is often, and wrongly, attributed to Dirichlet, who proved in 1842

a stronger result, namely that, under the assumption of Theorem 6.1 and for everyinteger Q

1, there exist integers p; q with 1

q

Q and

j

p=qj < 1=.qQ/.

Theorem 6.1 was proved long before 1842.

An easy covering argument shows that, for almost all numbers, the exponent of qin Theorem 6.1 cannot be improved.

Theorem 6.2. For every " > 0 and almost all real numbers , there exist only finitely

many rational numbers p=q with q 1 and ˇ

ˇ p

q

ˇ

ˇ <

1

q2C": (2.17)

Proof. Without loss of generality, we may assume that is in .0;1/. If there areinfinitely many rational numbers p=q with q 1 satisfying (2.17), then belongs tothe limsup set \

Q1

[qQ

q[pD0

p

q 1

q2C"; p

q C 1

q2C"

\ .0; 1/:

The Lebesgue measure of the latter set is, for every Q 1, at most equal to

XqQ

q 2

q2C";

which is the tail of a convergent series and thus tends to 0 as Q tends to infinity. Thisproves the theorem.

The case of algebraic numbers is of special interest and has a long history. First,we define the (naıve) height of an algebraic number.

Definition 6.3. Let be an irrational, real algebraic number of degree d and let

ad X d C C a1X C a0 denotes its minimal polynomial over Z .that is, the integer

polynomial of lowest positive degree, with coprime coefficients and positive leading

coefficient, which vanishes at /. Then, the height H./ of is defined by

H./ WD maxfja0j; ja1j; : : : ; jad jg:



46 Yann Bugeaud

We begin by a result of Liouville [51, 52] proved in 1844, and alluded to in Sec-

tion 1.

Theorem 6.4. Let be an irrational, real algebraic number of degree d and height at most H . Then, ˇ

p

q

ˇ 1

d 2H.1 C j j/d 1qd (2.18)

for all rational numbers p=q with q 1.

Proof. Inequality (2.18) is true when j p=qj 1. Let p=q be a rational numbersatisfying j p=qj < 1. Denoting by P.X/ the minimal defining polynomial of

over Z, we have P.p=q/ 6D 0 and jqd P.p=q/j 1. By Rolle’s Theorem, there

exists a real number t lying between and p=q such that

jP.p=q/j D jP./ P.p=q/j D j p=qj jP 0.t /j:Since jt j 1 and

jP 0.t /j d 2H.1 C j j/d 1;

the combination of these inequalities gives the theorem.

Thue [67] established in 1909 the first significant improvement of Liouville’s re-

sult. There was subsequent progress by Siegel, Dyson and Gelfond, until Roth [61]

proved in 1955 that, as far as approximation by rational numbers is concerned, the

irrational, real algebraic numbers do behave like almost all real numbers.

Theorem 6.5. For every " > 0 and every irrational real algebraic number , there

exist at most finitely many rational numbers p=q with q 1 and ˇ p

q

ˇ <

1

q2C": (2.19)

For a prime number ` and a non-zero rational number x, we set jxj` WD `u,

where u

2 Z is the exponent of ` in the prime decomposition of x . Furthermore, we

set j0j` D 0. The next theorem, proved by Ridout [59], extends Theorem 6.5.

Theorem 6.6. Let S be a finite set of prime numbers. Let be a real algebraic

number. Let " be a positive real number. The inequalityY`2S

jpqj` min

1;

ˇ p

q

ˇ <

1

q2C"

has only finitely many solutions in non-zero integers p; q.

Theorem 6.5 is ineffective, in the sense that its proofs do not allow us to computeexplicitly an integer q0 such that (2.19) has no solution with q greater than q0. Nev-

ertheless, we are able to bound explicitly the number of primitive solutions (that is,




of solutions in coprime integers p and q) to inequality (2.19). The first result in this

direction was proved in 1955 by Davenport and Roth [38]; see the proof of Theorem

12.1 for a recent estimate.

The Schmidt Subspace Theorem [62, 63, 64] is a powerful multidimensional ex-tension of the Roth Theorem, with many outstanding applications [20, 26, 69]. We

quote below a version of it which is suitable for our purpose, but the reader shouldkeep in mind that there are more general formulations.

Theorem 6.7. Let m 2 be an integer. Let S be a finite set of prime numbers. Let

L1;1; : : : ; Lm;1 be m linearly independent linear forms with real algebraic coeffi-

cients. For any prime ` in S , let L1;`; : : : ; Lm;` be m linearly independent linear

forms with integer coefficients. Let " be a positive real number. Then, there exist

an integer T and proper subspaces S 1; : : : ; S T of Qm such that all the solutions

x D .x1; : : : ; xm/ in Zm to the inequalityY`2S

mYiD1

jLi;`.x/j` mY

iD1

jLi;1.x/j .maxf1; jx1j; : : : ; jxmjg/" (2.20)

are contained in the union S 1 [ : : : [ S T .

Let us briefly show how Roth’s theorem can be deduced from Theorem 6.7. Let

be a real algebraic number and " be a positive real number. Consider the twoindependent linear forms X

Y and X . Theorem 6.7 implies that there are integers

T 1, x1; : : : ; xT , y1; : : : ; yT with .xi ; yi / 6D .0; 0/ for i D 1 ; : : : ; T , such that, forevery integer solution .p; q/ to

jqj jq pj < jqj";

there exists an integer k with 1 k T and xkp C yk q D 0. If is irrational, this

means that there are only finitely many rational solutions to j p=qj < jqj2",which is Roth’s theorem.

Note that (like Theorems 6.5 and 6.6) Theorem 6.7 is ineffective, in the sense that

its proof does not yield an explicit upper bound for the height of the proper rational

subspaces containing all the solutions to (2.20). Fortunately, Schmidt [65] was ableto give an admissible value for the number T of subspaces; see [42] for a commongeneralization of Theorems 6.6 and 6.7, usually called the Quantitative Subspace

Theorem, and [41] for the current state of the art.

7 Sketch of proof and historical comments

Before proving Theorems 3.1 and 3.2, we wish to highlight the main ideas and explain

how weaker results can be deduced from various statements given in Section 6. We

focus only on b-ary expansions.



48 Yann Bugeaud

The general idea goes as follows. Let us assume that (2.5) does not hold. Then,

the sequence of digits of our real number satisfies a certain combinatorial property.

And a suitable transcendence criterion prevents the sequence of digits of an irrational

algebraic numbers to fulfill the same combinatorial property.Let us see how transcendence results listed in Section 6 apply to get a combina-

torial transcendence criterion. We introduce some more notation. Let W be a finiteword. For a positive integer `, we write W ` for the word W : : : W (` times repeated

concatenation of the word W ) and W 1 for the infinite word constructed by concate-nation of infinitely many copies of W . More generally, for any positive real number

x, we denote by W x the word W bxcW 0, where W 0 is the prefix of W of length

d.x bxc/jW je. Here, de denotes the upper integer part function. In particular, wecan write

aabaaaabaaaa D .aabaa/12=5

D .aabaaaabaa/6=5

D .aabaa/log 10

:

Let a D .a`/`1 be a sequence of elements from A. Let w > 1 be a real number.We say that a satisfies Condition ./w if a is not

ultimately periodic and if there exist two sequences of finite words .Zn/n1, and

.W n/n1 such that:

(i) For every n 1, the word W nZwn is a prefix of the word a.

(ii) The sequence .jW nj=jZnj/n1 is bounded from above.

(iii) The sequence .jZnj/n1 is increasing.

We say that a satisfies Condition ./1 if it satisfies Condition ./w for everyw > 1.

Our first result is an application of Theorem 6.4 providing a combinatorial condi-

tion on the sequence b-ary expansion of a real number which ensures that this number

is trancendental.

Theorem 7.1. Let b 2 be an integer. Let a D .a`/`1 be a sequence of elements

from f0 ; 1 ; : : : ; b 1g. If a satisfies Condition ./1 , then the real number

WDC1

XD1

a`

b`

is transcendental.

Proof. Let w > 1 be a real number. By assumption, there exist two sequences of finite words, .Zn/n1, and .W n/n1, and an integer C , such that jW nj C jZnj and

W nZwn is a prefix of a for n 1. Let n 1 be an integer. In particular, is very

close to the rational number n whose b-ary expansion is the eventually periodic word

W nZ1n . A rapid calculation shows that there is an integer pn such that

n D pn

bjW nj.bjZnj 1/




and

j

n

j D ˇ

pn

bjW n

j.bjZn

j 1/ ˇ 1

bjW n

jCw

jZn

j 1

bjW nj.bjZnj 1/

.jW njCwjZnj/=.jW njCjZnj/:

Since jW nj C jZnj, the quantity .jW nj C wjZnj/=.jW nj C jZnj/ is bounded frombelow by .C C w/=.C C 1/. Consequently, we getˇ

pn

bjW nj.bjZnj 1/

ˇ 1

bjW nj.bjZnj 1/

.C Cw/=.C C1/

: (2.21)

Let d be the integer part of .C C w/=.C C 1/. Since (2.21) holds for every n 1, it

follows from Theorem 6.4 that cannot be algebraic of degree d 1. Since w canbe taken arbitrarily large, one deduces that must be transcendental.

It is apparent from the proof of Theorem 7.1 that, if instead of Liouville’s theoremwe use Roth’s (Theorem 6.5) or, even better, Ridout’s Theorem 6.6, then the assump-

tions of Theorem 7.1 can be substantially weakened. Indeed, Roth’s theorem is suffi-

cient to establish the transcendence of as soon as the exponent .C C w/=.C C 1/strictly exceeds 2, that is, if w > 2 C C . We explain below how Ridout’s theorem

yields a much better result.

Theorem 7.2. Let b

2 be an integer. Let a

D .a`/`

1 be a sequence of elements

from f0 ; 1 ; : : : ; b 1g. Let w > 2 be a real number. If a satisfies Condition ./w ,then the real number

WDC1X`D1

a`

b`

is transcendental.

Proof. We keep the notation of the proof of Theorem 7.1, where it is shown that, for

any n 1, we have

ˇ pn

bjW nj.bjZnj 1/ ˇ 1

bjW njCwjZnj ;

that is,

bjW njˇ pn

bjW nj.bjZnj 1/

ˇ 1

b2jW njCwjZnj

< 1

bjW nj.bjZnj 1/

.2jW njCwjZnj/=.jW njCjZnj/:

Observe that, for n 1,

2jW nj C wjZnjjW nj C jZnj D 2 C .w 2/jZnj

jW nj C jZnj 2 C w 2C C 1

:



50 Yann Bugeaud

Hence, if we take for S the set of prime divisors of b and set " WD .w 2/=.C C 1/,

we conclude that there are infinitely many rational numbers p=q such that

Y2S

jpqj` min1; ˇ pqˇ < 1

q2C" :

By Theorem 6.6, this shows that is transcendental, since w > 2.

We postpone to Section 9 the proof that the conclusion of Theorem 7.2 remains

true under the weaker assumption w > 1 (the reader can easily check that this corre-

sponds exactly to Theorem 4.1).

8 A combinatorial lemma

The purpose of this section is to establish a combinatorial lemma which allows us to

deduce Theorems 3.1 and 3.2 from Theorems 4.1 and 4.2.

Lemma 8.1. Let w D w1w2 : : : be an infinite word over a finite or an infinite alpha-

bet A such that

lim inf n!C1

p.n; w;A/

n < C1:

Then, the word w satisfies Condition ./ defined in Section 4.

Proof. By assumption, there exist an integer C 2 and an infinite set N of positive

integers such thatp.n; w;A/ C n; for every n in N : (2.22)

This implies, in particular, that w is written over a finite alphabet.Let n be in N . By (2.22) and the Schubfachprinzip, there exists (at least) one block

X n of length n having (at least) two occurrences in the prefix of length .C C 1/n of w. Thus, there are words W n, W 0n, Bn and B 0

n such that jW nj < jW 0nj and

w1 : : : w.C C1/n D W nX nBn D W 0nX nB 0n:If jW nX nj jW 0nj, then define V n by the equality W nX nV n D W 0n. Observe that

w1 : : : w.C C1/n D W nX nV nX nB 0n (2.23)

and jV nj C jW njjX nj C: (2.24)

Set U n WD X n.




If jW 0nj < jW nX nj, then, recalling that jW nj < jW 0nj, we define X 0n by W 0n DW nX 0n. Since X nBn D X 0nX nB 0

n and jX 0nj < jX nj, the word X 0n is a strict prefix of

X n and X n is the concatenation of at least two copies of X 0n and a (possibly empty)

prefix of X 0n. Let tn be the largest positive integer such that X n begins with 2tn copiesof X 0n. Observe that

2tnjX 0nj C 2jX 0nj jX 0nX nj;thus

n D jX nj .2tn C 1/jX 0nj 3tnjX 0nj:Consequently, W n.X 0n

tn /2 is a prefix of w such that

jX 0ntn j n=3

and jW njjX 0n

tnj 3

n .C C 1/n 2jX 0n

tn j 3C C 1: (2.25)

Set U n WD X 0ntn and let V n be the empty word.

It then follows from (2.23), (2.24), and (2.25) that, for every n in the infinite set

N ,

W nU nV nU n is a prefix of w

with

jW n

j C jV n

j .3C

C1/

jU n

j:

This shows that w satisfies Condition ./.

We are now in position to deduce Theorems 3.1 and 3.2 from Theorems 4.1

and 4.2.

Proof of Theorem 3.1. Let b 2 be an integer and be an irrational real number.Assume that p.n;;b/ does not tend to infinity with n. It then follows from Lemma

8.1 that the

infinite word composed of the digits of written in base b satisfies Condition ./.Consequently, Theorem 4.1 asserts that cannot be algebraic. By contraposition, we

get the theorem.

Proof of Theorem 3.2. Let ˛ be a real number not algebraic of degree at most two.

Assume that p.n; ˛/ does not tend to infinity with n. It then follows from Lemma 8.1that the

infinite word composed of the partial quotients of satisfies Condition ./. Fur-thermore, p.1; ˛/ is finite, thus the sequence of partial quotients of ˛ is bounded, say

by M . It then follows from Theorem 5.3 that q` .M C 1/` for ` 1, hence the

sequence .q1=`

` /`1 is bounded. Consequently, all the hypotheses of Theorem 4.2 are

satisfied, and one concludes that ˛ cannot be algebraic of degree at least three. By

contraposition, we get the theorem.



52 Yann Bugeaud

9 Proof of Theorem 4.1

We present two proofs of Theorem 4.1, which was originally established in [10].

Throughout this section, we set jU nj D un, jV nj D vn and jW nj D wn, for n 1.We assume that is algebraic and we derive a contradiction by a suitable application

of Theorem 6.7.

First proof. Let n 1 be an integer. We observe that the real number is quite close

to the rational number n whose b-ary expansion is the infinite word W n.U nV n/1.Indeed, there exists an integer pn such that

n D pn

bwn .bunCvn 1/

and j nj 1

bwnCvnC2un;

since and n have the same first wn C vn C 2un digits in their b-ary expansion.

Consequently, we have

jbwnCunCvn bwn pnj D jbwn .bunCvn 1/ pnj bun :

Consider the three linearly independent linear forms with real algebraic coefficients

L1;

1.X 1; X 2; X 3/

DX 1;

L2;1.X 1; X 2; X 3/ D X 2;

L3;1.X 1; X 2; X 3/ D X 1 X 2 X 3:

Evaluating them on the integer points xn WD .bwnCunCvn ; bwn ; pn/, we get thatY1j 3

jLj;1.xn/j b2wnCvn : (2.26)

For any prime number ` dividing b, consider the three linearly independent linearforms with integer coefficients

L1;`.X 1; X 2; X 3/ D X 1;

L2;`.X 1; X 2; X 3/ D X 2;

L3;`.X 1; X 2; X 3/ D X 3:

We get that Y`jb

Y1j 3

jLj;`.xn/j` b2wnunvn : (2.27)

Since a satisfies Condition ./, we have

lim inf n!C1

un

wn C un C vn

> 0:




It then follows from (2.26) and (2.27) that there exists " > 0 such that

Y1j 3 jLj;

1.xn/

j Yjb Y1j 3 jLj;`.xn/

j`

bun

maxfbwnCunCvn ; bwn ; png";

for every n 1.

Now we infer from Theorem 6.7 that all the points xn lie in a finite number of

proper subspaces of Q3. Thus, there exist a non-zero integer triple .z1; z2; z3/ and aninfinite set of distinct positive integers N 1 such that

z1bwnCunCvn C z2bwn C z3pn D 0; (2.28)

for any n in N 1.Dividing (2.28) by bwnCunCvn , we get

z1 C z2bunvn C z3

pn

bwnCunCvnD 0: (2.29)

Since un tends to infinity with n, the sequence .pn=bwnCunCvn /n1 tends to . Let-

ting n tend to infinity along N 1, we then infer from (2.29) that either is rational, or

z1 D z3 D 0. In the latter case, z2 must be zero, a contradiction. This shows that cannot be algebraic.

Second proof. Here, we follow an alternative approach presented in [2].Let pn and p0

n be the rational integers defined by

wnCvnC2unX`D1

a`

b` D pn

bwnCvnC2unand

wnCunX`D1

a`

b` D p0

n

bwnCun

Observe that there exist integers f n and f 0n such that

pn D awnCvnC2unC awnCvnC2un1b C C awnCvnCunC1bun1 C f nbun (2.30)

and

p0n D awnCun

C awnCun1b C C awnC1bun1 C f 0nbun : (2.31)

Since, by assumption,

awnCunCvnCj D awnCj ; for j D 1 ; : : : ; un;

it follows from (2.30) and (2.31) that pn p0n is divisible by an integer multiple of

bun . Thus, for any prime number ` dividing b , the `-adic distance between pn andp0

n is very small and we have

jpn p0nj` jbjun

` :



54 Yann Bugeaud

Furthermore, it is clear that

jbwnCun p0nj < 1 and jbwnCvnC2un pnj < 1:

Consider now the four linearly independent linear forms with real algebraic coef-

ficients

L1;1.X 1; X 2; X 3; X 4/ D X 1;

L2;1.X 1; X 2; X 3; X 4/ D X 2;

L3;1.X 1; X 2; X 3; X 4/ D X 1 X 3;

L4;1.X 1; X 2; X 3; X 4/ D X 2 X 4:

Evaluating them on the integer points xn WD .bwnCvnC2un ; bunCwn ; pn; p0n/, we get

that Y1j 4

jLj;1.xn/j b2wnCvnC3un : (2.32)

For any prime number p dividing b, we consider the four linearly independentlinear forms with integer coefficients

L1;`.X 1; X 2; X 3; X 4/ D X 1;

L2;` .X 1; X 2; X 3; X 4/ D X 2;

L3;`.X 1; X 2; X 3; X 4/ D X 3;

L4;` .X 1; X 2; X 3; X 4/

DX 4

X 3:

We get that Y`jb

Y1j 4

jLj;`.xn/j` b.2wnCvnC3un/ bun : (2.33)


lim inf n!C1

un

wn C 2un C vn

> 0:

It then follows from (2.32) and (2.33) that there exists " > 0 such that

Y1j 4

jLj;1.xn/j Yjb

Y1j 4

jLj;`.xn/j` bun

maxfbwnCvnC2un ; bunCwn ; pn; p0ng";

for every n 1.We then infer from Theorem 6.7 that all the points xn lie in a finite number of

proper subspaces of Q4. Thus, there exist a non-zero integer quadruple .z1; z2; z3; z4/and an infinite set of distinct positive integers N 1 such that

z1bwnCvnC2un

Cz2bunCwn

Cz3pn

Cz4p0

n

D0; (2.34)

for any n in N 1.




Dividing (2.34) by bwnCvnC2un , we get

z1

Cz2bunvn

Cz3

pn

bwnCvnC2un Cz4bunvn

p0n

bunCwn D0: (2.35)

Recall that un tends to infinity with n. Thus, the sequences .pn=bwnCvnC2un /n1

and .p0n=bunCwn /n1 tend to as n tends to infinity. Letting n tend to infinity along

N 1, we infer from (2.35) that either is rational, or z1 D z3 D 0. In the latter case,we obtain that is rational. This is a contradiction, since the sequence .a`/`1 is not

ultimately periodic. Consequently, cannot be algebraic.

Also, the Schmidt Subspace Theorem was applied similarly as in the first proof of Theorem 4.1 by Troi and Zannier [68] to establish the transcendence of the number

Pm2S 2

m, where S denotes the set of integers which can be represented as sums of

distinct terms 2k C 1, where k 1.

Moreover, in his short paper Some suggestions for further research published in

1984, Mahler [53] suggested explicitly to apply the Schmidt Subspace Theorem ex-

actly as in the first proof of Theorem 4.1 given in Section 9 to investigate whether the

middle third Cantor set contains irrational algebraic elements or not. More precisely,

he wrote:

A possible approach to this question consists in the study of the non-

homogeneous linear expressions

j3prCP r X 3pr X N r j: It may be that a p-adic form of Schmidt’s theorem on the rational ap-

proximations of algebraic numbers Œ10 holds for such expressions.

The reference [10] above is Schmidt’s book [64].

We end this section by mentioning an application of Theorem 4.1. Adamczewskiand Rampersad [12] proved that the binary expansion of an algebraic number contains

infinitely many occurrences of 7=3-powers. They also established that the ternaryexpansion of an

algebraic number contains infinitely many occurrences of squares, or infinitelymany occurrences of one of the blocks 010 or 02120.

10 Proof of Theorem 4.2

We reproduce the proof given in [29].

Throughout, the constants implied in depend only on ˛. Assume that the

sequences .U n/n

1, .V n/n

1, and .W n/n

1 occurring in the definition of Condition

./ are fixed. For n 1, set un D jU nj, vn D jV nj, and wn D jW nj. We assume



56 Yann Bugeaud

that the real number ˛ WD Œ0I a1; a2; : : : is algebraic of degree at least three. Set

p1 D q0 D 1 and q1 D p0 D 0.

We observe that ˛ admits infinitely many good quadratic approximants obtained

by truncating its continued fraction expansion and completing by periodicity. Pre-cisely, for every positive integer n, we define the sequence .b

.n/

k /k1 by

b.n/

h D ah for 1 h wn C un C vn,

b.n/

wnChCj.unCvn/ D awnCh for 1 h un C vn and j 0.

The sequence .b.n/

k /k1 is ultimately periodic, with preperiod W n and with period

U nV n. Set

˛n

D 0Ib

.n/1 ; b

.n/2 ; : : : ; b

.n/

k ; : : : and note that, since the first wn C 2un C vn partial quotients of ˛ and of ˛n are the

same, it follows from Corollary 5.2 that

j˛ ˛nj q2wnC2unCvn

: (2.36)

Furthermore, Lemma 5.8 asserts that ˛n is root of the quadratic polynomial

P n.X / WD .qwn1qwnCunCvn qwn

qwnCunCvn1/X 2

.qwn1pwnCunCvn qwn pwnCunCvn1

C pwn1qwnCunCvn pwn qwnCunCvn1/X C .pwn1pwnCunCvn pwn

pwnCunCvn1/:

By (2.10), we have

j.qwn1qwnCunCvnqwn

qwnCunCvn1/˛ .qwn1pwnCunCvnqwn

pwnCunCvn1/j qwn1jqwnCunCvn ˛ pwnCunCvn

j C qwnjqwnCunCvn1˛ pwnCunCvn1j

2 qwn q1wnCunCvn

(2.37)and, likewise,

j.qwn1qwnCunCvn qwn qwnCunCvn1/˛ .pwn1qwnCunCvn

pwn qwnCunCvn1/j qwnCunCvn

jqwn1˛ pwn1j C qwnCunCvn1jqwn˛ pwn

j 2 q1

wn qwnCunCvn :

(2.38)

Using (2.36), (2.37), and (2.38), we then get

jP n.˛/j D jP n.˛/ P n.˛n/jD j.qwn1qwnCunCvn

qwn qwnCunCvn1/.˛ ˛n/.˛ C ˛n/

.qwn1pwnCunCvn qwn pwnCunCvn1

C pwn1qwnCunCvn pwn qwnCunCvn1/.˛ ˛n/j




D j.qwn1qwnCunCvn qwn

qwnCunCvn1/˛

.qwn1pwnCunCvn qwn pwnCunCvn1/

C.q

wn1q

wnCunCvn q

wn

qwnCunCvn1

/˛

.pwn1qwnCunCvn pwn

qwnCunCvn1/

C .qwn1qwnCunCvn qwn qwnCunCvn1/.˛n ˛/j j˛ ˛nj

j˛ ˛nj qwn q1wnCunCvn

C q1wn

qwnCunCvnC qwn qwnCunCvn

j˛ ˛nj j˛ ˛njq1

wn qwnCunCvn

q1wn

qwnCunCvn q2

wnC2unCvn: (2.39)

We consider the four linearly independent linear forms

L1.X 1; X 2; X 3; X 4/

D˛2X 1

˛.X 2

CX 3/

CX 4;

L2.X 1; X 2; X 3; X 4/ D ˛X 1 X 2;

L3.X 1; X 2; X 3; X 4/ D ˛X 1 X 3;

L4.X 1; X 2; X 3; X 4/ D X 1:

Evaluating them on the 4-tuple

xn WD.qwn1qwnCunCvn qwn qwnCunCvn1;

qwn1pwnCunCvn qwn pwnCunCvn1;

pwn1qwnCunCvn pwn

qwnCunCvn1;

pwn1pwnCunCvn pwn pwnCunCvn1/;it follows from (2.37), (2.38), (2.39), and Theorem 5.3 thatY

1j 4

jLj .xn/j q2wnCunCvn

q2wnC2unCvn

2un

.qwn qwnCunCvn /ıun=.2wnCunCvn/;

if n is sufficiently large, where we have set

M D 1 C lim sup`!C1 q

1=`

` and ı D log 2

log M :


lim inf n!C1

un

2wn C un C vn

> 0:

Consequently, there exists " > 0 such thatY1j 4

jLj .xn/j .qwn qwnCunCvn

/"

holds for any sufficiently large integer n.



58 Yann Bugeaud

It then follows from Theorem 6.7 that the points xn lie in a finite union of proper

linear subspaces of Q4. Thus, there exist a non-zero integer quadruple .x1; x2; x3; x4/and an infinite set N 1 of distinct positive integers such that

x1.qwn1qwnCunCvn qwn qwnCunCvn1/

Cx2.qwn1pwnCunCvn qwn pwnCunCvn1/

Cx3.pwn1qwnCunCvn pwn qwnCunCvn1/

Cx4.pwn1pwnCunCvn pwn

pwnCunCvn1/ D 0;

(2.40)

for any n in N 1.

First case: we assume that there exist an integer ` and infinitely many integers

n in N 1 with wn D `.

By extracting an infinite subset of N 1 if necessary and by considering the real

number Œ0I a`C1; a`C2; : : : instead of ˛ , we may without loss of generality assumethat wn D ` D 0 for any n in N 1.

Then, recalling that q1 D p0 D 0 and q0 D p1 D 1, we deduce from (2.40)

thatx1qunCvn1 C x2punCvn1 x3qunCvn

x4punCvn D 0; (2.41)

for any n in N 1. Observe that .x1; x2/ 6D .0;0/, since, otherwise, by letting n tend to

infinity along N 1 in (2.41), we would get that the real number ˛ is rational. Dividing(2.41) by qunCvn , we obtain

x1

qunCvn1

qunCvn

C x2

punCvn1

qunCvn1

qunCvn1

qunCvn

x3 x4

punCvn

qunCvn

D 0: (2.42)

By letting n tend to infinity along N 1 in (2.42), we get that

ˇ WD lim N 13n!C1

qunCvn1

qunCvn

D x3 C x4˛

x1 C x2˛:

Furthermore, observe that, for any sufficiently large integer n in N 1, we have

ˇˇ qunCvn1

qunCvn

ˇ D ˇx3 C x4˛x1 C x2˛

x3 C x4punCvn =qunCvn

x1 C x2punCvn1=qunCvn1

ˇ 1

qunCvn1qunCvn

;

(2.43)

by (2.10). Since the rational number qunCvn1=qunCvn is in its reduced form and

un C vn tends to infinity when n tends to infinity along N 1, we see that, for everypositive real number and every positive integer N , there exists a reduced rational

number a=b such that b > N and jˇ a=bj =b . This implies that ˇ is irrational.

Consider now the three linearly independent linear forms

L01.Y 1; Y 2; Y 3/ D ˇY 1 Y 2; L0

2.Y 1; Y 2; Y 3/ D ˛Y 1 Y 3; L03.Y 1; Y 2; Y 3/ D Y 2:




Evaluating them on the triple .qunCvn; qunCvn1; punCvn

/ with n 2 N 1, we infer

from (2.10) and (2.43) that

Y1j 3

jL0j .qunCvn ; qunCvn1; punCvn /j q1unCvn :

It then follows from Theorem 6.7 that the points .qunCvn ; qunCvn1; punCvn / with

n 2 N 1 lie in a finite union of proper linear subspaces of Q3. Thus, there exista non-zero integer triple .y1; y2; y3/ and an infinite set of distinct positive integers

N 2 N 1 such that

y1qunCvn C y2qunCvn1 C y3punCvn

D 0; (2.44)

for any n in N

2. Dividing (2.44) by qunCvn and letting n tend to infinity along N

2,we get

y1 C y2ˇ C y3˛ D 0: (2.45)

To obtain another equation relating ˛ and ˇ, we consider the three linearly inde-

pendent linear forms

L001.Z1; Z2; Z3/ D ˇZ1 Z2; L00

2.Z1; Z2; Z3/ D ˛Z2 Z3;

L003.Z1; Z2; Z3/ D Z2:

Evaluating them on the triple .qun

Cvn ; qun

Cvn

1; pun

Cvn

1/ with n in N 1, we infer

from (2.10) and (2.43) thatY1j 3

jL00j .qunCvn ; qunCvn1; punCvn1/j q1

unCvn:

It then follows from Theorem 6.7 that the points .qunCvn; qunCvn1; punCvn1/ with

n 2 N 1 lie in a finite union of proper linear subspaces of Q3. Thus, there exist a non-

zero integer triple .z1; z2; z3/ and an infinite set of distinct positive integers N 3 N 2such that

z1qunCvn C z2qunCvn1 C z3punCvn1 D 0; (2.46)

for any n in N 3. Dividing (2.46) by qunCvn1 and letting n tend to infinity along N 3,we get

z1

ˇ C z2 C z3˛ D 0: (2.47)

We infer from (2.45) and (2.47) that

.z3˛ C z2/.y3˛ C y1/ D y2z1:

Since ˇ is irrational, we get from (2.45) and (2.47) that y3z3 6D 0. This shows that ˛is an algebraic number of degree at most two, which contradicts our assumption that

˛ is algebraic of degree at least three.



60 Yann Bugeaud

Second case: extracting an infinite subset N 4 of N 1 if necessary, we assume

that .wn/n2 N 4 tends to infinity.

In particular .pwn =qwn /n2 N 4 and .pwn

Cun

Cvn =qwn

Cun

Cvn /n

2 N 4 both tend to ˛

as n tends to infinity.We make the following observation. Let a be a letter and U; V ; W be three finite

words (V may be empty) such that a begins with W U V U and a is the last letter of

W and of U V . Then, writing W D W 0a, V D V 0a if V is non-empty, and U D U 0aif V is empty, we see that a begins with W 0.aU/V 0.aU/ if V is non-empty and with

W 0.aU 0/.aU 0/ if V is empty. Consequently, by iterating this remark if necessary, wecan assume that for any n in N 4, the last letter of the word U nV n differs from the last

letter of the word W n. Said differently, we have awn 6D awnCunCvn for any n in N 4.

Divide (2.40) by qwn qwnCunCvn1 and write

Qn WD .qwn1qwnCunCvn /=.qwn qwnCunCvn1/:

We then get

x1.Qn 1/ C x2

Qn

pwnCunCvn

qwnCunCvn

pwnCunCvn1

qwnCunCvn1

C x3

Qn

pwn1

qwn1

pwn

qwn

Cx4Qn

pwn1

qwn1

pwnCunCvn

qwnCunCvn

pwn

qwn

pwnCunCvn1

qwnCunCvn1 D0;

(2.48)

for any n in N 4. To shorten the notation, for any ` 1, we put R` WD ˛ p`=q` and

rewrite (2.48) as

x1.Qn 1/ C x2

Qn.˛ RwnCunCvn

/ .˛ RwnCunCvn1/

Cx3

Qn.˛ Rwn1/ .˛ Rwn

/

Cx4

Qn.˛ Rwn1/.˛ RwnCunCvn

/ .˛ Rwn/.˛ RwnCunCvn1/

D 0:

This yields

.Qn 1/x1 C .x2 C x3/˛ C x4˛2 D x2QnRwnCunCvn x2RwnCunCvn1

C x3QnRwn1 x3Rwn

x4QnRwn1RwnCunCvn

C x4Rwn RwnCunCvn1 C ˛.x4QnRwn1

C x4QnRwnCunCvn x4Rwn

x4RwnCunCvn1/:(2.49)

Observe that

jR`j q1

` q1

`C1; ` 1; (2.50)by (2.10).




We use (2.49), (2.50) and the assumption that awn 6D awnCunCvn

for any n in N 4to establish the following claim.

Claim. We havex1 C .x2 C x3/˛ C x4˛2 D 0:

Proof of the Claim. If there are arbitrarily large integers n in N 4 such that Qn 2or Qn 1=2, then the claim follows from (2.49) and (2.50).

Assume that 1=2 Qn 2 holds for every large n in N 4. We then derive from

(2.49) and (2.50) that

j.Qn 1/.x1 C .x2 C x3/˛ C x4˛2/j jRwn1j q1wn1q1

wn:

If x1 C .x2 C x3/˛ C x4˛2

6D 0, then we get

jQn 1j q1wn1q1

wn: (2.51)

On the other hand, observe that, by Theorem 5.4, the rational number Qn is the quo-

tient of the two continued fractions ŒawnCunCvnI awnCunCvn1; : : : ; a1 and

ŒawnI awn1; : : : ; a1. Since awnCunCvn

6D awn , we have either awnCunCvn awn

1 or awn

awnCunCvn 1. In the former case, we see that

Qn awnCunCvn

awn C 1

1 C 1

awn2 C 1

awn C 1

awn C awn

2

C1

awn2 C 2

1C 1

.awn

C1/.awn

2

C2/

:

In the latter case, we have

1

Qn

awn

C 1

awn1C1

awnCunCvnC1

1C 1

.awn1C1/.awnCunCvnC1/

1C 1

.awn1C1/awn

:

Consequently, in any case,

jQn 1j a1wn

minfa1wn2; a1

wn1g a1wn

q1wn1:

Combined with (2.51), this gives

awn qwn

awn qwn1;

which implies that n is bounded, a contradiction. This proves the Claim.

Since ˛ is irrational and not quadratic, we deduce from the Claim that x1 D x4 D0 and x2 D x3. Then, x2 is non-zero and, by (2.40), we have, for any n in N 4,

qwn1pwnCunCvn qwn pwnCunCvn1 D pwn1qwnCunCvn

pwn qwnCunCvn1:



62 Yann Bugeaud

Thus, the polynomial P n.X / can be simply expressed as

P n.X / WD .qwn1qwnCunCvn qwn qwnCunCvn1/X 2

2.qwn1pwnCunCvn qwn pwnCunCvn1/X C .pwn1pwnCunCvn pwn pwnCunCvn1/:


L0001 .T 1; T 2; T 3/ D ˛2T 1 2˛T 2 C T 3;

L0002 .T 1; T 2; T 3/ D ˛T 1 T 2;

L0003 .T 1; T 2; T 3/ D T 1:

Evaluating them on the triple

x0n WD

.qwn

1qwn

Cun

Cvn

qwn

qwn

Cun

Cvn

1;

qwn1pwnCunCvn qwn pwnCunCvn1;

pwn1pwnCunCvn pwn pwnCunCvn1/;

for n in N 4, it follows from (2.37) and (2.39) thatY1j 3

jL000j .x0n/j qwn qwnCunCvn q2

wnC2unCvn .qwn qwnCunCvn /";

with the same " as above, if n is sufficiently large.We then deduce from Theorem 6.7 that the points x0

n, n 2 N 4, lie in a finite union

of proper linear subspaces of Q3. Thus, there exist a non-zero integer triple .t1; t2; t3/and an infinite set of distinct positive integers N 5 included in N 4 such that

t1.qwn1qwnCunCvn qwn

qwnCunCvn1/

Ct2.qwn1pwnCunCvn qwn pwnCunCvn1/

Ct3.pwn1pwnCunCvn pwn pwnCunCvn1/ D 0;

(2.52)

for any n in N 5.

We proceed exactly as above. Divide (2.52) by qwn qwnCunCvn1 and set

Qn WD .qwn1qwnCunCvn/=.qwn

qwnCunCvn1/:

We then get

t1.Qn 1/ C t2

Qn

pwnCunCvn

qwnCunCvn

pwnCunCvn1

qwnCunCvn1

C t3

Qn

pwn1

qwn1

pwnCunCvn

qwnCunCvn

pwn

qwn

pwnCunCvn1

qwnCunCvn1

D 0;

(2.53)

for any n in N 5. We argue as after (2.48). Since pwn=qwn and pwnCunCvn=qwnCunCvn

tend to ˛ as n tends to infinity along N 5, we derive from (2.53) that

t1 C t2˛ C t3˛2 D 0;

a contradiction since ˛ is irrational and not quadratic. Consequently, ˛ must be tran-scendental. This concludes the proof of the theorem.




11 A transcendence criterion for quasi-palindromic

continued fractions

In this section, we present another combinatorial transcendence criterion for contin-

ued fractions which was established in [29], based on ideas from [5].For a finite word W WD w1 : : : wk, we denote by W WD wk : : : w1 its mirror

image. The finite word W is called a palindrome if W D W .Let a D .a`/`1 be a sequence of elements from A. We say that a satisfies

Condition .|/ if a is not

ultimately periodic and if there exist three sequences of finite words .U n/n1,

.V n/n1, and .W n/n1 such that:

(i) For every n 1, the word W nU nV nU n is a prefix of the word a.

(ii) The sequence .jV nj=jU nj/n1 is bounded from above.(iii) The sequence .jW nj=jU nj/n1 is bounded from above.

(iv) The sequence .jU nj/n1 is increasing.

Theorem 11.1. Let a D .a`/`1 be a sequence of positive integers. Let .p`=q`/`1

denote the sequence of convergents to the real number

˛ WD Œ0I a1; a2; : : : ; a`; : : : :

Assume that the sequence .q1=`

` /`1 is bounded. If a satisfies Condition .

|/ , then ˛

is transcendental.

A slight modification of the proof of Theorem 11.1 allows us to remove the as-

sumption on the growth of the sequence .q`/`1, provided that a stronger condition

than Condition .|/ is satisfied.

Theorem 11.2. Let a D .a`/`1 be a sequence of positive integers and set

˛ WD Œ0I a1; a2; : : : ; a`; : : : :

Assume that a

D .a`/`

1 is not eventually periodic. If there are arbitrarily large

integers ` such that the word a1 : : : a` is a palindrome, then ˛ is transcendental.

We leave to the reader the proof of Theorem 11.2, established in [5], and establish

Theorem 11.1.

Proof. Throughout, the constants implied in are absolute.

Assume that the sequences .U n/n1, .V n/n1, and .W n/n1 are fixed. Set wn DjW nj, un D jU nj and vn D jV nj, for n 1. Assume that the real number ˛ WDŒ0I a1; a2; : : : is algebraic of degree at least three.

For n 1, consider the rational number P n=Qn defined by

P nQn

WD Œ0I W nU nV nU n W n



64 Yann Bugeaud

and denote by P 0n=Q0n the last convergent to P n=Qn which is different from P n=Qn.

Since the first wn C 2un C vn partial quotients of ˛ and P n=Qn coincide, we deduce

from Corollary 5.2 that

jQn˛ P nj < Qnq2wnC2unCvn

; jQ0n˛ P 0nj < Qnq2

wnC2unCvn: (2.54)

Furthermore, it follows from Theorem 5.4 that the first wn C 2un C vn

partial quotients of ˛ and of Q0n=Qn coincide, thus

jQn˛ Q0nj < Qnq2

wnCun; (2.55)

again by Corollary 5.2, and, by Theorem 5.6,

Qn

2qwn qwnC2unCvn

2qwnCun qwnC2unCvn : (2.56)

Since

˛.Qn˛ P n/ .Q0n˛ P 0n/ D ˛Qn

˛ P n

Qn

Q0

n

˛ P 0n

Q0n

D .˛Qn Q0

n/

˛ P n

Qn

C Q0

n

P 0nQ0

n

P n

Qn

;

it follows from (2.54), (2.55), and (2.56) that

j˛2Qn

˛Q0

n ˛P n

CP 0

nj Qnq2

wnCun

q2

wnC2unCvn CQ1

n Q1n :

(2.57)

Consider the four linearly independent linear forms with algebraic coefficients

L1.X 1; X 2; X 3; X 4/ D ˛2X 1 ˛X 2 ˛X 3 C X 4;

L2.X 1; X 2; X 3; X 4/ D ˛X 2 X 4;

L3.X 1; X 2; X 3; X 4/ D ˛X 1 X 2;

L4.X 1; X 2; X 3; X 4/ D X 2:

We deduce from (2.54), (2.55), (2.56), and (2.57) thatY1j 4

jLj .Qn; Q0n; P n; P 0n/j Q2

n q2wnC2unCvn

q2wnCun

q2wn

q2wnCun

:

By combining Theorems 5.3 and 5.6 with (2.56), we have

q2wn

q2wnCun

2un Qıun=.2wnC2unCvn/n ;

if n is sufficiently large, where we have set

M D 1 C lim sup`!C1

q1=``

and ı D log 2log M

:




Since a satisfies Condition .|/, we have

lim inf

n!C1

un

2wn C 2un C vn

> 0:

Consequently, there exists " > 0 such thatY1j 4

jLj .Qn; Q0n; P n; P 0n/j Q"

n ;

for every sufficiently large n.

It then follows from Theorem 6.7 that the points .Qn; Q0n; P n; P 0n/ lie in a fi-

nite number of proper subspaces of Q4. Thus, there exist a non-zero integer 4-tuple

.x1; x2; x3; x4/ and an infinite set of distinct positive integers N 1 such that

x1Qn C x2Q0n C x3P n C x4P 0n D 0; (2.58)

for any n in N 1. Dividing by Qn, we obtain

x1 C x2

Q0n

Qn

C x3

P n

Qn

C x4

P 0nQ0

n

Q0n

Qn

D 0:

By letting n tend to infinity along N 1, we infer from (2.55) and (2.56) that

x1 C .x2 C x3/˛ C x4˛2 D 0:

Since .x1; x2; x3; x4/ 6D .0;0;0;0/ and since ˛ is irrational and not quadratic, we

have x1 D x4 D 0 and x2 D x3. Then, (2.58) implies that

Q0n D P n:

for every n in N 1. Thus, for n in N 1, we have

j˛2Qn 2˛Q0n C P 0nj Q1

n : (2.59)


L01.X 1; X 2; X 3/ D ˛2X 1 2˛X 2 C X 3;

L02.X 1; X 2; X 3/ D ˛X 2 X 3;

L03.X 1; X 2; X 3/ D X 1:

Evaluating them on the triple .Qn; Q0n; P 0n/ for n in N 1, it follows from (2.54), (2.56)

and (2.59) thatY1j 3

jL0j .Qn; Q0

n; P 0n/j Qnq2wnC2unCvn

qwn q1

wn

C2un

Cvn

qwn q1

wn

Cun

Q"=2

n ;

with the same " as above, if n is sufficiently large.



66 Yann Bugeaud

It then follows from Theorem 6.7 that the points .Qn; Q0n; P 0n/ lie in a finite num-

ber of proper subspaces of Q3. Thus, there exist a non-zero integer triple .y1; y2; y3/and an infinite set of distinct positive integers N 2 such that

y1Qn C y2Q0n C y3P 0n D 0; (2.60)

for any n in N 2.Dividing (2.60) by Qn, we get

y1 C y2

P n

Qn

C y3

P 0nQ0

n

P n

Qn

D 0: (2.61)

By letting n tend to infinity along N 2, it thus follows from (2.61) that

y1 C y2˛ C y3˛2 D 0:

Since .y1; y2; y3/ is a non-zero triple of integers, we have reached a contradiction.Consequently, the real number ˛ is transcendental. This completes the proof of the

theorem.

12 Complements

We collect in this section several results which complement Theorems 3.1 and 3.2.The common tool for their proofs (which we omit or just sketch) is the Quantitative

Subspace Theorem, that is, a theorem which provides an explicit upper bound for thenumber T of exceptional subspaces in Theorem 6.7.

We have mentioned at the beginning of Section 3 that the sequence of partial

quotients of an algebraic irrational number cannot grow too rapidly. More precisely,it can be derived from Roth’s Theorem 6.5 that

limn!C1

log log qn

n D 0; (2.62)

where .p`=q`/`1 denotes the sequence of convergents to . This is left as an exer-cise. The use of a quantitative form of Theorem 6.5 allowed Davenport and Roth [38]

to improve (2.62). Their result was subsequently strengthen [4, 25] as follows.

Theorem 12.1. Let be an irrational, real algebraic number and let .pn=qn/n1

denote the sequence of its convergents. Then, for any " > 0 , there exists a constant c ,

depending only on and " , such that

log log qn c n2=3C":

Proof. We briefly sketch the proof of a slightly weaker result. Let d be the degree of

. By Theorem 6.4, there exists an integer n0 such that

ˇ pn

qn

ˇ > 1qd C1

n

;




for n n0. Combined with Theorem 5.1, this gives

qnC1 qd n ; for n n0: (2.63)

On the other hand, a quantitative form of Theorem 6.5 (see e.g. [40]) asserts thatthere exists a positive number 0 < 1=5, depending only on , such that for every with 0 < < 0, the inequality ˇ

p

q

ˇ <

1

q2C;

has at most 4 rational solutions p=q with p and q co prime and q > 161=.Consequently, for every n 8=, with at most 4 exceptions, we have

qnC1 q1Cn : (2.64)

Let N .8=/2 be a large integer and set h WD dp N e. We deduce from (2.63) and

(2.64) thatlog qN

log qh

D log qN

log qN 1

log qN 1

log qN 2

log qhC1

log qh

.1 C /N d 4

;

whence

log log qN

log log qh

N

C4.log d /:

Observe that it follows from (2.62) that

log log qh N 1=2;

when N is sufficiently large. Choosing D N 1=5, we obtain the upper bound

log log qN log log qh C N 4=5.log 3d / 2N 4=5.log 3d/;

when N is large enough. A slight refinement yields the theorem.

We first observe that, if we assume a slightly stronger condition than

lim inf n!C1

p.n; w;A/

n < C1

in Lemma 8.1, namely, that

lim supn!C1

p.n; w;A/

n < C1;

then the word w satisfies a much stronger condition than Condition ./. Indeed, there

then exists an integer C

2 such that

p.n; w;A/ C n; for n 1;



68 Yann Bugeaud

instead of the weaker assumption (2.22). In the case of the first proof of Theorem 4.1

given in Section 9, this means that, keeping its notation, one may assume that, up to

extracting subsequences, there exists an integer c such that

2.un C vn C wn/ unC1 C vnC1 C wnC1 c.un C vn C wn/; n 1:

This observation is crucial for the proofs of Theorems 12.2 to 12.4 below.

Now, we mention a few applications to the complexity of algebraic numbers, be-

ginning with a result from [31]. Recall that the complexity function n 7! p.n;; b/has been defined in Section 2.

Theorem 12.2. Let b 2 be an integer and an algebraic irrational number. Then,

for any real number such that < 1=11 , we have

lim supn!C1

p.n;;b/

n.log n/ D C1:

The main tools for the proof of Theorem 12.2 are a suitable extension of the

Cugiani–Mahler Theorem and a suitable version of the Quantitative Subspace Theo-rem, which allows us to get an exponent of log n independent of the base b . Using

the recent results of [41] allows us to show that Theorem 12.2 holds for in a slightly

larger interval than Œ0; 1=11/.As briefly mentioned in Section 6, one of the main features of the theorems of

Roth and Schmidt is that they are ineffective, in the sense that we cannot produce anexplicit upper bound for the denominators of the solutions to (2.19) or for the height

of the subspaces containing the solutions to (2.20). Consequently, Theorems 3.1 and

3.2 are ineffective, as are the weaker results from [14, 43]. It is shown in [24] that, bymeans of the Quantitative Subspace Theorem, it is possible to derive an explicit form

of a much weaker statement.

Theorem 12.3. Let b 2 be an integer. Let be a real algebraic irrational number

of degree d and height at most H , with H ee . Set

M

Dexp

f10190.log.8d//2.log log.8d//2

g C232 log.240 log.4H//:

Then we have

p.n;;b/

1 C 1

M

n; for n 1:

Unfortunately, the present methods do not seem to be powerful enough to get aneffective version of Theorem 3.1.

We have shown that, if the b-ary or the continued fraction expansion of a real num-

ber is not ultimately periodic and has small complexity, then this number cannot bealgebraic, that is, the distance between this number and the set of algebraic numbers

is strictly positive. A natural question then arises: is it possible to get transcendencemeasures for , that is, to bound from below the distance between and any algebraic




number? A positive answer was given in [6], where the authors described a general

method to obtain transcendence measures by means of the Quantitative Subspace

Theorem. In the next statement, proved in [9], we say that an infinite word w written

on an alphabet A is of sublinear complexity if there exists a constant C such that thecomplexity function of w satisfies

p.n; w;A/ C n; for all n 1:

Recall that a Liouville number is an irrational real number such that for every

real number w, there exists a rational number p=q with j p=qj < 1=qw .

Theorem 12.4. Let be an irrational real number and b 2 be an integer. If the

b-ary expansion of is of sublinear complexity, then, either is a Liouville number,

or there exists a positive number C such that

j j > H./.2d/C loglog.3d/

;

for every real algebraic number of degree d .

In Theorem 12.4, the quantity H./ is the height of as introduced in Defini-

tion 6.3.The analogue of Theorem 12.4 for continued fraction expansions has been estab-

lished in [7, 28]. The analogues of Theorems 12.2 and 12.3 have not been written yet,

but there is little doubt that they hold and can be proved by combining the ideas of the proofs of Theorems 3.2, 12.2 and 12.3.

13 Further notions of complexity

For an integer b 2, an irrational real number whose b-ary expansion is given by

(2.1), and a positive integer n, set

NZ .n;;b/ WD # f` W 1 ` n; a` 6D 0g;

which counts the number of non-zero digits among the first n digits of the b-aryexpansion of .

Alternatively, if 1 n1 < n2 < denotes the increasing sequence of the indices

` such that a` 6D 0, then for every positive integer n we have

NZ .n;;b/ WD max fj W nj ng:

Let " > 0 be a real number and be an algebraic, irrational number. It follows

from Ridout’s Theorem 6.6 that nj C1 .1 C "/nj holds for every sufficiently large

j . Consequently, we get that

limn!C1

NZ .n;;b/log n

D C1:



70 Yann Bugeaud

For the base b D 2, this was considerably improved by Bailey, Borwein, Crandall,

and Pomerance [17] (see also Rivoal [60]), using elementary considerations and ideas

from additive number theory. A minor modification of their method allows us to get

a similar result for expansions to an arbitrary integer base. The following statementis extracted from [27] (see also [11]).

Theorem 13.1. Let b 2 be an integer. For any irrational real algebraic number of degree d and height H and for any integer n exceeding .20bd d 3H /d , we have

NZ .n;;b/ 1

b 1

n

2.d C 1/ad

1=d

;

where ad denotes the leading coefficient of the minimal polynomial of over the

integers.

The idea behind the proof of Theorem 13.1 is quite simple and was inspired by

a paper by Knight [48]. If an irrational real number has few non-zero digits, then itsinteger powers 2; 3; : : :, and any finite linear combination of them, cannot have too

many non-zero digits. In particular, cannot be a root of an integer polynomial of small degree. This is, in general, not at all so simple, since we have to take much care

of the carries. However, for some particular families of algebraic numbers, including

roots of positive integers, a quite simple proof of Theorem 13.1 can be given. Here,we follow [60] and (this allows some minor simplification) we treat only the case

b

D2.

For a non-negative integer x, let B.x/ denote the number of 1’s in the (finite)binary representation of x.

Theorem 13.2. Let be a positive real algebraic number of degree d 2. Let

ad X d C C a1X C a0 denote its minimal polynomial and assume that a1; : : : ; ad

are all non-negative. Then, there exists a constant c , depending only on , such that

NZ .n;;2/ B.ad /1=d n1=d c;

for n 1.

Proof. Observe first that, for all positive integers x and y , we have

B.x C y/ B.x/ C B.y/

and

B.xy/ B.x/ B.y/:

For simplicity, let us write NZ .n; / instead of NZ .n; ; 2/. Let and be posi-

tive irrational numbers (the assumption of positivity is crucial) and n be a sufficiently

large integer. We state without proof several elementary assertions. If C is irra-tional, then we have

NZ .n; C / NZ .n;/ C NZ .n; / C 1:




If is irrational, then we have

NZ .n;/ NZ .n;/ NZ .n;/ C log2. C C 1/ C 1;

where log2 denotes the logarithm in base 2. If m is an integer, then we have

NZ .n;m/ B.m/. NZ .n;/ C 1/:

Furthermore, for every positive integer A, we have

NZ .n;/ NZ .n; A=/ n 1 log2. C A= C 1//: (2.65)

Let be as in the statement of the theorem. The real number ja0j= is irrational,

as are the numbers aj j 1 for j D 2 ; : : : ; d provided that aj 6D 0. Since

ja0

j 1

Da1

Ca2

C Cad d 1

and NZ .n;/ tends to infinity with n, the various inequalities listed above imply that

NZ .n; ja0j 1/ d C B.a1/ C NZ .n;a2 / C C NZ .n;ad d 1/

d C B.a1/ C B.a2/. NZ .n;/ C 1/ C C B.ad /. NZ .n; d 1/ C 1/

B.ad / NZ .n;/d 1 C c1 NZ .n;/d 2;

(2.66)

where c1, like c2; c3; c4 below, is a suitable positive real number depending only on . By (2.65), we get

NZ .n; ja0j 1/ n

NZ .n;/ c2:

Combining this with (2.66), we obtain

B.ad / NZ .n;/d C c3 NZ .n;/d 1 n

and we finally deduce that

NZ .n;/ B.ad /1=d n1=d c4;

as asserted.

We may also ask for a finer measure of complexity than simply counting the num-

ber of non-zero digits and consider the number of digit changes.

For an integer b 2, an irrational real number whose b-ary expansion is given

by (2.1), and a positive integer n, we set

NBDC .n;;b/ WD # f` W 1 ` n; a` 6D a`C1g;

which counts the number of digits followed by a different digit, among the first ndigits in the b-ary expansion of . The functions n 7! NBDC .n;;b/ have beenintroduced in [23]. Using this notion for measuring the complexity of a real number,

Theorem 13.3 below, proved in [23, 31], shows that algebraic irrational numbers are‘not too simple’.



72 Yann Bugeaud

Theorem 13.3. Let b 2 be an integer. For every irrational, real algebraic number

, there exist an effectively computable constant n0.;b/ , depending only on and

b , and an effectively computable constant c , depending only on the degree of , such

that NBDC .n;;b/ c .log n/3=2 .log log n/1=2 (2.67)

for every integer n n0.;b/.

A weaker result than (2.67), namely that

limn!C1

NBDC .n;;b/

log n D C1; (2.68)

follows quite easily from Ridout’s Theorem 6.6. The proof of Theorem 13.3 depends

on a quantitative version of Ridout’s Theorem. We point out that the lower bound in(2.67) does not depend on b.

Further results on the number of non-zero digits and the number of digit changes

in the b-ary expansion of algebraic numbers have been obtained by Kaneko [ 45, 46].

Bibliography

[1] B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers, II. Continued

fractions. Acta Math. 195 (2005), 1–20.

[2] B. Adamczewski and Y. Bugeaud, On the independence of expansions of algebraic numbers

in an integer base. Bull. London Math. Soc. 39 (2007), 283–289.

[3] B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers I. Expansions in

integer bases. Ann. of Math. 165 (2007), 547–565.

[4] B. Adamczewski and Y. Bugeaud, On the Maillet–Baker continued fractions. J. reine angew.

Math. 606 (2007), 105–121.

[5] B. Adamczewski and Y. Bugeaud, Palindromic continued fractions. Ann. Inst. Fourier (Greno-

ble) 57 (2007), 1557–1574.

[6] B. Adamczewski and Y. Bugeaud, Mesures de transcendance et aspects quantitatifs de la

methode de Thue–Siegel–Roth–Schmidt. Proc. London Math. Soc. 101 (2010), 1–31.

[7] B. Adamczewski and Y. Bugeaud, Transcendence measures for continued fractions involving

repetitive or symmetric patterns. J. Europ. Math. Soc. 12 (2010), 883–914.

[8] B. Adamczewski and Y. Bugeaud, Diophantine approximation and transcendence. In Ency-

clopedia of Mathematics and its Applications 135, Cambridge University Press, 2010, pp.

424–465.

[9] B. Adamczewski and Y. Bugeaud, Nombres reels de complexite sous-lineaire : mesures

d’irrationalite et de transcendance. J. reine angew. Math. 658 (2011), 65–98.

[10] B. Adamczewski, Y. Bugeaud, and F. Luca, Sur la complexite des nombres algebriques. C. R. Acad. Sci. Paris 339 (2004), 11–14.



Bibliography 73

[11] B. Adamczewski and C. Faverjon, Chiffres non nuls dans le developpement en base entiere

d’un nombre algebrique irrationnel. C. R. Math. Acad. Sci. Paris 350 (2012), 1–4.

[12] B. Adamczewski and N. Rampersad, On patterns occurring in binary algebraic numbers. Proc.

Am. Math. Soc. 136 (2008), 3105–3109.

[13] R. Adler, M. Keane, and M. Smorodinsky, A construction of a normal number for the contin-

ued fraction transformation. J. Number Theory 13 (1981), 95–105.

[14] J.-P. Allouche, Nouveaux resultats de transcendance de reels a developpements non aleatoire.

Gaz. Math. 84 (2000), 19–34.

[15] J.-P. Allouche, J. L. Davison, M. Queffelec, and L. Q. Zamboni, Transcendence of Sturmian

or morphic continued fractions. J. Number Theory 91 (2001), 39–66.

[16] J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations.

Cambridge University Press, Cambridge, 2003.

[17] D. H. Bailey, J. M. Borwein, R. E. Crandall, and C. Pomerance, On the binary expansions of algebraic numbers. J. Th´ eor. Nombres Bordeaux 16 (2004), 487–518.

[18] D. H. Bailey and R. E. Crandall, Random generators and normal numbers. Experiment. Math.

11 (2002), 527–546.

[19] V. Becher and P. A. Heiber, On extending de Bruijn sequences. Inform. Process Lett. 111

(2011), 930–932.

[20] Yu. F. Bilu, The many faces of the subspace theorem Œafter Adamczewski, Bugeaud, Corvaja,

Zannier: : : . Seminaire Bourbaki. Vol. 2006/2007. Ast´ erisque 317 (2008), Exp. No. 967, vii,

1–38.

[21] E. Borel, Les probabilites denombrables et leurs applications arithmetiques. Rend. Circ. Math.

Palermo 27 (1909), 247–271.

[22] Y. Bugeaud, Approximation by Algebraic Numbers. Cambridge Tracts in Mathematics 160,

Cambridge, 2004.

[23] Y. Bugeaud, On the b -ary expansion of an algebraic number. Rend. Sem. Mat. Univ. Padova

118 (2007), 217–233.

[24] Y. Bugeaud, An explicit lower bound for the block complexity of an algebraic number. Atti

Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008), 229–235.

[25] Y. Bugeaud, On the convergents to algebraic numbers. In Analytic Number Theory, 133–143,

Cambridge Univ. Press, Cambridge, 2009.

[26] Y. Bugeaud, Quantitative versions of the Subspace Theorem and applications. J. Th´ eor. Nom-

bres Bordeaux 23 (2011), 35–57.

[27] Y. Bugeaud, Distribution Modulo one and Diophantine Approximation. Cambridge Tracts in

Mathematics 193, Cambridge, 2012.

[28] Y. Bugeaud, Continued fractions with low complexity: Transcendence measures and quadratic

approximation. Compos. Math. 148 (2012), 718–750.

[29] Y. Bugeaud, Automatic continued fractions are transcendental or quadratic. Ann. Sci. ´ Ecole

Norm. Sup. 46 (2013), 1005–1022.

[30] Y. Bugeaud, Transcendence of stammering continued fractions. In Number Theory and Re-

lated Fields: In Memory of Alf van der Poorten, J. M. Borwein et al. (eds.), Springer Proceed-ings in Mathematics & Statistics 43, 2013.



74 Yann Bugeaud

[31] Y. Bugeaud and J.-H. Evertse, On two notions of complexity of algebraic numbers. Acta Arith.

133 (2008), 221–250.

[32] J. Cassaigne, Sequences with grouped factors. In DLT’97, Developments in Language Theory

III, Thessaloniki, Aristotle University of Thessaloniki, 1998, pp. 211–222.

[33] J. W. S. Cassels, An Introduction to Diophantine Approximation. Cambridge Tracts in Math.

and Math. Phys., vol. 99, Cambridge University Press, 1957.

[34] D. G. Champernowne, The construction of decimals normal in the scale of ten. J. London

Math. Soc. 8 (1933), 254–260.

[35] A. H. Copeland and P. Erd˝ os, Note on normal numbers. Bull. Am. Math. Soc. 52 (1946),

857–860.

[36] K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers. Carus Mathematical Monographs,

29. Mathematical Association of America, Washington, DC, 2002. x+190 pp.

[37] H. Davenport and P. Erd˝ os, Note on normal decimals. Canadian J. Math. 4 (1952), 58–63.

[38] H. Davenport and K. F. Roth, Rational approximations to algebraic numbers. Mathematika 2

(1955), 160–167.

[39] L. Euler, De fractionibus continuis. Commentarii Acad. Sci. Imperiali Petropolitanae 9 (1737).

[40] J.-H. Evertse, The number of algebraic numbers of given degree approximating a given alge-

braic number. In Analytic Number Theory (Kyoto, 1996), 53–83, London Math. Soc. Lecture

Note Ser. 247, Cambridge Univ. Press, Cambridge, 1997.

[41] J.-H. Evertse and R. G. Ferretti, A further improvement of the Quantitative Subspace Theo-

rem. Ann. of Math. 177 (2013), 513–590.

[42] J.-H. Evertse and H. P. Schlickewei, A quantitative version of the Absolute Subspace Theo-rem. J. reine angew. Math. 548 (2002), 21–127.

[43] S. Ferenczi and Ch. Mauduit, Transcendence of numbers with a low complexity expansion. J.

Number Theory 67 (1997), 146–161.

[44] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th. edition, Claren-

don Press, 1979.

[45] H. Kaneko, On the binary digits of algebraic numbers. J. Austral. Math. Soc. 89 (2010), 233–

244.

[46] H. Kaneko, On the number of digit changes in base-b expansions of algebraic numbers. Uni-

form Distribution Theory 7 (2012), 141–168.

[47] A. Ya. Khintchine, Continued Fractions. The University of Chicago Press, Chicago Ill., Lon-

don, 1964.

[48] M. J. Knight, An ‘Ocean of Zeroes’ proof that a certain non-Liouville number is transcenden-

tal. Am. Math. Monthly 98 (1991), 947–949.

[49] A. N. Korobov, Continued fractions of some normal numbers. Mat. Zametki 47 (1990), 28–33,

158 (in Russian). English transl. in Math. Notes 47 (1990), 128–132.

[50] J. L. Lagrange, Additions au m´ emoire sur la r´ esolution des ´ equations num´ eriques, Mem. Berl.

24 (1770).

[51] J. Liouville, Remarques relatives a des classes tres-etendues de quantites dont la valeur n’est

ni algebrique, ni meme reductible a des irrationnelles algebriques, C. R. Acad. Sci. Paris 18(1844), 883–885.




[52] J. Liouville, Nouvelle demonstration d’un theoreme sur les irrationnelles algebriques, C. R.

Acad. Sci. Paris 18 (1844), 910–911.

[53] K. Mahler, Some suggestions for further research. Bull. Austral. Math. Soc. 29 (1984), 101–

108.

[54] G. Martin, Absolutely abnormal numbers. Am. Math. Monthly 108 (2001), 746–754.

[55] M. Morse and G. A. Hedlund, Symbolic dynamics. Am. J. Math. 60 (1938), 815–866.

[56] M. Morse and G. A. Hedlund, Symbolic dynamics II. Am. J. Math. 62 (1940), 1–42.

[57] O. Perron, Die Lehre von den Ketterbr¨ uchen, Teubner, Leipzig, 1929.

[58] A. J. van der Poorten, An introduction to continued fractions. In Diophantine Analysis (Kens-

ington, 1985), 99–138, London Math. Soc. Lecture Note Ser. 109, Cambridge Univ. Press,

Cambridge, 1986.

[59] D. Ridout, Rational approximations to algebraic numbers. Mathematika 4 (1957), 125–131.

[60] T. Rivoal, On the bits counting function of real numbers. J. Austral. Math. Soc. 85 (2008),

95–111.

[61] K. F. Roth, Rational approximations to algebraic numbers. Mathematika 2 (1955), 1–20; cor-

rigendum, 168.

[62] W. M. Schmidt, Simultaneous approximations to algebraic numbers by rationals. Acta Math.

125 (1970), 189–201.

[63] W. M. Schmidt, Norm form equations. Ann. of Math. 96 (1972), 526–551.

[64] W. M. Schmidt, Diophantine Approximation, Lecture Notes in Math. 785, Springer, Berlin,

1980.

[65] W. M. Schmidt, The subspace theorem in Diophantine approximation. Compositio Math. 69

(1989), 121–173.

[66] R. G. Stoneham, On the uniform "-distribution of residues within the periods of rational frac-

tions with applications to normal numbers. Acta Arith. 22 (1973), 371–389.

[67] A. Thue, Uber Annaherungswerte algebraischer Zahlen. J. reine angew. Math. 135 (1909),

284–305.

[68] G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers. II.

Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 2 (1999), 397–399.

[69] U. Zannier, Some Applications of Diophantine Approximation to Diophantine Equations, Ed-

itrice Forum, Udine, 2003.



Chapter 3

Multiplicative Toeplitz matrices

and the Riemann zeta function

Titus Hilberdink

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2 Toeplitz matrices and operators – a brief overview . . . . . . . . . . . . . . . . . . . . 802.1 C.T/, W .T/, and winding number . . . . . . . . . . . . . . . . . . . . . . . . 82

2.2 Invertibility and Fredholmness . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.3 Hankel matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.4 Wiener–Hopf factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3 Bounded multiplicative Toeplitz matrices and operators . . . . . . . . . . . . . . . . . 85

3.1 Criterion for boundedness on l 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.2 Viewing 'f as an operator on function spaces; Besicovitch space . . . . . . . . 89

3.3 Interlude on .s/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.4 Factorization and invertibility of multiplicative Toeplitz operators . . . . . . . . 95

4 Unbounded multiplicative Toeplitz operators and matrices . . . . . . . . . . . . . . . . 99

4.1 First measure – the function ˆ˛.N / . . . . . . . . . . . . . . . . . . . . . . . 99

4.2 Connections between ˆ˛ .N / and the order of j.˛ C i t/j . . . . . . . . . . . . 105

4.3 Second measure – '˛ on M2 and the function M .T / . . . . . . . . . . . . . 110

5 Connections to matrices of the form .f .ij =.i; j /2//i;j N . . . . . . . . . . . . . . . 118

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

1 Introduction

In this short course, we aim to highlight connections between a certain class of ma-trices and Dirichlet series, in particular the Riemann zeta function. The matrices we

study are of the form 0BBBBBB@

f .1/ f . 12

/ f . 13

/ f . 14

/ f .2/ f .1/ f . 2

3/ f . 1

2/

f .3/ f . 32

/ f .1/ f . 34

/ f .4/ f .2/ f . 4

3/ f .1/

::: ::: ::: ::: : : :

1CCCCCCA

; ()



78 Titus Hilberdink

i.e., with entries aij D f.i=j/ for some function f WQC ! C. They are a multiplica-

tive version of Toeplitz matrices which have entries of the form aij D aij . For this

reason we call them Multiplicative Toeplitz Matrices.

Toeplitz matrices (and operators) have been studied in great detail by many au-thors. They are most naturally studied by associating with them a function (or ‘sym-

bol’) whose Fourier coefficients make up the matrix. With aij D aij , this ‘symbol’is

a.t/ D1X

nD1ant n: t 2 T

Then properties of the matrix (or rather the operator induced by the matrix) implyproperties of the symbol and vice versa. For example, the boundedness of the opera-

tor is essentially related to the boundedness of the symbol, while invertibility of the

operator is closely related to a.t/ not vanishing on the unit circle.For matrices of the form () we associate, by analogy, the (formal) seriesXq2QC

f.q/qi t ;

where q ranges over the positive rationals. Note, in particular, that if f is supported

on the natural numbers, this becomes the Dirichlet seriesXn2N

f.n/ni t :

In the special case where f.n/ D n˛

, the symbol becomes .˛ i t /. It is quitenatural then to ask to what extent properties of these Multiplicative Toeplitz Matrices

are related to properties of the associated symbol. Rather surprisingly perhaps, these

type of matrices do not appear to have been studied much at all – at least not in this

respect. Finite truncations of them have appeared on occasions, notably Redheffer’s

matrix [29], the determinant of which is related to the Riemann Hypothesis. Denotingby An.f / the n n matrix with entries f.i=j/ if j ji and zero otherwise, it is easy to

see that

An.f /An.g/ D An.f g/; where f g is Dirichlet convolution;

since the ijth

entry on the left product isnX

rD1

An.f /i r An.g/rj DXj jrji

f i

r

g r

j

DX

d ji=j

f i=j

d

g.d/

if j ji by putting r D jd , and zero otherwise. With 1 and denoting the constant

1 and the Mobius functions, respectively, it follows that An.1/An./ D I n – theidentity matrix. Note also that det An.1/ D det An./ D 1. Redheffer’s matrix is

Rn D An.1/ C0B@0 1 1 10 0 0

0

0 0 0 0

1CA D An.1/ C En;



3 Multiplicative Toeplitz Matrices and the Riemann zeta function 79

say, where the matrix En has only 1s on the topmost row from the 2nd column on-

wards. Then, with M.n/ D

PnrD1 .r/,

RnAn./ D I n C0B@ M.n/ 1

0 0 0 0 0 0

1CA D0BB@ M.n/ 0 1 0

: : : 0 0 1

1CCA ;

so that det Rn D M.n/. The well-known connection between the Riemann Hypothe-

sis (RH) and M.n/ therefore implies that RH holds if and only if det Rn D O.n12C"/

for every " > 0. (See also [20] for estimates of the largest eigenvalue of Rn).

Briefly then, the course is designed as follows: In Section 2, we recall somebasic aspects of the theory of Toeplitz operators, in particular their boundedness and

invertibility. In Section 3, we study bounded multiplicative Toeplitz operators. Thisis partly based on some of Toeplitz’s own work [34], [35] and recent results from [17]and [18], but we also present new results, mainly in Section 3. Thus Theorem 3.1 is

new, generalising Theorem 2.1 of [18], which in turn is now contained in Corollary

3.2. Also Subsection 3.2 and parts of 3.4 are new.

Preliminaries and Notation

(a) The sequence spaces lp

.1 p < 1) consist of sequences .an/ for whichP1nD1 janjp converges. They are Banach spaces with the norm

k.an/kp D 1X

nD1

janjp

1=p

:

The space l1 is the space of all bounded sequences, equipped with the normk.an/k1 D supn2N janj. We shall also use l p.QC/, which is the space of se-

quences aq where q ranges over the positive rationals such thatP

q jaqjp < 1,with analogous norms and also for p D 1.

l2

and l2

.QC/ are Hilbert spaces with the inner products

ha; bi D1X

nD1

anbn and ha; bi DX

q2QCaqbq;

respectively.

(b) Let T D fz 2 C W jzj D 1g – the unit circle. We denote by L2.T/ the space of

square-integrable functions on T. L2.T/ is a Hilbert space with the inner product

and corresponding norm given by

hf; gi D Z T

f g D 12

Z 2

0

f .ei /g.ei /d; kf k D s Z T

jf j2:



80 Titus Hilberdink

The space L1.T/ consists of the essentially bounded functions on T with norm

kf k1 denoting the essential supremum of f . (Strictly speaking, L2 and L1consist of equivalence classes of functions satisfying the appropriate conditions,

with two functions belonging to the same class if they differ on a set of measurezero.)

Let n.t / D t n for n 2 Z. Then .n/n2Z is an orthonormal basis in L2.T/and L2.T/ is isometrically isomorphic to l 2.Z/ via the mapping f 7! .f n/n2Z,

where f n are the Fourier coefficients of f , i.e.,

f n D hf; ni DZ T

f n:

(c) A linear operator ' on a Banach space X is bounded if

k'x

k C

kx

k for all

x 2 X . In this case the operator norm of ' is defined to be

k'k D supx2X;x¤0

k'xkkxk D sup

kxkD1

k'xk:

The algebra of bounded linear operators on X is denoted by B.X /.

(d) An infinite matrix A D .aij / induces a bounded operator on a Hilbert space H if

there exists ' 2 B.H/ such that

aij D h'ej ; ei i;

where .ei / is an orthonormal basis of H . Note that not every infinite matrixinduces a bounded operator, and it may be difficult to tell when it does.

(e) For the later sections we require the usual O;o; ; notation. Given f; g de-

fined on neighbourhoods of 1 with g eventually positive, we write f.x/ Do.g.x// (or simply f D o.g/) to mean limx!1 f.x/=g.x/ D 0, f.x/ DO.g.x// to mean jf.x/j Ag.x/ for some constant A and all x sufficiently

large, and f .x/ g.x/ to mean limx!1 f .x/=g.x/ D 1.

The notation f g means the same as f D O.g/, while f . g means

f.x/ .1 C o.1//g.x/.

2 Toeplitz matrices and operators – a brief overview

Toeplitz matrices are matrices of the form

T D

0

BBBB@a0 a1 a2 a3 a1 a0 a1 a2 a2 a1 a0 a1 a

3 a

2 a

1 a

0 ::::::

::::::

: : :

1

CCCCA ; (3.1)




i.e., T D .tij /, where tij D aij . They are characterised by being constant on

diagonals.

For a Toeplitz matrix, the question of boundedness of T was solved by Toeplitz.

Theorem 2.1 (Toeplitz [34]). The matrix T induces a bounded operator on l 2 if and

only if there exists a 2 L1.T/ whose Fourier coefficients are an .n 2 Z/. If this is

the case, then kT k D kak1.

We refer to the function a as the ‘symbol’ of the matrix T , and we write T .a/.

Sketch of Proof. For a 2 L2.T/, the multiplication operator

M.a/W L2.T/ ! L2.T/; f 7! af

is bounded if and only if a 2 L1.T/. If bounded, then kM.a/k D kak1. The matrix

representation of M.a/ with respect to .n/n2Z is given by

hM.a/j ; ii D haj ; i i DZ T

aj i DZ T

aij D aij ;

i.e., by the so-called Laurent matrix

L.a/ WD 0BBBBB@ a0 a1 a2 a3 a4 a1 a0 a1 a2 a3 a2 a1 a0 a1 a2 a3 a2 a1 a0 a1

1CCCCCA : (3.2)

The matrix for T is just the lower right quarter of L.a/. We can therefore think of T as the compression PL.a/P , where P is the projection of l 2.Z/ onto l 2 D l 2.N/.An easy argument shows that T is bounded if and only if a 2 L1, and then kT k DkL.a/k D kak1.

Hardy space. Let H 2.T/ denote the subspace of L2.T/ of functions f whoseFourier coefficients f n vanish for n < 0. Let P be the orthogonal projection of L2 onto H 2, i.e., P .

Pn2Z f nn/ D Pn0 f nn. The operator f 7! P.af / has

matrix representation (3.1). For, with j 0 (so that j 2 H 2.T/),

hT.a/j ; i i DZ T

P.aj /i DZ T

P

Xn2Z

annCj

i D

Xn0

anj

Z T

ni D aij

if i 0, and zero otherwise. Hence, we can equivalently view T .a/ as the operator

T.a/W H 2.T/ ! H 2.T/; f 7! P .af /:



82 Titus Hilberdink

2.1 C.T/, W.T/, and winding number Let C.T/ denote the space of con-

tinuous functions on T. For a 2 C.T/ such that a.t/ ¤ 0 for all t 2 T, we de-

note by wind.a;0/ the winding number of a with respect to zero. More generally,

wind.a; / Dwind.a ;0/ denotes thewinding number with respect 2 C. Forexample, wind.n; 0/ D n.

The Wiener Algebra is the set of absolutely convergent Fourier series:

W .T/ D( 1X1

ann W1X1

janj < 1)

:

Some properties:

(i) W .T/ forms a Banach algebra under pointwise multiplication, with norm

kakW WD1

X1 janj:(ii) (Wiener’s Theorem) If a 2 W and a.t/ ¤ 0 for all t 2 T, then a1 2 W .

(iii) If a 2 W .T/ has no zeros and wind.a; 0/ D 0, then a D eb for some b 2 W .T/.

We have

W .T/ C.T/ L1.T/ L2.T/:

2.2 Invertibility and Fredholmness Let A be a bounded operator on a Banachspace X .

(i) A is invertible if there exists a bounded operator B on X such that AB D

BADI . As such, B is the unique inverse of A, and we write B D A1. The spectrum

of A is the set

.A/ D f 2 C W I A is not invertible in X g:

The kernel and image of A are defined by

Ker A D fx 2 X W Ax D 0g; Im A D fAx W x 2 X g:

(ii) The operator A is Fredholm if ImA is a closed subspace of X and both KerAand X=ImA are finite-dimensional. As such, the index of A is defined to be

Ind A D dim Ker A dim .X=Im A/:

For example, T .n/ is Fredholm with Ind T .n/ D n.

Equivalently, A is Fredholm if it is invertible modulo compact operators; thatis, there exists bounded operator B on X such that AB I and BAI are both

compact.

The essential spectrum of A is the set

ess.A/ D f 2 C W I A is not Fredholm in X g:

Clearly ess.A/ .A/. Note that A invertible implies A is Fredholm of indexzero. For Toeplitz operators, the converse actually holds (see [5], p. 12).




2.3 Hankel matrices These are matrices of the form

H D 0BBBB@a1 a2 a3 a4 a2 a3 a4 a5 a3 a4 a5 a6 a4 a5 a6 a7 :::

::::::

::: : : :

1CCCCA ; (3.3)

i.e., H D .hij /, where hij D aiCj 1. They are characterised by being constant on

cross diagonals. The boundedness of H was solved by Nehari, and the compactnessof H by Hartman.

Theorem 2.2 ([28], [14]). The matrix H generates a bounded operator on l 2 if and

only if there exists b 2 L1.T/ .with Fourier coefficients bn/ such that bn D an for n 1. Furthermore, the operator H is compact if and only if b 2 C.T/.

We refer to the function a as the ‘symbol’ of the matrix H , and we write H.a/.

Given a function a defined on T, let Qa be the function

Qa.t/ D a.1=t/ .t 2 T/:

Proposition 2.3. For a; b 2 L1.T/ ,

T.ab/ D T.a/T.b/ C H.a/H. Qb/H.ab/ D H.a/T. Qb/ C T.a/H.b/:

Proof. The matrix L.a/ in (3.2) is of the form

L.a/ D

T .a/ H. Qa/H .a/ T . a/

:

Since L.ab/ D L.a/L.b/, the result follows by multiplying the 2 2 matrices.

As a special case, we see that T .abc/ D T.a/T.b/T.c/ for a 2 H 1; b 2L1; c 2 H 1. The space H 1 is defined analogously to L1. (T.a/ is upper-

triangular and T .c/ is lower-triangular.)

By Theorem 2.2, if a; b 2 C.T/, then H.a/H. Qb/ is compact, so that T.ab/ T.a/T.b/ is compact. In particular, if a has no zeros on T, we can take b D a1 2C.T/. Then T .ab/ D T.1/ D I , so T .a/ is invertible modulo compact operators

(i.e., Fredholm) with ‘inverse’ T .a1/. This type of reasoning leads to:

Theorem 2.4 (Gohberg [9]). Let a 2 C.T/. Then T .a/ is Fredholm if and only if ahas no zeros on T , in which case

Ind T .a/ D wind.a; 0/:



84 Titus Hilberdink

Hence T.a/ is invertible if and only if a has no zeros on T and wind.a; 0/ D 0.

Equivalently, since T . a/ D I T.a/ for 2 C , we have

ess.T .a// D a.T/;.T.a// D a.T/ [ f 2 C n a.T/ W wind.a; / ¤ 0g:

Sketch of Proof. We have seen that a ¤ 0 on T implies T .a/ is Fredholm. In this

case, let wind.a;0/ D k. Then a is homotopic to k , and (since the index varies

continuously)Ind T .a/ D Ind T .k / D k D wind .a; 0/:

For the converse, suppose T.a/ is Fredholm with index k, but a has zeros on T.Then a can be slightly perturbed to produce two functions b; c 2 W .T/ without zeros

such that

ka

b

k1 and

ka

c

k1 are as small as we please, but wind.b; 0/ and

wind.c;0/ differ by one. As the index is stable under small perturbations, T .b/ andT.c/ are Fredholm with equal index. But Ind T.b/ D wind.b; 0/ and Ind T.c/ Dwind.c;0/ (by above), so wind.b;0/ wind.c;0/ D 0 — a contradiction.

2.4 Wiener–Hopf factorization Since W .T/ C.T/, Theorem 2.4 appliesto W .T/. However, for Wiener symbols we can obtain a quite explicit form for the

inverse when it exists. This is because Wiener functions can be factorized.

Denote by W C and W the subspaces of W consisting of functions

1

XnD0

ant

n

and

1

XnD0

antn

t 2 T

respectively, whereP janj < 1.

Theorem 2.5 (Wiener–Hopf factorization). Let a 2 W .T/ such that a has no zeros,

and let wind.a; 0/ D k. Then there exist a 2 W and aC 2 W C such that

a D k aaC:

Proof. We have wind.ak; 0/ D 0. So ak D eb for some b 2 W . But b Db C

bC

, where b 2

W

and bC 2

W C

. Hence, writing a D

e b and aC D

ebC

gives

ak D ebebC D aaC:

Theorem 2.6 (Krein [23]). Let a 2 W .T/. Then T .a/ is Fredholm if and only if ahas no zeros on T , in which case

Ind T .a/ D wind.a; 0/:

In particular, T.a/ is invertible if and only if a has no zeros on T and wind.a; 0/ D 0.

In this case

T.a/1

D T .a1

C /T.a1

/;where a D aCa is the Wiener–Hopf factorization of a.




Proof of second part. Note that if a 2 W , then H.a/ D 0, while if a 2 W C, then

H. Qa/ D 0. Suppose a has no zeros on T and wind.a; 0/ D 0. Then a factorizes as

a

Da

a

C with a

˙ 2W

˙. Applying Proposition 2.3 with a

and a

C in turn gives

T .a1 /T.a/ D T .a1

a/ D I D T .aa1 / D T .a/T.a1

/;

T .a1C /T.aC/ D T .a1

C aC/ D I D T .aCa1C / D T .aC/T.a1

C /;

so that T .a˙/ are invertible with T .a˙/1 D T .a1˙ /. But also T .a/ D T .aaC/ D

T .a/T.aC/ (by Proposition 2.3). Hence T.a/1 D T .aC/1T .a/1 DT .a1C /T.a1 /.

3 Bounded multiplicative Toeplitz matrices and operators

3.1 Criterion for boundedness on l2 Now we consider the linear operatorsinduced by matrices of the form ./, regarding them as operators on sequence spaces,

in particular l 2.

For a function f WQC ! C on the positive rationals, we define

Xq2QCf.q/ D lim

N !1

Xm; n N

.m;n/D 1

f m

n

; whenever this limit exists:

We shall sometimes abbreviate the left-hand sum byP

q f .q/. We say that f 2l 1.QC/ if X

q2QCjf.q/j

converges. In this case, the function

F.t/ D

Xq2QCf.q/qi t t 2 R

exists and is uniformly continuous on R. Note that for > 0,

limT !1

1

2T

Z T

T

F.t/i t dt D

f.q/; if D q 2 QC;0; otherwise.

(3.4)

Theorem 3.1. Let f 2 l 1.QC/ and let 'f denote the mapping .an/ 7! .bn/ where

bn

D

1

XmD1

f n

mam:



86 Titus Hilberdink

Then 'f is bounded on l 2 with operator norm

k'f

k Dsupt2R ˇ Xq2QC f.q/qi t ˇ D k

F

k1:

Proof. We shall first prove that 'f is bounded on l 2, showing k'f k kF k1 in the

process, and then show that kF k1 is also a lower bound.

For q 2 QC and N 2 N, let

b.N /q D

N XmD1

f q

m

am and bq D

1XmD1

f q

m

am:

Note that b.N /q

! bq as N

! 1 for every q

2 QC, whenever an is bounded. We

have the following formulae:

limT !1

1

2T

Z T

T

F.t/

ˇ N XnD1

anni t

ˇ2dt D

N XnD1

anb.N /n (3.5)

limT !1

1

2T

Z T

T

ˇF.t/

N XnD1

anni t

ˇ2dt D

Xq2QC

jb.N /q j2: (3.6)

(These hold for an bounded.) To prove these expand the integrand in a Dirichlet

series. For the first formula we have

1

2T

Z T

T

F.t/

ˇ N XnD1

anni t

ˇ2dt D

Xm;nN

aman

1

2T

Z T

T

F.t/ n

m

i t

dt

!X

m;nN

amanf n

m

D

N XnD1

anb.N /n

as T ! 1. For the second formula, note first that

F.t/

N XnD1

anni t D Xq2QC;nN

f.q/an.qn/i t

DX

r2QC

XnN

f r

n

an

r i t D

Xr2QC

b.N /r r i t ;

the series converging absolutely. Thus

1

2T

Z T

T

ˇˇF.t/

N

XnD1

anni t

ˇˇ

2

dt D

Xq1;q2

2QC

b.N /q1

b.N /q2

1

2T

Z T

T q1

q2i t

dt !

Xq2QCjb.N /

q j2

as T ! 1.




Since jF.t/j kF k1, we have

Xq2QC jb.N /

q j2

kF

k2

1 limT !1

1

2T Z T

T ˇN

XnD1

anni t ˇ2

dt D k

F k

2

1

N

XnD1 jan

j2:

Thus if a D .an/ 2 l 2, we haveP1

nD1 jb.N /n j2 kF k21kak2 for every N . Letting

N ! 1 shows that .bn/ 2 l 2 too (indeed .bq/ 2 l 2.QC/), and so 'f is bounded onl 2, with

k'f k kF k1:

Now we need a lower bound. By Cauchy–Schwarz,

ˇ1

XnD1

anbn ˇ2

1

XnD1

janj2 1

XnD1

jbnj2:

Thus k'f k jP1nD1 anbnj for every a D .an/ 2 l 2 with kak D 1. Choose an as

follows: let N 2 N (to be determined later) and put

an D ni tp d.N/

for njN , and zero otherwise.

Here d.N / is the number of divisors of N . Thus .an/ 2 l 2 and kak D 1. With this

choice,

bn

D

1

p d.N/ XmjN

f n

mmi t .

Db.N /

n /

and so1X

nD1

anbn D 1

d.N/

XnjN

ni tXmjN

f n

m

mi t D 1

d.N/

Xm;njN

f n

m

n

m

i t

D 1

d.N/

Xq2QC

f.q/qi t S q .N/;

where

S q .N / D Xm; njN nm D q

1:

Put q D kl , where .k;l/ D 1. Then n

m D k

l if and only if l n D km. Since .k;l / D 1,

this forces kjn and l jm. So, for a contribution to the sum, we need k; l jN , i.e., kl jN .Suppose therefore that kl jN . Then

S q .N / DX

m; njN

ln D km

1 DX

rk;rljN

1 m D r l ; n D rk with r 2 N

D Xrj N kl

1D

d N

kl :



88 Titus Hilberdink

Writing jqj D kl whenever q D kl

in its lowest terms, gives

1

XnD1

an

bn D Xq 2 QC

jqjjN

f.q/qi t d.N=jqj/d.N/

: (3.7)

The idea is now to choose N in such a way that it has all ‘small’ divisors whiled.N=jqj/

d.N/ is close to 1 for all such small divisors jqj. Take N of the form

N DY

pP

p˛p ; where ˛p Dh

log P log p

i.

Thus every natural number up to P is a divisor of N . Every q such that jqjjN is of

the form

jqj D QpP

pˇp (0

ˇp

˛p ), so that

d.N=jqj/d.N/

DY

pP

1 ˇp

˛p C 1

:

If we take jqj p log P , then pˇp p

log P for every prime divisor p of jqj.Hence, for such p, ˇp loglog P

2 log p and ˇp D 0 if p >

p log P . Thus

d.N=jqj/d.N/

DY

p

p log P

1 ˇp

˛p C 1

Y

p

p log P

1 log log P

2 log P

D

1 log log P

2 log P

.p log P /

;

where .x/ is the number of primes up to x . Since .x/ D O. xlog x

/, it follows that

for all P sufficiently large, the RHS above is at least

1 Ap log P

for some constant A.

Let " > 0. Then there exists n0 such that Pjqj>n0 jf.q/j < ". Choose P e

n20

so thatp

log P n0. Then the modulus of the sum in (3.7) can be made as close tokF k1 as we please by increasing P , for it is at leastˇ X

jqjp log P

f.q/qi t

ˇ Ap

log P

Xjqjp

log P

jf.q/j X

jqj>p log P

jf.q/j

ˇ

ˇ

Xq2QC

f.q/qi t

ˇ

ˇ Ap

log P

Xq2QC

jf.q/j 2X

jqj>p log P

jf.q/j

> jF.t/j A0p log P 2";




where " can be made as small as we please by making P large. Since this holds for

any t , we can choose t to make F.t/ as close as we like to kF k1. Hence k'f k

kF

k1 and so we must have equality.

In the special case where f 0, the supremum of jF.t/j is attained when t D 0,in which case kF k1 D F.0/ D kf k1;QC . Thus:

Corollary 3.2. Let f WQC ! C such that f 0. Then 'f is bounded on l 2 if and

only if f 2 l 1.QC/ , in which case k'f k D kf k1;QC .

Example. Take f .n/ D n˛ for n 2 N, and zero otherwise. Then F.t/ D .˛ i t /.

We shall denote 'f by '˛ in this case. Applying Corollary 3.2, we see that '˛ isbounded on l 2 if and only if ˛ > 1, and the norm is .˛/.

3.2 Viewing 'f as an operator on function spaces; Besicovitch spaceWe can view 'f as an operator on functions rather than sequences. For this we need

to construct the appropriate spaces.Let A denote the set of trigonometric polynomials; i.e., the elements of A are all

finite sums of the formnX

kD1

akei k t ;

where ak

2C and k

2R. The space A has an inner product and a norm given by

hf; gi D limT !1

1

2T

Z T

T

f g and kf k Dp

hf; f i Ds

limT !1

1

2T

Z T

T

jf j2:

Now let B2 (Besicovitch space) denote the closure of A with respect to this inner

product; i.e., f 2 B2 if kf f nk ! 0 as n ! 1 for some f n 2 A. We turn B 2 into

a Hilbert space by identifying f and g whenever kf gk D 0. (See [3], Chapter II.)

Now write .t / D i t ( > 0; t 2 R) and let Of ./ denote the Fourier coefficient

Of ./ D limT !1

1

2T Z T

T f D lim

T !11

2T Z T

T f.t/i t dt where it exists:

Denote by F the space of locally integrable f WR ! C such that Of ./ exists for all

> 0.

(a) Fourier coefficients and series For f 2 B2 , the Fourier coefficients Of ./ ex-

ist and Of ./ is non-zero on at most a countable set, say fngn2N. The function

f has the . formal/ Fourier seriesP

n1Of .n/i t

n .

(b) Uniqueness f; g 2 B2 have the same Fourier series if and only if kf gk D 0.



90 Titus Hilberdink

(c) Parseval For f 2 B2 ,

kf

k D lim

T !1s 1

2T Z T

T jf

j2

D s X j Of ./

j2; (3.8)

and, more generally, for f; g 2 B2 ,

hf; gi D limT !1

1

2T

Z T

T

f g DX

Of ./ Og./:

(d) Riesz–Fischer Theorem Given n > 0 and an 2 l 2 , there exists f 2 B2

such that f .t/ Pn1 ani t

n .

(e) Criterion for membership in B 2: With F as before, if f

2F and Parseval’s

identity (3.8) holds, then f 2 B2.

Indeed, the set of for which Of ./ ¤ 0 is necessarily countable and we

may write this as f1; 2; : : :g withP1

kD1 j Of .k /j2 D kf k2. Let f n.t / DPkn

Of .k/i tk

. Then

kf f nk2 D kf k2 kf nk2 DXk>n

j Of ./j2 ! 0 as n ! 1.

The analogues of the Hardy and Wiener spaces: B2QC

, B2N

, W QC , W N.

(a) Let B2QC

denote the subspace of B2 of functions with exponents D log q

for some q 2 QC. Alternatively, start with the subset of A consisting of finite

trigonometric polynomials of the formP

aqq , where q ranges over a finitesubset of QC, and take its closure.

(b) Let B 2N denote the subspace of B 2 of functions with exponents D log n for

some n 2 N. This is the analogue of the Hardy space.

(c) Let W QC denote the set of all absolutely convergent Fourier series in B 2QC

; i.e.

W QC

D n Xq2QC c.q/q

W Xq2QC jc.q/

j<

1o:

This is the analogue of the Wiener algebra. As we saw earlier, if

f DX

q2QCc.q/q 2 W QC ;

then Of .q/ D c.q/. With pointwise addition and multiplication, W QC becomes

an algebra. Further, W QC becomes a Banach algebra with respect to the norm

kf

kW

D Xq2QC j Of .q/

j:

Analogously, let W N denote the set of absolutely convergentseriesP1

nD1 anni t .




B2QC

has .q/q2QC as an orthonormal basis while B2N

has .n/n2N as an orthonor-

mal basis. The spaces B 2QC

and B 2N are isometrically isomorphic to l 2.QC/ and l 2,

respectively, via the mappings

f 7! f Of .q/gq2QC and f

7! f Of .n/gn1:

The operator M.f /. Let P be the projection from B 2QC

to B 2N; that is,

P X

q2QCc.q/qi t

DXn2N

c.n/ni t :

For f 2

W QC , we define the operator M.f /

WB2

N !B2

N

by g 7!

P.fg/.

The matrix representation of M.f / w.r.t. fngn2N is the multiplicative Toeplitz

matrix . Of .i=j //i;j 1. Indeed, if f D Pq

Of .q/q , then

P.fj / D P X

q

Of .q/qj

D P

Xq

Of .q/qj

D P

Xq

Of .q=j /q

D

1XnD1

Of .n=j /n:

Hence

hM.f /j ; i i D hP.fj /; ii D1X

nD1

Of .n=j /hn; i i D Of i

j

:

In terms of the operator ',M.f / D 1' Of

:

3.3 Interlude on .s/ Since this work concerns connections to Dirichlet seriesand the Riemann zeta function in particular, we recall a few relevant facts regarding

.s/.The Riemann zeta function is defined for <s > 1 by the Dirichlet series

.s/ D1X

nD1

1

ns:

In this half-plane .s/ is holomorphic and there is an analytic continuation to thewhole of C except for a simple pole at s D 1 with residue 1. Furthermore, .s/satisfies the functional equation

.s/ D .s/.1 s/



92 Titus Hilberdink

where

.s/ D 2s s1.1 s/ sin

s

2 D s 1

2. 1

2 s

2/

. s2

/ :

The connection to prime numbers comes from Euler’s remarkable product formula

.s/ DY

p

1

1 ps <s > 1

The order of .s/. Considering . C i t / as a function of the real variable t for

fixed (but arbitrary) , it is known that for jt j large

.t / def

D .

Ci t /

DO.

jt

jA/ for some A:

The infimum of such A is the order of and is called the Lindel¨ of function; i.e.,

./ D inf fA W . C i t / D O.jt jA/g:

From the general theory of functions it is known that the Lindelof function is convex

and decreasing. Since is bounded for > 1, but 6! 0, it follows that . / D 0for 1. By the functional equation and continuity of we then have

./

D 0; if 1;1

2 ; if 0.

(Þ)

For 0 < < 1, the value of ./ is not known, but it is conjectured that the two

line segments in (Þ) above extend to D 12

. This is the Lindel¨ of Hypothesis. It is

equivalent to the statement that

. 12 C i t / D O.t "/ for every " > 0:

The Lindelof Hypothesis is a major open problem and is a consequence of the Rie-

mann Hypothesis, which states that .s/ ¤ 0 for > 12

.

Upper and lower bounds for . Let

Z .T / D max1jt jT

j. C i t /j:

(The restriction jt j 1 is only added to avoid problems for the case D 1.) The

following results hold for large T .

(a) Z .T / ! ./ for > 1.

(b) For D 1, unconditionally it is known that Z1.T / D O..log T /23 /, while on

RHZ1.T / . 2e log log T:




On the other hand, Granville and Soundararajan [13] proved that

Z1.T / e .log log T C log log log T log log log log T /;

for some arbitrarily large T . They further conjectured that it equals the abovewith an O.1/ term instead of the quadruple log-term.

(c) For 12

< < 1, unconditionally one has Z .T / D O.T a/ for various a > 0,

while on RH

log Z .T / A .log T /22

.1 / loglog T

for some constant A. Montgomery [27] showed that

log Z .T /

p 1=2

20

.log T /1

.log log T /

and, using a heuristic argument, he conjectured that this is (apart from the con-

stant) the correct order of log Z .T /. In a recent paper (see [24]), Lamzourisuggests that

log Z .T / C./ .log T /1

.loglog T /

with C./ D G1. / 2 .1 / 1, where

G1.x/

D Z 1

0

log 1

XnD0

.u=2/2n

.nŠ/2 !

du

u1

C1=x

:

(d) For D 12

unconditionally one has Z 12

.T / D O.T 32205 .log T /c / D O.T 0:156::/

(see [19]), while on RH

log Z 12

.T / A log T

log log T

for some constant A. On the other hand, it is known that

log Z 12 .T / cs log T

log log T

(see [2], [31]). Using a heuristic argument, Farmer et al. ([8]) conjectured that

log Z 12

.T / r

1

2 log T log log T :

(e) For < 12

, the functional equation for .s/ reduces the problem to the case

> 12

. So, for example,

Z .T / .1 / T 21

2

for < 0.



94 Titus Hilberdink

Mean values. For 2 . 12

; 1/, the mean-value formula

limT !1

1

2T Z T

T j.

i t /

j2 dt

D

1

XnD1

1

n2 D.2/;

is well-known (see [33]). Furthermore,

limT !1

1

2T

Z T

T

j. i t /j4 dt D1X

nD1

d.n/2

n2 D .2/4

.4/ :

For higher powers, however, present knowledge is very patchy. It is expected that theabove formulas extend to all higher moments, i.e.,

limT !11

2T Z T

T j. i t /j2k

dt D1

XnD1

d k.n/2

n2 :

This is equivalent to the Lindelof Hypothesis.

Examples.

(a) The mean values for and 2 imply that ; 2 2 B 2N for 2 . 1

2; 1/. Note that

this also implies j j2 2 B2QC

.

For higher powers, however, only partial results are known. For example, it is

known that k 2 B2N if 2 .1 1

k; 1/. Slightly better bounds are available,

especially for particular values of k, but it is expected that much more holds,

namely: k 2 B 2N for every k 2 N and all 2 . 1

2; 1/. This is (equivalent to) the

Lindelof Hypothesis.

(b) Let g.s/ D P1nD1

an

ns and h.s/ D P1nD1

bn

ns be two Dirichlet series which con-

verge absolutely for <s > 0 and <s > 1, respectively. Let ˛ > 0 and ˇ > 1and put f .t/ D g.˛ it/h.ˇ C i t /. Then f 2 W QC with

Of .q/ D 1

m˛nˇ

1Xd D1

amd bnd

d Cˇ for q D m

n with .m;n/ D 1.

We can prove this by multiplying out the series for g.˛ i t /

and h.ˇ C i t /

. We

have

f.t/ D1X

mD1

am

m˛i t

1XnD1

bn

nˇCi t D

Xm;n1

ambn

m˛nˇ

m

n

i t

D1X

d D1

Xm; n 1

.m;n/D d

ambn

m˛nˇ

m

n

i t D1X

d D1

1

d Cˇ

Xm; n 1

.m;n/D 1

amd bnd

m˛nˇ

m

n

i t

D Xm; n 1

.m;n/D 1

1

m˛nˇ 1

Xd D1

amd bnd

d Cˇ m

n i t

:




Further Properties of W QC and W N. Notation. For a unital Banach algebra A,

denote by GA the group of invertible elements of A, and by G0A, the connected

component of GA which contains the identity. If A is commutative, then

a 2 G0A” a D eb for some b 2 A.

(a) Wiener’s Theorem for W QC and W N (see [15], Theorems 1 and 2):

Let f 2 W QC . Then 1=f 2 W QC if and only if inf R jf j > 0; i.e., GW QC Dff 2 W QC W inf R jf j > 0g:

Let f .t/ D P1nD1 anni t 2 W N and put F .s/ D P1

nD1an

ns for <s 0. Then

1=f 2 W N if and only if there exists ı > 0 such that jF.s/j ı for all <s 0.

The example f .t/ D 2i t shows that the condition jf.t/j ı > 0 for all t 2 R is

not sufficient for 1=f 2 W N.(b) Let f 2 GW QC . Then the average winding number 1 w.f /, defined by

w.f / D limT !1

arg f . T / arg f .T /

2T

exists, and w.f / D log q for some q 2 QC (see [21], Theorem 27).

It is easy to see that (i) w.fg/ D w.f / C w.g/, and (ii) w.q / D log q.

(c) G0W N D G W N; i.e. for f 2 W N, we have 1=f 2 W N if and only if f D eg for

some g

2W N

3.4 Factorization and invertibility of multiplicative Toeplitz operatorsThe analogue of the factorization T.abc/ D T.a/T.b/T.c/ for a 2 H 1; b 2L1; c 2 H 1 for Toeplitz matrices holds for Multiplicative Toeplitz matrices.

Let W N denote the set of absolutely convergent seriesP1

nD1 anni t .

Theorem 3.3. Let f 2 W N , g 2 W QC , and h 2 W N. Then

M.fgh/ D M.f /M.g/M.h/:

Proof. We show the matrix entries agree. By Proposition 3.1, fgh 2 W QC andM.f /M.g/M.h/

ij

DX

k;l1

M.f /i kM.g/kl M.h/lj

DX

k; l 1

ijk and j jl

Of i

k

Ogk

l

Oh l

j

DX

m;n1

Of 1

m

Ogmi

nj

Ohn

1

:

1Also known as mean motion.



96 Titus Hilberdink

On the other hand, since all QC-series converge absolutely we have

f.t/g.t/h.t/

D Xq1;q2;q32QC

Of .q1/

Og.q2/ Oh.q3/.q1q2q3/i t

DX

q2QC

Xq1;q3

Of .q1/ Og q

q1q3

Oh.q3/

qi t

DX

q2QCbfgh.q/qi t :

Hence

M.fgh/ij

Dbfgh

i

j D Xq1;q3 Of .q1/

Og

i

j q1q3 Oh.q3/

D M.f /M.g/M.h/ij

;

since 1=q1 and q3 must range over N.

In view of Theorem 3.3, it is of interest to know when a given f 2 W QC factorises

as f D gh with g 2 W N and h 2 W N. For then M.f / D M.g/M.h/ and the

invertibility of M.f / follows from knowing when M.g/ and M.h/ are invertible.

Thus, if h is invertible in W N, then

M.h/M.h1/ D M.hh1/ D M.1/ D I D M.1/ D M.h1h/ D M.h1/M.h/;

so that M.h/1 D M.h1/. Similarly, if g is invertible in W N, then M.g/1 DM.g1/. It follows then that M.f /1 D M.h/1M.g/1.

Let F W QC denote the set of functions in W QC which factorise as

f D f qf C (3.9)

where f 2 GW N, f C 2 GW N, and q 2 QC.

Note that with f as above, then 1=f D f 1 .1=q/f 1C , so 1=f 2 F W QC . Inparticular,F W QC GW QC . Note that M.q / is invertible if and only if q D 1.

Theorem 3.4. Let f

2F W QC . Then M.f / is invertible if and only if w.f /

D0. If

this is the case, then M.f /1 D M.f 1C /M.f 1 / , with f as in (3.9).

Proof. Write f D f qf C as in (3.9). Then M.f / D M.f /M.q /M.f C/. Now

M.f / and M.f C/ are invertible, with inverses M.f 1 / and M.f 1C /, respectively.Thus M.f / is invertible if and only if M.q / is invertible. But this happens if and

only if q D 1.Since w.f / D w.f / C w.q / C w.f C/ D w.q/ D log q, we see that M.f /

is invertible if and only if w.f / D 0.

Now suppose w.f / D 0. Then the above gives

M.f /1 D M.f /M.f C/1 D M.f C/1M.f /1 D M.f 1C /M.f 1 /:




Multiplicative coefficients and Euler products.

Definition.

(a) A function aWQC ! C is multiplicative if a.1/ D 1 and

a.pa1

1 pak

k / D a.p

a1

1 / a.pak

k /;

for all distinct primes pi and all ai 2 Z. We say a is completely multiplicative if,

in addition to the above,

a.pk / D a.p/k and a.pk/ D a.p1/k ;

for all primes p and k 2 N.

(b) For a subset S of F , let MF denote the set of f 2 S for which Of ./ is multi-plicative.

(c) Let f 2 F and p prime. Suppose thatP

k2Z j Of .pk/j converges. Then define

f p DXk2Z

Of .pk /pk :

Note that f p is periodic with period 2log p

. Define f ]p WT ! C by

f ]p .ei / D f p.= log p/ D Xk2Z

Of .pk /eki for 0 < 2.

Further, denote by W QC;p the set of those f 2 W QC whose QC-coefficients are

supported on fpk W k 2 Zg. (Thus f p 2 W QC;p by definition.)

Note that, for fixed p, there is a one-to-one correspondence between W QC;p and

W .T/ via the mapping ] .

In [16], the Euler product formulas

f D Xq2QC

Of .q/q DYp

f p and M.f / D Yp

M.f p/;

were shown to hold whenever f 2 MW QC .

The analogue of the Wiener–Hopf factorization holds forMW QC -functions with-out zeros.

Theorem 3.5. Let f 2 MW QC such that f has no zeros. Then f 2 F W QC .

Proof. We have f D Qp f p , where f p 2 W QC

;p and each is non-zero. Hencef

]p 2 W .T/ and has no zeros. Let kp D wind .f

]p ; 0/. Note that kp D 0 for all



98 Titus Hilberdink

sufficiently large p, since

jf ]p .t /

1

j Xm¤0 j Of .pm/

j Xjqjp j Of .q/

j !0 as p

! 1.

Let q DQp pkp (a finite product).

By Section 2.1(iii), we have

f ]p D ]

pkpeg

]p ;

for some g]p 2 W .T/. Hence

f p D pkp egp ;

with gp 2 W QC;p. Thus for P so large that kp D 0 for p > P , we haveYpP

f p D q exp

XpP

gp

:

Now f p.t / ! 1 as p ! 1 uniformly in t , so can choose gp so that gp.t / ! 0 as

p ! 1 (uniformly in t ). Hence, for all sufficiently large p (and all t ), jf p 1j Djegp 1j 1

2jgpj, so that

jgp

j 2

jf p

1

j 2 Xm¤0 j O

f .pm/

j:

Let g.n/ D Ppn gp. Then fg.n/g is a Cauchy sequence in W QC: for n > m

jg.n/ g.m/j X

m<pn

jgpj 2X

m<pn

Xm¤0

j Of .pm/j 2Xr >m

j Of .k/j ! 0

as m ! 1. Thus g.n/ ! g 2 W QC . But each g.n/ is of the form hn C kn with

hn 2 W N and kn 2 W N (since gp 2 W QC;p). Thus g D h C k with h 2 W N, k 2 W N.

It follows that f D qeh

ek

, which is of the required form. Note that, as such,

w.f / DX

p

w.kpp / D

Xp

kpw.p/ DX

p

wind.f ]p ; 0/ log p;

where the sum is finite.

Corollary 3.6. Let f 2 MW QC such that f has no zeros and w.f / D 0. Then

M.f / is invertible.

Example. M. / is invertible for ˛ > 1 with M. /1 D M.1= /.




4 Unbounded multiplicative Toeplitz operators and

matrices

The last section has, in various ways, been a relatively straightforward extension of

the theory of bounded Toeplitz operators to the multiplicative setting. The theory of unbounded Toeplitz operators is rather less satisfactory and not easy to generalise.

Our particular concern is in fact the operator M. / for ˛ 1, since we expect

a connection with the Riemann zeta function. The hope is that if a satisfactory theoryis developed for this case, it can be generalised to other symbols.

In this section, we shall therefore concentrate on the particular case when f .n/ Dn˛, when the symbol is .˛ i t /. Throughout we write '˛ (equivalently, M. /)

for 'f .

From Section 3 we see that for ˛

1, '˛ is unbounded. It is interesting to see to

what extent properties of '˛ are related to properties of . The above theory is only

valid for absolutely convergent Dirichlet series, when the symbols are bounded. But

.˛ i t / is unbounded for ˛ 1.How to measure unboundedness? We shall investigate two different measures.

The first case can be considered as restricting the range, while in the second case we

shall restrict the domain.

4.1 First measure – the function ˆ˛.N / With bn defined by a D .an/ '˛7!

.bn/, i.e., bn DPd jn d ˛an=d , let

ˆ˛ .N / D supkakD1

N XnD1

jbnj21=2

:

Theorem 4.1. We have the following asymptotic formulae for large N :

ˆ1.N / D e log log N C O.1/ .˛ D 1/

log ˆ˛.N / .log N /1˛

log log N

1

2 < ˛ < 1

log ˆ 12 .N / log N

log log N 12

: ˛ D 1

2Sketch of proof. .For the proof see [17]/. We start with upper bounds.

First we note that for any positive arithmetical function g.n/,

ˆ˛ .N /2 X

nN

g.n/

n˛

maxnN

Xd jn

1

g.d/d

: (3.10)

This is because

jbnj2 D ˇXd jn

1p g.d/d

˛2

p g.d/an=d

d ˛

2ˇ2 X

d jn

1

g.d/d Xd jn

g.d/jan=d

j2

d ;



100 Titus Hilberdink

by Cauchy–Schwarz. Writing G.n/ D Pd jn g.d /1d ˛ , we have

XnN jbn

j2

XnN

G.n/Xd jn

g.d/jan=d j2

d maxnN

G.n/ Xd N

g.d/

d XnN=d jan

j2:

Taking kak2 D 1 yields (3.10). The idea is to choose g appropriately, so that the

RHS of (3.10) is as small as possible.For 1

2 < ˛ 1, choose g.n/ to be the following multiplicative function: for

a prime power pk let

g.pk / D(

1; if pk M;

. M pk /ˇ ; if pk > M .

Here M D .2˛ 1/ log N and ˇ is a constant satisfying 1 ˛ < ˇ < ˛. Note thatg.pk / g.p/ for every k 2 N and p prime.

We estimate the expressions in (3.10) separately. FirstXnN

g.n/

n˛

Yp

1 C

1XkD1

g.pk /

pk˛

Y

p

1 C g.p/

p˛ 1

exp

nXp

g.p/

p˛ 1

o(3.11)

and, after some manipulations using the Prime Number Theorem, one finds for the

case ˛ < 1

log XnN

g.n/

n˛

. ˇM 1˛

.1 ˛/.˛ C ˇ 1/ log M : (3.12)

Now consider G.n/, which is multiplicative because g.n/ is. At the prime powers wehave

G.pk / DkX

rD0

1

p˛r g.pr / D

Xr 0

pr M

1

p˛r C 1

M

Xr k

pr > M

1

p.˛ˇ /r

1 C 1

p˛ 1 C 1

M .1 pˇ˛ /:

(Here we require ˇ < ˛.) Note that this is independent of k. It follows that

G.n/ exp

Xpjn

1

p˛ 1 C 1

M

Xpjn

1

1 pˇ˛

:

The right-hand side is maximised when n is as large as possible (i.e. N ) and N is of the form N D 2 : 3 : : : P . For such a choice, log N D .P/ P , so that (using the

prime number theorem)

log max

nN

G.n/ . XpP

1

p˛

1 C

1

M ˛ XpP

1

.log N /1˛

.1 ˛/ log log N C

log N

M ˛

log log N

:

(3.13)




From (3.12) and (3.13) it follows that M should be of order log N for optimality. So

taking M D log N (with > 0), (3.10), (3.12) and (3.13) then imply

log ˆ˛.N / . ˇ1

˛

2.1 ˛/.˛ C ˇ 1/ C 1

2.1 ˛/ C 1

2˛.log N /

1

˛

log log N

for every ˇ 2 .1 ˛;˛/ and > 0. Since ˇ˛Cˇ1

decreases with ˇ, the optimal

choice is to take ˇ arbitrarily close to ˛. Hence we require inf >0 h./, where

h./ D ˛1˛

.1 ˛/.2˛ 1/ C 1

.1 ˛/ C 1

˛:

Since h0./ D ˛˛C1 .

2˛1 1/, we see that the optimal choice is indeed D 2˛ 1.

Substituting this value of gives

log ˆ˛.N / . .1 C .2˛ 1/˛/

2.1 ˛/

.log N /1˛

log log N :

For ˛ D 1, we use the same function g.n/ as before (though with possibly differ-ent values of M and ˇ). From (3.11) it follows thatX

nN

g.n/

n

YpM

1

1 1p

Y

p>M

1 C M ˇ

pˇ .p 1/

:

By Mertens’ Theorem, thefirst productis e log M CO.1/,while M ˇ Pp>M p1ˇ D

O.1= log M /, and soXnN

g.n/

n

e log M C O.1/

expfO.1= log M /g D e log M C O.1/: (3.14)

For the G.n/ term we have, as for the ˛ < 1 case,

G.pk / 1

1

1p

C 1

M.1

pˇ1/

:

Thus, with N D 2 : 3 : : : P ,

G.N/ Y

pP

1

1 1p

1 C 1 1=p

M.1 pˇ1/

D

e log P C O.1/

1 C O P

M log P

:

Taking M D log N and noting that P log N , the right-hand side is e log log N CO.1/. Combining with (3.14) shows that

ˆ1.N / e log log N C O.1/:




The case ˛ D 12

. The function g as chosen for ˛ 2 . 12

; 1 is not suitable for

an upper bound as we would require 12

< ˇ < 12

! Instead we take g to be the

multiplicative function following: for a prime power pk , let

g.pk / D minn

1; M

pk .log p/2

12o

:

Here M > 0 is independent of p and k and will be determined later. Thus g.pk / D 1if and only if pk .log p/2 M . Note that g.pk/ g.p/ 1 for all k 1 and allprimes p. Thus (3.11) holds with ˛ D 1

2 and (using the prime number theorem)

log

XnN

g.n/p n.

Xp.

M

.log M /2

1p p

1 C

p M

Xp&

M

.log M /2

1

p log p

p M

log M : (3.15)

(The first sum is of orderp

M =.log M /2 and the main contribution comes from thesecond term.)

Regarding G.n/, this time we have2

G.n/ DY

pkkn

G.pk / Y

pkkn

1 C

kXrD1

1

pr=2 C 1p

M

kXrD1

log p

;

so that

log G.n/ Xpjn

1p p 1

C 1p M

Xpkkn

k log p log np M

CXpjn

1p p 1

:

The right-hand side above is maximal when n D N D 2 : 3 : : : P , hence

log maxnN

G.n/ . log N p

M CX

pP

1p p log N p

M C 2

p log N

log log N :

Combining with (3.15), (3.10) then gives

log ˆ 12

.N / .p M

2 log M C log N

2p

M C p log N

log log N :

The optimal choice for M is easily seen to be M D log N log log N , and this givesthe upper bound in (iii).

Now we proceed to give lower bounds.

For a fixed n 2 N, let

ad D 1

p d.n/if d jn, and zero otherwise.

2Here as usual, p kjjn means p kjn but pkC1 6jn.




Then kak2 D 1, while for d jn

bd

D 1

p d.n/ Xcjd

1

c˛ D ˛.d /

p d.n/

:

Hence for N n,XkN

jbkj2 Xd jn

b2d D

1

d.n/

Xd jn

˛.d /2 DW ˛ .n/:

With this notation ˆ˛ .N / maxnN

p ˛.n/, and the lower bounds follow from

the maximal order of ˛ .n/. For 12

< ˛ < 1 this can be found easily. Since ˛ .p/ D1

Cp˛

C 12

p2˛ for p prime, we find for n

D2 : 3 : : : P (so that log n

P ) that

˛.n/ DY

pP

1 C 1

p˛ C O

1

p2˛

D exp

.1 C o.1//

XpP

1

p˛

D exp

.1 C o.1//P 1˛

.1 ˛/ log P

D exp

.1 C o.1//.log n/1˛

.1 ˛/ log log n

:

Now, if tk is the kth number of the form 2:3 P (i.e., tk D p1 pk ), then log tk k log k log tkC1. Hence for tk N < tkC1, log N k log k. It follows that

ˆ˛ .N / p ˛.tk / exp .1Co.1//.log tk /1

˛

2.1 ˛/ log log tk D exp .1Co.1//.log N /

1

˛

2.1 ˛/ log log N :

For ˛ D 1, we have to be a little subtler to obtain maxnN

p 1.n/ D e log log N C

O.1/. We omit the details, which can be found in [17].

For the case ˛ D 12

, the above choice doesn’t give the correct order and we lose

a power of log log N . Instead, we follow an idea of Soundararajan [31]. Let f be themultiplicative function supported on the squarefree numbers whose values at primes

p is

f.p/

D . M p

/1=2 1log p

; for M p R;

0; otherwise.

Here M D log N.log log N / as before and log R D .log M /2.

Now take an D f.n/F.N/1=2, where F. N/ D PnN f.n/2 so thatPnN a

2n D1. Then by Holder’s inequality N X

nD1

b2n

1=2

N X

nD1

anbn D 1

F. N/

N XnD1

f.n/p n

Xd jn

p df. d/

D 1

F. N/ XnN

f.n/

p n Xd N=n

.n;d/ D 1

f. d/2: (3.16)




Now using ‘Rankin’s trick’3 we have, for any ˇ > 0

XnN

f.n/

n1=2 Xd N=n

.n;d/D 1

f. d/2

D XnN

f.n/

n1=2 Xd 1

.n;d/ D 1

f. d/2

Xd > N=n

.n;d/ D 1

f. d/2DX

nN

f.n/

n1=2

Yp−n

1 C f.p/2

C O

n

N

ˇ Yp −n

1 C pˇ f.p/2

:

(3.17)

The O -term in (3.17) is at most a constant times

1

N ˇ XnN

f.n/nˇ1=2

Yp−n1

Cpˇ f.p/2

1

N ˇ Yp 1

Cpˇ f.p/2

Cpˇ1=2f.p/;

while the main term in (3.17) is (using Rankin’s trick again)

Yp

1 C f.p/2 C f .p/

p1=2

C O

1

N

Yp

1 C f.p/2 C pˇ1=2 f.p/

:

Hence (3.16) implies

N XnD1

b2n1=2

1

F. N/Yp1 C f.p/2 C

f .p/

p1=2 CO

1

N

Yp

1 C pˇ f.p/2 C pˇ1=2f.p/

:

The ratio of the O -term to the main term on the right is less than

exp

ˇ log N C

XM

p

R

.pˇ 1/

f.p/2 C f.p/

p1=2

;

which equals

exp

ˇ log N C

XM pR

.pˇ 1/

M

p.log p/2 C M 1=2

p.log p/

:

Take ˇ D .log M /3. The term involving M 1=2 is at most .log N /1=2C" for every

" > 0, while the remaining terms in the exponent are (by the

3

If cn > 0, then for any ˇ > 0, Pn>x cn xˇ P1

nD1 nˇ

cn.




prime number theorem in the form .x/ D li.x/ C O.x.log x/A/ for all A)

ˇ log N

CM Z

R

M

t ˇ 1

t .log t /3

dt C

O log N

.log log N /AD ˇ log N C ˇM

Z R

M

dt

t .log t /2 C O

ˇ2M

Z R

M

dt

t log t

ˇ

log N log log log N

log log N ;

after some calculations.

Finally, since F.N / Qp.1 C f.p/2/, this implies

ˆ1=2.N / 1

2 YM pR

1 C f.p/

p1=2.1 C f.p/2/;

for all N sufficiently large. Hence

log ˆ1=2.N / & M 1=2X

M pR

1

p.log p/ log N

log log N

1=2

;

as required.

Remark. The result for 1

2

< ˛ < 1 is

.log N /1˛

2.1 ˛/ log log N . log ˆ˛ .N / .

.1 C .2˛ 1/˛/

2.1 ˛/

.log N /1˛

log log N :

It would be nice to obtain an asymptotic formula for log ˆ˛.N /. Indeed, it is possible

to improve the lower bound at the cost of more work by using the method for the case

˛ D 12

, but we have not been able to obtain the same upper and lower limits.

4.2 Connections between ˆ˛.N / and the order of j.˛Cit/j The lower

bounds obtained for ˆ˛.N / for 1

2

< ˛

1 can be used to obtain information regard-

ing the maximum order of .s/ on the line <s D ˛.

Proposition 4.2. With bn DP

d jn d ˛an=d we have, for any ,

XnN

jbnj2 DX

m;nN

aman.m;n/2˛

m˛n˛

Xk N

Œm;n

1

k2˛:

Proof. We have

jbnj2 D bnbn D 1

n2˛ Xcjn;d jn

c˛d ac ad D 1

n2˛ XŒc;djn

c˛d acad ;




since cjn; d jn if and only if Œc;d jn. Hence

XnN jbn

j2

D Xc;d N

c˛d ˛ acad XnN;Œc;djn

1

n2˛ D Xc;d N

c˛d ˛ ac ad

Œc;d2˛ Xk N Œc;d

1

k2˛

;

by writing n D Œc;dk. Since .c; d/Œc; d D cd , the result follows.

It follows that if an 0 for all n and ˛ > 12

, then

XnN

jbnj2 .2˛/X

m;nN

aman.m; n/2˛

.mn/˛ : (3.18)

Theorem 4.3. Let 1

2 < ˛ 1 and let a 2 l 2 with kak2 D 1. Let AN .t / DPN nD1 anni t . Let N T , where 0 < < 2

3.˛ 1

2/. Then for some > 0 ,

1

T

Z T

1

j.˛ C i t /j2jAN .t /j2 dt D .2˛/X

m;nN

aman.m; n/2˛

.mn/˛ C O.T /: (3.19)

Proof. We shall assume 12

< ˛ < 1, adjusting the proof for the case ˛ D 1 after-

wards. For ˛ ¤ 1, we can integrate from 0 to T since the error involved is at most

O.N=T/

DO.T /.

Starting from the approximation .˛ C i t / D Pnt n˛i t C O.t ˛/, we have

j.˛ C i t /j2 DˇX

nt

1

n˛Ci t

ˇ2C O.t 12˛/:

Let k; l 2 N such that .k;l / D 1. Let M D maxfk; lg < T . The above givesZ T

0

j.˛ C i t /j2k

l

i t

dt DZ T

0

ˇˇXnt

1

n˛Ci t

ˇˇ

2k

l

i t

dt C O.T 22˛/:

The integral on the right isZ T

0

Xm;nt

1

.mn/˛

km

ln

i t

dt DX

m;nT

1

.mn/˛

Z T

maxfm;ng

km

ln

i t

dt:

The terms with k m D ln (which implies m D r l ; n D rk with r integral) contribute

1

.kl/˛ XrT =M

T rM

r 2˛ D .2˛/

.kl/˛ T C O

M 2˛1T 22˛

.kl/˛ :




The remaining terms contribute at most

2 Xm; n T km ¤ ln

1

.mn/˛j log kmln j 2M 2˛ Xm; n T

km ¤ ln

1

.kmln/˛j log kmln j

2M 2˛X

m1 kT;n1 lT m1 ¤ n1

1

.m1n1/˛j log m1

n1j

2M 2˛X

m1; n1 M T m1 ¤ n1

1

.m1n1/˛j log m1

n1j

D O.M

2˛

. M T /

2

2˛

log.MT//D O.M 2T 22˛ log T /;

using Lemma 7.2 from [33]. HenceZ T

0

j.˛ C i t /j2k

l

i t

dt D .2˛/

.kl/˛ T C O.M 2T 22˛ log T /:

It follows that for any positive integers m; n < T ,

Z T

0 j.˛ C i t /j2m

n i t

dt D .2˛/.m; n/2˛

.mn/˛ T C O.maxfm; ng2

T 2

2˛

log T /:

Thus, with AN .t / D PN nD1 anni t ,Z T

0

j.˛ C i t /j2jAN .t /j2 dt DX

m;nN

aman

Z T

0

j.˛ C i t /j2m

n

i t

dt

D .2˛/T X

m;nN

aman.m; n/2˛

.mn/˛

C OT 22˛ log T Xm;nN

maxfm; ng2jamanj:

The sum in the O -term is at most N 2.P

nN janj/2 N 3, using Cauchy–Schwarz.Hence

1

T

Z T

0

j.˛ C i t /j2jAN .t /j2 dt D .2˛/X

m;nN

aman.m; n/2˛

.mn/˛ C O

N 3 log T

T 2˛1

:

Since N 3

T 3

and 3 < 2˛ 1, the error term is O.T

/ for some > 0.




If ˛ D 1 we integrate from 1 to T instead and the O -term above will contain an

extra log T factor, but this is still O.T /.

We note that with more care the N

3

could be turned into an N

2

, so that we cantake < ˛ 12

in the theorem. This is however not too important for us.

Corollary 4.4. Let 12

< ˛ 1. Then for every " > 0 and N sufficiently large,

maxtN

j.˛ C i t /j ˆ˛ .N 23

.˛ 12

/"/ C O.N / (3.20)

for some > 0.

Proof. Let an 0 be such that kak2 D 1, and take N D T with < 23

.˛ 12

/. By

(3.18) and (3.19),XnN

jbnj2 1

T

Z T

0

j.˛ C i t /j2jAN .t /j2 dt C O.T /

maxtT

j.˛ C i t /j2 1

T

Z T

0

jAN .t /j2 dt C O.T /

D maxtT

j.˛ C i t /j2X

nN

janj2.1 C O.N=T // C O.T /

using the Montgomery and Vaughan mean value theorem. The implied constants inthe O -terms depend only on T and not on the sequence fang. Taking the supremum

over all such a, this gives

ˆ˛ .N /2 D supkak2D1

XnN

jbnj2 maxtT

j.˛ C i t /j2 C O.T /;

for some > 0, and (3.20) follows.

In particular, this gives the (known) lower bounds

maxtT

j.˛ C i t /j expnc.log T /

1

˛

log log T ofor 1

2 < ˛ < 1 and maxtT j.1 C i t /j e log log T CO.1/ (obtained by Levinson

in [25]).

Morever, we can say more about how often j.˛ C i t /j is as large as this. ForA 2 R and c > 0, let

F A.T / D

t 2 Œ1; T W j.1 C i t /j e log log T A

: (3.21)

F ;c .T / D t 2 Œ0; T W j.˛ C i t /j expnc.log T /1

˛

log log T o: (3.210)




We outline the argument in the case 12

< ˛ < 1. We have

XnN jbn

j2

1

T Z F ;c .T / C Z Œ0;TnF ;c .T /j.˛

Ci t /

j2

jAN .t /

j2 dt

CO.T /:

(3.22)

The second integral on the right is at most

expn2c.log T /1˛

log log T

o 1

T

Z T

0

jAN .t /j2 dt D O

expn2c.log T /1˛

log log T

o;

while, by choosing an D d.N/1=2 for njN and zero otherwise, the LHS of (3.22) isat least ˛ .N /. Every interval ŒT =3; T contains an N of the form 2:3:5 : : :P . As

such, ˛ .N /

expn c0.log T /1˛

loglog T o for some c 0 > 0. Hence, for 2c < c0,

1

T

Z F ;c .T /

j.˛ C i t /j2jAN .t /j2 dt expnc0.log T /1˛

2 log log T

o:

We have j.˛ C i t /j D O.T / for some and jAN .t /j2 d.N/ D O.T "/, so

1

T

Z F ;c .T /

j.˛ C i t /j2jAN .t /j2 dt T 21C".F ;c .T //:

Thus .F ;c .T// T 12" for all c sufficiently small.

In particular, since < 1

˛3 , we have:

Theorem 4.5. For all A sufficiently large .and positive/ ,

.F A.T// T exp

a

log T

log log T

for some a > 0 , and for all c sufficiently small, .F ;c .T // T .1C2˛/=3 for all

sufficiently large T . Furthermore, under the Lindel¨ of Hypothesis, the exponent can

be replaced by 1 ".

Addendum: There has been much recent activity in this field. Aistleitner [1], inessence restricting () to N rows and columns (rather than just the first N ) used

an ingenious idea based on GCD-sums to improve these -results for j.˛ C i t /j and

Theorem 3.5 to expn

c .log T /1˛

.loglog T /˛

o for 1

2 < ˛ < 1, thereby improving Montgomery’s

result. More recently still, Bondarenko and Seip [4] used a similar method for thecase ˛ D 1

2, obtaining

Z 12

.T / exp

nc

s log T log log log T

log log T ofor any c < 1p

2.




4.3 Second measure – '˛ onM2 and the function M .T / For ˛ 1,

'˛ is unbounded on l 2 and so '˛ .l 2/ 6 l 2 (by the closed graph theorem4 – see [32],

p. 183). However, if we restrict the domain to M2, the set of multiplicative elements

of l 2, we find that '.M2/ l 2. More generally, if f is multiplicative then, as weshall see, 'f .M

2/ l 2 in many cases (and hence 'f .M2/ M2).

Notation. LetM2 andM2c denote the subsets of l 2 of multiplicative and completely

multiplicative functions, respectively. Further, writeM2C for the non-negative mem-bers of M2, and similarly for M2

c C.

Let M20 denote the set of M2 functions f for which f g 2 M2 whenever

g 2M2; that is,

M20

D ff

2M2

Wg

2M2

H) f

g

2M2

g:

Thus for f 2 M20, 'f .M

2/ M2.

It was shown in [18] thatM2 is not closed under Dirichlet convolution, soM20 ¤

M2. A criterion for f 2 M2 to be in M20 was given, namely:

Proposition 4.6. Let f 2 M2 be such that P1

kD1 jf .pk /j converges for every prime

p and that P1

kD1 jf .pk /j A for some constant A independent of p. Then f 2 M20.

On the other hand, if f 2 M2 with f 0 and for some prime p0 , f .pk0 /

decreases with k and

P1kD1 f .pk

0 / diverges, then f 62 M20.

The proof is based on the following necessary and sufficient condition: Let f; g 2M2 be non-negative. Then f g 2M2 if and only if

Xp

Xm;n1

1XkD0

f .pm/g.pn/f.pmCk /g.pnCk/ converges:

This can be proven in a direct manner.Thus, in particular, M2

c M20. For f 2 M2

c if and only if jf.p/j < 1 for all

primes p and Pp jf.p/j2 < 1. Thus

1XkD1

jf .pk /j D jf.p/j1 jf.p/j A;

independent of p.For example, .n˛/ 2 M2

0 for ˛ > 12

.

The “quasi-norm” M f .T /. Let f 2 M20. The discussion above shows that

'f .M2/ M2 but, typically, 'f is not ‘bounded’ on M2 (if f 62 l 1) in the sense

4Being a ‘matrix’ mapping, '˛ is necessarily a closed operator, and so '˛ .l2/ l2 implies '˛ is bounded.




that5 k'f ak=kak is not bounded by a constant for all a 2 M2. A natural question

is: how large can k'f ak become as a function of kak? It therefore makes sense to

define, for T

1,

M f .T / D supa 2M2

kak D T

k'f akkak :

We shall consider only the case f .n/ D n˛ , although the result below can be ex-

tended without any significant changes to f completely multiplicative and such that

f jP is regularly varying of index ˛ with ˛ > 1=2 in the sense that there exists

a regularly varying function Qf (of index ˛) with Qf .p/ D f .p/ for every prime p.

We shall write M f .T / D M .T / in this case.

Theorem 4.7. As T

! 1M 1.T / D e .log log T C log log log T C 2 log 2 1 C o.1//; (3.23)

while for 12

< ˛ < 1 ,

log M .T / D

B. 1˛

; 1 12˛

/˛

.1 ˛/2˛ C o.1/

.log T /1˛

.log log T /˛: (3.24)

Here B.x; y/ denotes the Beta function

R 1

0 t x1.1 t /y1dt .

Sketch of proof for the case 12 < ˛ < 1 (for the full proof see [18]). We consider first

upper bounds. The supremum occurs for a 0, which we now assume. Write

a D .an/, '˛a D b D .bn/. Define ˛p and ˇp for prime p by

˛p D1X

kD1

a2pk and ˇp D

1XkD1

b2pk :

By the multiplicativity of a and b , T 2 D kak2 D Qp .1 C ˛p/ and kbk2 D Q

p.1 Cˇp/. Thus

k'˛akkak D Y

p

s 1 C ˇp

1 C ˛p

:

Now for k 1

bpk DkX

rD0

pr ˛apkr D apk C p˛ bpk1 ;

whence

b2pk D a2

pk C 2p˛ apk bpk1 C p2˛ b2pk1 :

5Here, and throughout this section k k D k k2 is the l 2-norm.




Summing from k D 1 to 1 and adding 1 to both sides gives

1 C ˇp D 1 C ˛p C 2p˛

1

XkD1

apk bpk1 C p2˛

.1 C ˇp/: (3.25)

By Cauchy–Schwarz,

1XkD1

apk bpk1 1X

kD1

a2pk

1XkD1

b2pk1

1=2

Dq

p.1 C ˇp/;

so, on rearranging,

.1 C ˇp/ 2p˛p p.1 C ˇp/1 p2˛

1 C ˛p

1 p2˛:

Completing the square we obtainq 1 C ˇp p˛p

p

1 p2˛

2

1 C ˛p

.1 p2˛/2:

The term on the left inside the square is non-negative for every p since 1 C ˇp 1C˛p

1

p2˛ , which is greater than

p2˛ ˛p

.1

p2˛/2 for ˛ > 1

2. Rearranging givess

1 C ˇp

1 C ˛p

1

1 p2˛

1 C 1

p˛

r ˛p

1 C ˛p

:

Let p Dq

˛p

1C˛p. Taking the product over all primes p gives

k'˛ akkak .2˛/

Yp

1 C p

p˛

.2˛/ exp

nXp

p

p˛

o: (3.26)

Note that 0 p < 1 and Qp1

1 2pD T 2. The idea is to show now that the main

contribution to the above sum comes from the range p log T log log T .Let " > 0 and put P D log T log log T . We split up the sum on the right-hand

side of (3.26) into p aP , aP < p AP , and p > AP (for a small and A large).

First, XpaP

p˛ p X

paP

p˛ a1˛P 1˛

.1 ˛/ log P < "

.log T /1˛

.log log T /˛; (3.27)




for a sufficiently small. Next, using the fact that log T 2 D logQ

p1

1 2p Pp 2p ,

we have

Xp>AP

p˛ p Xp>AP

p2˛ Xp>AP

2p1=2

. 2A

1

2˛

P

1

2˛

log T .2˛ 1/ log P

1=2

.log T /1˛.log log T /˛

A˛1=2p 1=2

< " .log T /1˛

.log log T /˛ (3.28)

for A sufficiently large. This leaves the range aP < p AP and the problem

therefore reduces to maximising XaP<pAP

p

p˛

subject to 0 p < 1 and Qp1

1 2pD T 2. The maximum clearly occurs for p

decreasing (if p0 > p for primes p < p0, then the sum increases in value if weswap p and p0). Thus we may assume that p is decreasing.

By interpolation we may write p D g. pP

/, where gW .0; 1/ ! .0; 1/ is con-tinuously differentiable and decreasing. Of course, g will depend on P . Let h Dlog 1

1g2 , which is also decreasing. Note that

2 log T DX

p

h p

P

X

paP

h p

P

h.a/.aP / cah.a/ log T;

for P sufficiently large, for some constant c > 0. Thus h.a/ C a (independently of

T ).Now, for F W .0; 1/ ! Œ0; 1/ decreasing, it follows from the Prime Number

Theorem in the form .x/ D li.x/ C O. x.log x/2 / that

Xax<pbx

F p

x

D x

log x

Z b

a

F C O xF.a/

.log x/2

; (3.29)

where the implied constant is independent of F (and x). Thus

2 log T XaP<pAP

h p

P

P

log P

Z A

a

h log T Z A

a

h:

Since a and A are arbitrary,R 1

0 h must exist and is at most 2. Also, by (3.29),X

aP<pAP

p

p˛ D 1

P

XaP<pAP

g p

P

p

P

˛

P 1˛

log P

Z A

a

g.u/

u˛ du:

As a; A are arbitrary, it follows from above and (3.26), (3.27), (3.28) that

log k'˛ akkak Z 10

g.u/u˛

du C o.1/ .log T /1˛

.log log T /˛:




Thus we need to maximiseR 1

0 g.u/u˛du subject to

R 10

h 2 over all decreasing

gW .0; 1/ ! .0; 1/. Since h is decreasing, one finds that xh.x/ ! 0 as x ! 1 and

as x

!0C.

For the supremum, we can consider only those g (and h) which are continuouslydifferentiable and strictly decreasing, since we can approximate arbitrarily closely by

such functions. On writing g D s ı h, where s.x/ D p 1 ex , we haveZ 1

0

g.u/

u˛ du D

hg.u/u1˛

1 ˛

i10

1

1 ˛

Z 10

g0.u/u1˛ du

D 1

1 ˛

Z 10

s0.h.u//h0.u/u1˛ du

D 1

1 ˛ Z h.0C/

0

s0.x/l.x/1˛ dx;

where l D h1, sincep

ug.u/ ! 0 as u ! 1. The final integral is, by Holder’s

inequality, at most Z h.0C/

0

s01=˛

˛Z h.0C/

0

l

1˛

: (3.30)

ButR h.0C/

0 l D R 10 uh0.u/du D R 1

0 h 2, so

Z 1

0

g.u/

u˛

du

21˛

1 ˛Z 1

0

s01=˛

˛

:

A direct calculation shows that6 R 1

0 .s0/1=˛ D 21=˛B. 1˛

; 1 12˛

/. This gives the

upper bound.The proof of the upper bound leads to the optimal choice for g and the lower

bound. We note that we have equality in (3.30) if l=.s0/1=˛ is constant, i.e., l.x/ Dcs0.x/1=˛ for some constant c > 0 — chosen so that

R 10 l D 2. This means that we

take

h.x/ D .s0/1x

c

˛D log

1

2 C 1

2

r 1 C

c

x

2˛

;

from which we can calculate g . In fact, the required lower bound can be found bytaking an completely multiplicative, with ap for p prime defined by

ap D g0

p

P

;

where P D log T log log T and g0 is the function

g0.x/ Dvuut

1 2

1 C

q 1 C . c

x/2˛

;

6The integral is 21=˛R 1

0 ex=˛ .1 ex /1=2˛ dx D 21=˛R 1

0 t 1=˛1.1 t /1=2˛dt .




with c D 21C1=˛=B. 1˛

; 1 12˛

/. As such, by the same methods as before, we have

kak D T 1Co.1/ and

log k'˛akkak D Xp

p˛ g0 pP C O.1/ P

1

˛

log P Z 1

0

g0.u/u˛

du:

By the choice of g0, the integral on the right isB. 1

˛; 1 1

2˛/˛

.1 ˛/2˛ , as required.

Remark. These asymptotic formulae bear a strong resemblance to the (conjectured)maximal order of j.˛ C iT /j. It is interesting to note that the bounds found here are

just larger than what is known about the lower bounds for Z˛.T / (see the interlude

on upper and lower bounds on .s/, especially items (b) and (c)). We note that the

constant appearing in (3.24) is not Lamzouri’s C.˛/ since, for ˛ near 12 , the former is

roughly 1q 1

2

, while C.˛/ 1p 2˛1

.

Lower bounds for '˛ and some further speculations. We can study lower bounds

of '˛ via the function

m˛.T / D inf a 2M2

kakD T

k'˛akkak :

Using very similar techniques, one obtains results analogous to Theorem 4.7:

1

m1.T / D 6e

2 .log log T C log log log T C 2 log 2 1 C o.1//

and

log1

m˛.T / B. 1

˛; 1 1

2˛/˛.log T /1˛

.1 ˛/2˛.log log T /˛ for 1

2 < ˛ < 1:

We see that m˛.T / corresponds closely to the conjectured minimal order of j.˛ CiT /j (see [13] and [27]).

The above formulae suggest that the supremum (respectively infimum) of

k'˛ak=kak with a 2 M2 and kak D T are close to the supremum (resp. infi-mum) of j j on Œ1; T . One could therefore speculate further that there is a close

connection between k'˛ ak=kak (for such a) and j.˛ C iT /j.Heuristically, we could argue as follows. Consider

1

T

Z T

0

j.˛ i t /j2

ˇ 1XnD1

anni t

ˇ2dt: (3.31)

This is less than

Z˛.T /2 1T Z T

0ˇ 1X

nD1

anni t ˇ2 dt Z˛.T /2kak2;




by the Montgomery–Vaughan mean value theorem (under appropriate conditions).

On the other hand, (3.31) is expected to be approximately

1T Z T

0

ˇ 1XnD1

bnni t ˇ2 dt 1XnD1

jbnj2 D k'˛ ak2:

Putting these together givesk'˛ akkak . Z˛.T/:

The left-hand side, as a function of kak, can be made as large as F .kak/, where

F.x/ D expfc˛.log x/1˛

.loglog x/˛ g. If the above continues to hold for kak as large as T , then

M .T /

Z˛.T / would follow. Even if it holds for

ka

kas large as a smaller power

of T , one would recover Montgomery’s -result.Alternatively, considering (3.31) over ŒT; 2T ,

1

T

Z 2T

T

j.˛ i t /j2

ˇ 1XnD1

anni t

ˇ2dt D j.˛ Ci t0/j2 1

T

Z 2T

T

ˇ 1XnD1

anni t

ˇ2dt (3.32)

for some t0 2 ŒT;2T and, using the Montgomery–Vaughan mean value theorem

(assuming it applies), this is approximately

j.˛ C i t0/j2

1

XnD1 janj

2

D j.˛ C i t0/j2

kak2

:

On the other hand, formally multiplying out the integrand, by writing .˛ i t / DP1nD1

nit

n˛ and formally multiplying out the integrand, the left-hand side of (3.32)becomes (heuristically)

1

T

Z 2T

T

ˇ

ˇ

1XnD1

bnni t

ˇ

ˇ

2

dt 1X

nD1

jbnj2 D k'˛ ak2:

Equating these gives k'˛ akkak j.˛ C i t0/j:

Clearly there are a number of problems with this. For a start, we need '˛ a 2l 2. More importantly, the error term in the Montgomery–Vaughan theorem containsP1

nD1 njanj2, which may diverge. Also, an and hence kak may depend on T , andfinally, the series for .˛ i t / doesn’t converge for ˛ 1.

If an D 0 for n > N , the above argument can be made to work, even for N a (small) power of T (see for example [17]), but difficulties arise for larger powersof T .

There seem to be some reasons to believe that the error from the Montgomery–Vaughan theorem should be much smaller when considering products. These occur




when an is multiplicative. For example (with Q DQpP p so that log Q D .P/

P by the Prime Number Theorem),

12T

Z T

T

ˇ YpP

.1 C pi t /ˇ2 dt D 12T

Z T

T

ˇXd jQ

d i t ˇ2 dt D Xd jQ

1 C O 1T Xd jQ

d :

The ‘main term’ is d.Q/ D 2.P/, while the error is at least QT

. However the left-

hand side is trivially at most 4.P/, so the error dominates the other terms if P >.1 C ı/ log T . If, say, P is of order log T log log T (which is the range of interest),

then .P / log T , so 2.P/ is like a power of T , but Q is roughly like T loglog T –

far too large.Thus it may be that for an completely multiplicative, it holds that

1

T

Z 2T

T

j.˛ i t /j2ˇ Y

pP

1

1 appi t

ˇ2dt j.˛ C i t0/j

YpP

1

1 japj2

for P up to c log T log log T . This suggests the following might be true:

(a) given a 2 M2 with kak D T , there exists t 2 ŒT;2T such that

k'˛akkak j.˛ C i t /j:

(b) given T 1 , there exists a 2 M2 with kak D T such that

k'˛ akkak j.˛ C iT /j:

Here, means something like log-asymptotic, , or possibly even D. Thus (a)

implies M .T / . Z˛.T /, while (b) implies the opposite. Together they would imply

we can encode real numbers into M2-functions with equal l 2-norm, such that '˛ hasa similar action as .

Closure of MB2N

? We finish these speculations with a final plausible conjecture

regarding the closure under multiplication of functions in B 2N with multiplicative co-

efficients.

Let M0B2N denote the subset MB2

N of functions f for which Of 2 M0. Recall

that M20 is the subset of M2 for which g 2M2 ) f g 2 M2. This suggests the

following conjecture:

Conjecture. Let f 2 MB2N and g 2M0B2

N. Then fg 2MB2N.

In particular, MB2NMcB

2N D MB

2N. Since 2 McB

2N for ˛ >

1

2 , this wouldimply k˛ 2 MB2

N for every k 2 N and ˛ > 12

, which implies the Lindelof hypothesis.




5 Connections to matrices of the form

.f .ij =.i; j /2//i;jN

The asymptotic formulae for ˆ˛.N / in Theorem 4.1 can be used to obtain informa-

tion on the largest eigenvalue of certain arithmetical matrices. Various authors havediscussed asymptotic estimates of eigenvalues and determinants of arithmetical ma-

trices (see for example [6], [7], [26] to name just a few).

Let AN .f / denote the N N matrix with ij th-entry f.i=j/ if j ji and zero other-wise. As noted in the introduction, these matrices behave much like Dirichlet series

with coefficients f .n/; namely,

AN .f /AN .g/ D AN .f g/:

In particular, AN .f / is invertible if f has a Dirichlet inverse, i.e., f.1/ ¤ 0, in whichcase AN .f /1 D AN .f 1/.

Suppose for simplicity that f is a real arithmetical function. (For complex valueswe can easily adjust.) Let

ˆf .N / D supkak2D1

N XnD1

jbnj2

12

where bn D Pd jn f.d/an=d . Observe that ˆf .N /2 is the largest eigenvalue of the

matrix AN .f /T AN .f /:

Indeed, we have

b2n D

Xi;j jn

f n

i

f n

j

ai aj ;

so that, on noting i; j jn if and only if Œi ; j jn (where Œ i ; j denotes the lcm of i and j )

N

XnD1

b2n D

N

XnD1 XŒi;jjn

f

n

i f

n

j ai aj D Xi;j N

b.N /ij ai aj ;

where (using .i; j/Œi;j D ij )

b.N /ij D

Xk N

Œi;j

f ki

. i; j/

f kj

. i; j/

:

But b.N /ij is also the ij th-entry of AN .f /T AN .f /, as an easy calculation shows. Thus

ˆf .N /

2

D supa21CCa2

N D1 Xi;j N

b

.N /

ij ai aj (3.33)




is the largest eigenvalue of AN .f /T AN .f /, i.e., ˆf .N / the largest singular value7

of AN .f /. Thus an equivalent formulation of Corollary 3.2 for f supported on N is:

For f

2l 1 non-negative, the largest singular value of AN .f / tends to

kf

k1.

Now if f is completely multiplicative, then

b.N /ij D f

ij

. i; j/2

Xk N

Œi;j

f.k/2;

which for large N is roughly kf k22f . ij

.i;j/2 / for f 2 l 2. This suggests that the matrixf . ij

.i;j/2 /

i;j N has its largest eigenvalue close to ˆf .N /2=kf k2

2. This is indeed

the case.

Corollary 5.1. Let f 2 l 2 be non-negative and completely multiplicative. Let ƒN

denote the largest eigenvalue of

f . ij

.i;j/2 /

i;j N . Then

ˆf .N /2

kf k22

ƒN ˆf .N 3/2PN kD1 f.k/2

:

In particular, for f 2 l 1 ,

limN !1

ƒN D kf k21

kf

k22

:

Proof. We have

ƒN D supa2

1CCa2N D1

Xi;j N

f ij

. i; j/2

ai aj (3.34)

When f 0, the supremums in (3.33) and (3.34) are reached for an 0. Thus,

ˆf .N /2 kf k22ƒN

follows immediately.

On the other hand, for i; j N , Œi ; j N 2

so

ˆf .N 3/2 X

i;j N

f ij

. i; j/2

ai aj

XkN

f.k/2:

Taking the supremum over all such an gives, ˆf .N 3/2 ƒN

PkN f.k/2, as

required.

Finally, if f 2 l 1, then ˆf .N / ! kf k1 and so ƒN ! kf k21

kf k22

follows.

7

The singular values of a matrix A are the square roots of the eigenvalues of AT

A (or A

A if A hascomplex entries).




The approximate formulae for ˆ˛.N / in Theorem 4.1 lead to:

Corollary 5.2. Let f.n/

D n˛ and let ƒN .˛/ denote the largest eigenvalue of f . ij .i;j/2 /i;j N . Then

ƒN .1/ D 6

2.e log log N C O.1//2;

log ƒN .˛/ .log N /1˛

log log N for 1

2 < ˛ < 1;

log ƒN

1

2

s

log N

log log N :

Bibliography

[1] C. Aistleitner, Lower bounds for the maximum of the Riemann zeta function along vertical

lines, arXiv:1409.6035.

[2] R. Balasubramanian and K. Ramachandra, On the frequency of Titchmarsh’s phenomenom

for .s/. III, Proc. Indian Acad. Sci. 86 A (1977), 341–351.

[3] A. S. Besicovitch, Almost Periodic Functions. Dover Publications, 1954.

[4] A. Bondarenko and K. Seip, Large GCD sums and extreme values of the Riemann zeta func-tion, arXiv:1507.05840v1.

[5] A. Bottcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices. Springer-

Verlag, 1999.

[6] K. Bourque and S. Ligh, Matrices associated with classes of arithmetical functions. J. Number

Theory 45 (1993), 367–376.

[7] D. A. Cardon, Matrices related to Dirichlet series. J. Number Theory 130 (2010), 27–39.

[8] D. W. Farmer, S. M. Gonek and C. P. Hughes, The maximum size of L-functions. J. Reine

Angew. Math. 609 (2007), 215–236.

[9] I. Gohberg, On an application of the theory of normed rings to singular integral equations.

Usp. Mat. Nauk 7 (1952), 149–156.

[10] I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of Linear Operators Vol. I . Birkhauser,

1990.

[11] S. M. Gonek, Finite Euler products and the Riemann Hypothesis, Trans. Amer. Math. Soc. 364

(2012), no. 4, 2157–2191.

[12] S. M. Gonek and J. P. Keating, Mean values of finite Euler products. J. London Math. Soc. 82

(2010), 763–786.

[13] A. Granville and K. Soundararajan, Extreme values of j.1 C i t/j. In Ramanujan Math. Soc.

Lect. Notes Ser 2, Ramanujan Math. Soc., Mysore, 2006, 65–80.

[14] P. Hartman, On completely continuous Hankel operators. Proc. Am. Math. Soc. 9 (1958),862–866.




[15] E. Hewitt and J. H. Williamson, Note on absolutely convergent Dirichlet series. Proc. Am.

Math. Soc. 8 (1957), 863–868.

[16] T. W. Hilberdink, Determinants of multiplicative Toeplitz matrices. Acta Arith. 125 (2006),

265–284.

[17] T. W. Hilberdink, An arithmetical mapping and applications to -results for the Riemann zeta

function. Acta Arith. 139 (2009), 341–367.

[18] T. W. Hilberdink, ‘Quasi’-norm of an arithmetical convolution operator and the order of the

Riemann zeta function. Funct. Approx. 49 (2013), 201–220.

[19] M. Huxley, Exponential sums and the Riemann zeta function V. PLMS 90 (2005), 1–41.

[20] T. J. Jarvis, A dominant negative eigenvalue of a matrix of Redheffer. Linear Algebra Appl.

142 (1990), 141–152.

[21] B. Jessen and H. Tornehave, Mean motions and zeros of almost periodic functions. Acta Math.

77 (1945), 137–279.

[22] J. Keating and N. Snaith, Random matrix theory and some zeta-function moments. Lecture at

the Erwin Schrodinger Institute (1998).

[23] M. Krein, Integral equations on a half-line with kernel depending upon the difference of the

arguments. Am. Math. Soc. Transl. (2) 22 (1962), 163–288.

[24] Y. Lamzouri, On the distribution of extreme values of zeta and L-functions in the strip 12 <

< 1. Int. Math. Res. Nat., to appear; arXiv:1005.4640v2 [math.NT].

[25] N. Levinson, -theorems for the Riemann zeta function. Acta Arith. 20 (1972), 319–332.

[26] P. Lindqvist and K. Seip, Note on some greatest common divisor matrices. Acta Arith. 84

(1998), 149–154.[27] H. L. Montgomery, Extreme values of the Riemann zeta-function. Comment. Math. Helv. 52

(1977), 511–518.

[28] Z. Nehari, On bounded bilinear forms. Ann. Math. 65 (1957), 153–162.

[29] R. M. Redheffer, Eine explizit losbare Optimierungsaufgabe. Int. Schriftenr. Numer. Math. 36

(1977).

[30] L. Rodman, I. M. Spitkowsky and H. J. Woerdeman, Fredholmness and Invertibility of

Toeplitz operators with matrix almost periodic symbols. Proc. Am. Math. Soc. 130 (2001),

1365–1370.

[31] K. Soundararajan, Extreme values of zeta and L-functions. Math. Ann. 342 (2008), 467–486.[32] A. E. Taylor, Introduction to Functional Analysis. Wiley and Sons, 1958.

[33] E. C. Titchmarsh, The Theory of the Riemann Zeta-function. Second edition, Oxford Univer-

sity Press, 1986.

[34] O. Toeplitz, Zur Theorie der quadratischen und bilinearen Formen von unendlichvielen

Veranderlichen. Math. Ann. 70 (1911), 351–376.

[35] O. Toeplitz, Zur Theorie der Dirichletschen Reihen. Am. J. Math. 60 (1938), 880–888.



Chapter 4

Arithmetical topics in algebraic graph theory

Jurgen Sander

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2 Some (algebraic) graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

2.1 Basics and some specialities on graphs . . . . . . . . . . . . . . . . . . . . . . 124

2.2 Some algebraic graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3 The prime number theorem for graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.1 Primes in friendly graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.2 Ihara’s zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

3.3 The determinant formulae for Ihara’s zeta function . . . . . . . . . . . . . . . . 149

3.4 The prime number theorem for graphs . . . . . . . . . . . . . . . . . . . . . . 155

3.5 Some remarks on Ramanujan graphs . . . . . . . . . . . . . . . . . . . . . . . 158

4 Spectral properties of integral circulant graphs . . . . . . . . . . . . . . . . . . . . . . 159

4.1 Basics on integral circulant graphs . . . . . . . . . . . . . . . . . . . . . . . . 159

4.2 Ramanujan integral circulant graphs . . . . . . . . . . . . . . . . . . . . . . . 1624.3 The energy of a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

1 Introduction

In comparison with geometry, number theory, analysis, or even algebra and topology,graph theory is a rather young mathematical discipline. It has undergone crucial

developments with important theoretical advances as well as closer connections to

other mathematical fields and applications in extra-mathematical areas only withinthe last 80 years. The term graph was introduced by Sylvester in 1878 in an article

published in “Nature”, the by far most influential scientific magazine worldwide.

By now graph theory is without any doubt an important mathematical discipline

in its own right with prestigious topics, results and open questions, as for instance

the four color problem, just to name the most famous one. At the same time, itis influential for many other mathematical fields, not only in discrete mathematics,

but also for areas such as algebra, topology and even probability theory. Naturally,

the development of graph theory benefited from the rapid progress in computers andcomputer science in the 20th century.

However, graph theory has yet another facet to it, namely it may serve as a math-ematical “playground” in the following sense. Quite a few mathematical concepts in



124 Jurgen Sander

other fields, as for instance topological items or the variety of zeta functions, related

to number theory, algebra, differential geometry, or other subjects, have a counterpart

in the world of (finite) graphs. Looking at and experimenting with these somewhat

discrete relatives sometimes helps to better understand the original objects.Algebraic graph theory – nomen est omen – connects algebra and graph the-

ory. Cayley graphs, encoding structural features of groups in correlated graphs, area prominent example studied in this rather young mathematical field. Besides the nec-

essary basics of algebraic graph theory, these lecture notes will present three topicsconnecting graph theory and arithmetic.

2 Some (algebraic) graph theory

Here we give a short introduction to the concepts of graph theory that will be used

in the sequel. Besides some standard basics like cycle graphs, trees, regular graphs,

spanning subgraphs, etc., which can be found in almost any introductory book ongraph theory (e.g. [12]) and are mainly included to set up the notation, the reader will

find more specialized results on circulant graphs and Cayley graphs. The introductionto algebraic graph theory deals with the adjacency matrix and the spectrum of graphs,

in particular the spectrum of circulant and integral circulant graphs.

2.1 Basics and some specialities on graphs

Definition 2.1.

A . finite/ graph G D .V; E/ consists of a (finite) non-empty set V of vertices

and a set E V V of edges. The order of G is the number of vertices jV j in

G .

We usually write v1v2 rather than .v1; v2/, and for v1v2 2 E we say that v1

and v2 are adjacent or neighbors in G , written v1 v2.

The graph G is called undirected if E is symmetric, i.e., if v1 v2 , v2 v1 for all v1; v2 2 V , and otherwise directed . For undirected graphs we shallnot distinguish between v1v2 and v2v1 (here v1v2 represents a set rather thanan ordered pair).

Edges of type vv 2 E are called loops. A loopfree graph has no loops.

For several applications it is useful to allow graphs with multiple edges betweenvertices, so-called multigraphs. We call graphs without loops or multiple edges sim-

ple.

Assumption. Unless explicitly stated otherwise, the graphs G D .V;E/ we consider

are always simple, finite, and undirected.



126 Jurgen Sander

Figure 4.2. Circulant Cayley graph .Z10; Ef1;3;7;9g/

Figure 4.3. Graph with (disconnected) spanning subgraph (solid lines)



4 Arithmetical Topics in Algebraic Graph Theory 127

An important type of subgraph in a given graph is a clique, i.e., a subgraph which

is a complete graph.

Definition 2.4. Let G D .V;E/ be a (directed) graph.

A (directed) path or walk from u 2 V to v 2 V of length k in G is a sequence of

k C 1 vertices u D v0; v1; v2; : : : ; vk1; vk D v 2 V such that vi1vi 2 E fori D 1; 2 ; : : : ; k (or, equivalently, the sequence of the corresponding k edges).

The path is called simple if vi ¤ vj for i D 1 ; 2 ; : : : ; k 1 and all j . For

a path P D .v0; v1; v2; : : : ; vk1; vk/, its length is denoted by .P / WD k.

A (directed) cycle or closed walk in G is a (directed) path from v to v for some

v 2 V .

If C

D.v0; v1; : : : ; vk1; v0/ is a cycle in G , then

C n WD .v0; v1; : : : ; vk1; v0; v1; : : : ; vk1; : : : ; v0; v1; : : : ; vk1„ ƒ‚ …n times

; v0/

is called the n-th power of C .

Definition 2.5. Let G D .V;E/ be a graph.

G is called connected if there is a path from u to v for any vertices u ¤ v in G ;otherwise G is called disconnected .

If G is disconnected, its induced connected subgraphs G i

D.V i ; Ei / with pair-

wise disjoint V i V satisfying SV i D V are called the components of G .

All graphs in Example 2.2 are connected and contain cycles.

Example 2.6. The graph .Z10; Ef2;8g/ is disconnected with two components.

If a graph G contains a cycle, we call G cyclic, otherwise we call it acyclic. Weintroduce the very important subclass of graphs which are connected, but acyclic.

Definition 2.7. A tree is a connected acyclic graph (cf. Fig. 4.5).

Exercise 2.8. Show that the following assertions are equivalent for a graph T D.V;E/:

(i) T is a tree.

(ii) There is a unique path in T between any two distinct vertices of T .

(iii) T is maximally acyclic, i.e., T is acyclic, but T 0 WD .V; E [ feg/ is cyclic for

all e 2 .V V / n E.

(iv) T is minimally connected , i.e., T is connected, but T 00 WD .V;E n feg/ is

disconnected for every e 2 E. This means that jEj D jV j 1.



128 Jurgen Sander

Figure 4.4. Disconnected Cayley graph .Z10; Ef2;8g/ with two components

Proposition 2.9. Every connected graph G contains a spanning tree , which by defi-

nition means that there is a spanning subgraph of G which is a tree.

Proof. G contains a connected spanning subgraph, e.g., G itself. Now assume that G 0is a connected spanning subgraph of G with a minimal number of edges. Then theequivalence of (i) and (iv) in Exercise 2.8 tells us that G 0 is a tree.

Definition 2.10. Let G D .V;E/ be a graph.

The degree of v 2 V is d.v/ WD jfu 2 V W u vgj, i.e., the number of

neighbors of v.

A vertex of degree 1 is called a leaf . If all vertices in V have the same degree (let’s say k), then G is called regular

or, more precisely, k-regular .

Exercise 2.11. Every tree with more than two vertices has at least two leaves.

Example 2.12.

(i) Each cycle graph C n is 2-regular.

(ii) The complete graphKn is .n 1/-regular.

(iii) The circulant graph .Z10; E/ from Example 2.2 is 4-regular.




Figure 4.5. Tree

Definition 2.13. Let G be a finite group and S G. We usually assume that S is

a generating set of G and that S is symmetric, i.e., s 2 S implies s1 2 S .

(i) Then the corresponding Cayley graph Cay.G;S/ D .V;E/ is defined by V WDG and E WD f.g;gs/ W g 2 G; s 2 S g.

(ii) A Cayley graph Cay.Z

n;Zn/ is called unitary, where

Zn WD

Z=nZ

is theadditive group of residues mod n for some positive integer n and Zn D f1

a n W .a; n/ D 1g is the multiplicative unit group in the residue class ringZn.

Example 2.14. Clearly, .Z10; Ef1;3;7;9g/ D Cay.Z10; f˙1; ˙3g/ (cf. Example 2.2

(iii)) is a unitary Cayley graph, and .Z10; Ef2;8g/ D Cay.Z10; f˙2g/ (cf. Example

2.6) is a disconnected Cayley graph.

Exercise 2.15. Show that

(i) Cay.G;S/ is connected if and only if S is a generating set.

(ii) Cay.G;S/ is undirected if and only if S is symmetric.

(iii) Cay.G;S/ is regular of degree jS j.Definition 2.16. A graph G D .V;E/ is called bipartite if V is the union of two

disjoint subsets V 1 and V 2 such that E .V 1 V 2/ [ .V 2 V 1/ (cf. Fig. 4.6).

Exercise 2.17.

(i) Show that a bipartite graph cannot contain a cycle of odd length.

(ii) Prove that bipartite graphs are characterized by the property in (i).

Hint: Assume the graph is connected and consider a spanning tree.



130 Jurgen Sander

Figure 4.6. A bipartite graph

2.2 Some algebraic graph theory Much more than what we need from alge-

braic graph theory can be found in [7] of [14]. A tool which uniquely determinesa graph and, more importantly, enables us to apply (linear) algebra to examine the

graph and its properties is the adjacency matrix.

Definition 2.18. Let G D .V; E/ be a graph with V D fv1; : : : ; vng, which may bedirected and is allowed to have loops. The adjacency matrix AG D .aij /nn of G isdefined by setting

aij D8<: 0; if vi 6 vj ,

1; if vi vj for i ¤ j ,

2; if vi vi for i D j .

Example 2.19. Consider the Cayley graph Cay.Z10; f˙1; ˙3g/ with Z10 'f0 ; 1 ; 2 ; : : : ; 9g from Example 2.2 (iii). Then

ACay.Z10;f˙1;˙3g/ D

0BBBBBBBBBBBBB@

0 1 0 1 0 0 0 1 0 11 0 1 0 1 0 0 0 1 00 1 0 1 0 1 0 0 0 11 0 1 0 1 0 1 0 0 00 1 0 1 0 1 0 1 0 00 0 1 0 1 0 1 0 1 00 0 0 1 0 1 0 1 0 11 0 0 0 1 0 1 0 1 00 1 0 0 0 1 0 1 0 11 0 1 0 0 0 1 0 1 0

1CCCCCCCCCCCCCA

Exercise 2.20. Let G be an undirected graph, hence AG is a real symmetric matrix.




Figure 4.7. Two non-isomorphic cospectral graphs

(i) Let x and y be eigenvectors of AG corresponding to different eigenvalues.

Show that x and y are orthogonal.

(ii) Show that all eigenvalues of AG are real.

(iii) Verify that all rational eigenvalues of AG are integers.

Definition 2.21. Let G D .V;E/ be a graph. The eigenvalues of G are the eigenval-

ues of AG. We call the multiset (= set with repetitions) of all eigenvalues of G the

spectrum Spec.G

/ WD Spec.AG/ of G

(where Spec.G

/ R

by Exercise 2.20).

We have seen in Section 2.1 that renaming the vertices of a graph G 1 gives an

isomorphic graph G 2. Usually AG1 ¤ AG2

, but we have

Proposition 2.22. If two graphs G 1 and G 2 are isomorphic, then Spec.G 1/ DSpec.G 2/.

Proof. Isomorphic graphs have the same number of vertices, thus assume that the

sets V 1 and V 2 of vertices of G 1 and G 2, respectively, are V 1 D fv1; : : : ; vng and

V 2 D fu1; : : : ; ung. Since G 1 ' G 2, we have ui D v.i/ for i D 1; : : : ; n with some

permutation on f1 ; 2 ; : : : ; ng. If P denotes the n n permutation matrix of ,we have P t AG1

P D AG2. So AG1

and AG2 are similar matrices, and therefore they

have the same characteristic polynomial, hence the same eigenvalues.

Exercise 2.23.

(i) Show that Spec.G 1/ D Spec.G 2/ (therefore G 1 and G 2 are called cospectral, orisospectral), but G 1 6' G 2 for the two graphs G 1 and G 2 in Fig. 4.7. This means

that the spectrum of a graph does not determine the structure of the graph. In

other words, “you cannot hear the shape of a graph”, varying a famous questionof Marc Kac posed in 1966.

In particular, the graph on the right obviously is planar , i.e., it can be embeddedin the plane, while the graph on the left is not planar.



132 Jurgen Sander

(ii) What was the famous question of Marc Kac referred to in (i)?

(iii) Look at some examples to find out that isomorphic graphs usually do not have

the same eigenspaces!

A nice property of the adjacency matrix and its powers for a directed graph is

given by the following result, which is proved by straightforward induction.

Proposition 2.24. If G D .V;E/ is a directed graph, then the number of (simple and

nonsimple) directed paths of length ` from u 2 V to v 2 V is .A`G/uv.

Exercise 2.25. Let G be a graph (loopfree and undirected) with e edges and t triangles.

(i) Show that

P2Spec.G/ D trAG D 0, where tr denotes the trace of a matrix.

The sum of the absolute values of the eigenvalues E .G / WD P2Spec.G/ jj iscalled the energy of G .

(ii) Use Proposition 2.24 to prove that trA2G D 2e and trA3

G D 6t .

(iii) What about the corresponding problem for quadrilaterals?

The spectrum of a graph reflects quite a few of its properties. We are particularlyinterested in regular graphs.

Proposition 2.26. Let G be a k-regular graph.

(i) Then k 2

Spec.G / , and jj

k for all 2

Spec.G /.

(ii) If G is connected, then k has multiplicity 1 in Spec.G /.

(iii) If G is connected, then k 2 Spec.G / if and only if G is bipartite.

Proof. Let G D .V;E/ with V D fv1; : : : ; vng. Since G is k-regular, every row of

AG contains k entries 1 and n k entries 0. Hence

AG

0

BB@11:::1

1

CCAD

0

BB@kk:::k

1

CCAD k

0

BB@11:::1

1

CCA;

i.e., k 2 Spec.G /.Let 2 Spec.G / be arbitrary, hence AGx D x for some x D .x1; : : : ; xn/ ¤ 0.

Assume that jxmj is the largest among all jxi j. Then

jxmj D j.AGx/mj Dˇ X

vivm

xi

ˇ kjxmj; (4.1)

thus

j

j k, which proves (i).

To verify (ii), it remains to show that k has multiplicity 1 in Spec.G /. As-sume that AGx D kx for some x ¤ 0. Let again jxmj be the largest and w.l.o.g.




xm > 0 (otherwise consider x). It follows from (4.1) (without absolute values) that

Pvivm

xi D kxm. Since vm has k neighbors, the sum has k terms and thus all

corresponding summands xi must be equal to xm. Since G is connected, iteration of

the argument finally yields xi D xm for all i . This means that .1; 1; : : : ; 1/t spans theeigenspace corresponding to D k.

We are left with (iii). First assume that AGx D kx for some x ¤ 0. Againwe may assume w.l.o.g. that some xm > 0 has maximal jxmj. Moreover, we may

also assume that xi > 0 for 1 i s and xi < 0 for t C 1 i n, whilexsC1 D xsC2 D : : : D xt D 0. As in (4.1) we conclude that

kxm DsX

iD1vivm

xi CnX

iDtC1vivm

xi nX

iDtC1vivm

xi kxm:

This means we have in fact equality everywhere, hence vi 6 vm for 1 i sand xi D xm for all vi vm. Iterating this argument, each neighbor vj , say, of

a neighbor vi of vm has again xj D xi D xm. Since G is connected, V splits intotwo sets, V 1 WD fvi W xi D xmg and V 2 WD fvi W xi D mg, such that u 6 v for any

u; v 2 V 1 or any u; v 2 V 2. This means that G is bipartite.

To prove the opposite direction of (iii) we observe that after reordering the verticesa connected bipartite graph G typically has an adjacency matrix of type

AG D

0BBBBBBBBBBBBB@

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1CCCCCCCCCCCCCA;

where the entries in the positions are 0 or 1. Now it is easy to see that

AG . 1 ; : : : ; 1 ; 1 ; : : : ; 1/t D .k ; : : : ; k ; k ; : : : ; k /t

D .k/.1;:::;1; 1 ; : : : ; 1/t ;

i.e., k 2 Spec.G /.

Beyond the results of Proposition 2.26 the following facts are well known:

A k-regular graph has as many components as the multiplicity of its eigenvalue

k.

Spec.G / is symmetric about the origin, i.e., the multiplicities of and in Spec.G / are the same, if and only if G is bipartite (this uses the Perron–

Frobenius Theorem: see e.g. [14], Theorem 8.8.1 on the eigenvalues and eigen-vectors of nonnegative matrices; also cf. Theorem 3.24).



134 Jurgen Sander

Exercise 2.27. We know that k 2 Spec.G / for a k-regular graph G . Show that G

necessarily is connected if k has multiplicity 1 in Spec.G /.

Definition 2.28. A graph G is called circulant if AG is a circulant matrix (briefly:a circulant , cf. [11]), i.e. a square matrix whose rows are obtained by cylic right shifts

of the first row.

Example 2.29. Clearly, Cay.Z10; f˙1; ˙3g/, the cycle graphs C n and the completegraphs Kn are examples of circulant graphs.

Proposition 2.30 (Spectrum of C n). Let n be a positive integer and !j WD exp. 2ij n

/ for j D 0 ; 1 ; : : : ; n 1. We have

(i) Spec.C n/

D ˚!j

C 1!j

W j

D0 ; 1 ; : : : ; n

1.

(ii) The largest eigenvalue of C n is 2 .with multiplicity 1/ , and the second largest

is 2 cos .n1/n

.with multiplicity 2/. The smallest eigenvalue is 2 .with mul-

tiplicity 1/ for even n , and it is 2 cos .n1/

n .with multiplicity 2/ for odd n.

(iii) C n is bipartite if and only if n is even.

Proof. A helpful observation is that ACn D P C P 1, where P is the permutation

matrix of the permutation defined by a cyclic left shift. If ! is any n-th root of unity,then it is obvious that the vector .1; !; !2; : : : ; !n1/t is an eigenvector of P with

eigenvalue ! as well as an eigenvector of P 1 with eigenvalue 1!

. Since there are

exactly n n-th roots of unity, namely ! D !j for j D 0; 1; : : : ; n 1, identity (i)follows.

By (i), the largest eigenvalue is !0 C 1!0

D 2 (with multiplicity 1), and the

second largest is 2 cos .n1/

n D !1 C 1

!1D !n1 C 1

!n1(with multiplicity 2).

The smallest eigenvalue is 2 (with multiplicity 1) for even n, and it is 2 cos .n1/n

(with multiplicity 2) for odd n.(iii) follows from (ii) and Proposition 2.26 (iii).

Arguing as in the preceding Proposition 2.30 one can show

Proposition 2.31. If A is a circulant nn matrix with first row a0; a1; : : : ; an1 , say,

then the eigenvalues j are given by

j WDn1XkD0

ak!kj .j D 0 ; 1 ; : : : ; n 1/;

with corresponding eigenvectors vj WD .1;!j ; !2j ; : : : ; !n1

j /t , where !j WDexp. 2ij

n /.

A class of graphs which are interesting with respect to arithmetic consists of theso-called integral graphs.



136 Jurgen Sander

Exercise 2.35. Which of the cycle graphs C n and the complete graphs Kn are Ra-

manujan?

3 The prime number theorem for graphs

We denote by P the set of prime numbers and, as usual, by

.x/ WDX

px; p2P1

the function counting the prime numbers up to x in the set of positive integers. There

is indeed a concept of primality in graphs, and this will be defined for arbitrary finite,undirected and connected graphs G . We introduce a corresponding prime counting

function G.x/ and relate it to the Ihara zeta function. This reveals analogies to otherzeta functions, in particular the classical Riemann zeta function with its application

to .x/ (cf. [4], [23] or [37]). We shall derive determinant formulae and functional

equations and consider a Riemann hypothesis, which is related to the Ramanujan

property of a graph (cf. Def. 2.34). Finally, the Ihara zeta function is used to prove

a graph-theoretic prime number theorem. Most of the material covered in this section

is taken from Audrey Terras’ book “Zeta Functions of Graphs – A Stroll through theGarden” [48].

3.1 Primes in friendly graphs

Assumption. The graphs G D .V;E/ we consider always are finite, undirected, and

connected.

Unless otherwise stated, we also assume that G has no leaves (= vertices of de-

gree 1) and is not a cycle graph with or without leaves (i.e., G n fleavesg ¤ C n forall n, in other words G is not unicyclic).

A graph is called friendly if it satisfies the above assumption.

Exercise 3.1. Show that every friendly graph with at least one edge contains a cycleand, therefore, is multicyclic.

Let G D .V; E/ be a friendly graph with jEj D m. We orient the m edges of G arbitrarily to obtain directed edges e1; e2; : : : ; em (with arbitrary labelling). Then we

label the inverse edges by setting emCj WD e1j , where e1

j D .v; u/ for ej D .u;v/.

The graph EG D .V; EE/ with EE WD E [ E1 is called an orientation of G .




Figure 4.8. An orientation of K4 e

Example 3.2 (An orientation of K4 e).

Definition 3.3. Let G D .V; E/ be a graph, and let EG D .V; EE/ be an orientation of

G . For m WD jEj D 12j EEj, the 2m 2m matrix W EG D .wij / is defined as follows:

Set wij

D 1 if ei ej is a directed path (of length 2) in

EG , but not e1

i

D ej , and

otherwise wij WD 0. Then W EG is called the edge adjacency matrix of G .

Definition 3.4. Let C D e1e2 : : : es be a path of edges in an orientation EG of a graph G .

C is said to have a backtrack if ej C1 D e1j for some j 2 f1 ; 2 ; : : : ; s 1g or

es D e11 .

If C is closed but not the power of a shorter cycle, then we call it a primitive

or prime path if it has no backtracks.

Two cycles are said to be equivalent if we get one from the other just by chang-ing the starting vertex. For C closed,

ŒC WD fC; e2e3 : : : ese1; e3 : : : es e1e2; : : : ; ese1e2 : : : es1gis called the equivalence class of C .

A prime in G is an equivalence class ŒP of prime paths, i.e., P is a primitivepath.

ŒP WD .P/ is called the length of the prime ŒP .



138 Jurgen Sander

Example 3.5.

(i) Primes in K4 e are (Ex. 3.2 cont’d) :

ŒC 1 for C 1 WD e2e3e5 and Œe10e8e7 D ŒC 1

1 with lengths ŒC 1 DŒC 11 D 3;

ŒC 2 for C 2 WD e1e2e3e4 with length ŒC 2 D 4;

ŒC n2 C 3 for C 3 WD e1e10e4 and each n 0, hence there are infinitely

many primes in K4 e.

(ii) In naughty graphs one could also look for primes, but the example at the be-

ginning of this section shows that there are only two primes, namely the twoorientations of the unique cycle.

Exercise 3.6.(i) Check that equivalence between cycles is in fact an equivalence relation.

(ii) Let C be a prime path. Show that each closed path equivalent with C is also

a prime path.

(iii) Verify that powers of primes are the only non-primes among cycles.

(iv) If C is a cycle, we can factorize it into subcycles C 1 and C 2 if C 1 and C 2 havea common vertex v and, starting from v, we obtain C by concatenation of C 1and C 2. Give examples of the fact that we do not have unique factorization of cycles into primes, e.g., in Example 3.5.

Definition 3.7. Let G be a friendly graph, and let PG be the set of primes in G . Then

G.k/ WD #fŒP 2 PG W ŒP D kgis called the prime counting function. (Observe the D sign as opposed to in the

prime counting function for integers).

An important parameter will be the greatest common divisor of the prime path

lengths

G

WDgcd

fŒP

W ŒP

2PG

g (4.2)

in a friendly graph G .

Proposition 3.8. Let G be a friendly graph. Then G.k/ D 0 if G − k.

Proof. Since G j ŒP for each prime ŒP , there is no prime ŒP with ŒP D k for

G − k.

3.2 Ihara’s zeta function The Riemann zeta function encodes informationabout the distribution of primes in the set of positive integers and can be used to

prove the prime number theorem (cf. [4], [23] or [37]). Other zeta functions havebeen defined and studied in order to obtain information about mathematical objects




like prime numbers in arithmetic progressions (Dirichlet L-functions), prime ideals

in rings of algebraic integers (Dedekind zeta functions), primitive closed geodesics

in manifolds (Selberg zeta function), and many more (see [24], [27], [33], [38]). We

shall look at a zeta function related to primes in graphs and will uncover analogieswith the other zeta functions.

The Ihara zeta function was first defined by Yasutaka Ihara [21] in the 1960s in thecontext of discrete subgroups of the two-by-two p-adic special linear group. Jean-

Pierre Serre suggested 1977 in his book “Arbres, Amalgames, SL2” (D“Trees” [43])that Ihara’s original definition could be reinterpreted graph-theoretically, namely as

a zeta function related to closed geodesics in graphs (compare with the Selberg zeta

function). In 1985 Toshikazu Sunada [47] put this suggestion into practice.

Definition 3.9. Let G be a friendly graph. The Ihara zeta function is the complex

function G.u/ D .u;G / WD Y

ŒP 2PG

1 uŒP1

; (4.3)

where u 2 C is supposed to have sufficiently small absolute value. Recall that wedistinguish between the primes ŒP and ŒP 1. The question of convergence will be

postponed until after Proposition 3.11.

Example 3.10. Although the cycle graph C n is not friendly (with only two primes =

cycle in two orientations), we still can evaluate its Ihara zeta function:

Cn .u/ D .1 un/2 D 1XkD0

uk n!2

:

Let us explain at the outset why the Ihara zeta function of a friendly graph G is

useful in dealing with primes in G . To this end, let N m.G / be the number of cycles of

length m without backtracks in G .

Proposition 3.11. For all u 2 C with sufficiently small absolute value juj it holds

that

log G.u/

D

1

XmD1

N m.G /

m

um: (4.4)

Proof. Taking logarithms in (4.3) we obtain

log G.u/ D logY

ŒP 2PG

1 uŒP

1

D X

ŒP 2PGlog

1 uŒP

DX

ŒP 2PG

1Xj D1

1

j ujŒP D

XP primitive

1

.P/

1Xj D1

1

j uj.P/

D XP primitive

1

Xj D1

1

.P j / u.P j /

;



140 Jurgen Sander

since there are exactly .P / elements in a prime ŒP and trivially .P j / D j.P /.

As seen in Exercise 3.6 (iii), any cycle without backtrack in G is a power of some

primitive cycle. Hence

log G.u/ D XC cycle w/o backtr.

1.C/

u.C/

D1X

mD1

N m.G /

m um:

Now let us verify that G.u/ does indeed converge inside some circle in the com-

plex plane with positive radius centered at 0. By Proposition 2.24, .AmG /jj equals the

number of directed paths of length m from vj 2 V to vj , which in turn is the number

of directed cycles of length m containing vj . Hence .AmG /jj is greater than or equal

to the number of cycles of length m containing vj . Therefore, setting n WD jV j forthe number of vertices in G , we have

nXj D1

.AmG /jj N m.G /:

Since AG has only non-negative entries 0 and 1, it is easy to see that

.AmG /ij .

m/ij ;

where denotes the n n-matrix with all entries 1. Obviously,

m D nm1 , hence

N m.G / nX

j Dnm1 D nm:

For juj < 1n

the sum1X

mD1

N m.G /

m um

converges absolutely. By Proposition 3.11, this implies that log G.u/ is well definedfor juj < 1

n, and thus the same is true for G.u/ itself.

We introduce the notion of the radius of convergence RG such that G.u/ con-

verges inside juj < RG and has a singularity on the border juj D RG, called circle of

convergence.One might like to compare formula (4.4) with related identities for the classi-

cal Riemann zeta function and the von Mangoldt function or other characteristic

functions for primes in Z (cf. [4] or [23]).We are lucky to have explicit formulae which allow us to compute G.u/. One of

them is called the two-term determinant formula (Theorem 3.31) and relates G.u/ to




the determinant of the edge adjacency matrix W EG (cf. Definition 3.3). This formula

can be used to obtain the so-called Ihara three-term determinant formula (Theorem

3.15).

In order to formulate the latter result, we should consider the fundamental groupof a graph and its rank. The fundamental group, in general associated to any given

pointed topological space, provides a way to determine when two paths, starting andending at a fixed base point, hence called loops, can be continuously deformed into

each other. Such loops are considered equivalent. Two loops can be combined by justwandering around the first one and afterwards around the second one. The elements

of the fundamental group are the equivalence classes of loops and the operation is the

combination of these.One of the simplest examples is the fundamental group of the circle, which is

obviously isomorphic to the group .Z;

C/. This corresponds to the fundamental group

of any of the cycle graphs C n (with or without leaves). In general, it turns out that thefundamental group .G ; v1/ of a graph G D .V;E/ relative to the base vertex v1 2 V is a free group on r generators, where r is the rank of the group. .G ; v1/ and, inparticular, its rank r can easily be determined by the following

Algorithm 3.12. Let G D .V; E/ be a graph (possibly directed with multi-edges andloops).

(1) Choose any orientation of G to obtain EG .(2) Choose any spanning tree T D .V;E0/ of G (which exists by Proposition 2.9).

(3) For each edge vi vj 2 E n E 0, construct a path from v1 to vi and from vj back to v1 (all these paths are unique by Exercise 2.8) to obtain a cycle starting from

v1 via vi and vj back to v1.

Proposition 3.13. Let G D .V;E/ be a graph with v1 2 V .

(i) The equivalence classes of the cycles obtained in (3) of Algorithm 3.12 form

a free basis of .G ; v1/.

(ii) The rank r of .G ; v1/ satisfies r D jEj jV j C 1.

Proof. The fundamental group .G ; v1/ consists of sequences of cycles (more pre-cisely, equivalence classes of cycles) starting from v1. The spanning tree constructed

in step (2) of Algorithm 3.12 does not have any cycles. However, each edge in E nE0completes a unique cycle, and different edges in E n E 0 yield inequivalent cycles.

This proves (i).

By (i) we know that r D jEj jE 0j, where E 0 is the edge set of a spanning treeof G . By Exercise 2.8, a tree on jV j vertices has jV j 1 edges. This proves (ii).

Example 3.14. We determine a free basis and the rank of the fundamental group of the graph G below:




(1) We first choose a vertex v1, any orientation of G , and any spanning tree T of

G .

(2) For each directed edge e in G not belonging to T , we construct the uniquedirected cycle in G starting from v1 through e.

(3) These cycles are a free basis of .G ; v1/ and their number is the rank of the

fundamental group.

Now we are able to formulate the three-term determinant formula, first proved by

Y. Ihara [21] in 1966 and later generalized by K. Hashimoto [18] in 1989 and H. Bass

[5] in 1992.

Theorem 3.15 (Ihara’s three-term determinant formula). Let G D .V;E/ be a friendly

graph with V D fv1; : : : ; vng. Let QG D .qij / be the n n diagonal matrix withdiagonal entries qjj D d.vj / 1 , where d.vj / is the degree of vj . Then

G.u/1 D .1 u2/r1 det.I AGu C QGu2/;

where I is the unit matrix and r D jEjjV jC1 is the rank of the fundamental group

of G .

Before proving Theorem 3.15 in Section 3.3, let us deduce a variety of conse-

quences of the three-term determinant formula.

Example 3.16. (cf. Fig. 4.10 and 4.11) Consider the complete graph K4 with r D6 4 C 1 D 3,

AK4 D

0B@0 1 1 11 0 1 11 1 0 11 1 1 0

1CAand QK4

D 2I , because K4 is 3-regular. The three-term determinant formula yields

K4 .u/1 D .1 u2/2 det0BB@2u2 C 1 u u u

u 2u2

C 1 u uu u 2u2 C 1 uu u u 2u2 C 1

1CCAD .1 u2/2.1 u/.1 2u/.1 C u C 2u2/3:

Hence the five poles of K4.u/ with different multiplicities are located as follows:

three real poles at ˙1 and 12

, and two complex poles at 14

.1 ˙ ip

7/ (with absolute

valuep

22

). Since the pole closest to the origin lies at 12

, we have RK4 D 1

2 as radius

of convergence.



144 Jurgen Sander

Figure 4.10. Ihara’s zeta function for K4 in the u-variable: z D j K4.x C iy/j1. This figure is

taken from [48] with permission of the author.

Figure 4.11. Ihara’s zeta function for K4 in the s-variable: z D j K4.2.xCiy//j1. This figure is

taken from [48] with permission of the author.

Exercise 3.17. Show that the irregular graph K4 e (cf. Example 3.2) satisfies

K4e.u/1 D .1 u2/.1 u/.1 C u2/.1 C u C 2u2/.1 u2 2u3/

with nine roots: ˙1; ˙i; 14 .1 ˙ ip 7/ and three cubic roots u1; u2; u3, say, satis-fying js1j 0:657 and js2j D js3j 0:872. Hence the radius of convergence is

RK4e 0:657.

We shall now specialize on regular graphs. For this graph class a lot of the con-

cepts we are interested in are partly easier to understand, but most results can begeneralized to irregular graphs.

Reconsider the regular graph K4 (cf. Example 3.16) with

K4.u/1 D .1 u2/2.1 u/.1 2u/.1 C u C 2u2/3:

We take a look at z D j K4 .x C iy/j1 (Fig. 4.10), depicting the five zeros (notcounting multiplicities).




Figure 4.12. Riemann’s zeta function z D j.x C iy/j. This figure is taken from [48] with

permission of the author.

To catch another (logarithmic) sight of z D j K4.u/j1, we make the change

of complex variable u 7! 12u and display the landscape of z D j K4

.2.xCiy//j1

(Fig. 4.11).

This bears some resemblance to the landscape z D j.x C iy/j of Riemann’s zetafunction .s/ (Fig. 4.12).

In fact, the resemblance between the zeta functions of Riemann and Ihara is strik-ing. The classical Riemann hypothesis (see e.g. [23] or [37]) conjectures the loca-

tion of the non-trivial zeros of the Riemann zeta function.

Definition 3.18. Let G be a friendly .q C1/-regular graph and consider the Ihara zetafunction G.qs/ as a complex function in s 2 C. We say that G.qs / satisfies the

Riemann hypothesis if G.qs/1 can vanish in the critical strip 0 < Re s < 1 onlyfor Re s D 1

2, i.e., for juj D jqsj D 1p

q.

Exercise 3.19. Check that K4.2s/ satisfies the Riemann hypothesis (cf. Example

3.16).

Theorem 3.20. Let G be a friendly .q C 1/-regular graph. Then G.qs/ satisfies the

Riemann hypothesis if and only if G is a Ramanujan graph .cf. Definition 2.34/.

Proof. By the determinant formula in Theorem 3.15, we have

G.qs/1 D .1 q2s /r1 det.I AGqs C QGq2s /: (4.5)

Since AG is a real symmetric matrix, it is diagonalizable (even by means of orthonor-mal matrices). Hence there is an invertible matrix T such that T

1AGT

DLG , where



146 Jurgen Sander

LG is a diagonal matrix with the eigenvalues of G on the diagonal. Since G is assumed

to be .q C 1/-regular, we have QG D qI . Altogether

det.I AGqs C QGq2s / D detT I T 1 T .qsLG/T 1 C T .q12s I /T 1D det T det

I qs LG C q12s I

det T 1

D det

I qsLG C q12s I

DY

2Spec.G/

1 qs C q12s

;

hence

G.qs

/1

D .1 q2s

/

r

1 Y

2Spec.G/1 q

s

C q

1

2s:

Fix some 2 Spec.G / and write

1 qs C q12s D .1 ˛1qs/.1 ˛2qs/;

i.e., ˛1 and ˛2 are reciprocals of poles of G.qs/ satisfying ˛1 C ˛2 D and

˛1˛2 D q. This implies the quadratic equation ˛21 ˛1 D q, leading to

˛1

D 1

2 Cp 2

4q and ˛2

D 1

2 p 2

4q ; (4.6)

or vice versa.

By Proposition 2.26 we know that q C 1 2 Spec.G / and jj q C 1 for all

2 Spec.G /. We distiguish three cases, namely D ˙.q C 1/, jj 2p

q, and2p

q < jj < q C 1.

Case 1: D ˙.q C 1/.

By (4.6), we immediately obtain ˛1 D ˙q and ˛2 D ˙1, or vice versa.

Case 2: jj 2p

q.

It follows that 2 4q 0, thus ˛1; ˛2 2 C and p 2 4q D ˙i p 4q 2. This implies that

j˛1j D 1

2

ˇ ˙ i

p 4q 2

ˇ D 1

2

p 2 C .4q 2/ D p

q;

and similarly j˛2j D p q.

Case 3: 2p

q < jj < q C 1.

Now ˛1; ˛2 2 R. Observing that Cp

2 4q is increasing in , we obtain

by (4.6) for positive , i.e., for 2p q < < q C 1, that p q < ˛1 < q. Since˛1˛2 D q, this implies 1 < ˛2 < p q. Checking the vice versas and < 0,

we conclude that 1 < j˛i j < q, but j˛i j ¤ p q for i D 1; 2.




Figure 4.13. Possible locations for poles of G.u/ for regular G . This figure is taken from [48] with

permission of the author.

To complete the proof we just have to remember that the poles of G.qs/, other

than those coming from the first factor in (4.5) and having< s D 0, are the reciprocals

of some ˛i , i.e., ˛i D qs and j˛i j D jq< sCi= sj D q< s. The eigenvalues ˙.q C 1/ 2Spec.G / are maximal in absolute value and therefore irrelevant for the Ramanujan

property. At the same time, they do not affect the Riemann hypothesis, because they

correspond to poles having < s D 1 or < s D 0 (see Case 1). The eigenvaluesconsidered in Case 2 satisfy the Ramanujan condition and the corresponding ˛i all

have < s D 12

, satisfying the Riemann hypothesis, as shown in Case 2. If there existedan eigenvalue

2 Spec.G / belonging to Case 3, it would violate the Ramanujan

condition. At the same time, Case 3 has revealed the existence of a pole with 0 << s < 1, but < s ¤ 1

2, thus contradicting the Riemann hypothesis.

The preceding proof immediately implies

Corollary 3.21. Let G be a friendly .q C 1/-regular graph.

(i) The figure above shows the possible locations for the poles of G.u/. Poles

satisfying the Riemann hypothesis are those on the circle.

(ii) The radius of convergence satisfies RG D 1q

.

Setting again u D qs, there are several functional equations for the Ihara zetafunction on a regular graph, relating the value at s to that at 1 s (corresponding to

u $ 1qu

), just as in the case of the Riemann zeta function.

Theorem 3.22 (Functional equations for Ihara zeta functions). Let G be friendly

.q C 1/-regular graph of order n and rank r of its fundamental group. Then, among

other similar identities, we have

(i) .1/G .u/ WD .1 u2/r1Cn

2 .1 q2u2/n2 G.u/ D .1/n

.1/G . 1

qu/;

(ii) .2/

G .u/ WD .1 C u/r

1

.1 u/r

1

Cn

.1 qu/n

G.u/ D .2/

G . 1qu /;

(iii) .3/G .u/ WD .1 u2/r1.1 C qu/n G.u/ D

.3/G . 1

qu/.



148 Jurgen Sander

Proof of (i): The determinant formula in Theorem 3.15 and the transformation u 7!1

qu imply

.1/G .u/ D .1 u2/ n2 .1 q2u2/ n2 det.I AGu C qu2I /1

D

q2

q2u2 1

n2

1

q2u2 1

n2

det

I 1

qu AG C q

q2u2 I

1

D .1/n .1/G

1

qu

:

Exercise 3.23.

(i) Check (ii) and (iii) in Theorem 3.22.

(ii) Show that G has a pole at 1qu if it has one at u. Verify that 1qu is the complexconjugate of u if u lies on the circle of radius 1p

q, and that 1

qu lies in the interval

. 1q

; 1p q

/ for u 2 . 1p q

; 1/.

Most of the results in this section can be generalized to irregular graphs. All we

shall present here is a theorem showing the possible location of the poles of G forirregular G .

Theorem 3.24 (Motoko Kotani and Toshikazu Sunada [25], 2000). Let G D .V; E/

be a friendly graph with p C 1 WD minfd.v/ W v 2 V g and q C 1 WD maxfd.v/ Wv 2 V g , and let RG be the radius of convergence of G .

(i) We have 1q RG 1

p , and every pole u of G.u/ satisfies RG juj 1.

(ii) Every non-real pole u 2 C of G.u/ lies in the ring 1p q juj 1p

p.

(iii) The poles u of G.u/ lying on the circle juj D RG have the form

u D RGe2ik

G .k D 1 ; 2 ; : : : ; G/; (4.7)

where G

Dgcd

fŒP

W ŒP

2PG

g , as defined in (4.2).

The most interesting part of the preceding theorem is the characterization in (iii).

It is a rather straightforward consequence of an advanced result in linear algebra

called the Perron–Frobenius theorem (cf. [20]), which essentially states:

Let M be an nn matrix with non-negative real entries and irreducible ,i.e. .I C M /n1 has only positive entries. Then the eigenvalues k 2Spec.M/ .k D 1; : : : ; K / , say, of maximal absolute value are given by

k D max Spec.M / e2ik

K .

Applying this to the edge adjacency matrix M WD W EG proves (iii).




Example 3.25. The irregular graph G on the left below is obtained by adding four

vertices to each edge of the complete graphK5. Clearly, p C 1 D 2 and q C 1 D 4.

On the right, all poles u ¤ ˙1 of G.u/ are depicted (observe 5-fold rotational

symmetry). The circles centered at 0 have inverse radii 1p q D

p 3

3 0:577,

p RG Dr

5

q 13 0:896, and 1p

p D 1, coming from G.u/1 D 1

K5.u5/ D .1 u10/5

.1 3u5/.1 u5/.1 C u5 C 3u10/. (The figures are taken from [48] with permissionof the author.)

Exercise 3.26. Prove that in our case K

DG (cf. [48], p. 95).

We shall see in Exercise 3.34 how to produce pictures like Figure 4.14.

3.3 The determinant formulae for Ihara’s zeta function A proof of Ihara’s three-term determinant formula has still to be given. We shall proceed by

first showing the two-term determinant formula (Theorem 3.31) mentioned above

and derive Theorem 3.15 from it. First we have to introduce the exponential functionand the logarithmic function for square matrices (as one usually does in the theory of

Lie groups).

Definition 3.27. For an n n matrix X over R or C, the matrix

exp X WD1X

kD0

1

kŠX k

is called the matrix exponential of X .

Proposition 3.28. Let X; Y be n n matrices over R or C.

(i) The series defining exp X converges absolutely.

(ii) For the zero matrix O we have exp O D I .



150 Jurgen Sander

Figure 4.14. Locations of poles of G.u/ for an irregular random graph G with 800 vertices:

– violet circle of radiusp

p, blue circle of radiusp

q;

– green (Riemann hypothesis) circle of radius 1p RG

.

RH says spectrum should lie inside green circle – almost true!

This figure is taken from [48] with permission of the author.

(iii) If X D .xij / is diagonal, then exp X D . Qxij / with Qxi i D exi i and Qxij D 0 for

i ¤ j .

(iv) If X Y D

YX , then .exp X /.exp Y /D

.exp Y /.exp X /D

exp.X C

Y /.

(v) If Y is invertible, then exp.Y 1X Y / D Y 1.exp X /Y .

(vi) det.exp X / D etrX .

Proof of (vi). For any X there is an invertible matrix Y such that YX Y 1 is an upper

triangular matrix T , say (see any book on linear algebra). Hence

det.exp X / D det.exp.Y 1T Y / / .v/D det.Y 1 .exp T / Y /

D det.Y 1/ det.exp T / det Y D det.exp T /:

Since T D .tij / is upper triangular, so is T k D .t .k/ij / for all k . In particular, t .k/

i i Dt k

i i for all i and k. This implies, by the definition of the matrix exponential, that

exp T D .Qtij / is upper diagonal with Qti i D eti i for all i . Hence

det.exp T / D et11 etnn D etrT D etrX ;

since trT D tr.Y.XY 1// D tr..XY 1/Y / D trX .

Exercise 3.29. Prove Proposition 3.28 (i)–(v).

Definition 3.30. If the n n matrices X and Y over R or C satisfy exp Y D X , thenY D log X is called a matrix logarithm of X .




log X does not exist for every matrix X , and if it exists it is usually not unique.

For X D .xij /,

kX kFrob WDvuut nXiD1

nXj D1

jxij j2

is called the Frobenius norm or Schur norm of X . If kX kFrob < 1, then

log.I X / D 1X

kD1

1

kX k (4.8)

is a matrix logarithm of X .

Theorem 3.31 (Two-term determinant formula). Let G be a friendly graph and let EG be an orientation of G with edge adjacency matrix W EG. Then

G.u/1 D det.I W EGu/:

Proof. Let G D .V;E/ and m WD jEj. Recall that N m.G / is the number of back-trackless cycles of length m in G . We first claim that

N m.G / D tr W mEG : (4.9)

To prove this, let W EG D .wij / and W kEG D .w

.k/ij / for all positive k. By induction,

w.m/ij D

X`1

X`2

: : :X

`m1

wi `1w`1`2

: : : w`m1j ;

hence

tr W mEG DX

i

w.m/i i D

Xi

X`1

X`2

: : :X

`m1

wi `1w`1`2

: : : w`m1i :

Since we have wi `1w`1`2

: : : w`m1i D 1 if and only if ei e`1e`2

: : : e`m1 is a closed

path of length m starting with edge ei , tr W mEG apparently counts all these paths, and

just the ones without backtracks due to the definition of wij . This proves (4.9).

By virtue of identities (4.4) in Proposition 3.11 and (4.9), we have for sufficientlysmall juj

log G.u/ D1X

mD1

N m.G /

m um D

1XmD1

um

m tr W mEG D tr

1XmD1

1

m .W EGu/m

!; (4.10)

where the last identity follows from the fact that tr is continous and linear.



152 Jurgen Sander

If juj is sufficiently small, we clearly have kW EGukFrob < 1, and consequently

1

XmD1

1

m .W EGu/m

D log.I W EGu/

by (4.8). This and (4.10) imply

G.u/1 D etr

log.I W EGu/

D det

exp

log.I W EGu/

D det.I W EGu/; (4.11)

thanks to Proposition 3.28 (vi).

Corollary 3.32. For a friendly graph G with orientation EG and edge adjacency matrix

W EG , we have G.u/1 D

Y2Spec.W EG/

.1 u/:

In particular, the poles of G.u/ are the reciprocals of the eigenvalues of W EG .

Proof. det.zI W EG/ is the monic characteristic polynomial of W EG , i.e.,

det.zI W EG/ D

Y

2Spec.W EG/

.z /:

For the number m of edges in G , we consequently have

det.I W EGu/ D det

u 1

uI W EG

D u2m

Y2Spec.W EG/

1

u

D

Y2Spec.W EG/

.1 u/:

The last ingredients we need to verify Theorem 3.15 are a few identities for the

following block matrices related to an orientation EG of a graph G D .V;E/ with

n WD jV j and m WD jEj: The 2m 2m matrix

J WD

Om I mI m Om

;

where Om denotes the m m zero matrix and I m is the m m unit matrix;

the n 2m start matrix S EG D .sij /, derived by

sij WD 1; if vi is the starting vertex of ej in EG ,0; otherwiseI




the n 2m terminal matrix T EG D .tij / defined by

tij WD 1; if vi is the terminal vertex of ej in

EG ,

0; otherwise:

Note that, upon numbering directed edges in our orientation EG of G by the rule

ej Cm WD e1j , we have S D .M jN / and T D .N jM / for two suitable n m

matrices M and N .

Proposition 3.33. Let G D .V;E/ be a graph with n WD jV j and m WD jEj , and

let EG be an orientation of G . Let J , S EG , and T EG be defined as above, AG be the

adjacency matrix of G , W

EG

be the edge adjacency matrix of

EG , and QG be as defined

in Theorem 3.15. Then

(i) S EGJ D T EG and T EGJ D S EG;

(ii) AG D S EGT tEG and QG C I n D S EGS tEG D T EGT tEG;

(iii) W EG C J D T tEGS EG.

Partial proof. The proofs of (i) and the second identity in (ii) are left as exercises. By

the definition of matrix multiplication,

S EGT tEG

i k D

2mXj D1

sij tkj ; (4.12)

where sij tkj D 1 if and only if the edge ej has starting vertex vi and terminal vertexvk, and sij tkj D 0 otherwise. Hence the right-hand side of (4.12) equals the number

of oriented edges from vi to vk , and precisely this is the entry Aij . This proves the

first identity in (ii).

Similarly, we have

T t

EG

S

EGi k

D

n

Xj D1

tj i sj k;

where tj i sj k D 1 if and only if the edge ei feeds via the vertex vj directly into theedge ek (and this is even true in case ek D e1

i ), and tj i sj k D 0 otherwise. Hence

nXj D1

tj i sj k D

1; if ei feeds directly into ek ,

0; otherwise. (4.13)

By definition, this equals the entry wi k of W EG for ek ¤ e1i . Recalling our labeling

convention ej Cm WD

e

1

j in

EG , we see that the right-hand side of (4.13) equals w

i k C1

in case ek D e1i . Putting everything together, (iii) follows.



154 Jurgen Sander

Exercise 3.34.

(i) Prove (i) and the relation QG C I n D S EGS tEG D T EGT tEG from Proposition 3.33.

(ii) Show that AG D S EGT t

EG even holds if G is allowed to have loops.(iii) Use Prop. 3.33 to create random graphs (see figure at the end of Section 3.2):

Take random permutation matrices P 1; : : : ; P k and define M WD P 1 P k.

Similarly, build N , and then define S EG WD .M jN / and T EG WD .N jM /. Finally,obtain W EG by Proposition 3.33 (iii).

Proof of the three-term determinant formula – Theorem 3.15. Let n D jV j and m DjEj, and let A WD AG, Q WD QG, W WD W EG , S WD S EG and T WD T EG . All four-block

matrices in the following calculations are of size .n C 2m/ .n C 2m/ with blocks

of size n

n in the upper left corner. Then

M 1 WD

I n OT t I 2m

I n.1 u2/ S uO I 2m W u

D

I n.1 u2/ SuT t .1 u2/ T t S u C I 2m W u

and

M 2 WD

I n Au C Qu2 S uO I 2m C J u

I n OT t S t u I 2m

D I n Au C Qu2 C S T t u S S t u2 S u

.I 2m C J u/.T t S t u/ I 2m C J u :

We claim that M 1 D M 2. For the upper left corner of M 2 we have, by Proposition

3.33,

I n Au C Qu2 C ST t u SS t u2 .ii/D I n ST t u C .S S t I n/u2 C ST t u SS t u2

D I n.1 u2/:

Similarly, the lower left corner of M 2 equals

.I 2m C J u/.T t S t u/ D T t S t u C J T t u JS t u2

D T t S t u C .TJ/t u .SJ/t u2

.i/D T t S t u C S t u T t u2 DD T t .1 u2/;

and finally the lower right corner of M 1 is

T t S u

CI 2m

W u

.iii/

D T t S u

CI 2m

.T t S

J /u

D I 2m C J u;




which proves our claim. Consequently, det M 1 D det M 2 with

det M 1 D det.I n.1 u2// det.I 2m W u/

D .1 u2/n det.I 2m W u/ D .1 u2/n G.u/1

by the two-determinant formula Theorem 3.31. On the other hand,

det M 2 D det.I n Au C Qu2/ det.I 2m C Ju/;

which implies

G.u/1 D .1 u2/n det M 1 D .1 u2/n det M 2

D .1 u2/n det.I n Au C Qu2/ det.I 2m C Ju/:(4.14)

Observe that I m OI mu I m

.I 2mCJ u/ D

I m OI mu I m

I m I muI mu I m

D

I m I muO I m.1 u2/

;

hence

det.I 2m C J u/ D det

I m I muO I m.1 u2/

D .1 u2/m:

By (4.14), this yields

G.u/1

D .1 u2

/m

n

det.I n Au C Qu2

/ D .1 u2

/r

1

det.I n Au C Qu2

/

for connected graphs, by Proposition 3.13 (ii).

3.4 The prime number theorem for graphs For a friendly graph G , we de-fined PG as the set of primes in G , the greatest common divisor G D gcdfŒP WŒP 2 PGg and the prime counting function

G.k/ WD #fŒP 2 PG W ŒP D kg:

Recall that RG denotes the radius of convergence of G and is the reciprocal of the

modulus of the largest eigenvalue of W EG (cf. Corollary 3.32), while the second largest

modulus of the eigenvalues is ƒ.W EG/ WD maxfj jW 2 Spec.W EG/; j j< R1G g

(cf. Definiton 2.34).

Theorem 3.35. Let G D .V; E/ be a friendly graph, and let k be a positive integer.

If G − k , then G.k/ D 0. For G j k , we have

G.k/ D G

RkG

k C O

ƒ.W EG/k

k C O

R

k2

G

k : (4.15)

In particular, we have G.k/ GRkG

k asymptotically for G j k with k ! 1.



156 Jurgen Sander

Proof. This proof somewhat imitates the proof of the analogous result for zeta func-

tions of function fields introduced by Emil Artin — where the Riemann hypothesis

is known to be true by works of Helmut Hasse, Andre Weil and Pierre Deligne (cf.

[38]).By Proposition 3.8, the case G − k has already been treated, and we may assume

that G j k. By our definitions,

G.u/ DY

ŒP 2PG

1 uŒP

1

D1Y

kD1

1 uk

G.k/

:

This implies that

log G.u/

D

1

XkD1

G.k/ log.1

uk /

D1X

kD1

G.k/

1X`D1

uk`

`

k`DmD1X

mD1

umXkjm

G.k/ k

m;

and consequently

u d

du log G.u/ D u

1XmD1

mum1Xkjm

G.k/ k

m D

1XmD1

umXkjm

kG.k/: (4.16)

On the other hand, we know from (4.4) in Proposition 3.11 that

u d

du log G.u/ D u

d

du

1XmD1

N m.G /

m um

D

1XmD1

N m.G /um; (4.17)

which combined with (4.16) yields

N m.G / DXkjm

kG.k/: (4.18)

Using Mobius’ inversion formula, we obtain that

G.k/ D 1kXd jk

kd N d .G / D 1

kN k.G / C X

d jk; d k2

kd N d .G /

: (4.19)

Applying the two-term determinant formula in terms of Corollary 3.32, we deduce

a third identity for u d du

log G.u/, namely

u d

du log G.u/ D u

d

du log

Y2Spec.W EG/

.1 u/1

D u

d

du X2Spec.W EG/

log.1 u/




D u d

du

X2Spec.W EG/

1XmD1

.u/m

m

D X2Spec.W EG/

1XmD1

.u/m

D1X

mD1

X2Spec.W EG/

m

um:

By comparing with (4.17), we see that

N m.G /D X2Spec.W EG/

m: (4.20)

For large m the dominant terms in the last sum are the ones coming from 2Spec.W EG/ with maximal absolute value, i.e., from the poles u of G.u/ with minimal

absolute value, as follows from Corollory 3.32. By Theorem 3.24 (i), these haveabsolute value jj D R1

G . By Theorem 3.24 (iii), we even know them exactly and

they have the form

D R1G e

2ikG .k D 1 ; : : : ; G/:

Hence, by (4.20),

N m.G / DGX

kD1

RmG e

2ikmG C

X2Spec.W EG/; jjƒ.W EG/

m: (4.21)

For the main term in (4.21) we obtain

G

XkD1

RmG e

2ikmG

DRmG

G

XkD1

e2ikm

G

D Rm

G G; if G j m,

0; if G − m,

(4.22)

and the error term can be bounded asˇ X2Spec.W EG/; jjƒ.W EG /

m

ˇ jSpec.W EG/jƒ.G /m D 2jEjƒ.W EG/m: (4.23)

Combining (4.21), (4.22), and (4.23), we obtain in case G j k, as we may assume,that

N k .G / D RkG G C 2jEjƒ.W EG/k D RkG G C O.ƒ.W EG/k /; (4.24)




The asymptotic formula (4.15) has a “good” error term if the second largest

eigenvalue of the edge adjacency matrix of the graph G is “small”. For .q C 1/-

regular graphs, in particular, this is true when the graph is a Ramanujan graph with

ƒ.G / 2p q, and this in turn is equivalent with the fact that the correspondingRiemann hypothesis is satisfied (see Theorem 3.20). The following theorem in fact

shows that in this case the error term will be best possible if we consider a family of .q C 1/-regular graphs whose vertex numbers go to infinity.

Theorem 3.38 (Alon [2] and Boppana [8], 1986–87). Let .G n/n2N be a sequence of

.q C 1/-regular graphs G n D .V n; En/ with limn!1 jV nj D 1. Then

limn!1 inf

mnƒ.G m/ 2

p q:

In 1988 Lubotzky, Phillips, and Sarnak [31] constructed such an infinite familyof .q C 1/-regular Ramanujan graphs for q 1 mod 4 prime. Their proof uses the

Ramanujan conjecture, which led to the name of Ramanujan graphs. Morgenstern

[34] extended the construction of Lubotzky, Phillips, and Sarnak to all prime powers

in 1994. The problem is open for general q.In Section 4.2 we shall take up the topic of Ramanujan graphs once again.

4 Spectral properties of integral circulant graphsA graph is called integral if its spectrum is integral (cf. Definition 2.32). Up to now

no handy criterion for this property is known (cf. [16]). Yet many classes of integralgraphs have been identified and studied recently, in particular in the case when the

graph is circulant (cf. Definition 2.28), meaning its adjacency matrix is circulant (see

e.g. [1], [3], [22], [42], [45]). The class of integral circulant graphs displays richalgebraic, arithmetic and combinatorial features. In this section we shall focus on

some arithmetical aspects of spectra of these graphs, including open problems andconjectures. Our main topics will be the Ramanujan property (see Definition 2.34)

discussed in Section 3.5, and the concept of the energy of a graph.

4.1 Basics on integral circulant graphs By Theorem 2.33 we know that inte-

gral circulant graphs ICG.n;D/ D .Zn; E/ are Cayley graphs characterized by theirorder n and a set D of positive divisors of n and

E WD f.a;b/ W a; b 2 Zn; .a b;n/ 2 Dg:

The unitary Cayley graphs (cf. Definition 2.13 (ii)) are the integral circulant graphs

with D D f1g.Let us recall some general facts about ICG.n;D/ for arbitrary positive integers n

and arbitrary divisor sets D D.n/ WD fd > 0 W d j ng.



160 Jurgen Sander

Figure 4.15. Subclasses of graphs among Cayley graphs

Proposition 4.1.

(i) Since ICG.n;D/ is circulant, its eigenvalues k .n;D/ .1 k n/ , say, can

be calculated by using Proposition 2.31 , yielding

k.n;D/ DXd 2D

c

k; n

d

.1 k n/; (4.26)

where

c.k;n/ WDXj mod n

.j;n/D1

exp

2i kj

n is the well-known Ramanujan sum .cf. [4] or [44]).

(ii) ICG.n;D/ is regular, more precisely ˆ.n;D/-regular, where

ˆ.n;D/ WD n.n;D/ DXd 2D

' n

d

; (4.27)

with Euler’s totient function '. This follows from Exercise 2.15 (iii) , saying

that a Cayley graph Cay.G;S/ is

jS

j-regular.




(iii) By the observation of So [46] that ICG.n;D/ with D D fd 1; : : : ; d rg , say,

is connected if and only if gcd.d 1; : : : ; d r / D 1 , connectivity can readily be

checked. If ICG.n;D/ is connected, then ˆ.n;D/ is the largest eigenvalue

of ICG.n;D/ , the so-called spectral radius of the graph, and it occurs withmultiplicity 1 (cf. Proposition 2.26 (ii)).

Exercise 4.2.

(i) Show that ICG.n;D/ has loops if and only if n 2 D. For that reason, we shall

usually require that D D.n/ WD f0 < d < n W d j ng.

(ii) Deduce (4.26) from Proposition 2.31.

(iii) Prove Proposition 4.1 (ii).

Recently, Le and the author [28] observed that (4.26) can be rewritten as

k .n;D/ D . D c.k; //.n/ .1 k n/; (4.28)

where

is the constant function and D denotes the so-calledD-convolution of arith-

metic functions introduced by Narkiewicz [36], which is a generalisation of the clas-sical Dirichlet convolution. This representation of the eigenvalues will be of great

importance for our purpose.

In general, given non-empty sets A.n/ D.n/ for all positive integers n, the

(arithmetical) convolution A or the A-convolution of two arithmetic functions f; g 2CN is defined as

.f A

g/.n/D Xd 2A.n/

f.d/g n

d :

All convolutions considered in the literature are required to be regular , a property

which basically guarantees that f A g is multiplicative for multiplicative functions

f and g, and the inverse of with respect to A, i.e., an analogue of the Mobius

function , exists. Narkiewicz [36] proved that regularity of an A-convolution is,besides some minor technical requirements, essentially equivalent with the following

two conditions:

(i) A is multiplicative, i.e., A.mn/ D A.m/A.n/ WD fab W a 2 A.m/; b 2 A.n/gfor all coprime m; n

2N.

(ii) A is semi-regular , i.e., for every prime power ps with s 1 there exists a divisort D tA.ps / of s, called the type of ps , such that A.ps / D f1; pt ; p2t ; : : : ; pjt gwith j D s

t .

Both of these properties will occur in a natural fashion along our way, but for ourpurposes concerning the eigenvalues of ICG.n;D/ the multiplicativity of the divisor

sets D will be the guiding feature.

The product of non-empty sets A1; : : : ; At of integers is defined as

t

YiD1

Ai WD fa1 at W ai 2 Ai .1 i t /g:



162 Jurgen Sander

For infinitely many such sets A1; A2; : : :, we require that Ai D f1g for all but finitely

many i and define1

YiD1

Ai WD1

YiD1

Ai¤f1gAi

Let us call a set A of positive integers a multiplicative set if it is the product of non-empty finite sets Ai f1; pi ; p2

i ; p3i ; : : :g, 1 i t say, with pairwise distinct

primes p1; : : : ; pt . In other words, a given set A is multiplicative if and only if

A D Qp2PAp, where Ap WD fpep .a/ W a 2 Ag for each prime p, and ep.a/

denotes the order of the prime p in a. Observe that Ap ¤ f1g only for those finitelymany primes dividing at least one of the a 2 A.

If D is a multiplicative divisor set, then this property is extended to the spectrum

of the corresponding integral circulant graph.

Proposition 4.3 (Le and Sander [28], 2012). For a multiplicative divisor set D DQpjn Dp D.n/ ,

k .n;D/ DY

p2P; pjnk .pep .n/;Dp/; (4.29)

where for each prime p dividing n the integers k .pep .n/;Dp/ , 1 k pep .n/ , are

the eigenvalues of ICG.pep .n/;Dp/ , and k.pep .n/;Dp/ is defined for all integers k

by periodic continuation mod pep .n/.

4.2 Ramanujan integral circulant graphs In 2010 Droll [13] classified allRamanujan unitary Cayley graphs ICG.n; f1g/. We extend Droll’s result by drawing

up a complete list of all graphs ICG.ps ;D/ having the Ramanujan property for eachprime power ps and arbitrary divisor set D. In order to do this we shall have to check

for which D D.ps/ the Ramanujan condition

ƒ.ps ;D/ WD ƒ.ICG.ps;D// 2p

.ps ;D/ 1 (4.30)

is satisfied (cf. Definition 2.34 and Proposition 4.1 (ii)). Since Ramanujan graphs

are required to be connected, we necessarily have 1 2 D for a Ramanujan graph

ICG.ps ;D/ due to the criterion of So (cf. Proposition 4.1 (iii)). While (4.27) provides

an explicit formula for ˆ.ps ;D/, it is just as important to have such a formula for

ƒ.ps ;D/. As a tool to prove such a formula we first show

Lemma 4.4. Let ps be a prime power and D D fpa1 ; pa2 ; : : : ; par1 ; par g with

integers 0 a1 < a2 < < ar1 < ar s. Then

k.ps ;D/ DrX

iD1ai

s

j

'.psai / rX

iD1ai

Ds

j

1

psai1 (4.31)

for k 2 f1 ; 2 ; : : : ; psg , where j WD ep.k/.




Proof. By (4.26) in Proposition 4.1, we have for k 2 f1 ; 2 ; : : : ; psg that

k

.ps ;D/D Xd 2D

ck; ps

d D

r

XiD1

c.k;ps

ai /: (4.32)

We use two well-known properties of Ramanujan sums (cf. [4] or [44]), namelyc.k;n/ D c.gcd.k;n/;n/ for all k and n, and

c.pu; pv / D8<: '.pv/; if u v,

pv1; if u D v 1,

0; if u v 2,

for primes p and non-negative integers u and v. On setting m WD k

pj , i.e. k D pj m

with 0 j s and m 1; p − m, it follows that

c.k;psai / D c.pj m; psai /

D c.pminfj;saig; psai / D8<: '.psai /; if j s ai ,

psai1; if j D s ai 1,

0; if j s ai 2.

Inserting this into (4.32), we obtain (4.31).

Exercise 4.5. Check the identities for Ramanujan sums stated in the preceding proof.

Proposition 4.6. Let ps 3 be a prime power, and let D D.ps/ with 1 2 D.

Then ƒ.2s ; f1g/ D 0 for s 2 , and in all other cases

ƒ.ps ;D/ D ps1 Xd 2Dd ¤1

'ps

d

;

where the empty sum for D D f1g vanishes.

Proof. Since 1

2 D, we have D

D fpa1 ; pa2 ; : : : ; par

g, say, for suitable integers

0 D a1 < a2 < < ar1 < ar s. Writing k D pj m; p − m; with suitableintegers j , 0 j s, and m 1 for each k 2 f1 ; 2 ; : : : ; psg, we obtain, by Lemma4.4 (D 4.2 in [29]), that

k .ps ;D/ DrX

iD1aisj

'.psai / rX

iD1aiDsj 1

psai1:(4.33)

We distinguish several cases. If j D s, that is k D ps , we have k .ps;D/ Dˆ.ps ;D/, which is the largest eigenvalue of ICG.ps ;D/ (see (iii)) and thus irrelevant

for the determination of ƒ.ps ;D/. Hence we are left with the following three cases,the last two of which correspond with the cases in the proof of Proposition 4.1 in [29]:



164 Jurgen Sander

Case 0: 0 j s ar 2.

Observe that this only occurs if ar s 2. Then (4.33) implies

k .ps ;D/ D 0:

Case 1: s a` j s a`1 2 for some 2 ` r .

Then

k.ps ;D/ DrX

iD1aia`

'.psai / D ˆ.ps ;D.pa` // > 0

with D.x/ WD fd 2 D W d xg.

Case 2: j D s a` 1 for some 1 ` r .Then, on setting arC1 WD s C 1 and consequently ˆ.ps ;D.parC1 // D 0,

we have

k .ps ;D/ DrX

iD1aia`C1

'.psai / psa`1

D ˆ.ps ;D.pa`C1 // psa`1 0;

such that jk.p

s

;D/j D p

s

a`

1

ˆ.p

s

;D.p

a`C1

//.

Let us start by looking at the special case r D 1, i.e., D D f1g, a1 D 0 and s 1. This means that Case 1 never occurs. Case 2 does occur for j D s 1, with

corresponding k .ps ;D/ D ps1. This value equals ˆ.2s; f1g/ D '.2s/ D 2s1 in

case p D 2 (and s 2, since s D 1 is outruled by our condition ps 3), hence

ƒ.2s ; f1g/ D 0 (originating from Case 0). For any p 3 and all s 1, we have

ps1 < ˆ.ps ; f1g/, which proves the proposition in this situation.From now on, we assume that r 2. We define

m DmD WD 8<: 1; if D is uni-regular,

min2`r

a`a`12

`; otherwise. (4.34)

FormD D 1, Case 1 does not occur at all. Obviously,

ˆ.ps ;D/ ˆ.ps ;D.pa` // > ˆ.ps ;D.pa`C1 // > 0

for 1 ` r 1. If mD < 1, Case 1 occurs for ` D mD, but for no smaller `.

Therefore, the only candidate for ƒ.ps;D/ originating from Case 1 is ˆ.ps;D.pam//.

On the other hand, it is easily seen that

ˆ.ps ;D/ > psa`1 ˆ.ps ;D.pa`C1 // psa`C11 ˆ.ps ;D.pa`C2 // 0




for 1 ` r 1. Hence, the only candidate for ƒ.ps ;D/ originating from Case 2

is psa11 ˆ.ps ;D.pa2 // D ps1 ˆ.ps ;D.pa2 //. Now let us compare the two

candidates for ƒ.ps ;D/ found above. By the definition of m

DmD, we obtain

ˆ.ps ;D.pa2 // C ˆ.ps ;D.pam// DrX

iD2

'.psai / CrX

iDm'.psai /

Dm1XiD2

'.psai / C 2

rXiDm

'.psai /

D psa2 psam11 C 2

rXiDm

'.psai /

ps

a2

ps

am1

1

C 2ps

am

ps

1

:

This implies ps1 ˆ.ps ;D.pa2 // ˆ.ps ;D.pam//, and we have

ƒ.ps ;D/ D ps1 ˆ.ps ;D.pa2 // D ps1 Xd 2Dd ¤1

'ps

d

:

Corollary 4.7. Let ps 3 be a prime power, and let D D.ps/ with 1 2 D.

(i) Then

ˆ.ps ;D/ C ƒ.ps ;D/ D 2s

1

; if p D 2 , s 2 and D D f1g;ps ; otherwise. (4.35)

(ii) For all odd primes p , we have ƒ.ps ;D/ D 0 if and only if D D D.ps/.

Proof. The identity (4.35) follows right away from (4.27) and Proposition 4.6. The

second assertion is another consequence of Proposition 4.6, becauseXd 2Dd

¤1

'ps

d

ps1;

with equality if and only if D D D.ps/.

Corollary 4.8. Let ps 3 be a prime power, and let D D.ps/ with 1 2 D. Then

ICG.ps;D/ is Ramanujan if and only if either p D 2 , s 2 and D D f1g or, in all

other cases, Xd 2Dd ¤1

'ps

d

ps1 2

p ps C 2; (4.36)

where the empty sum for D

D f1

gvanishes.



166 Jurgen Sander

Proof. The special case p D 2, s 2 and D D f1g is an immediate consequence of

Proposition 4.6. From (4.30) and Proposition 4.6 it follows in all other situations that

ICG.ps ;D/ is Ramanujan if and only if ˆ2

WDPd

2D; d

¤1 '. ps

d / satisfies

ps1 ˆ2 2p

.ps ;D/ 1 D 2p

'.ps / C ˆ2 1:

A short calculation reveals that this is equivalent with (4.36).

Expanding our definition of D.n/, we set D.nI m/ WD fd 2 D.n/ W d mgfor any positive integer m. Now we are able to show precisely which ICGs of prime

power order do have the Ramanujan property.

Theorem 4.9 ([30]). Let p be a prime and s a positive integer such that ps 3. Let

D

D.ps / be an arbitrary divisor set of ps . Then ICG.ps;D/ is a Ramanujan

graph if and only if D lies in one of the following classes:(i) D D D.pd s

2 e1/ [D0 for some D0 D.ps1I pd s2 e/;

(ii) D D f1g in case p D 2 and s 3;

(iii) D D D.ps3

2 / [D0 such that jDj 2 for some D0 D.ps1I psC1

2 / in case

p 2 f2; 3g and s 3 odd;

(iv) D D D.2s4

2 / [ D0 for some D0 D.2s1I 2s2 / satisfying ; ¤ D0 ¤ f2s1g

in case p D 2 and s 4 even;

(v) D D f1; 22; 23; 24g in case p D 2 and s D 5;

(vi) D D D.5s3

2 / [ f5sC1

2 g [ D0 for some D0 D.5s1I 5sC3

2 / in case p D 5and s 5 odd;

(vii) D D D.2s5

2 / [ f2s1

2 g [D0 for some D0 D.2s1I 2sC1

2 / satisfying

3 2p

2 C 1

2s3

2

2s1

2

Xd 02D0

1

d 0 (4.37)

in case p D 2 and s 5 odd.

Theorem 4.9 tells us for each prime power in a simple way how to choose divisorsets to obtain an integral circulant graph that is Ramanujan, except for the final case

(vii), where the construction is a little more intricate. In this situation, the binary

expansion of the left-hand side of (4.37) is obtained by adding 2 s32 to the binary

expansion of the real constant 3 2p

2 D .0:0010101111101100001 : : : /2. Obvi-

ously, the sum on the right-hand side of (4.37) is a binary expansion by construction,and in order to generate a Ramanujan graph we just have to pick D0 appropriately.

Let us sketch an example that illustrates what to do explicitly in case (vii):

Consider the prime power 229, for which condition (4.37) turns into

3 2p 2 C 1213 D .0:0010101111110100001 : : : /2

rXiD15

12ai14 ;




with D0 D f2a15 ; 2a16 ; : : : ; 2ar g, say. The parameter r 15 may be selected arbi-

trarily so that ar 28. Now for each choice of the ai , 15 i r , it is completely

obvious whether the corresponding divisor set D0 generates a Ramanujan graph or

not. For instance, the choice a15 D 17, a16 D 19, a17 D 21, a18 D 22, a19 D 23,a20 D 24, a21 D 25, a22 D 26, a23 D 27 generates the Ramanujan graph

ICG.229;D.212/ [ f214; 217; 219; 221; 222; 223; 224; 225; 226; 227g/;

while the divisor set of

ICG.229;D.212/ [ f214; 217; 219; 221; 222; 223; 224; 225; 226; 228g/

violates (4.37), and thus the graph is not Ramanujan.

Theorem 4.9 (i) immediately implies the following statement.

Corollary 4.10. For each prime power ps 3 and all divisor sets D Df1 ; p ; : : : ; pr1g with s

2 r s , the graph ICG.ps;D/ is Ramanujan. In par-

ticular, there is an integral circulant Ramanujan graph ICG.ps ;D/ for each prime

power ps 3.

We like to make the reader aware of the fact that the divisor sets ensuring theRamanujan property in Corollary 4.10 are uni-regular , as introduced in [29], where

a subset of D.ps / is called uni-regular if it consists of successive powers of p.

Proof of Theorem 4.9. Let us assume that D D fpa1 ; pa2 ; : : : ; par g for suitable in-

tegers 0 a1 < a1 < a2 < < ar1 < ar s 1. Since Ramanujan graphs haveto be connected by definition, i.e., 1 2 D, we necessarily have a1 D 0.

We consider the case r D 1, i.e., D D f1g separately. We know by Corollary 4.8that ICG.2s; f1g/ is Ramanujan for each s 2. For p 3, the Ramanujan property

requires ps1 2ps=2 C 2 0 by condition (4.36) of Corollary 4.8. It is easy to

check that this inequality holds if and only if s D 1 or s D 2. For r D 1, we thushave exactly the following types of Ramanujan ICGs:

ps D

A 2s

.s 2/ f1gB p .p 3/ f1gC p2 .p 3/ f1g

In the sequel we may assume that r 2. Since ar s 1, we trivially have

rXiD2

'.psai / .psa2psa21/C.psa21psa22/CC.p1/ D psa21:

Hence, by Corollary 4.8, a necessary condition for ICG.ps;D/ to be Ramanujan is

thatpsa2 ps1 2

p ps C 2: (4.38)




Then, by Corollary 4.8, ICG.ps ;D/ is Ramanujan if and only if

p

s

am1

1

C 2 2p ps

Cr

XiDm'.p

s

ai

/: (4.40)

For s 2.am1 C1/, i.e., s2 s am1 1, the Ramanujan condition (4.40) is satis-

fied for all primes p. By the definition of m DmD, we have am1 Dm 2. HenceICG.ps;D/ is Ramanujan for any prime power ps and D D f1 ; p ; : : : ; pm2; pam ;: : : ; par g, if 3 m r satisfies s 2.m 1/ and am m, which requires s 4.

Setting ` WDm 1, we have found the next type of Ramanujan ICGs:

ps D

I ps .p 2 P; s 4/ f1 ; p ; : : : ; p`1; pa`C1 ; : : : ; par g . s2 ` a`C1 1/

For s 2.am1 C 3/, i.e. s2 s am1 3, we obtain for all primes p

p C 2

psam11 > 2 p

s2C1

psam12 C 1 2p

s2

psam12 C 1;

hence

psam11 C 2 D psam12p C 2psam12

> psam12

2p

s2

psam12 C 1

!D 2p

s2 C psam12

2ps2 C psam > 2

p ps C

rXiDm

'.psai /:

This contradicts (4.40), which means that there are no Ramanujan graphs in this situ-

ation.It remains to consider the three special cases s D 2.am1 C 1/ C j with j 2

f1; 2; 3g. This turns (4.40) into the Ramanujan condition

pj2 2 1

pam1C1C j2

rX

iDm'.psai / 2

!: (4.41)

By the definition of m DmD, we know that am am1 C 2. Since

rXiDm

'.ps

ai / p

s

am 1 p2.am

1C1/C

j

.am1C

2/

1 D pam

1Cj

1;



170 Jurgen Sander

careful calculations reveal that condition (4.41) can only be satisfied for the pairs

.j;p/ 2 f.1;2/;.1;3/;.1;5/;.2;2/;.3;2/g. For the first two pairs, corresponding

with am

1

D s3

2 , the left-hand side of (4.41) is negative, while the right-hand

side is non-negative except for the special case p D 2 and am D ar D s 1. However, a closer look shows that in this latter situation (4.41) is still satis-

fied, and we obtain Ramanujan graphs ICG.ps;D/ for p 2 f2; 3g, where D Df1 ; p ; : : : ; p

s32 ; p

a sC12 ; : : : ; par g with a sC1

2D am am1 C 2 D s3

2 C 2 D sC1

2 ,

thus s 5. Inserting the third pair j D 1 and p D 5 into (4.41), a short computationdiscloses that the corresponding ICGs are Ramanujan if and only if am D am1 C 2,

and then D D ff1 ; 5 ; : : : ; 5s3

2 ; 5sC1

2 ; : : : ; 5ar g with s 5. To sum up, the Ramanu-

jan ICGs for .j; p/ 2 f.1;2/;.1;3/;.1;5/g are the following:

ps D

J ps .p 2 f2; 3g; s 5; 2 − s/ f1 ; p ; : : : ; ps3

2 ; pa sC1

2 ; : : : ; par g .a sC12

sC12 /

K 5s .s 5; 2 − s/ f1 ; 5 ; : : : ; 5s3

2 ; 5sC1

2 ; 5a sC3

2 ; : : : ; 5ar g

For j D p D 2, the Ramanujan condition (4.41) readsPr

iDm '.psai / 2, and

this always holds unless am D ar D s 1, i.e., we obtain Ramanujan ICGs of type

ps

DL 2s .s 6; 2 j s/ f1 ; 2 ; : : : ; 2

s42 ; 2

a s2 ; : : : ; 2ar g . s

2 a s

2 s 2/

We are left with the case j D 3 and p D 2, where s D 2am1 C 5 D 2.m 2/ C5 D 2mC 1, and our Ramanujan condition (4.41) becomes

2p

2 2 1

2am1C52

rX

iDm'.2sai / 2

!

D 12m

p 2 rX

iDm2sai1 2! D 1p

2 rX

iDm1

2aim 12m1

! :

Apparently, this is equivalent to 4 2p

2 PriDm

12aim

12m1 ; where we know

that ai i for all i m. This immediately implies that the Ramanujan condition

necessarily requires that am Dm. Then our condition reads

3 2p

2 rX

iDmC1

1

2aim 1

2m1 D

rXiD sC1

2

1

2ai s12

1

2s3

2

; (4.42)




and we have found our last type of Ramanujan ICGs:

ps D

M 2s .s 5; 2 − s/ f1 ; 2 ; : : : ; 2 s52 ; 2 s1

2 ; 2a sC12 ; : : : ; 2ar g

.a sC12

; : : : ; ar satisfy (4.42)/

To complete the proof of Theorem 4.9, it remains to show that the Ramanujan ICGsof types A–M exactly constitute the Ramanujan graphs listed in the assertion. The

following table provides the interrelations between types A–M on one hand and cases

(i)–(vii) on the other hand.

(i) A .s D 2/, B (s D 1), C (s D 2), G (s 2), I (s 4)(ii) A (s 3)

(iii) D (s D 3), H (s 5 odd), J (s 5 odd)(iv) E (s D 4), L (s 6 even)

(v) F (s D 5)

(vi) K (s 5 odd)

(vii) M (s 5 odd)

Applying the concept of multiplicative divisor sets introduced at the end of Sec-

tion 4.1, Corollary 4.10 can be extended to integral circulant graphs whose order isan arbitrary composite number. It has recently been shown that for every even integer

n 4 and a positive proportion of the odd integers n, namely those having a “domi-nating” prime power factor, there exists a divisor setD D.n/ such that ICG.n;D/is Ramanujan. Unfortunately, the corresponding assertion cannot be confirmed for all

odd n by use of the tool of multiplicative divisor sets. In fact, it will transpire thatthe set of odd n for which no Ramanujan graph ICG.n;D/ with multiplicative divisor

set D exists has positive density. This means that one would definitely need to study

integral circulant graphs with non- multiplicative divisor sets in order to finally settlethe matter.

At the end of this section we just state and discuss some further results and finallyask a couple of questions.

Theorem 4.11 ([39] (2013)). For each even integer n 3 there is a .multiplicative/divisor set D D.n/ such that ICG.n;D/ is a Ramanujan graph.

If n is an odd integer, a multiplicative divisor set D D.n/ such that ICG.n;D/is Ramanujan can only exist if n has a comparatively large prime power factor. This

will be specified in detail by the following theorem. To this end, we define for every

integer n > 1 its largest prime power factor PP.n/ WD maxp2P; pjn pep .n/, and we

shall say that n has a dominating prime power factor if PP.n/ n3=2

Cn

2.n1/ .



172 Jurgen Sander

Theorem 4.12 ([39] (2013)). Let n 3 be an odd integer.

(i) If n has a dominating prime power factor, then there is a .multiplicative/ divi-

sor set D

D.n/ such that ICG.n;D/ is a Ramanujan graph.

(ii) If n 8259 does not have a dominating prime power factor, then there is nomultiplicative divisor set D D.n/ such that ICG.n;D/ is a Ramanujan

graph.

There are a little less than 200 odd integers n < 8259 without dominating prime

power factor, the smallest being 315 D 32 5 7 and the largest 8211 D 3 7 17 23.

For each of these n we assured ourselves that ICG.n;D/ is not Ramanujan for the“canonical” multiplicative candidate D WD Qp2P; pjn Dp to produce a Ramanujan

integral circulant graph, namely by choosing Dq D D.qeq .n// for the largest prime

power qeq .n/

D PP.n/ of n and Dp

D D.pep .n// for all other primes p

j n. Check-

ing, however, all possible multiplicative divisor sets for all those n would require anenormous computational effort, which is not justified by whatever the result is.

While Theorem 4.11 confirms the existence of a Ramanujan graph ICG.n;D/ for

all even n, Theorem 4.12 leaves unanswered the question if for every odd n there isa Ramanujan integral circulant graph of order n. However, Theorem 4.12 (ii) does tell

us that in order to deal with this problem one would have to study integral circulant

graphs with non-multiplicative divisor sets D for infinitely many n. Our final resultdiscloses the proportion between the odd n for which ICG.n;D/ is Ramanujan for

some multiplicative divisor set D D.n/ and those odd n without this property.

More precisely, it turns out that both cases occur with positive density among all odd

integers.For this purpose, let .x/ denote

#fn x W 2 − n; ICG.n;D/ Ramanujan for some multiplicative D D.n/g:

It can be shown that limx!1 .x/x

exists and is positive. Notice that the set of odd

prime powers below x, for each of which there is some Ramanujan ICG by Corollary

1.1 in [30] (see also Proposition 4.10), does not suffice to warrant this result, since ithas density zero in the set of positive integers. By use of methods and results from

analytic number theory, one can show

Theorem 4.13 ([39] (2013)). For x ! 1 ,

.x/ D log 2

2 x C O

x

log x

:

Corollary 4.14. We have

limx!1

.x/x2

D log 2;

i.e., the density of odd positive integers n for which ICG.n;D/ is Ramanujan for

some multiplicative divisor set D D.n/ in the set of all odd positive integers

equals log 2 , while the corresponding density of odd positive integers n such that noICG.n;D/ with multiplicative D is Ramanujan equals 1 log 2.




Open problems:

Given a (sufficiently large) positive integer n, is there some D D.n/ such

that ICG.n;D/ is Ramanujan?

How do non-multiplicative divisor sets such that the corresponding integralcirculant graphs are Ramanujan look?

4.3 The energy of a graph In 1978 Gutman [15] introduced the mathematical

concept of the energy

E.G / WDX

2Spec.G/

jj:

of a graph G , though this concept is rooted in chemistry way back in the 1930s (see[32] for connections between H uckel molecular orbital theory and graph spectral

analysis, and [9] for a mathematical survey).We denote the energy of an integral circulant graph by

E .n;D/ WD E.ICG.n;D// DX

2Spec.ICG.n;D//

jj DnX

kD1

jk .n;D/j: (4.43)

It is of particular interest to determine for any fixed positive integer n the extremal

energies

E min.n/ WD minfE .n;D/ W D D.n/gand

E max.n/ WD maxfE .n;D/ W D D.n/g;

as well as the divisor sets producing these energies. For prime powers n D ps thisproblem was completely settled by the author and T. Sander in [40], Theorem 3.1,

and [41], Theorem 1.1, respectively. The basis of these results was an explicit for-

mula evaluating E .ps ;D/ (cf. [40], Theorem 2.1). Due to the lack of a comparableenergy formula for arbitrary n, it seems much more difficult to deal with E min.n/ and

E max.n/ in general. Unfortunately, the energy reveals no sign of multiplicativity with

respect to n. We shall overcome that deficiency by using the concept of multiplica-tive divisor sets (cf. Section 4.1 and [28]), which is closely linked with the eigenvalue

representation in (4.28). This will at least provide us with good bounds for E min.n/and E max.n/ as well as divisor sets producing the corresponding energies. In case

of the minimal energy, we even conjecture to have found E min.n/ and its associated

divisor sets for all n.By use of (4.28) and Proposition 4.3, Le and the author proved in [28], Theo-

rem 4.2, that given a multiplicative divisor set D D.n/, we have for the energy of

ICG.n;D/, as defined in (4.43), that

E .n;D/ D Yp2P; pjn

E .pep .n/;Dp/; (4.44)



174 Jurgen Sander

where the sets Dp are defined as in Proposition 4.3. We shall investigate minimal and

maximal energies of integral circulant graphs with respect to multiplicative divisor

sets. To this end, we define

QE min.n/ WD minfE .n;D/ W D D.n/ multiplicativeg

and

QE max.n/ WD maxfE .n;D/ W D D.n/ multiplicativeg:

Clearly,E min.n/ QE min.n/ QE max.n/ E max.n/ (4.45)

for all positive integers n. Moreover, for prime powers ps any divisor set D

D.ps/is trivially multiplicative, hence QE min.ps/ D E min.ps / and QE max.ps / D E max.ps/.

We shall first state all our results and provide the corresponding proofs later.

Theorem 4.15 ([29] (2012)). Let n 2 be an integer with prime factorisation n Dp

s1

1 pstt . Then

QE max.n/ DtY

iD1

.psi

i /;

where for any prime power ps

.ps / WD( 1

.pC1/2.s C 1/.p2 1/ps C 2.psC1 1/ ; if 2 − s ,

1.pC1/2

s.p2 1/ps C 2.2psC1 ps1 C p2 p 1/

; if 2 j s.

Moreover, for multiplicative setsD D.n/ , we have E .n;D/ D QE max.n/ if and only

if D DQtiD1 D

.i / with

D.i / D

8<:f1; p2

i ; p4i ; : : : ; p

si3i ; p

si1i g; if 2−si ; pi 3 ,

f1; 22

; 24

; : : : ; 2si

3

; 2si

1

g or f1; 2; 23

; : : : ; 2si

4

; 2si

2

; 2si

1

g; if 2−si ; pi D 2 ,f1; p2i ; p4

i ; : : : ; psi4i ; p

si2i ; p

si1i g or f1; pi ; p3

i ; : : : ; psi3i ; p

si1i g; if 2 j si .

It should be observed that the maximising factor divisor sets D.i / are (almost)

semi-regular (cf. (ii) at the end of Section 4.1).In order to describe the multiplicative divisor sets minimising the energy, let us

call D D.ps/ for a prime power ps uni-regular , if D D fpu; puC1; : : : ; pv1; pvgfor some integers 0 u v s , which is a special case of semi-regularity if u D 0and v D s.




Theorem 4.16 ([29] (2012)). Let n 2 be an integer with prime factorisation n Dp

s1

1 pstt and p1 < p2 < < pt . Then

QE min.n/ D 2n1 1p1

:

Moreover, for multiplicative sets D D.n/ , we have E .n;D/ D QE min.n/ if and only

if D D QtiD1 D

.i / with D.1/ D fpu1 g for some u 2 f0 ; 1 ; : : : ; s1 1g and arbitrary

uni-regular sets D.i / with psi

i 2 D.i / for 2 i t .

Remarks 4.17.

(i) According to Proposition 4.1 (iii), ICG.n;D/ is connected if and only if the

elements of D are co prime. Assuming connectivity in Theorem 4.16, min-

imising sets D D QtiD1 D

.i / then necessarily have D.1/ D f1g and D.i / Df1; pi ; p2

i ; : : : ; psi

i g for 2 i t , i.e., all D.i / with 2 i t have to be

semi-regular sets of type 1.

(ii) The proof of Theorem 4.16 reveals that for D D.n/, i.e., possibly n 2 D,

we would have QE min.n/ D n with minimising sets D DQtiD1 D

.i /, where each

D.i / is an arbitrary uni-regular set containing psi

i .

(iii) One could prove Theorem 4.16 by using the energy formula for ICG.ps ;D/containing loops (cf. [28], Proposition 5.1). Instead we shall take a closerlook at the second largest

j

jwith

2 Spec.ICG.n;D//. This provides more

insight into the underlying structure.

As a consequence of (4.45), we immediately obtain from Theorems 4.15 and 4.16

the desired bounds for the extremal energies of integral circulant graphs with arbitrary

divisor sets.

Corollary 4.18. Let n 2 be an integer with prime factorisation n D ps1

1 pstt

and p1 < p2 < < pt . Then

(i) E max.n/ QE max.n/ DQ

tiD1 .p

si

i /;

(ii) E min.n/ QE min.n/ D 2n1 1p1 .Examples show that, while equality between E max.n/ and QE max.n/ does occur oc-

casionally, we usually haveE max.n/ > QE max.n/ (cf. Example 4.20). This phenomenon

is due to the fact that maximising divisor sets normally are not multiplicative. YetQE max.n/ falls short of E max.n/ by less than a comparatively small factor. To state this

result, we denote by ' Euler’s totient function, by .n/ the number of positive divi-

sors of n, and by !.n/ the number of distinct prime factors of n. As before, ep.n/denotes the order of the prime p in n.



176 Jurgen Sander

Theorem 4.19 ([29] (2012)). Let n be a positive integer. Then

(i) E max.n/ nXd jn

'.d/.d/

d

D nY

p2P; pjn

12

1 1

p

.ep.n/ C 1/.ep.n/ C 2/ C 1

p

I

(ii) E max.n/ <

34

!.n/n.n/2I

(iii) E max.n/ QE max.n/ .n/.

The proof of Theorem 4.19 (ii) will show that

14!.n/

.n/2

< Xd jn

'.d/.d/

d < 34!.n/

.n/2

; (4.46)

and (ii) is deduced from (i) by use of the upper bound in (4.46). We like to pointout that the constants 1

4 and 3

4 in the lower and upper bound of (4.46), respectively,

cannot be improved for all n, although we expectP

d jn'.d/.d/

d

12

!.n/.n/2 for

most n. Yet, each of the bounds is sharp for integers with certain arithmetic properties

(cf. Remark 1).

As far as the magnitude of E max.n/ is concerned, it is well known that .n/ DO.n"/ for any real " > 0. In fact the true maximal order of .n/ is approximately

n log 2loglog n . On average, .n/ is of order log n, but for almost all n it is considerably

smaller, because the normal order of .n/ is roughly .log n/log 2 .log n/0:693. Forall these results as well as for the average and normal order of !.n/ the reader is

referred to [17], 18.1–18.2, 22.13, and 22.11.

As opposed to sets maximising the energy of integral circulant graphs, numericalcalculations suggest that divisor sets minimising the energy are always multiplicative.

According to that, equality in Corollary 4.18 (ii) should hold for all n. We shall

specify this observation in two conjectures at the end of this section.

Now we start to prove all results stated so far in this section.

Proof of Theorem 4.15. Let n D ps

11 ps

tt be fixed, and let D D.n/ be amultiplicative set satisfying E .n;D/ D QE max.n/. By (4.44) (cf. [28], Theorem 4.2),

QE max.n/ D E .n;D/ DY

p2P; pjnE .pep .n/;Dp/ D

tYiD1

E .psi

i ;Dpi/:

This implies that E .psi ;Dpi/ D E max.p

si

i / for 1 i t . From [41], Theorem 1.1,

we conclude that E max.psi

i / D .psi

i / for all i , and the only corresponding divisor

sets D.i / are just the ones listed in our assertion.




While equality between E max.n/ and QE max.n/ does occur occasionally, we usually

have E max.n/ > QE max.n/. This is illustrated by

Example 4.20. We have .p/ D 2.p 1/ for each prime p. By use of Theorem 4.15,we easily obtain QE max.6/ D .2/ .3/ D 8, QE max.105/ D .3/ .5/ .7/ D 384, andQE max.21/ D .3/ .7/ D 48. Numerical evaluation of (4.43) yields E max.6/ D 10,E max.105/ D 520, but E max.21/ D 48 D QE max.21/.

In order to be able to compare E max.n/ and QE max.n/ and finally prove Theorem

4.19 completely, we start by establishing an upper bound for E max.n/.

Proof of Theorem 4.19 (i) & (ii). (i) By (4.43), (4.28), and the well-known Holder

identity for Ramanujan sums (cf. [4], chapt. 8.3-8.4), we have for any n and any

divisor setD

D.n/

E .n;D/ DnX

kD1

ˇXd 2D

n

.n;kd/

'. nd

/

'

n.n;kd/

ˇ ;

where denotes the Mobius function. Let n=D WD f nd W d 2 Dg D.n/ be the set

of complementary divisors of all d 2 D with respect to n. Then

E .n;D/ Dn

XkD1

ˇ

ˇ Xd

2n=D

d

.d;k/

'.d/

' d

.d;k/

ˇ

ˇ

nXkD1

Xd 2n=D

'.d/

'

d .d;k/

D Xd 2n=D

'.d/Xgjd

1

'. d g

/

nXkD1

.k;d/Dg

1 :

Since

nXkD1

.k;d/Dg

1 D n

d

d XrD1

.r;d/Dg

1 D n

d

dgX

rD1.r; d

g/D1

1 D n

d 'd

g

;

we conclude that

E .n;D/ nX

d 2n=D

'.d/.d/

d n

Xd jn

'.d/.d/

d :

Since this holds for all n, the inequality in (i) follows. Since ', and the identity

function id are all multiplicative, this property is carried over to ' id

and its summatory

function

f.n/ WDXd jn

'.d/.d/

d .n 2 N/: (4.47)



178 Jurgen Sander

Given a prime power ps , we obviously have

f .ps /D

s

Xj D0

'.pj / .pj /

pj D1

C

s

Xj D1

.pj pj 1/.j C 1/

pj

D 1

2

1 1

p

.s C 1/.s C 2/ C 1

p:

(4.48)

This completes the proof of (i).

(ii) Taking into account (i), it suffices to prove the upper bound in (4.46), but in

order to justify the remark following Theorem 4.19 we shall verify the lower boundas well. By (4.48), we have for all primes p and all positive integers s that

1

4 1

2 1 1

p <

f .ps /

.s C 1/2 3

4 1 2

3p <

3

4 ; (4.49)

thus 14

.ps/2 < f .ps / < 34

.ps /2. By the multiplicativity of .n/ and the additivity

of !.n/, which implies the multiplicativity of c !.n/ for any positive constant c , thisproves (4.46).

Remark 1. It is easy to see that f .ps /

.sC1/2 comes close to the lower bound 14

in (4.49) for

p D 2 and large s and close to the upper bound 34

for large p and s D 1. Hence, the

lower bound in (4.46) is approached for integers n that are high powers of 2, while

the upper bound is approached for squarefree integers n having large prime factors.

Proposition 4.21. Let p be a prime, and let s be a positive integer. Then

g.ps / WD ps f .ps /

QE max.ps /(

pC12p

.s C 2/ for 2 − s ,pC1

2p .sC1/.sC2/

s for 2 j s

for the function f defined in (4.47). More precisely, we have in particular g.2/ D 2 ,

g.p2/ 3 and g.24/ D 6417

.

Proof. By [41], Theorem 1.1, we know that

QE max.ps/

D E max.ps/

D .ps / as de-

fined in Theorem 4.15.

Case 1: 2 − s.

By use of (4.48) we obtain

g.ps /

12

1 1

p

.s C 1/.s C 2/ C 1

p

.p C 1/2

.s C 1/.p2 1/ C 2p 2ps

1 1

p

.s C 1/.s C 2/.p C 1/2

2.sC

1/.p2

1/

D p C 1

2p .s C 2/:

Observe that the second inequality is an identity for p D 2 and s D 1.




Case 2: 2 j s.

Applying again (4.48), we conclude

g.ps / 1

21 1

p.s C 1/.s C 2/ C 1

p.p C 1/2

s.p2 1/ C 4p 2p C 2

ps2 2ps1 2

ps

1 1p

.s C 1/.s C 2/.p C 1/2

2s.p2 1/ D p C 1

2p .s C 1/.s C 2/

s :

The special values for g.ps/ are the results of straightforward computations.

Corollary 4.22. Suppose that n is a positive integer with prime factorisation n

Dps11 pstt . Then we have

E max.n/

QE max.n/ .n/

tYiD1

pi C 1

2pi

tYiD12−si

1 C 1

si C 1

tYiD12jsi

1 C 2

si

:

Proof. By Theorem 4.19 (i), Theorem 4.15, and Proposition 4.21,

E max.n/

QE max.n/ nf .n/

QE max.n/ D

t

YiD1

g.psi

i /

tY

iD1

pi C 1

2pi

tYiD12−si

.si C 2/

tYiD12jsi

.si C 1/.si C 2/

si

:

(4.50)

Since .n/ D QtiD1.si C 1/, the corollary follows.

Proof of Theorem 4.19 (iii). We cannot use Corollary 4.22 directly, but by (4.50) weknow that

E max.n/

QE max.n/ Yp2P; pjn

g.pep .n//:

Since .n/ D Qp2P; pjn.ep .n/ C 1/, it suffices to show that g.ps/ s C 1 for any

prime power ps . This will be verified by use of the different bounds obtained inProposition 4.21.

Case 1: 2 − s.

We have g.ps / pC12p

.s C 2/ s C 1 for all p and s except for p D 2,

s D 1, but g.2/ D 2 closes the gap.

Case 2: s D 2.This case is settled by the fact that g.p2/ 3.



180 Jurgen Sander

Case 3: 2 j s and s 4.

Here we have g.ps/ pC12p

.sC1/.sC2/s

s C 1 for all p and s except for

p D 2, s D 4, but we know that g.24

/ D 64

17 5.

Besides multiplicativity, our proofs concerning E min.n/ will be based on knowl-

edge about the second largest modulus of the eigenvalues of ICG.n;D/ (cf. Re-mark 4.17 (iii)), i.e., about

ƒ.n;D/ WD maxfjj W 2 Spec.ICG.n;D//; jj < ˆ.n;D/g; (4.51)

which we only define if ICG.n;D/ has eigenvalues differing in modulus. It will be

crucial for us to gather some facts about ƒ.n;D/. The first step is

Lemma 4.23. Let n be a positive integer and D D.n/ with n 2 D.

(i) Then E .n;D/ n.

(ii) ICG.n;D/ has a negative eigenvalue if and only if E .n;D/ > n.

Proof. From linear algebra we know thatPn

kD1 k.n;D/ equals the trace of the adjacency matrix of ICG.n;D/. Since n 2 D by hypothesis, ICG.n;D/ has a loop

at every vertex, i.e., all diagonal entries of its adjacency matrix are 1. Hence,

PnkD1 k .n;D/ D n, and consequently

E .n;D/ D nXkD1

jk .n;D/j ˇ nXkD1

k .n;D/ˇ D n; (4.52)

which proves (i). Equality in (4.52) only holds if all k .n;D/ have the same sign.

We know that n.n;D/ D ˆ.n;D/ > 0, and this implies (ii).

Proposition 4.24. Let ps be a prime power and D D.ps/ with ps 2 D , and set

r WD jDj.(i) For r D 1 , i.e.,D D fpsg , we have Spec.ICG.n;D// D f1g.

(ii) Let r 2. Then D is uni-regular if and only if ƒ.ps ;D/ D 0. In this case themaximal eigenvalue ˆ.ps ;D/ D psa1 D pr1 has multiplicity pa1 .

(iii) If r 2 and D is not uni-regular, then E .ps;D/ > ps .

Proof. Let D D fpa1 ; pa2 ; : : : ; par1 ; par g with 0 a1 < a2 < < ar1 <ar D s. Hence in case r D 1, that is D D fpsg, we trivially have k .ps ;D/ Dc.k;1/ D 1 for all k, which proves (i).

Henceforth we assume r 2. On setting j WD ep.k/, we apply Lemma 4.4 and

distinguish two cases.




Case 1: s a` j s a`1 2 for some 1 ` r , where a0 WD 2.

Using the notation D.x/ WD fd 2 D W d xg, we obtain from Lemma 4.4that

k.ps ;D/ D rXiD1

aia`

'.psai / D ˆ.ps ;D.pa` //; (4.53)

with ˆ.n;D/ as defined in (4.27).

Case 2: j D s a` 1 for some 1 ` r 1.

Now Lemma 4.4 yields

k.ps ;D/

D

r

XiD1aia`C1

'.psai /

psa`1

Dˆ.ps ;D.pa`C1 //

psa`1:

(4.54)

In order to prove (ii), we first assume that D D fpsrC1; psrC2; : : : ; ps1; psgis uni-regular. We observe that for a` D a`1 C 1 (2 ` s) the corresponding

interval considered in Case 1 is empty. Hence Case 1 occurs only if ` D 1, i.e., for

s a1 j s, and then

k .ps ;D/ D ˆ.ps ;D.pa1 // D ˆ.ps ;D/ D ps .ps;D/; (4.55)

which is the largest eigenvalue. This reflects the phenomenon that the largest eigen-value has multiplicity greater than 1 if a1 > 0, that is, the elements of D are not co

prime or, equivalently, ICG.n;D/ is disconnected (see Remark 4.17 (i)). More pre-

cisely, we have for each j D s u, 0 u a1, exactly '.pu/ integers k D pj mwith p − m and 1 k ps . Hence the multiplicity of the largest eigenvalue

ˆ.ps ;D/ is preciselyPa1

uD0 '.pu/ D pa1 . By (4.55) we know that ˆ.ps ;D/ Dˆ.ps ;D.pa1 //, and since a1 D s r C 1 in D D fpsrC1; psrC2; : : : ; ps1; psg,the asserted formulas for ˆ.ps ;D/ in (ii) follow.

By the argument above, eigenvalues other than the largest one can only appear inCase 2. For D

D fps

r

C1; ps

r

C2; : : : ; ps

1; ps

gand j

D r

`

1 (1

`

r ),

we obtain by (4.54)

k.ps ;D/ D ˆ.ps ;D.psrC`C1// pr`1 Dr`1X

iD0

'.pi / pr`1 D 0:

This proves ƒ.ps ;D/ D 0.

To complete the proof of (ii), it remains to show that ƒ.ps ;D/ ¤ 0 for any non-

regular setD. SuchadivisorsetcanbewrittenasD D fpa1 ; : : : ; pa` ; pa`C1 ; : : : ; par g



182 Jurgen Sander

with 0 a1 < a2 < < ar D s, a`C1 a` 2, and aiC1 ai D 1 for some

1 ` r 1 and all i D ` C 1 ; : : : ; r 1. Then for all k D psa`1m, p − m, i.e.

j

Ds

a`

1, have by (4.54) in Case 2

k.ps ;D/ D ˆ.ps ;D.pa`C1 // psa`1 DrX

iD`C1

'.psai / psa`1 < 0;

(4.56)hence ƒ.ps ;D/ ¤ 0.

It remains to verify (iii). Since D is not uni-regular, we have negative eigenvalues

by (4.56), and Lemma 4.23 (ii) proves E .ps ;D/ > ps .

It was shown in Œ40, Theorem 3.1, that for a prime power ps

E min.ps / D 2ps1 1p: (4.57)

Observe that the minimum is extended over divisor sets D D.ps /, i.e., over

loopless graphs. Moreover, the ps -minimal divisor sets were identified precisely asthe singleton sets D D fpj g with 0 j s 1. For our purpose we shall require

a corresponding result for graphs ICG.ps;D/ containing loops, that is with ps 2 D.

Proposition 4.25. Let ps be a prime power. Then

OE min.ps

/ WD minfE .ps

;D/ W ps

2 D D.ps

/g D ps

; (4.58)

where the minimising divisor sets are exactly the uni-regular ones.

The reader might notice that (4.57) implies OE min.ps / E min.ps/, where equality

only holds in case p D 2.

Proof of Proposition 4.25. For r D 1, the assertion follows immediately from

Proposition 4.24(i). Hence assume that r 2. By Proposition 4.24(iii), it suf-fices to show that we have E .ps ;D/ D ps for each uni-regular divisor set D D

fpa1 ; pa1

C1; : : : ; ps

1; ps

g. We know from Proposition 4.24(ii) that ICG.n;D/ has

only two different eigenvalues, namely ˆ.ps ;D/ D psa1 with multiplicity pa1 and

0 (consequently with multiplicity ps pa1 ). Therefore, E .ps;D/ D pa1 psa1 Dps , as required.

Proof of Theorem 4.16. Let D D.n/ be a multiplicative set such that E .n;D/ DQE min.n/. Then D D Qt

iD1D.i / for certain divisor sets D.i / D.p

si

i / (1 i t ),

and QE min.n/ D QtiD1 E .p

si

i ;D.i // by [28], Corollary 4.1(ii). By the minimality of E .n;D/ it follows from [40], Theorem 3.1, and our Proposition 4.25 that for 1 i t

E .psi

i ;D.i /

/ D 2psi

i .1

1pi

/; if psi

i

…D.i /,

psi

i ; if psi

i 2 D.i /,



Bibliography 183

where D.i / is either a singleton set fpui

i g for some 0 ui si 1, or a uni-regular

set containing psi

i , respectively. This yields

QE min.n/ D n

tYiD1

psii …D.i/

21 1

pi; (4.59)

and our assumption n … D implies that psi

i … D.i / for at least one i . Under this

restriction, and since p1 is the smallest of the primes involved, it is easily seen that

the right-hand side of (4.59) becomes minimal if ps1

1 … D.1/ and psi

i 2 D.i / for

2 i t with the corresponding divisor sets D.i / just mentioned.

We confined our study of the energies of integral circulant graphs to the rather

restricted class having multiplicative divisor sets. Yet, somewhat unexpectedly, this

led to good bounds for E min.n/ and E max.n/. On top of that, the results obtainedby the study of multiplicative divisor sets, combined with some numerical evidence,encourage us to make the following two conjectures.

Conjecture 4.26. For each integer n 2 , we have E min.n/ D 2n

1 1p1

, where p1

denotes the smallest prime factor of n.

Conjecture 4.27. Let n 2 be an arbitrary integer. Then E .n;D/ D E min.n/ implies

that D is a multiplicative divisor set.

Observe that Conjecture 4.26 is a consequence of Conjecture 4.27 by Theorem

4.16.In 2005 it was conjectured by So that, given a positive integer n, two graphs

ICG.n;D1/ and ICG.n;D2/ are cospectral, that is Spec.ICG.n;D1// DSpec.ICG.n;D2//, if and only if D1 D D2. For n D ps this follows immediatelyfrom (4.27) by straightforward comparison of the largest eigenvalues ˆ.ps ;D1/ and

ˆ.ps ;D2/ of the two graphs. So’s conjecture was also confirmed for the slightly more

general case n D ps qt with primes p ¤ q and t 2 f0; 1g (cf. [10] for details), butits proof required the study of eigenvalues other than the largest one as well as their

multiplicities. For arbitrary positive integers n and arbitrary divisor sets the problem

is still open. Therefore, we suggest to study the following weaker conjecture, which

might be more accessible.

Conjecture 4.28. Let n be a positive integer, and let D1;D2 D.n/ be two multi-

plicative sets. If ICG.n;D1/ and ICG.n;D2/ are cospectral, then D1 D D2.

Bibliography

[1] O. Ahmadi, N. Alon, I. F. Blake and I. E. Shparlinski, Graphs with integral spectrum. Linear

Algebra Appl. 430 (2009), 547–552.[2] N. Alon, Eigenvalues and expanders. Combinatorica, 6 (1986), 83–96.



184 Jurgen Sander

[3] R. J. Angeles-Canul, R. Norton, M. Oppermann, C. Paribello, M. Russel and C. Tamon, Per-

fect state transfer, integral circulants and join of graphs. Quantum Inf. Comput. 10 (2010),

325–342.

[4] T. M. Apostol, Introduction to Analytic Number Theory, Springer Verlag, Berlin –Heidelberg – New York, 1976.

[5] H. Bass, The Ihara-Selberg zeta function of a tree lattice. Intl. J. Math. 3 (1992), 717–797.

[6] F. Bien, Construction of telephone networks by group representations. Notices Am. Math. Soc.

36 (1989), No. 1, 5–22.

[7] N. L. Biggs, Algebraic Graph Theory. 2nd edition, Cambridge University Press, Cambridge,

1993.

[8] R. B. Boppana, Eigenvalues and graph bisection: an average case analysis. In Proc. 28th Ann.

Symp. Found. Comp. Sci., IEEE 1987, 280–285.

[9] R. A. Brualdi, Energy of a graph. AIM Workshop Notes, 2006.[10] C. F. Cusanza, Integral circulant graphs with the spectral ´ Ad´ am property. Master’s Thesis.

Paper 2875. http://scholarworks.sjsu.edu/etd theses/2875

[11] P. J. Davis, Circulant Matrices. John Wiley & Sons, New York – Chichester – Brisbane, 1979.

[12] R. Diestel, Graph Theory. Springer-Verlag, New York, 1997.

[13] A. Droll, A classification of Ramanujan unitary Cayley graphs. Electron. J. Combin 17 (2010),

#N29.

[14] C. Godsil and G. Royle, Algebraic Graph Theory. Vol. 207 of Graduate Texts in Mathematics,

Springer-Verlag, New York, 2001.

[15] I. Gutman, The energy of a graph. Ber. Math.-Stat. Sekt. Forschungszent. Graz 103, 1978.[16] F. Harary and A. J. Schwenk, Which graphs have integral spectra? In Graphs and Combina-

torics (Ed. R. Bari and F. Harary). Berlin: Springer-Verlag, pp. 45–51, 1974.

[17] G. H. Hardy, E. M. Wright and J. H. Silverman, An Introduction to the Theory of Numbers.

Oxford University Press, Oxford, 2008.

[18] K. Hashimoto, Zetas functions of finite graphs and representations of p-adic groups. Ad-

vanced Studies in Pure Math., vol. 15, Academic Press, New York, 1989, 211–280.

[19] Sh. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications. Bull. Amer.

Math. Soc. 43 (2006), no. 4, 439–561.

[20] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge University Press, 1991.

[21] Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields.

J. Math. Soc. Japon. 18 (1966), 219–235.

[22] A. Ilic, Distance spectra and distance energy of integral circulant graphs. Linear Algebra Appl.

433 (2010), 1005–1014.

[23] A. Ivic, The Riemann Zeta Function: The Theory of the Riemann Zeta-Function with Appli-

cations. John Wiley & Sons Inc., New York, 1985.

[24] H. Iwaniec and E. Kowalski, Analytic Number Theory. AMS Publications, Providence, 2004.

[25] M. Kotani and T. Sunada, Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo 7 (2000),

7–25.

[26] T. Y. Lam and K. H. Leung On vanishing sums of roots of unity. J. Algebra 224 (2000),91–109.

http://scholarworks.sjsu.edu/etd_theses/2875

http://scholarworks.sjsu.edu/etd_theses/2875




[27] S. Lang, Algebraic Number Theory. Addison-Wesley, Reading MA, 1968.

[28] T. A. Le and J. W. Sander, Convolutions of Ramanujan sums and integral circulant graphs.

Intl. J. Number Theory 7 (2012), 1777–1788.

[29] T. A. Le and J. W. Sander, Extremal energies of integral circulant graphs via multiplicativity.

Lin. Algebra Appl. 437 (2012), 1408–1421.

[30] T. A. Le and J. W. Sander, Integral circulant Ramanujan graphs of prime power order. Elec-

tronic J. Combinatorics 20(3) (2013), #P35, 12pp.

[31] A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs. Combinatorica 8 (1988), 261–277.

[32] R. B. Mallion, Some chemical applications of the eigenvalues and eigenvectors of certain

finite, planar graphs. In Applications of Combinatorics, R. J. Wilson (Ed.), Shiva Publishing

Ltd., Nantwich, Cheshire (England, United Kingdom), 1982, 87–114.

[33] S. J. Miller and R. Takloo-Bigash, An Invitation to Modern Number Theory. Princeton Uni-

versity Press, Princeton, 2006.

[34] M. Morgenstern, Existence and explicit constructions of q C 1 regular Ramanujan graphs for

every prime power q . J. Combinatorial Theory, Series B 62 (1994), 44–62.

[35] M. R. Murty, Ramanujan graphs. J. Ramanujan Math. Soc. 18 (2003), 33–52.

[36] W. Narkiewicz, On a class of arithmetical convolutions. Colloq. Math. 10 (1963), 81–94.

[37] S. J. Patterson, An Introduction to the Theory of the Riemann Zeta Function. Cambridge Uni-

versity Press, Cambridge, 1988.

[38] M. Rosen, Number Theory in Function Fields. Springer-Verlag, New York, 2002.

[39] J. W. Sander, Integral circulant Ramanujan graphs via multiplicativity and ultrafriable inte-

gers, Linear Algebra Appl. 477 (2015), 21–41.

[40] J. W. Sander and T. Sander, The energy of integral circulant graphs with prime power order.

App. Anal. Discrete Math. 5 (2011), 22–36.

[41] J. W. Sander and T. Sander, The exact maximal energy of integral circulant graphs with prime

power order. Contr. Discr. Math. 8 (2013), 19–40.

[42] N. Saxena, S. Severini and I. E. Shparlinski, Parameters of integral circulant graphs and peri-

odic quantum dynamics. Int. J. Quantum Inf. 5 (2007), 417–430.

[43] J.-P. Serre, Trees. Springer-Verlag, New York, 1980.

[44] W. Schwarz and J. Spilker, Arithmetical Functions. Lond. Math. Soc., Lect. Notes Math. 184,

Cambridge University Press, 1994.

[45] I. Shparlinski, On the energy of some circulant graphs. Linear Algebra Appl. 414 (2006),

378–382.

[46] W. So, Integral circulant graphs. Discrete Math. 306 (2005), 153–158.

[47] T. Sunada, Riemannian coverings and isospectral manifolds. Ann. Math. 121 (1985), 169–186.

[48] A. Terras, Zeta Functions of Graphs. Cambridge Studies in Advanced Mathematics, Cam-

bridge University Press, Cambridge, 2011.



Index

acyclic, 127adjacency matrix, 124, 130, 132, 133,

137, 141, 148, 151–153, 159,

180adjacent, 14, 124

algebraic, 5, 14, 31, 34, 37, 38, 45–49,

51–56, 59, 63, 64, 66, 68–72,124, 130, 139, 159

almost holomorphic modular form, 26

almost mock modular form, 26

Andrews-Dragonette Conjecture, 20average winding number, 95

backtrack, 137, 139, 140, 151Bernoulli number, 3

bipartite, 129, 130, 132–134

Cayley graph, 124, 125, 129, 135, 159,

160, 162

circle of convergence, 140circulant, 124, 125, 128, 134, 135, 159,

160, 162, 166, 167, 171–176,

183

circulant graph, 124, 125, 128, 134, 135,

159, 162, 171, 173, 174

circulant matrix, 134

clique, 127closed walk, 127

complete graph, 125, 128, 134, 136, 143,

149completely multiplicative, 97, 110, 111,

114, 117, 119

complexity, 34, 36, 37, 68, 69, 71

component, 95, 127, 128, 133connected, 127–129, 132–134, 136, 155,

161, 162, 167, 175

continuant, 42

continued fraction, 31, 33, 34, 36–38, 40,43, 44, 56, 61, 68, 69

continued fraction expansion, 31, 33, 34,36–38, 40, 43, 44, 69

convergent, 7, 39, 41, 43–45, 63, 64, 66,82, 90, 95, 99

convolution, 78, 110, 161

cospectral, 131, 183

crank, 20

cusp form, 2, 5–7, 19

cycle, 127, 129, 136–141, 143, 151

cycle graph, 124, 125, 128, 134, 136,

139, 141

cyclic, 127, 134, 135

de Bruijn word, 39

Dedekind -function, 12

degree, 34, 38, 45, 46, 49, 51, 56, 59, 63,

66, 68–70, 72, 128, 129, 136,

143

directed, 124, 130, 132, 136, 137, 140,

141, 143, 153

Dirichlet inverse, 118

disconnected, 126–129, 181dominating prime power factor, 171, 172

edge, 124, 125, 127, 128, 132, 135–137,141, 143, 148, 149, 151–153,

159

edge adjacency matrix, 137, 141, 148,

151–153, 159

eigenvalue, 5, 79, 118–120, 131–135,146–148, 152, 155, 159–163,

173, 180–183Eisenstein series, 2–6, 9, 18

energy, 132, 159, 173–176

equivalent, 137, 138, 141

essential spectrum, 82

Euler product formula, 97

expander graph, 135, 158

Farey fractions, 14

Fourier coefficient, 2, 4–7, 9, 11, 21, 25,

27, 78, 80, 81, 83, 89Fourier expansion, 2–4, 6, 9, 23, 28



188 Index

Fredholm, 82–84

friendly, 136, 138, 139, 143, 145, 147,

148, 151, 152, 155

Frobenius norm, 151fundamental domain, 6–8

fundamental group, 141–143, 147

gamma function, 10, 19

incomplete, 19

graph, 110, 123–139, 141, 144, 147, 149,151–155, 158–160, 162, 166–

173, 175, 176, 182, 183

Cayley, 124–126, 128–130, 135,

159, 160, 162circulant, 124–126, 128, 134, 135,

159, 160, 162, 171, 173, 174

complete, 125, 128, 134, 136, 143,

149

cycle, 124, 125, 128, 134, 136, 139,

141

expander, 135, 158

integral, 135

integral circulant, 135, 159, 160,

162, 167isomorphic, 131

Ramanujan, 135, 145, 147, 158,

159, 162, 167, 172

Hankel loop integral formula, 17

Hankel maTrices, 83

Hecke bound, 5

holomorphic modular form, 2, 4

Hurwitz class number, 18

hyperbolic Laplacian, 17

Ihara three-term determinant formula,

141, 143

Ihara zeta function, 136, 138, 139, 145,147

infinite word, 34–37, 39, 48, 50, 51, 69

integral circulant graph, 135, 159, 160,

162, 167

integral graph, 135

isomorphic graph, 131isospectral, 131

J -Bessel function, 10, 13

k-regular, 128, 132, 134, 135

Kloosterman sums, 9, 13, 24, 27

Laurent matrix, 81

leaf, 128

length of the prime, 137

Lindelof function, 92

Lindelof Hypothesis, 92, 94, 109, 117

Liouville’s Theorem, 46

Lipschitz summation formula, 4

loop, 17, 124, 125, 130, 141, 154, 161,

175, 180, 182loopfree, 124, 132

Maass form, 18–22, 25, 26

almost harmonic, 26

Maass-Poincare series, 21, 22

matrix exponential, 149, 150

matrix logarithm, 150, 151

mean motion, 95

meromorphic at the cusps, 1

mirror formula, 41mixed mock modular form, 26, 27

mock modular form, 17, 19, 25

mock theta function, 19–21, 26

modular form, 1, 2, 4, 7, 8, 11, 12, 17–19,25–27

almost holomorphic, 26

almost mock, 26

holomorphic, 2, 4

mixed mock, 2, 25–27

mock, 17, 19, 25weakly holomorphic, 11, 18

multicyclic, 136

multigraph, 124, 125

multiple edges, 124

multiplicative, 9, 78, 79, 91, 95, 97, 99,

100, 102, 103, 110, 111, 114,117, 119, 129, 161, 162, 171–

176, 182, 183

multiplicative set, 162, 174–176, 182,

183multiplicative setting, 99



Index 189

Multiplicative Toeplitz Matrices, 78, 95

Nebentypus character, 22

neighbor, 124normal number, 32, 33

operator norm, 80, 86

order, 10, 13, 14, 27, 34, 39, 92, 93, 101–103, 105, 115, 117, 124, 125,

135, 138, 141, 147, 159, 162,

166, 171, 172, 174–178, 181orientation, 136–139, 141, 143, 151–153

palindrome, 63

Parseval’s identity, 90partial quotient, 31, 33, 34, 36, 37, 41,

44, 51, 64, 66partition, 2, 11–13, 18, 20, 21, 25

without sequences, 25

path, 11, 14, 127, 132, 137, 138, 140,141, 151

paths, 16

period integral, 21Perron–Frobenius theorem, 133, 148

Petersson coefficient formula, 7Petersson inner product, 6, 7, 9

Petersson slash operator, 3

planar, 131Poincare series, 6–9, 18, 23, 25

Poisson summation formula, 10

prime, 33, 44, 46, 47, 50, 52–54, 67,88, 92, 97, 100, 102–104, 110–

114, 135, 137–140, 155, 158,

159, 162, 163, 165–169, 171–

175, 177–183prime counting function, 136, 138Prime Number Theorem, 100, 102, 104,

113, 117, 136, 138, 155

prime path, 137, 138primitive path, 137

principal part, 18, 28

quotient, partial, 31, 33, 34, 36, 37, 41,44, 51, 64, 66

radius of convergence, 140, 143, 144,147, 148, 155

Ramanujan graph, 135, 145, 158, 159,

162, 167, 172

Ramanujan property, 136

Ramanujan sum, 135, 160, 163, 177Ramanujan-Petersson Conjecture, 5

Redheffer’s matrix, 78

regular, 124, 128, 129, 132, 135, 144,

147, 160, 161

regularized inner product, 7, 19

Ridout’s Theorem, 46

Riemann Hypothesis, 78, 79, 92, 136,145, 147, 150, 156, 158, 159

Riemann zeta function, 3, 77, 91, 99, 136,

145, 147Riesz–Fischer Theorem, 90

Roth’s Theorem, 46

Salie sum, 24

Schmidt Subspace Theorem, 47

Schur norm, 151

semi-regular, 161, 174, 175

sequence space, 79, 85

shadow, 19, 27

simple, 124, 127, 132spanning subgraph, 124–126

spanning tree, 128, 143

spectrum, 82, 124, 131, 132, 150, 159,162

Sturmian word, 36

subgraph, 124–128

symmetric, 1, 124, 129, 130, 133, 145

Toeplitz Matrices, 78, 80, 85, 95

transcendental, 14, 31, 38, 39, 48–50, 62,63, 66

tree, 124, 127–129, 141, 143

two-term determinant formula, 140, 149,

151, 156

ultimately periodic, 34–38, 43, 48, 55,

56, 63, 68

undirected, 124, 129, 130, 132, 136

unitary, 129, 159, 162

Valence formula, 8



190 Index

vertices, 124, 125, 127, 128, 131, 133,

136, 140, 141, 149, 150

walk, 127weakly holomorphic modular form, 11,

18

Whittaker function, 23, 24

Wiener Algebra, 82, 90

Wiener–Hopf factorization, 84, 97

winding number, 82, 95

average, 95



Kathrin Bringmann Yann BugeaudTitus HilberdinkJürgen Sander

Four Faces of Number Theory

Thi b k i f i t th I t ti l S S h l

(ems series of lectures in mathematics) kathrin bringmann, yann bugeaud, titus hilberdink, jurgen...

Documents