aspects of analytic number theory: the universality of the...

Computational Methods and Function TheoryVolume 00 (0000), No. 0, 1–? CMFT-MS XXYYYZZ

Aspects of Analytic Number Theory:

The Universality of the Riemann Zeta-Function

Jorn Steuding

Abstract. These notes deal with Voronin’s universality theorem which states,roughly speaking, that any non-vanishing analytic function can be uniformlyapproximated by certain shifts of the Riemann zeta-function. We start witha brief introduction to the classical theory of the zeta-function. Then we givea self-contained proof of the universality theorem. We conclude with severalinteresting applications of this remarkable property and discuss some relatedproblems and extensions.

Keywords. Riemann zeta-function, universality, value-distribution.

2000 MSC. 11M06, 11M26, 11M99, 30E10.

-4

-2

0

2

4

-100

1020

3040

0

1

2

3

4

0

Figure 1. The reciprocal of the absolute value of ζ(σ + it) for σ ∈[−4, 4], t ∈ [−10, 40]. The zeros of ζ(s) appear as poles.

Version CMFT-Workshop January 2008, Guwahati.I wish to express my gratitude to the organizers of the workshop Computational Methods

and Function Theory at Guwahati, Assam, in particular Meenaxi Bhattacharjee and StephanRuscheweyh. I am also very grateful to the audience for their interest in the topic and for theextraordinarily good atmosphere at Guwahati as well as to the anonymous referee for her orhis remarks how to make the content more accessible to the reader. Last but not least, I wantto thank my wife Rasa for her careful reading of the script.

ISSN 1617-9447/$ 2.50 c© 20XX Heldermann Verlag

2 Jorn Steuding CMFT

The theme of this course is an astonishing approximation property of the famous Rie-mann zeta-function, so the topic is settled in the intersection of complex analysis andanalytic number theory. Arithmetical problems may often sound simple in their formu-lation; however, their treatment often needs sophisticated machinery and challengingideas. Since the path-breaking works of Dirichlet and Riemann from the middle ofthe nineteenth century, analytic methods have become an important tool in numbertheory. The proof of the celebrated prime number theorem by investigating the distri-bution of zeros of the zeta-function is just one example. One of the most spectacularproperties of the zeta-function is Voronin’s universality theorem which states that anynon-vanishing analytic function can be uniformly approximated by certain shifts ofthe zeta-function. Here we give a (more or less) complete proof of this remarkableresult and discuss some of its applications, e.g., hypertranscendence and a criterion forthe truth of the famous yet unproved Riemann hypothesis. Finally, we discuss someextensions and related open problems.

These self-contained lecture notes are mainly based on the original paper of Voronin[67], resp. the presentation in the monograph [33] of Karatsuba & Voronin with slightmodifications. Thanks to Bagchi [1], Reich [56], and Laurincikas [36], there is another,more sophisticated probabilistic approach to universality which allows slightly moregeneral results. For the sake of simplicity we have chosen the down to earth approachof Voronin. Many of the additional results can be found in [61]. For the backgroundin zeta-function theory (and for help with respect to the exercises) we refer to theclassical monograph [63] of Titchmarsh and the online notes [62].

Jorn Steuding, Wurzburg, February 2009.

Contents

1. Introduction 3

1.1. The Riemann zeta-function is universal 3

1.2. Survey on value-distribution theory 5

1.3. A weak approximation theorem 9

2. Zeta-function theory 11

2.1. Primes and zeros 11

2.2. The approximate functional equation 16

2.3. The functional equation 23

2.4. The mean-square and applications 26

2.5. A density theorem 31

2.6. The prime number theorem 35

3. Universality theorems 42

00 (0000), No. 0 The Universality of the Riemann Zeta-Function 3

3.1. Voronin’s universality theorem 42

3.2. Rearrangement of conditionally convergent series 43

3.3. Finite Euler products 48

3.4. Diophantine approximation 55

3.5. Approximation in the mean — end of proof 59

3.6. Reich’s discrete universality theorem and other related results 63

4. Applications, extensions, and open problems 66

4.1. Functional independence 66

4.2. Self-recurrence and the Riemann hypothesis 69

4.3. The effectivity problem 72

4.4. L-functions and joint universality 78

4.5. The Linnik-Ibragimov conjecture 84

References 87

1. Introduction

Here we introduce the main actor, the Riemann zeta-function, and present first clas-

sical results on its amazing value-distribution due to Bohr as well as the remarkable

universality theorem of Voronin. For historical details we refer to [61].

1.1. The Riemann zeta-function is universal. The Riemann zeta-functionis a function of a complex variable s = σ + it,∗ for σ > 1 given by

ζ(s) =∞∑

n=1

1

ns=∏

p

(

1 − 1

ps

)−1

;(1.1)

here and in the sequel the letter p always denotes a prime number and theproduct is taken over all primes. The series and the product are prototypes ofso-called Dirichlet series, resp. Euler products. They both converge absolutelyin the half-plane σ > 1 and uniformly in each compact subset. The identitylinking both, the series and the product was discovered by Euler in 1737 and canbe regarded as an analytic version of the unique prime factorization of integers.The Euler product gives a first glance on the intimate connection between thezeta-function and the distribution of prime numbers. An immediate consequenceis Euler’s proof of the infinitude of the primes. Assuming that there were onlyfinitely many primes, the product in (1.1) is finite, and therefore convergent fors = 1, contradicting the fact that the Dirichlet series defining ζ(s) reduces tothe divergent harmonic series as s → 1+. Hence, there exist infinitely many

∗This mixture of latin and greek letters is tradition in analytic number theory.


prime numbers. This fact is well known since Euclid’s elementary proof, but theanalytic access gives deeper knowledge on the distribution of the prime numbersas we shall see in the second chapter.

However, the main theme of these notes is a remarkable approximation propertyof Riemann’s zeta-function, called universality.

By Weierstrass’ celebrated approximation theorem we know that any continu-ous function, defined on a closed interval, can be uniformly approximated bypolynomials. The set of continuous functions is rather big whereas the set ofpolynomials is comparably small. This makes the Weierstrass theorem remark-able. One may not believe that it is possible to approximate any continuousfunction on a bounded interval by a single function! Actually, the Riemann zeta-function has this astonishing approximation property! More precisely, shifts ofits logarithm s 7→ log ζ(s+ iτ) can approximate any continuous function definedon a bounded interval. Of course, this approximation cannot be realized in thehalf-plane of absolute convergence of the zeta defining series. For our purposewe note that our protagonist, ζ(s), can be analytically continued to the wholecomplex plane except for a simple pole at s = 1, e.g.,

(1.2) ζ(s) = (1 − 21−s)−1

∞∑

n=1

(−1)n+1

ns;

here the series on the right converges for σ > 0 (see also Exercise 1 below).Voronin’s famous universality theorem states that any non-vanishing analyticfunction g can be approximated uniformly by certain purely imaginary shifts ofthe zeta-function in the vertical strip 1

2< σ < 1. (A precise formulation will

be given below.) For instance, for any positive ǫ, there exists a real number τ ,which might be extremely large, such that the inequality

|ζ(s+ 34

+ iτ) − g(s)| < ǫ

holds on any disk |s| ≤ r, where 0 < r < 14

is fixed. For an illustrative exampletake g = exp and see the figure in Section 4.3. The same statement holds whenthe closed disk is replaced by a closed line segment on the imaginary axis andin this case we only need that the function g is continuous and has no zeros.This follows from a slightly more advanced version of Voronin’s theorem (seeTheorem 3.11). And if we want to get rid of the non-vanishing assumption,we can approximate by the logarithm of the zeta-function and this leads tothe aforementioned extension of Weierstrass’ approximation theorem. Anotherapplication of universality is related to the famous Riemann hypothesis, oneof the seven millennium problems, about the distribution of zeros of the zeta-function (Theorem 4.3).

The first universal object in the mathematical literature was discovered by Feketein 1914/15; he proved the existence of a real power series with the propertythat for any continuous function g on the unit interval, there exists a sequence


of partial sums which approximates g uniformly. In 1926, G.D. Birkhoff [5]proved the existence of an entire function f with the property that to any givenentire function g, there exists a sequence of complex numbers an such that f(z+an) → g(z) uniformly on compacta in C, as n → ∞. Universality is a frequentphenomenon in analysis, often appearing when analytical processes diverge orbehave irregularly in some sense. The Riemann zeta-function and its relativesare so far the only known explicit examples of universal objects. In the followingsection we shall give a precise statement; however, we start with a brief look howthis surprising and deep result has been developed.

1.2. Survey on value-distribution theory. The zeros of the zeta-functionare of special interest (for reasons we will explain in the following chapter). Itseems rather difficult to localize zeros or any other concrete values taken bythe zeta-function, whereas it is much easier to study how often the values liein a given set. Having this idea in mind, Harald Bohr refined former studieson the value-distribution of the Riemann zeta-function by applying diophantine,geometric, and probabilistic methods.

In the half-plane of absolute convergence σ > 1 we have

(1.3) 0 < |ζ(s)| ≤ ζ(σ).

Thus the values of ζ(s) in half-planes σ ≥ σ0 > 1 are lying in the disk of radiusζ(σ0) centered in the origin. It can be shown that ζ(s) assumes quite many ofthe complex values inside this disk when t varies in R. On the other side ζ(σ)tends to infinity as σ → 1+, and indeed Bohr [6] succeeded in proving that inany strip 1 < σ < 1 + ǫ, ζ(s) takes any non-zero value infinitely often. Wesketch his argument. Define log ζ(s) for any s ∈ C by choosing the principalbranch of the logarithm on the intersection of the real axis with the half-plane ofabsolute convergence, and for other points s = σ+ it let log ζ(σ+ it) be the valueobtained from log ζ(2) by continuous variation along the line segments [2, 2 + it]and [2 + it, σ + it], provided that the path does not cross a zero or pole of ζ(s);if it does, then take log ζ(σ + it) = limǫ→0+ log ζ(σ + i(t+ ǫ)). For σ > 1,

log ζ(s) = −∑

p

log

(

1 − 1

ps

)

= −∑

p

∞∑

k=1

1

kpsk.

For any fixed prime p and σ > 1, the set of values taken by the inner sum inthe series representation on the right-hand side is a convex curve while t runsthrough R. Adding up all these curves and using some facts from the theory ofdiophantine approximation, it follows that log ζ(s) takes any complex value in1 < σ < 1 + ǫ which leads to Bohr’s result.


The situation to the left of the abscissa of convergence is much more complicated.Here, Bohr studied finite Euler products

ζM(s) :=∏

p≤M

(

1 − 1

ps

)−1

.

As M tends to infinity, these products do not converge any longer but they ap-proximate ζ(s) in the mean (we will meet this ingenious idea later again). Thevalue-distribution of finite Euler products is treatable by the theory of diophan-tine approximation, and by their approximation property this leads to informa-tion on the values taken by the zeta-function. In a series of papers Bohr andhis collaborators discovered that the asymptotic behaviour of ζ(s) is ruled byprobability laws on every vertical line to the right of σ = 1

2. In particular, Bohr

& Courant [8] proved that for any fixed σ ∈ (12, 1] the set of values ζ(σ+ it) with

t ∈ R lies dense in the complex plane. Later, Bohr refined these results signifi-

-2 -1 1 2 3

-2

-1

1

2

Figure 2. ζ(3

5+ it) for t ∈ [0, 60]. The curve visits any neighbourhood of

any point in the complex plane as t runs through the set of real numbers, and

so the picture would be completely black in the end.

cantly by applying probabilistic methods. Let R be an arbitrary fixed rectanglein the complex plane whose sides are parallel to the real and the imaginary axes,and let G be the half-plane σ > 1

2where all points are removed which have the

same imaginary part as and smaller real part than one of the possible zeros ofζ(s) in this region. Then a remarkable limit theorem due to Bohr & Jessen [9, 10]states that for any fixed σ > 1

2the limit

limT→∞

1

Tmeas τ ∈ [0, T ] : σ + iτ ∈ G, log ζ(σ + iτ) ∈ R

exists. Here and in the sequel measA stands for the Lebesgue measure of ameasurable set A. This limit value may be regarded as the probability howmany values of log ζ(σ + it) belong to the rectangle R. Next, for any complexnumber c, denote by Nc(σ1, σ2, T ) the number of c-values of ζ(s), i.e., the roots


of the equation ζ(s) = c, inside the region σ1 < σ < σ2, 0 < t ≤ T (countingmultiplicities). From the limit theorem mentioned above Bohr & Jessen deduced

Theorem 1.1. Let c be a complex number 6= 0. Then, for any σ1 and σ2 satis-fying 1

2< σ1 < σ2 < 1, the limit limT→∞

1TNc(σ1, σ2, T ) exists and is positive.

In 1935, Jessen & Wintner proved limit theorems similar to the one above byusing more advanced methods from probability theory (infinite convolutions ofprobability measures). We do not mention further developments of Bohr’s ideasby his successors Borchsenius, Jessen, and Wintner but refer for more details onBohr’s contribution and results of his collaborators to the monograph of Lau-rincikas [36] and the survey of Matsumoto [48]. Bohr’s line of investigationsappears to have been almost abandoned for some time. Only in 1972, Voronin[66] obtained some significant generalizations of Bohr’s denseness result.

Theorem 1.2. For any fixed numbers s1, . . . , sn with 12< Re sk < 1 for 1 ≤ k ≤

n and sk 6= sℓ for k 6= ℓ, the set

(ζ(s1 + it), . . . , ζ(sn + it)) : t ∈ Ris dense in Cn. Moreover, for any fixed number s with 1

2< σ < 1,

(ζ(s+ iτ), ζ ′(s+ iτ), . . . , ζ (n−1)(s+ iτ)) : τ ∈ Ris dense in Cn.

What about the value-distribution of the zeta-function on the line σ = 12? It

is conjectured but yet unproved that also the set of values of ζ(s) taken on thisvertical line is dense in C. However, Garunkstis & Steuding [18] have shown thatthe second statement of Theorem 1.2 is false for σ = 1

2whenever n ≥ 2. Selberg

(unpublished) proved that the values taken by an appropriate normalization ofthe Riemann zeta-function on this line are normally distributed: let R be anarbitrary fixed rectangle in the complex plane whose sides are parallel to the realand the imaginary axes, then

limT→∞

1

Tmeas

t ∈ (0, T ] :log ζ

(

12

+ it)

√

12log logT

∈ R

=1

2π

∫∫

Rexp

(

−12(x2 + y2)

)

dx dy.

The value-distribution on the line σ = 12

is somehow special for several reasons(more about that in the following chapter).

In 1975, Voronin [67] proved his remarkable universality theorem:

Theorem 1.3. Let 0 < r < 14

and suppose that g(s) is a non-vanishing continu-ous function on the disk |s| ≤ r, which is analytic in the interior. Then, for any


ǫ > 0, there exists a positive real number τ such that

max|s|≤r

∣

∣ζ(

s + 34

+ iτ)

− g(s)∣

∣ < ǫ.

Moreover, the set of such τ has positive lower density:

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max|s|≤r

∣

∣ζ(

s+ 34

+ iτ)

− g(s)∣

∣ < ǫ

> 0.

Thus, any suitable target function can be approximated as good as we pleasean infinity of times. We say that ζ(s) is universal since appropriate shifts ap-proximate uniformly any element of a huge class of functions. We may interpretthe absolute value of an analytic function as an analytic landscape over thecomplex plane. Then the universality theorem states that any finite analyticlandscape can be found (up to an arbitrarily small error) in the analytic land-scape of the Riemann zeta-function. This is indeed a remarkable property of thezeta-function!

0.5

0.6

0.7

0.8

0.9

1

116118

120122

0

1

2

3

4

0

Figure 3. Some summits of the Himalaya or the analytic landscape of ζ(s)

for σ ∈ [ 12, 1], t ∈ [115, 122].

We shall give a more or less self-contained proof of Voronin’s universality theorem

in Chapter 3. A reader who is familiar with the basic theory of the Riemann zeta-

function may directly jump to Chapter 3; for anybody else we provide the essentials of

this theory in the following chapter. However, for the remaining part of this chapter we

shall investigate a weaker, nevertheless still interesting approximation property than

universality.


1.3. A weak approximation theorem. What might have been Voronin’s in-tention for his studies which had led him to the discovery of this astonishinguniversality property?† One reason for Voronin’s investigations might have beenBohr’s concept of almost periodicity and its applications to the Riemann hypoth-esis (see Section 4.2). Another starting point for Voronin could have been thewish to extend Theorem 1.2; the universality theorem can be seen as an infinitedimensional analogue of the second part of this theorem. To illustrate this, wesketch how Theorem 1.2 can be used to obtain some weak form of the universalitytheorem.

Assume we are given an analytic target function g(s) defined on |s| ≤ r, wherer is a positive real number. Our main tool is the Taylor series expansion

g(s) =

∞∑

k=0

g(k)(0)

k!sk,

valid for all s with |s| ≤ r. By Cauchy’s formula,

g(k)(0) =k!

2πi

∮

|s|=r

g(s)

sk+1ds,

where the integral is taken over the circle |s| = r in counterclockwise direction.Hence,

|g(k)(0)| ≤ k!Mr−k,

where M := max|s|=r |g(s)|. Let δ ∈ (0, 1). Then∣

∣

∣

∣

g(k)(0)

k!sk∣

∣

∣

∣

≤ Mδk for |s| ≤ δr.

For any positive ǫ, there exists a positive integer n such that

(1.4) Σ1 :=

∣

∣

∣

∣

∣

g(s) −∑

0≤k<n

g(k)(0)

k!sk

∣

∣

∣

∣

∣

<ǫ

3for |s| ≤ δr.

By Theorem 1.2 there exists a positive real number τ such that

(1.5) |ζ (k)(34

+ iτ) − g(k)(0)| < ǫ

3for 0 ≤ k < n.

Thus,

Σ2 :=

∣

∣

∣

∣

∣

∑

0≤k<n

ζ (k)(34

+ iτ)

k!sk −

∑

0≤k<n

g(k)(0)

k!sk

∣

∣

∣

∣

∣

<ǫ

3

∑

0≤k<n

(δr)k

k!<ǫ

3exp(δr) for |s| ≤ δr.(1.6)

†Voronin died in 1996.


Of course, we also have the Taylor series expansion

ζ(s+ 34

+ iτ) =∞∑

k=0

ζ (k)(34

+ iτ)

k!sk

for |s| ≤ r. PutM(τ) = max

|s|=r|ζ(s+ 3

4+ iτ)|.

Then, again by Cauchy’s formula,∣

∣

∣

∣

∣

ζ (k)(34

+ iτ)

k!sk

∣

∣

∣

∣

∣

≤ M(τ)δk for |s| ≤ δr.

Hence,

Σ3 :=

∣

∣

∣

∣

∣

ζ(s+ 34

+ iτ) −∑

0≤k<n

ζ (k)(34

+ iτ)

k!sk

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∞∑

k=n

ζ (k)(34

+ iτ)

k!sk

∣

∣

∣

∣

∣

≤M(τ)δn

1 − δfor |s| ≤ δr.(1.7)

Putting (1.4)-(1.7) together, we find

|ζ(s+ 34

+ iτ) − g(s)| ≤ Σ1 + Σ2 + Σ3 <ǫ

3+ǫ

3exp(δr) +M(τ)

δn

1 − δ.

Now choose δ > 0 such that

(1.8) M(τ)δn

1 − δ=ǫ

3(2 − exp(δr));

this is possible since the left-hand side tends to zero as δ → 0, while the right-hand side tends to ǫ

3> 0, resp. when δ → 1 the left-hand side tends to infinity,

but the right-hand side remains finite. We thus have proved

Theorem 1.4. Let g be analytic for |s| ≤ r. Then, for any ǫ > 0 there exist realnumbers τ > 0 and δ = δ(ǫ, g, τ) ∈ (0, 1) such that

max|s|≤δr

|ζ(s+ 34

+ iτ) − g(s)| < ǫ.

It is remarkable that there is no restriction on g to be non-vanishing on the disk|s| ≤ r as in the universality theorem. Indeed, the statement contradicts theRiemann hypothesis if g is not identically vanishing but has a zero inside thedisk |s| ≤ δr (since any such zero would generate via Rouche’s theorem manyzeros of ζ(s+ 3

4+ iτ), as we shall show in a later section). However, it seems that

there is an inner mechanism which prevents to obtain such an extraordinarilygood approximation of the target function. We observe that a small ǫ leads to abig τ and the smaller we have to choose δ via (1.8). This follows from the factthat the zeta-function is unbounded on any vertical line with real part σ < 1and so the quantity M(τ) is increasing to infinity as τ → ∞.


A quantitative version of Theorem 1.4 can be found in the recent article byGarunkstis et al. [17]; the main tool to obtain explicit bounds for the values τand δ is another result of Voronin, a so-called multidimensional Ω-theorem, whichcombines analytic and diophantine approximation properties (see also [33]). Itis believed that Voronin himself was aware about statements like Theorem 1.4.For more details on the history of Voronin’s theorem we refer to the nice surveyarticles of Laurincikas [39] and Matsumoto [49].

Mathematics is not a spectator sport! The following exercises may help to dive deeperinto zeta-function theory. Although ζ(s) is defined as an absolutely convergent seriesin the half-plane σ > 1, the distribution of values taken near the vertical line σ = 1 isinteresting:

Exercise 1. For σ > 0 prove the representation (1.2) and deduce that ζ(s) < 0 fors ∈ (0, 1).

Exercise 2. Prove inequality (1.3). Can you make use of formula (1.2) to estimatethe growth of ζ(σ + it) for fixed σ > 0 as t → ∞?

2. Zeta-function theory

In this chapter we give some hints for the importance of the Riemann zeta-function for

analytic number theory. We start with a survey on the remarkable link between prime

numbers and zeros of ζ(s). Later we prove the prime number theorem as well as density

estimates for the number of hypothetical zeros off the critical line σ = 12 . Besides we

develop parts of the machinery which is needed to prove Voronin’s universality theorem.

For historical details and more references we refer to [53].

2.1. Primes and zeros. It was the young Gauss who conjectured in 1791 forthe number π(x) of primes p ≤ x the asymptotic formula‡

(2.1) π(x) ∼ li(x),

where the logarithmic integral is given by

li(x) = limǫ→0+

∫ 1−ǫ

0

+

∫ x

1+ǫ

du

log u=

∫ x

2

du

log u− 1.04 . . . .

Gauss’ conjecture states that, in first approximation, the number of primes ≤ xis asymptotically x

log x. By elementary means, Chebyshev proved around 1850

that 0.921 . . . ≤ π(x) log xx

≤ 1.055 . . . for sufficiently large x. Furthermore, heshowed that if the limit

limx→∞

π(x)log x

x

‡We write f(x) ∼ g(x), if limx→∞ f(x)/g(x) = 1.


exists, the limit is equal to one, which supports conjecture (2.1). Riemann wasthe first to investigate the Riemann zeta-function as a function of a complexvariable. In his only one but outstanding paper [58] on number theory from1859 he outlined how Gauss’ conjecture could be proved by using the functionζ(s). However, at Riemann’s time the theory of functions was not developedsufficiently far, but the open questions concerning the zeta-function pushed theresearch in this field quickly forward. We shall briefly discuss Riemann’s memoir;some of the sketched results will later be proved in detail.

First of all, by partial summation,

ζ(s) =∑

n≤N

1

ns+N1−s

s− 1+ s

∫ ∞

N

[u] − u

us+1du;(2.2)

here and in the sequel [u] denotes the maximal integer less than or equal to u.This gives an analytic continuation for ζ(s) to the half-plane σ > 0 except for asimple pole at s = 1 with residue 1. This process can be continued to the lefthalf-plane and shows that ζ(s) is analytic throughout the whole complex planeexcept for s = 1. Riemann discovered the functional equation

π− s2 Γ(s

2

)

ζ(s) = π− 1−s2 Γ

(

1 − s

2

)

ζ(1 − s),(2.3)

where Γ(s) denotes Euler’s Gamma-function. In view of the Euler product (1.1)it is easily seen that ζ(s) has no zeros in the half-plane σ > 1. It follows from thefunctional equation and from basic properties of the Gamma-function that ζ(s)vanishes in σ < 0 exactly at the so-called trivial zeros s = −2n with n ∈ N. All

-14 -12 -10 -8 -6 -4 -2

-0.15

-0.1

-0.05

0.05

0.1

Figure 4. ζ(s) in the range s ∈ [−14.5, 0.5].

other zeros of ζ(s) are said to be nontrivial, and we denote them by ρ = β + iγ.Obviously, they have to lie inside the so-called critical strip 0 ≤ σ ≤ 1, and it iseasily seen that they are non-real. The functional equation (2.3) and the identity

ζ(s) = ζ(s) show some symmetries of ζ(s). In particular, the nontrivial zeros of


ζ(s) are distributed symmetrically with respect to the real axis and to the verticalline σ = 1

2. It was Riemann’s ingenious contribution to number theory to point

out how the distribution of these nontrivial zeros is linked to the distribution ofprime numbers. Riemann conjectured the asymptotics for the number N(T ) ofnontrivial zeros ρ = β + iγ with 0 < γ ≤ T (counted according multiplicities).This conjecture was proved in 1895 by von Mangoldt [46, 47] who found moreprecisely§

N(T ) =T

2πlog

T

2πe+O(logT ).(2.4)

Riemann worked with the function t 7→ ζ(12

+ it) and wrote that very likely all

roots t are real, i.e., all nontrivial zeros lie on the so-called critical line σ = 12. This

is the famous, yet unproved Riemann hypothesis which we rewrite equivalentlyas

Riemann’s hypothesis. ζ(s) 6= 0 for σ > 12.

In support of his conjecture, Riemann calculated some zeros; the first one withpositive imaginary part is ρ = 1

2+ i14.134 . . ..¶ Furthermore, Riemann conjec-

tured that there exist constants A and B such that

12s(s− 1)π− s

2 Γ(s

2

)

ζ(s) = exp(A +Bs)∏

ρ

(

1 − s

ρ

)

exp

(

s

ρ

)

,

where the product on the right is taken over all nontrivial zeros (the trivialzeta zeros are cancelled by the poles of the Gamma-factor). This was shownby Hadamard [22] in 1893 (on behalf of his theory of product representationsof entire functions). Finally, Riemann conjectured the so-called explicit formulawhich states that

π(x) +

∞∑

n=2

π(x1n )

n= li(x) −

∑

ρ=β+iγγ>0

(

li(xρ) + li(x1−ρ))

(2.5)

+

∫ ∞

x

du

u(u2 − 1) log u− log 2

for any x ≥ 2 not being a prime power (otherwise a term 12k

has to be added on

the left-hand side, where x = pk); the appearing integral logarithm is defined by

li(xβ+iγ) =

∫ (β+iγ) log x

(−∞+iγ) log x

exp(z)

z + δiγdz,

§We write f(x) = O(g(x)), if lim supx→∞ |f(x)|/g(x) < ∞; equivalently, we also write

f ≪ g.¶In 1932, Siegel published an account of Riemann’s work on the zeta-function found in

Riemann’s private papers in the archive of the university library in Gottingen. It becameevident that behind Riemann’s speculation there was extensive analysis and computation.


where δ = +1 if γ > 0 and δ = −1 otherwise. The explicit formula was provedby von Mangoldt [46] in 1895 as a consequence of both product representationsfor ζ(s), the Euler product (1.1) and the Hadamard product.

Building on these ideas, Hadamard [23] and de la Vallee-Poussin [64] found (in-dependently) in 1896 the first proof of Gauss’ conjecture (2.1), the celebratedprime number theorem. For technical reasons it is of advantage to work with thelogarithmic derivative of ζ(s) which is for σ > 1 given by

ζ ′

ζ(s) = −

∞∑

n=1

Λ(n)

ns,

where the von Mangoldt Λ-function is defined by

(2.6) Λ(n) =

log p if n = pk with k ∈ N,0 otherwise.

A lot of information concerning the prime counting function π(x) can be recov-ered from information about

(2.7) ψ(x) :=∑

n≤xΛ(n) =

∑

p≤xlog p +O

(

x12 log x

)

.

Partial summation gives π(x) ∼ ψ(x)log x

. First of all, we shall express ψ(x) in terms

of the zeta-function. If c is a positive constant, then

1

2πi

∫ c+i∞

c−i∞

xs

sds =

1 if x > 1,0 if 0 < x < 1.

(2.8)

This yields the so-called Perron formula: for x 6∈ Z and c > 1,

ψ(x) = − 1

2πi

∫ c+i∞

c−i∞

ζ ′

ζ(s)

xs

sds.(2.9)

Moving the path of integration to the left, we find that the latter expression isequal to the corresponding sum of residues, that are the residues of the integrandat the pole of ζ(s) at s = 1, at the zeros of ζ(s), and at the additional pole ofthe integrand at s = 0. The main term turns out to be

Res s=1

−ζ′

ζ(s)

xs

s

= lims→1

(s− 1)

(

1

s− 1+O(1)

)

xs

s= x,

whereas each nontrivial zero ρ gives the contribution

Res s=ρ

−ζ′

ζ(s)

xs

s

= −xρ

ρ.

By the same reasoning, the trivial zeros altogether contribute∞∑

n=1

x−2n

2n= 1

2log

(

1 − 1

x2

)

.


Incorporating the residue at s = 0, this leads to the exact explicit formula

ψ(x) = x−∑

ρ

xρ

ρ− 1

2log

(

1 − 1

x2

)

− log(2π),

which is equivalent to Riemann’s formula (2.5). This formula is valid for any x 6∈Z. Notice that the right-hand side of this formula is not absolutely convergent.If ζ(s) would have only finitely many nontrivial zeros, the right-hand side wouldbe a continuous function of x, contradicting the jumps of ψ(x) for prime powersx. Going on it is more convenient to cut the integral in (2.9) at t = ±T whichleads to the truncated version

ψ(x) = x−∑

|γ|≤T

xρ

ρ+O

( x

T(log(xT ))2

)

,(2.10)

valid for all values of x. Next we need information on the distribution of thenontrivial zeros. Already the non-vanishing of ζ(s) on the line σ = 1 yields theasymptotic relations ψ(x) ∼ x, resp. π(x) ∼ li (x), which is Gauss’ conjecture(2.1) and sufficient for many applications. However, more precise asymptoticswith a remainder term can be obtained by a zero-free region inside the criticalstrip. The largest known zero-free region for ζ(s) was found by Vinogradov [65]and Korobov [35] (independently) in 1958 who proved

ζ(s) 6= 0 in σ ≥ 1 − c

(log(|t| + 3))13 (log log(|t| + 3))

23

,

where c is some positive absolute constant. In combination with the Riemann-von Mangoldt formula (2.4) one can estimate the sum over the nontrivial zerosin (2.10). Balancing out T and x, we obtain the prime number theorem with thesharpest known remainder term:

Theorem 2.1. There exists an absolute positive constant C such that for suffi-ciently large x

π(x) = li (x) +O

(

x exp

(

−C (log x)35

(log log x)15

))

.

We shall give a complete proof of the prime number theorem with a slightlyweaker remainder term in Section 2.6.

By the explicit formula (2.10) the impact of the Riemann hypothesis on theprime number distribution becomes visible. In 1900, von Koch [34] showed thatfor fixed θ ∈ [1

2, 1)

π(x) − li (x) ≪ xθ+ǫ ⇐⇒ ζ(s) 6= 0 for σ > θ ;(2.11)

equivalently, one can replace the left-hand side by ψ(x) − x. Here and in thesequel ǫ stands for an arbitrary small positive constant, not necessarily the sameat each appearance. With regard to known zeros of ζ(s) on the critical line


it turns out that an error term with θ < 12

is impossible. Thus, the Riemannhypothesis states that the prime numbers are as uniformly distributed as possible!

Many computations were done to find a counterexample to the Riemann hypoth-esis. Van de Lune, te Riele & Winter [45] localized the first 1 500 000 001 zeros,all lying without exception on the critical line; moreover they all are simple! Byobservations like this it is conjectured, that all or at least almost all zeros of thezeta-function are simple. This is the so-called essential simplicity hypothesis.

A classical density theorem due to Bohr & Landau [11] states that most of thezeros lie arbitrarily close to the critical line. Denote by N(σ, T ) the number ofzeros ρ = β+ iγ of ζ(s) for which β > σ and 0 < γ ≤ T (counting multiplicities).Bohr & Landau proved

(2.12) N(σ, T ) ≪ T = o(N(T ))

for any fixed σ > 12.‖ Hence, almost all zeros are clustered around the critical

line. The strongest unconditional estimate that holds throughout the right halfof the critical strip is due to Gritsenko [21]:

Theorem 2.2. For any fixed σ with 12< σ < 1,

N(σ, T ) ≪ T125

(1−σ)(logT )915 .

Comparing with Theorem 1.1, we see that zero is an exceptional value of thezeta-function. The location of zeros appears to be completely different from anyother value.

On the other hand, Hardy [24] showed that infinitely many zeros lie on thecritical line. Refining a mollifying technique of Selberg, Levinson [43] localizedmore than one third of the nontrivial zeros of the zeta-function on the criticalline, and as Heath-Brown [26] and Selberg (unpublished) discovered, they are allsimple. Introducing Kloosterman sums, Conrey [13] was able to choose longermollifiers in order to show that more than two fifths of the zeros are simple andon the critical line.

In the remainder of this chapter we shall prove the prime number theorem as well as a

density theorem, both not as sharp as those mentioned above. Besides, we introduce

much of the analytic machinery needed for the proof of Voronin’s universality theorem

in the following chapter.

2.2. The approximate functional equation. By the Riemann integral con-vergence criterion the series defining zeta converges absolutely for σ > 1. Since,

‖Here we write f(x) = o(g(x)), if limx→∞ f(x)/g(x) = 0.


for σ ≥ σ0 > 1,∣

∣

∣

∣

∣

∞∑

n=1

1

ns

∣

∣

∣

∣

∣

≤∞∑

n=1

1

nσ0≤ 1 +

∞∑

n=2

∫ n

n−1

du

uσ0(2.13)

= 1 +

∫ ∞

1

u−σ0 du = 1 +1

σ0 − 1,

the series in question converges uniformly in any half-plane σ ≥ σ0 with σ0 > 1.Thus, by a well-known theorem of Weierstrass, ζ(s), being the limit of a uniformlyconvergent sequence of analytic functions, is analytic in its half-plane of absoluteconvergence.

Lemma 2.3. ζ(s) is analytic for σ > 1 and satisfies identity (1.1), i.e.,

ζ(s) =∞∑

n=1

1

ns=∏

p

(

1 − 1

ps

)−1

.

Proof. It remains to show the identity between the series and the product. Bythe geometric series expansion and the unique prime factorization of the integers,

∏

p≤x

(

1 − 1

ps

)−1

=∏

p≤x

(

1 +1

ps+

1

p2s+ . . .

)

=∑

np|n⇒p≤x

1

ns;

as usual, we write d|n if the integer d divides the integer n, and d ∤ n otherwise.Since

∣

∣

∣

∣

∣

∣

∞∑

n=1

1

ns−

∑

np|n⇒p≤x

1

ns

∣

∣

∣

∣

∣

∣

≤∑

n>x

1

nσ≤∫ ∞

x

u−σ du =x1−σ

σ − 1

tends to zero as x→ ∞, we get the desired identity by sending x → ∞. •Next we shall derive not only an analytic continuation for ζ(s) to the half-planeσ > 0 but also a rather good approximation which will be very useful later on.At s = 1 the zeta-function defining series reduces to the harmonic series. Toobtain an analytic continuation for ζ(s) we have to seperate this singularity. Forthat purpose we apply Abel’s partial summation:

Lemma 2.4. Let λ1 < λ2 < . . . be a divergent sequence of real numbers, definefor αn ∈ C the function A(u) =

∑

λn≤u αn, and let F : [λ1,∞) → C be acontinuous differentiable function. Then

∑

λn≤xαnF (λn) = A(x)F (x) −

∫ x

λ1

A(u)F ′(u) du.

Proof. We have

A(x)F (x) −∑

λn≤xαnF (λn) =

∑

λn≤xαn(F (x) − F (λn)) =

∑

λn≤x

∫ x

λn

αnF′(u) du.


Since λ1 ≤ λn ≤ u ≤ x, interchanging integration and summation yields theassertion. •Next we apply partial summation to the partial sums of the Dirichlet seriesdefining zeta. Let N < M be positive integers and σ > 1. Then, application ofLemma 2.4 with F (u) = u−s, αn = 1 and λn = n yields

∑

N<n≤M

1

ns= M1−s −N1−s + s

∫ M

N

[u]

us+1du

=1

s− 1(N1−s −M1−s) + s

∫ M

N

[u] − u

us+1du.

The integral exists for σ > 0. Sending M → ∞ we obtain

Theorem 2.5. For σ > 0,

ζ(s) =∑

n≤N

1

ns+N1−s

s− 1+ s

∫ ∞

N

[u] − u

us+1du.

In particular, ζ(s) has an analytic continuation to the half-plane σ > 0 exceptfor a simple pole at s = 1 with residue 1.

PuttingN = 1 in the formula of Theorem 2.5, we obtain the analytic continuation(2.2) for ζ(s). Our next aim is to derive from the representation of the theorema very useful approximation of ζ(s) inside the critical strip.

Let f(u) be any function with continuous derivative on the interval [a, b]. Usingthe lemma on partial summation with αn = 1 if n ∈ (a, b], and αn = 0 otherwise,we get

∑

a<n≤bf(n) = ([b] − [a])f(b) −

∫ b

a

([u] − [a])f ′(u) du

= [b]f(b) − [a]f(a) −∫ b

a

[u]f ′(u) du.

Obviously,

−∫ b

a

[u]f ′(u) du =

∫ b

a

(

u− [u] − 12

)

f ′(u) du−∫ b

a

(

u− 12

)

f ′(u) du.

Applying partial integration to the last integral on the right-hand side, we deduceEuler’s summation formula:

Lemma 2.6. Assume that f : [a, b] → R has a continuous derivative. Then

∑

a<n≤bf(n) =

∫ b

a

f(u) du+

∫ b

a

(

u− [u] − 12

)

f ′(u) du

+(

a− [a] − 12

)

f(a) −(

b− [b] − 12

)

f(b).


Next, we replace in Euler’s summation formula the function u − [u] − 12

by itsFourier series expansion.

Lemma 2.7. For u ∈ R \ Z,∣

∣

∣

∣

∣

∣

∣

u− [u] − 12−∑

|m|≤Mm6=0

exp(−2πimu)

2πim

∣

∣

∣

∣

∣

∣

∣

≤ 1

2πM(u− [u]),

and, for u ∈ R,∞∑

m=−∞m6=0

exp(−2πimu)

2πim=

u− [u] − 12

if u 6∈ Z,0 if u ∈ Z,

where the terms with ±m have to be added together; the partial sums are uni-formly bounded in u and M .

Proof. By symmetry and periodicity it suffices to consider the case 0 < u ≤ 12.

Since∫ 1

2

u

exp(−2πimx) dx =(−1)m+1 + exp(−2πimu)

2πimfor 0 6= m ∈ Z,

we obtain

∑

|m|≤Mm6=0

exp(−2πimu)

2πim− u+ 1

2=

∫ 12

u

∑

|m|≤Mexp(2πimx) dx

=

∫ 12

u

sin((2M + 1)πx)

sin(πx)dx.(2.14)

By the mean-value theorem there exists ξ ∈ (u, 12) such that the latter integral

equals∫ ξ

u

sin((2M + 1)πx)

sin(πu)dx.

This implies both formulas of the lemma. It remains to show that the partialsums of the Fourier series are uniformly bounded in u and M . Substitutingy = (2M + 1)πx in (2.14), we get∫ 1

2

u

sin((2M + 1)πx)

sin(πx)dx =

∫ 12

u

sin((2M + 1)πx)

πxdx

+

∫ 12

u

sin((2M + 1)πx)

(

1

sin(πx)− 1

πx

)

dx

≪∫ ∞

0

sin(y)

ydy +

∫ 12

0

∣

∣

∣

∣

1

sin(πx)− 1

πx

∣

∣

∣

∣

dx


with an implicit constant not depending on u and M . Both integrals on the rightexist, which gives the uniform boundedness. •Further, we need the following estimate of exponential integrals.

Lemma 2.8. Assume that F : [a, b] → R has a continuous non-vanishing de-rivative and that G : [a, b] → R is continuous. If G/F ′ is monotonic on [a, b],then

∣

∣

∣

∣

∫ b

a

G(u) exp(iF (u)) du

∣

∣

∣

∣

≤ 4

∣

∣

∣

∣

G

F ′ (a)

∣

∣

∣

∣

+ 4

∣

∣

∣

∣

G

F ′ (b)

∣

∣

∣

∣

.

Proof. First, we assume that F ′(u) > 0 for a ≤ u ≤ b. Since (F−1(v))′ =F ′(F−1(v))−1, substituting u = F−1(v) leads to

∫ b

a

G(u) exp(iF (u)) du =

∫ F (b)

F (a)

G(F−1(v))

F ′(F−1(v))exp(iv) dv.

By the monotonicity of G/F ′, application of the mean-value theorem gives

Re

∫ F (b)

F (a)

G(F−1(v))

F ′(F−1(v))exp(iv) dv

=G

F ′ (F (a))

∫ ξ

F (a)

cos v dv +G

F ′ (F (b))

∫ F (b)

ξ

cos v dv

with some ξ ∈ [F (a), F (b)]. This gives the desired estimate. The same ideaapplies to the imaginary part. The case F ′(u) < 0 can be treated analogously. •Now we are in the position to prove the van der Corput summation formula:

Theorem 2.9. For any given η > 0 there exists a positive constant C = C(η),depending only on η, with the following property: if f : [a, b] → R is a functionwith continuous derivative, g : [a, b] → [0,∞) is differentiable, and f ′, g and |g′|are all monotically decreasing, then

∑

a<n≤bg(n) exp(2πif(n)) =

∑

f ′(b)−η<m<f ′(a)+η

∫ b

a

g(u) exp(2πi(f(u) −mu)) du+ E ,

where|E| ≤ C(η) (|g′(a)| + g(a) log(|f ′(a)| + |f ′(b)| + 2)) .

Van der Corput’s summation formula looks rather technical but the idea is simpleas we shall shortly explain. The integral

∫ b

a

g(u) exp(2πi(f(u) −mu)) du

is (up to a constant factor) the Fourier transform of g(u) exp(2πif(u)) at u = m;therefore, one may interpret Theorem 2.9 as an approximate version of Poisson’ssummation formula (see (2.16) below).


Proof of Theorem 2.9. We apply Euler’s summation formula with the functionF (u) = g(u) exp(2πif(u)). Using the Fourier series expansion of Lemma 2.7, weget

∑


∫ b

a

g(u) exp(2πif(u)) du+O(g(a))

+

∫ b

a

∑

m6=0

exp(−2πimu)

2πim(g(u) exp(2πif(u)))′ du.

The series on the right-hand side converges uniformly on each compact subset,which is free of integers. Moreover, the partial sums are uniformly bounded.Hence, we may interchange summation and integration. This yields

∑


∫ b

a

g(u) exp(2πif(u)) du

+∑

m6=0

1

m

(

I1(m) +1

2πiI2(m)

)

+O(g(a)),(2.15)

where

I1(m) :=

∫ b

a

f ′(u)g(u) exp(2πi(f(u) −mu)) du,

I2(m) :=

∫ b

a

g′(u) exp(2πi(f(u) −mu)) du.

Partial integration gives

I1(m) =

[

exp(2πi(f(u) −mu))g(u)

2πi

]b

u=a

−∫ b

a

exp(2πif(u))

2πi(g(u) exp(−2πimu))′ du,

= O(g(a)) − 1

2πiI2(m) +m

∫ b

a

g(u) exp(2πi(f(u) −mu)) du.


Thus,

∑

f ′(b)−η<m<f ′(a)+ηm6=0

1

m

(

I1(m) +1

2πiI2(m)

)

=∑

f ′(b)−η<m<f ′(a)+ηm6=0

∫ b

a

g(u) exp(2πi(f(u) −mu)) du

+O

∑

f ′(b)−η<m<f ′(a)+ηm6=0

g(a)

|m|

.

Now assume that m > f ′(a) + η and f ′(b) > 0. Then f ′(u) > 0 for a ≤ u ≤ b.Using Lemma 2.8 with F (u) = 2π(f(u) −mu) and G = gf ′, we find

I1(m) ≪∣

∣

∣

∣

g(a)f ′(a)

f ′(a) −m

∣

∣

∣

∣

.

Hence,

∑

m>f ′(a)+ηm6=0

∣

∣

∣

∣

I1(m)

m

∣

∣

∣

∣

≪ g(a)∑

0<m≤2|f ′(a)|

1

m+ g(a)

∑

m>|f ′(a)|

|f ′(a)|m2

.

The contribution arising from m < f ′(b)− η can be treated similarly. This gives

∑

m6∈[f ′(b)−η,f ′(a)+η]m6=0

∣

∣

∣

∣

I1(m)

m

∣

∣

∣

∣

≪ g(a) log(|f ′(a)| + |f ′(b)| + 2).

Now assume m > f ′(a) + η and m 6= 0. Then, by the mean-value theorem, weget for the-real part

Re I2(m) = −∫ b

a

|g′(u)| cos 2π(f(u) −mu) du = g′(a)

∫ ξ

a

cos 2π(f(u) −mu) du

with some ξ ∈ (a, b). Partial integration yields∫ ξ

a

cos 2π(f(u) −mu) du =

[

−Reexp(2πi(f(u) −mu))

2πim

]ξ

u=a

+Re1

m

∫ ξ

a

f ′(u) exp(2πi(f(u) −mu)) du

≪ 1

|m|

(

1 +|f ′(a)|

|f ′(a) −m|

)

.

Therefore,∑

m>f ′(a)+η,m6=0

∣

∣

∣

∣

ReI2(m)

m

∣

∣

∣

∣

≪ g′(a).


With slight modifiactions this method applies also to the imaginary part of I2(m)and the case m ≤ f ′(b) − η. Further, if 0 6∈ [f ′(b) − η, f ′(a) + η], then Lemma2.8 gives

∫ b

a

g(u) exp(2πif(u)) du≪ g(a).

In view of (2.15) the theorem follows from the above estimates under the con-dition f ′(b) > 0. If this condition is not fulfilled, then we may argue withf(u) − (1 − [f ′(b)])u in place of f(u). •

Now we apply van der Corput’s summation formula to the zeta-function. Letσ > 0. By Theorem 2.5 we have

ζ(s) =∑

n≤x

1

ns+∑

x<n≤N

exp(−it log n)

nσ+N1−s

s− 1+ s

∫ ∞

N

[u] − u

us+1du.

Setting g(u) = u−σ and f(u) = − t2π

log u, we get f ′(u) = − t2πu

. Assume that

|t| ≤ 4x, then |f ′(u)| ≤ 78. With the choice η = 1

10the interval (f ′(b)−η, f ′(a)+η)

contains only the integer m = 0. Thus Theorem 2.9 yields

∑

x<n≤N

exp(−it log n)

nσ=

∫ N

x

u−s du+O(x−σ) =N1−s − x1−s

1 − s+O(x−σ).

In addition with

s

∫ ∞

N

[u] − u

us+1du ≪ |s|N−σ

and letting N → ∞, we deduce

Theorem 2.10. We have, uniformly for σ ≥ σ0 > 0, |t| ≤ 4x,

ζ(s) =∑

n≤x

1

ns+x1−s

s− 1+O

(

x−σ)

.

This so-called approximate functional equation was found by Hardy & Little-wood; the name comes from the appearing quantities s and 1 − s as in thefunctional equation (2.3). There are better approximate functional equationsknown, where the approximation is realized by shorter sums with a smaller errorterm.

2.3. The functional equation. Now we shall prove the functional equation(2.3) for Riemann’s zeta-function:

Theorem 2.11. For any s ∈ C,

π− s2 Γ(s

2

)

ζ(s) = π− 1−s2 Γ

(

1 − s

2

)

ζ(1 − s).


Riemann [58] himself gave two proofs of the functional equation. In the meantimeseveral rather different proofs were found. Here we follow Riemann’s originalapproach which relies on the functional equation of the theta-function which isgiven by the infinite series

θ(x) =∑

n∈Z

exp(−πxn2).

Recall the Poisson summation formula: if f : R → R is twice differentiable withf(z) ≪ z−2 as z → ±∞, and f ′′ is integrable over R, then

(2.16)∑

n∈Z

f(n+ α) =∑

m∈Z

f(m) exp(2πimα)

for all α ∈ R, where f denotes the Fourier transform of f . We shall apply thePoisson summation formula with the function f(z) := exp(−π z2

x), where x > 0.

First, we compute the Fourier transform by quadratic substitution:

f(y) =

∫ +∞

−∞exp(−π( z

2

x+ 2iyz)) dz

= x exp(−πxy2)

∫ +∞

−∞exp(−πx(w + iy)2) dw.(2.17)

To gon on we evaluate the integral

I(λ) :=

∫ +∞

−∞exp(−πx(w + λ)2) dw,

where λ is any complex number. For this aim we consider the integral∫

Rexp(−xω2) dω,

where R is the rectangular contour with vertices ±r,±r + iImλ, and r is apositive real number. By Cauchy’s theorem, the integral is equal to zero. Onthe line Reω = r, the integrand tends uniformly to zero as r → ∞. Hence,I(λ) = I(0), and thus the integral I(λ) does not depend on λ. This gives in(2.17)

f(y) = x exp(−πxy2)

∫ +∞

−∞exp(−πxw2) dw = C

√x exp(−πxy2),

where

C :=

∫ +∞

−∞exp(−πz2) dz.

Now applying Poisson’s summation formula leads to∑

n∈Z

exp(−π (n+α)2

x) = C

√x∑

m∈Z

exp(−πxm2 + 2πimα).


Choosing α = 0 and x = 1, both sums are equal; thus, C = 1 and we have justproved the functional equation for the theta-function:

Theorem 2.12. For any x > 0,

θ(x) =1√xθ

(

1

x

)

.

Now we are ready to give the

Proof of Theorem 2.11. For Re z > 0, the Gamma-function may be definedby Euler’s integral

Γ(z) =

∫ ∞

0

uz−1 exp(−u) du.

Substituting u = πn2x leads to

(2.18) Γ(s

2

)

π− s2

1

ns=

∫ ∞

0

xs2−1 exp(−πn2x) dx.

Summing up over all n ∈ N yields

π− s2 Γ(s

2

)

∞∑

n=1

1

ns=

∞∑

n=1

∫ ∞

0

xs2−1 exp(−πn2x) dx.

On the left-hand side we find the Dirichlet series defining ζ(s); in view of itsrange of convergence, the latter formula is valid only for σ > 1. On the right-hand side we may interchange summation and integration, justified by absoluteconvergence. Thus we obtain

π− s2 Γ(s

2

)

ζ(s) =

∫ ∞

0

xs2−1

∞∑

n=1

exp(−πn2x) dx.

We split the integral at x = 1 and get

(2.19) π− s2 Γ(s

2

)

ζ(s) =

∫ 1

0

+

∫ ∞

1

xs2−1ω(x) dx,

where the series ω(x) is given in terms of the theta-function:

ω(x) :=

∞∑

n=1

exp(−πn2x) = 12(θ(x) − 1) .

In view of the functional equation for the theta-function,

ω

(

1

x

)

= 12

(

θ

(

1

x

)

− 1

)

=√xω(x) + 1

2(√x− 1),

we find by the substitution x 7→ 1x

that the first integral in (2.19) is equal to∫ ∞

1

x−s2−1ω

(

1

x

)

dx =

∫ ∞

1

x−s+12 ω(x) dx+

1

s− 1− 1

s.


Substituting this in (2.19) yields

(2.20) π− s2 Γ(s

2

)

ζ(s) =1

s(s− 1)+

∫ ∞

1

(

x−s+12 + x

s2−1)

ω(x) dx.

Since ω(x) ≪ exp(−πx), the last integral converges for all values of s, and thus(2.20) holds by analytic continuation throughout the complex plane. The right-hand side remains unchanged by s 7→ 1− s. This proves the functional equationfor zeta. •

To indicate the power of the functional equation we consider the growth of thezeta-function on vertical lines. A standard application of the Phragmen–Lindelofprinciple (see [61, 63]) to the entire function

(2.21) 12s(s− 1)π− s

2 Γ(s

2

)

ζ(s)

in combination with Stirling’s formula shows that for any vertical strip σ1 ≤ σ ≤σ2 of bounded width there exists a positive constant c such that

(2.22) ζ(σ + it) ≪ tc as t→ ∞.

In the particular case of the critical line this easily yields the bound

ζ(

12

+ it)

≪ t14+ǫ as t→ ∞.

Better estimates are known. Using rather advanced methods (lattice points,estimates for exponential series, etc.), Huxley [30] obtained the exponent 32

205+ ǫ.

The yet unproved Lindelof hypothesis states

(2.23) ζ(

12

+ it)

≪ tǫ as t→ ∞;

note that the truth of the Riemann hypothesis would imply the latter estimate.

Another application of the functional equation yields a proof of the Riemann–vonMangoldt formula (2.4). For this aim one applies the argument principle to thefunction given by (2.21). However, in the sequel we are mainly concerned withzero-counting functions in rectangles to the right of the critical line.

2.4. The mean-square and applications. Using the approximate functionalequation, we shall derive a mean-square formula for ζ(s) in the half-plane σ > 1

2.

Such mean-square formulae are important tools in the theory of the Riemannzeta-function. For example, they provide information on the number of hypo-thetical zeros off the critical line as we shall see below.

Theorem 2.13. For σ > 12,

∫ T

1

|ζ(σ + it)|2 dt = ζ(2σ)T +O(T 2−2σ log T ).


Proof. By the approximate functional equation,

ζ(σ + it) =∑

n<t

1

nσ+it+O(t−σ).

Using ζ(s) = ζ(s), we get

∫ T

1

∣

∣

∣

∣

∣

∑

n<t

1

nσ+it

∣

∣

∣

∣

∣

2

dt =

∫ T

1

∑

m,n<t

1

nσ+itmσ−it dt =∑

m,n<T

1

(mn)σ

∫ T

τ

(m

n

)it

dt

with τ := maxm,n. The diagonal terms m = n give the contribution

∑

n<T

T − n

n2σ= T

(

ζ(2σ) −∑

n≥T

1

n2σ

)

−∑

n<T

1

n2σ−1= ζ(2σ)T +O(T 2−2σ).

The non-diagonal terms m 6= n contribute

∑

m,n<Tm6=n

1

(mn)σ

(

mn

)iT −(

mn

)iτ

i log nm

≪∑

0<m<n<T

1

(mn)σ log nm

.

If m < n2

then log nm> log 2 > 0, and hence

∑

n<T

∑

m<n2

1

(mn)σ log nm

≪(

∑

n<T

1

nσ

)2

≪ T 2−2σ.

If m ≥ n2

we write n = m+ r with 1 ≤ r ≤ n2. By the Taylor series expansion of

the logarithm,

logn

m= − log

(

1 − r

n

)

>r

n.

This gives∑

n<T

∑

r≤n2

1

(mn)σ log nm

≪∑

n<T

n1−2σ∑

r≤n2

1

r≪ T 2−2σ logT.

Collecting together, the assertion of the theorem follows. •

In view of the simple pole of the zeta-function, the mean-square formula abovecannot hold on the critical line because ζ(2σ) is unbounded as σ → 1

2+. Hardy

& Littlewood [25] have shown∫ T

0

|ζ(12

+ it)|2 dt = T log T +O(T ).

The asymptotics of the fourth power moment were found by Ingham; the asymp-totics of the sixth moment and all higher moments are unsettled.

From the above theorem we can deduce some remarkable information on thedistribution of zeros of ζ(s). This observation dates back to Littlewood [44]. For


this purpose we need the following integrated version of the argument principle,also known as Littlewood’s lemma:

Lemma 2.14. Let A < B and let f(s) be analytic on R := s ∈ C : A ≤ σ ≤B, |t| ≤ T. Suppose that f(s) does not vanish on the right edge σ = B of R.Let R′ be R minus the union of the horizontal cuts from the zeros of f in R tothe left edge of R, and choose a single-valued branch of log f(s) in the interior ofR′. Denote by ν(σ, T ) the number of zeros ρ = β+ iγ of f(s) inside the rectanglewith β > σ including zeros with γ = T but not those with γ = −T . Then

∫

∂Rlog f(s) ds = −2πi

∫ B

A

ν(σ, T ) dσ.

We give a sketch of the simple proof. Cauchy’s theorem implies∫

∂R′ log f(s) ds =0, and so the left-hand side of the formula of the lemma,

∫

∂R, is minus the sumof the integrals around the paths hugging the cuts. Since the function log f(s)jumps by 2πi across each cut (assuming for simplicity that the zeros of f in Rare simple and have different height; the general case is no harder),

∫

∂R is −2πitimes the total length of the cuts, which is the right-hand side of the formula inthe lemma.

Littlewood’s lemma can be used in various ways to obtain estimates for thenumber of zeros of the zeta-function in certain regions of the complex plane. Westart with a weak version of the Riemann-von Mangoldt formula (2.4) for thenumber N(T ) of nontrivial zeros ρ = β + iγ with imaginary part γ ∈ (0, T ].

Theorem 2.15. For sufficiently large T ,

N(T + 1) −N(T ) ≪ log T.

Proof. Jensen’s formula states that if f(s) is an analytic function for |s| ≤ Rwith zeros s1, . . . , sm (according their multiplicities) and f(0) 6= 0, then

(2.24)1

2π

∫ 2π

0

log |f(r exp(iθ))| dθ = logrm|f(0)|

|s1 · . . . · sm|

for r < R (this is a variant of the Poisson integral formula). This applied withf(s) = ζ(2 + iT + s) leads to the bound

logrm|f(0)|

|s1 · . . . · sm|≪ log T,

where r ∈ [3, 4] is chosen such that ζ(2 + iT + s) is non-zero. Since any zero ρof ζ(s) with |γ − T | ≤ 1 has distance at most r >

√5 to 2 + iT , it follows from


(2.22) that

N(T + 1) −N(T ) ≤∑

|γ−T |≤1

1 =∑

|γ−T |≤1

logr

|ρ− 2 − iT |1

log r√5

≪∑

|γ−T |≤1

logr

|ρ− 2 − iT | ≪ log T.

The theorem is proved. •

Next we are interested in an estimate for the number N(σ, T ) of zeros ρ = β+ iγof ζ(s) with β > σ, 0 < γ ≤ T . Application of Littlewood’s lemma with fixedσ0 >

12

yields

2π

∫ 1

σ0

N(σ, T ) dσ =

∫ T

0

log |ζ(σ0 + it)| dt−∫ T

0

log |ζ(2 + it)| dt

+

∫ σ0

2

arg ζ(σ + iT ) dσ −∫ σ0

2

arg ζ(σ) dσ.(2.25)

The main contribution comes from the first integral on the right-hand side. Thelast integral does not depend on T and so it is bounded. Since ζ(s) has an Eulerproduct representation, the logarithm has a Dirichlet series representation:

(2.26) log ζ(s) = −∑

p

log

(

1 − 1

ps

)

=∑

p,k

1

kpksfor σ > 1,

where k runs through the positive integers; here we choose that branch of thelogarithm which is real on the positive real axis. We obtain∫ T

0

log |ζ(2 + it)| dt = Re

∑

p,k

1

kp2k

∫ T

0

exp(−itk log p) dt

≪∞∑

n=2

1

n2≪ 1.

It remains to estimate arg ζ(σ+ iT ). We may assume that T is not the ordinateof any zero. Since arg ζ(2) = 0 and

arg ζ(s) = arctan

(

Im ζ(s)

Re ζ(s)

)

,

where

Re ζ(2 + it) =

∞∑

n=1

cos(it log n)

n2≥ 1 −

∞∑

n=2

1

n2> 1 −

∫ ∞

1

du

u2= 0,

we have by the argument principle

| arg ζ(2 + iT )| ≤ π2.

Now assume that Re ζ(σ + iT ) vanishes q times in the range 12≤ σ ≤ 2. Devide

the interval [12

+ iT, 2 + iT ] into q + 1 parts, throughout each of which Re ζ(s)


is of constant sign. Hence, again by the argument principle, in each part thevariation of arg ζ(s) does not exceed π. This gives

| arg ζ(s)| ≤(

q + 32

)

π for σ ≥ 12.

Further, q is the number of zeros of the function

g(z) = 12(ζ(z + iT ) + ζ(z − iT ))

for Im z = 0 and 12≤ Re z ≤ 2. Thus, q ≤ n(3

2), where n(r) denotes the number

of zeros of g(z) for |z − 2| ≤ r. Obviously,∫ 2

0

n(r)

rdr ≥

∫ 2

32

n(r)

rdr ≥ n

(

32

)

∫ 2

32

dr

r= n

(

32

)

log 43.

By Jensen’s formula (2.24),∫ 2

0

n(r)

rdr =

1

2π

∫ 2π

0

log |ζ(2 + r exp(iθ))| dθ − log |ζ(2)|.

In view of (2.22) we obtain

q ≤ n(

32

)

≤ 1

log 43

∫ 2

0

n(r)

rdr ≪ log T.

This yields

arg ζ(σ + iT ) ≪ log T uniformly for σ ≥ 12,

and, consequently, the same bound holds by integration with respect to 12≤ σ ≤

2. The restriction on T not to be an imaginary-part of a zero of ζ(s) can beremoved by considerations of continuity. Therefore, we may replace (2.25) by

(2.27)

∫ 1

σ0

N(σ, T ) dσ =1

2π

∫ T

0

log |ζ(σ0 + it)| dt+O(logT ).

Now we need another fact due to Jensen, namely the Jensen’s inequality, whichstates that for any continuous function f(u) on [a, b],

1

b− a

∫ b

a

log f(u) du ≤ log

(

1

b− a

∫ b

a

f(u) du

)

.

Hence, we obtain∫ T

0

log |ζ(σ + it)| dt ≤ T

2log

(

1

T

∫ T

0

|ζ(σ + it)|2 dt

)

≪ T

by applying Theorem 2.13. Thus, for any fixed σ0 >12,

∫ 1

σ0

N(σ, T ) dσ ≪ T.


Let σ1 = 12

+ 12(σ0 − 1

2), then we get

N(σ0, T ) ≤ 1

σ0 − σ1

∫ σ0

σ1

N(σ, T ) dσ ≤ 2

σ0 − 12

∫ 1

σ1

N(σ, T ) ≪ T.

Because of (2.27) we have proved estimate (2.12) from Section 2.1.

2.5. A density theorem. In the last section we have proved a first estimatefor the number of hypothetical zeros to the right of the critical line. Now we givethe proof of a stronger density theorem due to Hoheisel [29]:

Theorem 2.16. For any fixed σ ∈ (12, 1),

N(σ, T ) ≪ T 4σ(1−σ)(log T )10.

For the proof we need the following simple but powerful lemma, also calledGallagher’s lemma:

Lemma 2.17. Let f(t) be a continuously differentiable complex-valued functionon the interval [a, b]. Let t0 = a < t1 < . . . < tk−1 < tk = b and denote by δ theminimum of all differences tj+1 − tj. Then

k∑

j=1

|f(tj)|2 ≤1

δ

∫ b

a

|f(t)|2 dt+ 2

(∫ b

a

|f(t)|2 dt

∫ b

a

|f ′(t)|2 dt

)

12

.

Proof. Denote by χj(t) the characteristic function on the interval [tj , tj+1], i.e.,χj(t) = 1 for t ∈ [tj , tj+1] and χj(t) = 0 otherwise. Further let

λj(t) =1

tj+1 − tj

∫ t

a

χj(τ) dτ.

Then, by partial integration,∫ tj+1

tj

λj(t)(

|f(t)|2)′

dt = λj(t)|f(t)|2∣

∣

∣

tj+1

t=tj− 1

tj+1 − tj

∫ tj+1

tj

|f(t)|2χj(t) dt

= |f(tj+1)|2 −1

tj+1 − tj

∫ tj+1

tj

|f(t)|2 dt.

It follows that

|f(tj+1)|2 ≤1

δ

∫ tj+1

tj

|f(t)|2 dt+ 2

∫ tj+1

tj

|f(t)| |f ′(t)| dt.

Now the assertion of the lemma follows from summation over j and applicationof the Cauchy–Schwarz inequality. •

Now we are in the position to give the


Proof of Theorem 2.16. For 2 ≤ V ≤ T let N1(σ, V ) count the zeros ρ = β+iγof ζ(s) with β ≥ σ and 1

2V < γ ≤ V . Taking x = V in Theorem 2.10, we have

ζ(s) =∑

k≤V

1

ks+V 1−s

s− 1+O

(

V −σ)

for 12V < t ≤ V and 1

2≤ σ ≤ 1. Now define the Dirichlet polynomial

MX(s) :=∑

m≤X

µ(m)

ms,

where X = V 2σ−1 and µ(m) is the Mobius µ-function, defined by the represen-tation

(2.28) ζ(s)−1 =∏

p

(

1 − 1

ps

)

=∞∑

m=1

µ(m)

ms,

valid for σ > 1. In particular, it follows that µ(m) is equal to (−1)ℓ if m is theproduct of ℓ different primes, and equal to zero otherwise. Now let

ζ(s)MX(s) = P (s) +R(s),

where

P (s) :=∑

m≤X

µ(m)

ms

∑

k≤V

1

ks=∑

n≤XV

a(n)

ns

with

(2.29) a(n) :=∑

m|nm≤X,n≤mV

µ(m) =

1 if n = 1,0 if 1 < n ≤ X,

and

R(s) ≪ |MX(s)|V −σ.

Note that MX(s) mollifies ζ(s)−1. We shall use P (s) as a zero-detector. Lets = ρ = β + iγ be a zero of the zeta-function with 1

2V < γ ≤ V . Then,

1 ≤∣

∣

∣

∣

∣

∑

X<n≤XV

a(n)

nρ

∣

∣

∣

∣

∣

+O(|MX(ρ)|V −β),

1 ≪∣

∣

∣

∣

∣

∑

X<n≤XV

a(n)

nρ

∣

∣

∣

∣

∣

2

+O(|MX(ρ)|2V −2β).

Then, summing up both sides of the latter inequality over all zeros leads to

(2.30) N1(V ) ≪∑

σ≤β≤112 V <γ≤V

∣

∣

∣

∣

∣

∑

X<n≤XV

a(n)

nρ

∣

∣

∣

∣

∣

2

+ |MX(ρ)|2V −2σ

.


Now we divide the interval [12V, V ] into subintervals of length 1 of the form

[2m + n − 1, 2m + n], where n = 1, 2 and 14V − 1 ≤ m ≤ 1

2V . Then, we may

continue as follows

∑

σ≤β≤112 V <γ≤V

≤∑

14V−1≤m≤ 1

2V

2∑

n=1

∑

2m+n−1<γ≤2m+n

≤ 2 max1≤n≤2

∑

14V−1≤m≤ 1

2V

∑

2m+n−1<γ≤2m+n

;

here we have omitted the terms to be summed for better readability. By Theorem2.15 there are only ≪ log V many zeros with 2m+n−1 < γ ≤ 2m+n. Now denoteby∑′

ρ the largest of the related sums according to 2m + n − 1 < γ ≤ 2m + n.Then

∑

σ≤β≤112 V <γ≤V

≪ log V∑

ρ

′ ,

resp. in (2.30)

(2.31) N1(V ) ≪ log V∑

ρ

′

∣

∣

∣

∣

∣

∑

X<n≤XV

a(n)

nρ

∣

∣

∣

∣

∣

2

+

∣

∣

∣

∣

∣

∑

m≤X

µ(m)

mρ

∣

∣

∣

∣

∣

2

V −2σ

.

First of all we shall give a bound for

S(Y ) :=∑

ρ

′

∣

∣

∣

∣

∣

∑

Y <n≤U

b(n)

nρ

∣

∣

∣

∣

∣

2

,

where U ≤ 2Y , V ≥ Y ≥ 1 and

(2.32) b(n) ≪∑

d|n1 =: d(n);

the arithmetic function d(n) is called the divisor function since it counts thenumber of positive divisors of n. By partial summation, for fixed ρ = β + iγ,

∑

Y <n≤U

b(n)

nρ=

∫ U

Y

C(u) du−β with C(u) :=∑

Y <n≤u

b(n)

niγ.

Applying the Cauchy-Schwarz inequality we obtain∣

∣

∣

∣

∣

∑

Y <n≤U

b(n)

nρ

∣

∣

∣

∣

∣

≪ Y −β−1

∫ U

Y

|C(u)| du+ Y −β|C(U)|,∣

∣

∣

∣

∣

∑

Y <n≤U

b(n)

nρ

∣

∣

∣

∣

∣

2

≪ Y −2β−1

∫ U

Y

|C(u)|2 du+ Y −2β|C(U)|2.


This leads to

S(Y ) ≪ Y −2σ∑

ρ

′

∣

∣

∣

∣

∣

∑

Y <n≤W

b(n)

niγ

∣

∣

∣

∣

∣

2

,

where W ≤ U is such that the latter expression is maximal. Since all differencesγr+1 − γr of imaginary parts of counted zeros ρr = βr + iγr are ≥ 1, we deducefrom Lemma 2.17 the estimate

S(Y ) ≪ Y −2σ(I1 +√

I1I2),

where

I1 :=

∫ V

12V

∣

∣

∣

∣

∣

∑

Y <n≤Wb(n)nit

∣

∣

∣

∣

∣

2

dt , I2 :=

∫ V

12V

∣

∣

∣

∣

∣

∑

Y <n≤Wb(n) log n · nit

∣

∣

∣

∣

∣

2

dt.

Taking (2.29) into account, |a(n)| satisfies condition (2.32) on b(n). By elemen-tary estimates one can show that

(2.33)∑

n≤xd(n)k ≪k x(log x)2k−1,

where the implicit constant depends only on k. This yields

I1 ≪ (V + Y ) log V∑

Y <n≤2Y

d(n)2 ≪ (V Y + Y 2)(log V )5,

I2 ≪ (V Y + Y 2)(log V )7.

Now dividing the first sum on the right-hand side of (2.31) into ≪ log V sums(as above), application of the latter estimates yields

log V∑

ρ

′

∣

∣

∣

∣

∣

∑

X<n≤V X

a(n)

nρ

∣

∣

∣

∣

∣

2

≪ (V X1−2σ + (V X)2−2σ)(log V )9.

Similarly, we get for the second term

V −2σ(log T )2∑

ρ

′

∣

∣

∣

∣

∣

∑

m≤X

µ(m)

mρ

∣

∣

∣

∣

∣

2

≪ V −2σ(V +X2−2σ)(log V )9.

Substituting this in (2.31) with X = V 2σ−1, we obtain

N1(V ) ≪ V 4σ(1−σ)(log V )9.

Using this with V = T 1−n and summing up over all n ∈ N, finishes the proof ofthe theorem. •

The density hypothesis states

N(σ, T ) ≪ T (2+ǫ)(1−σ)

for all ǫ > 0 and sufficiently large T . Gritsenko’s theorem 2.2 falls not too farbehind this open conjecture. One can show that the Lindelof hypothesis (2.23)


implies the density hypothesis. However, already Theorem 2.16 can serve in quitemany applications as substitute for the Riemann hypothesis.

2.6. The prime number theorem. Now we shall prove the prime numbertheorem with a slightly weaker remainder term than in Theorem 2.1. For thisaim we need to establish a zero-free region for ζ(s) inside the critical strip. Wemay argue only for s = σ + it from the upper half-plane, since the zeros aresymmetrically distributed with respect to the real axis.

Lemma 2.18. For t ≥ 8, 1 − 12(log t)−1 ≤ σ ≤ 2,

ζ(s) ≪ log t and ζ ′(s) ≪ (log t)2.

Proof. Let 1 − (log t)−1 ≤ σ ≤ 3. If n ≤ t, then

|ns| = nσ ≥ n1−(log t)−1

= exp

((

1 − 1

log t

)

logn

)

≫ n.

Thus, the approximate functional equation, Theorem 2.10, implies

ζ(s) ≪∑

n≤t

1

n+ t−1 ≪ log t

(the bound for the sum is an easy exercise in analysis; in Exercise 5 belowone shall prove an asymptotic formula (2.39)). The estimate for ζ ′(s) followsimmediately from Cauchy’s formula,

ζ ′(s) =1

2πi

∮

|z−s|=r

ζ(z)

(z − s)2dz

with r > 0 sufficiently small, or alternatively, by (carefull) differentiation of theformula of Theorem 2.5. •

In view of the Euler product representation of zeta we find for σ > 1

|ζ(σ + it)| = exp(Re log ζ(s)) = exp

(

∑

p,k

cos(kt log p)

kpkσ

)

.

Since

17 + 24 cosα + 8 cos(2α) = (3 + 4 cosα)2 ≥ 0,

it follows that

ζ(σ)17|ζ(σ + it)|24|ζ(σ + 2it)|8 ≥ 1.(2.34)

This inequality is the main idea for our following observations. By the approxi-mate functional equation, Theorem 2.10, we have

ζ(σ) ≪ 1

σ − 1


for sufficiently small σ > 1. Assuming that ζ(1 + it) has a zero for t = t0 6= 0,we have |ζ(σ + it0)| ≪ σ − 1 as σ → 1+, which leads to

limσ→1+

ζ(σ)17|ζ(σ + it0)|24 = 0,

contradicting (2.34). Thus ζ(1 + it) 6= 0. A simple refinement of this argumentallows a lower estimate for the modulus of ζ(1 + it): for t ≥ 1 and 1 < σ < 2,we deduce from (2.34) and Lemma 2.18

1

|ζ(σ + it)| ≤ ζ(σ)1724 |ζ(σ + 2it)| 13 ≪ (σ − 1)−

1724 (log t)

13 .

Furthermore, with Lemma 2.18,

ζ(1 + it) − ζ(σ + it) = −∫ σ

1

ζ ′(u+ it) du≪ |σ − 1|(log t)2.(2.35)

Hence

|ζ(1 + it)| ≥ |ζ(σ + it)| − c1(σ − 1)(log t)2

≥ c2(σ − 1)1724 (log t)−

13 − c1(σ − 1)(log t)2,

where c1, c2 are certain positive constants. Chosing a constant B > 0 such thatA := c2B

1724 − c1B > 0 and putting σ = 1 +B(log t)−8, we obtain

|ζ(1 + it)| ≥ A

(log t)6.(2.36)

This gives an estimate on the left of the line σ = 1. It also allows an estimateinside the critical strip:

Lemma 2.19. For t ≥ 8, there exists a positive constant δ such that

ζ(s) 6= 0 for σ ≥ 1 − δmin1, (log t)−8.

Proof. In view of Lemma 2.18 estimate (2.35) holds for 1 − δ(log t)−8 ≤ σ ≤ 1.Using (2.36), it follows that

|ζ(σ + it)| ≥ A− c1δ

(log t)6,

where the term on the right is positive for sufficiently small δ. •Now we are in the position to prove the prime number theorem. We shall workwith the logarithmic derivative of ζ(s). Since ζ(s) does not vanish in the half-

plane σ > 1, the logarithmic derivative ζ′

ζ(s) is analytic for σ > 1. Partial

summation gives

−ζ′

ζ(s) = s

∫ ∞

1

ψ(x)dx

xs+1.

For the definition of ψ see (2.7). We would like to isolate ψ(x) from this for-mula. For this purpose we shall prove Formula (2.8) by some kind of Fouriertransformation.


Lemma 2.20. Let c and y be positive and real. Then

1

2πi

∫ c+i∞

c−i∞

ys

sds =

0 if 0 < y < 1,12

if y = 1,1 if y > 1.

Proof. If y = 1, then the integral in question equals

1

2π

∫ ∞

−∞

dt

c+ it=

1

πlimT→∞

∫ T

0

c

c2 + t2dt =

1

πlimT→∞

arctan(T/c) = 12,

by well-known properties of the arctan-function. Now assume that 0 < y < 1and r > c. Since the integrand is analytic in σ > 0, Cauchy’s theorem implies,for T > 0,

∫ c+iT

c−iT

ys

sds =

∫ r−iT

c−iT+

∫ r+iT

r−iT+

∫ c+iT

r+iT

ys

sds.

It is easily seen that∫ c±iT

r±iT

ys

sds ≪ 1

T

∫ c

r

yσ dσ ≪ yc

T | log y| ,∫ r+iT

r−iT

ys

sds ≪ yr

r+ yr

∫ T

1

dt

t≪ yr

(

1

r+ log T

)

.

Now sending first r and then T to infinity, the first case follows. Finally, if y > 1,then we bound the corresponding integrals over the rectangular contour withcorners c ± iT,−r ± iT , analogously. Now the pole of the integrand at s = 0with residue

Res s=0ys

s= lim

s→0

ys

s· s = 1

gives the values 2πi for the integral in this case. •

Now we continue our study of the logarithmic derivative of the zeta-function.For x 6∈ Z and c > 1 we have

∫ c+i∞

c−i∞

∞∑

n=1

Λ(n)

nsxs

sds =

∞∑

n=1

Λ(n)

∫ c+i∞

c−i∞

(x

n

)s ds

s;

here interchanging integration and summation is allowed by the absolute conver-gence of the series. In view of Lemma 2.20 it follows that

∑

n≤xΛ(n) =

1

2πi

∫ c+i∞

c−i∞

∞∑

n=1

Λ(n)

nsxs

sds,

resp.

ψ(x) =1

2πi

∫ c+i∞

c−i∞

(

−ζ′

ζ(s)

)

xs

sds;


this is known as Perron’s formula (2.9). Since

∫ c±i∞

c±iT

ys

sds =

ys

s log y

∣

∣

∣

∣

∣

c±i∞

s=c±iT

+1

log y

∫ c±i∞

c±iT

ys

s2ds≪ yc

T | log y|

for 0 < y 6= 1 and T > 0, it follows that∫ c±i∞

c±iT

( ∞∑

n=2

Λ(n)

ns

)

xs

sds≪ xc

T

∞∑

n=2

Λ(n)

nc∣

∣log xn

∣

∣

≪ xc

T

∣

∣

∣

∣

ζ ′

ζ(c)

∣

∣

∣

∣

+x(log x)2

T+ log x.

This yields

ψ(x) = − 1

2πi

∫ c+iT

c−iT

ζ ′

ζ(s)

xs

sds

+O

(

xc

T

∣

∣

∣

∣

ζ ′

ζ(c)

∣

∣

∣

∣

+x(log x)2

T+ log x

)

,(2.37)

which holds for arbitrary x. To find an asymptotic formula for the integral abovewe move the path of integration to the left. Here we may get contributions fromthe poles of the integrand, i.e., the residues at the nontrivial zeros of ζ(s), and atthe pole of ζ(s) at s = 1. For our purpose it is sufficient to exclude the zeros ofthe zeta-function. In view of the zero-free region of Lemma 2.19 we put c = 1+λwith λ = δ(logT )−8, where δ is given by Lemma 2.19, and integrate over theboundary of the rectangle R given by the corners 1±λ± iT . By this choice ζ(s)does not vanish in and on the boundary of R. Hence,

∫ c+iT

c−iT

(

−ζ′

ζ(s)

)

xs

sds

=

∫ 1−λ−iT

1+λ−iT+

∫ 1−λ+iT

1−λ−iT+

∫ 1+λ−iT

1−λ+iT

(

−ζ′

ζ(s)

)

xs

sds

+2πiRes s=1

(

−ζ′

ζ(s)

)

xs

s.

For the logarithmic derivative of ζ(s) we have

−ζ′

ζ(s) = − d

dslog ζ(s) =

1

s− 1+O(1)

as s→ 1. Thus, we obtain for the residue at s = 1

Res s=1

(

−ζ′

ζ(s)

)

xs

s= x.

It remains to bound the integrals. For the horizontal integrals we deduce fromLemma 2.19 that

∫ 1+λ±iT

1−λ±iT

(

−ζ′

ζ(s)

)

xs

sds≪ x1+λ

T.


Further, for the vertical integral,∫ 1+λ+iT

1−λ−iT

(

−ζ′

ζ(s)

)

xs

sds≪ x1−λ(log T )9.

Collecting together, we deduce from (2.37)

ψ(x) = x+O

(

x1+λ

Tλ+ x1−λ(logT )9 +

x(log x)2

T+ log x

)

.

Choosing T = exp(δ110 (log x)

19 ), we arrive at

ψ(x) = x+O(

x exp(−c(log x)19 ))

.

Setting

θ(x) :=∑

p≤xlog p,

sinceψ(x) − θ(x) =

∑

pk≤xk≤2

log p≪ x12 (log x)2,

it follows thatθ(x) = x+O

(

x exp(−c(log x)19 ))

.

Applying now partial summation, Lemma 2.4, we find

π(x) =∑

p≤xlog p · 1

log p=

θ(x)

log x−∫ x

2

θ(u)

(

1

log u

)′du

=x

log x−∫ x

2

u

(

1

log u

)′du

+O(

x exp(

−c(log x)19

))

.

Now partial integration leads to the prime number theorem with remainder term:

Theorem 2.21. There exists a positive constant c such that for x ≥ 2

π(x) = li (x) +O(

x exp(

−c(log x)19

))

.

Thus, the simple pole of the zeta-function is not only the key in Euler’s proof ofthe infinitude of primes but also gives the main term of the asymptotic formulain the prime number theorem. We see that the primes are not too irregularlydistributed. For example, the prime number theorem implies that, if pn denotesthe n-th prime number (in ascending order), then pn ∼ n log n.

We conclude this section by giving a sketch of von Koch’s equivalent (2.11) forthe Riemann hypothesis. By partial summation we obtain for σ > 1

−ζ′

ζ(s) =

s

s− 1+ s

∫ ∞

1

ψ(u) − u

us+1du.


Figure 5. This is Ulam’s spiral: the first 65 000 positive integers are listed

in a spiral in ascending order, the primes are coloured white, the composite

numbers black.

If ψ(x) − x ≪ xθ+ǫ, then the integral above converges for σ > θ, giving ananalytic continuation for

ζ ′

ζ(s) − 1

s− 1

to the half-plane σ > θ, and, in particular, ζ(s) does not vanish there. For theconverse implication we assume that all nontrivial zeros ρ = β+ iγ satisfy β ≤ θ.Then it follows from (2.10) that

(2.38) ψ(x) − x≪ xθ∑

|γ|≤T

1

|γ| +x

T(log(xT ))2.

By Theorem 2.15 we have N(T + 1) −N(T ) ≪ log T , and therefore

∑

|γ|≤T

1

|γ| ≪[T ]+1∑

m=1

logm

m≪ (log T )2.

Substituting this in (2.38) leads to

ψ(x) − x≪ xθ(log T )2 +x

T(log(xT ))2.

Now the choice T = x1−θ finishes the proof of this implication. So the Riemannhypothesis is true if and only if the error term in the prime number theorem isO(x

12+ǫ). In this case there cannot be too long intervals free of primes. Note

that it is an open question whether there is always a prime in between twoconsecutive squares. Some examples may convince the reader that this is a rea-sonable conjecture, however, this statement does not even follow from Riemann’shypothesis.


Practice makes perfect! We continue with some exercises. Already Euler found anexplicit formula for the values of the zeta-function at all positive even integers interms of Bernoulli numbers (see [63]).

Exercise 3. Recall the product representation

sin(πz) = πz

∞∏

k=1

(

1 − z

k2

)

and deduce some zeta-values: ζ(2) = π2

6 , ζ(4) = π4

90 , . . . by expanding the infiniteproduct.

On the contrary, not too much is known about zeta-values at positive odd integers. In1978, Apery proved that ζ(3) is irrational, however, it is not known whether this valueis transcendental or whether ζ(5) is irrational.

It is conjectured that all zeros of the zeta-function are simple. A classical theorem ofSpeiser states that the Riemann hypothesis is true if and only if ζ ′(s) is non-vanishingin 0 < σ < 1

2 .

Exercise 4. Prove that any zero of ζ ′(s) on the critical line is a multiple zero of ζ(s).

The following three exercises can be solved with Abel’s partial summation (Lemma2.4).

Exercise 5. Prove the following asymptotic formulas∑

n≤x

1

n= log x + γ + O

(

1

x

)

,(2.39)

∑

p≤x

1

p= log log x + O(1);(2.40)

here γ is the Euler–Mascheroni constant γ := limN→∞1N (∑N

n=11n− log N) = 0.557 . . ..

Exercise 6. Prove formula (2.33). For this one may count the lattice points (a, b) ∈ Z2

under a hyperbola:∑

n≤x d(n) =∑

ab≤x 1. For the second moment observe that∑

n≤xd(n)2 =

∑

ab≤xd(ab) ≤

∑

a≤xd(a)

∑

b≤x/ad(b).

Another approach uses contour integration of the function ζ(s)k xs

s , following the linesof proof of the prime number theorem.

The next exercise is about twin primes, that are pairs of primes of the form p, p + 2.It is unknown whether there are infinitely many twin primes. Brun showed for thenumber π2(x) of twin primes p, p + 2 with p ≤ x the estimate π2(x) ≪ x(log x)−2.

Exercise 7. Deduce from Brun’s estimate that the sum over the reciprocals of alltwin primes converges although the sum of the reciprocals over all primes diverges (see(2.40)).


Relevant information about the zeta-function is contained in its order of growth alongvertical lines as well as in the distribution of its zeros. For the next exercises one mayconsult [62, 63]:

Exercise 8. Apply the Phragmen-Lindelof principle in order to prove estimate (2.22)with an explicit constant c and give the details for the proof of Theorem 2.15.

Exercise 9. Prove the Riemann-von Mangoldt formula (2.4).

3. Universality theorems

In this chapter we shall prove the famous universality theorem of Voronin; besides

we indicate how to derive other remarkable universality theorems by similar means

(e.g., Reich’s universality theorem 3.11 below). The method of proof is a mixture of

techniques from function theory, analytic number theory, and basic functional analysis.

3.1. Voronin’s universality theorem. Now we are going to prove Voronin’suniversality theorem, that is Theorem 1.3 from the introduction: Let 0 < r < 1

4be fixed and suppose that g(s) is a non-vanishing continuous function on the disk|s| ≤ r which is analytic in the interior. Then, for any ǫ > 0,

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max|s|≤r

∣

∣ζ(

s+ 34

+ iτ)

− g(s)∣

∣ < ǫ

> 0.

The Euler product for the zeta-function is the key to prove the universality the-orem in spite of the fact that it does not converge in the region of universality.However, as already Bohr observed, an appropriate truncated Euler product ap-proximates ζ(s) in a certain mean-value sense inside the critical strip; this isrelated to the use of modified truncated Euler products in Voronin’s proof (see(3.23) below). Another important tool in the proof are approximation theorems,one for numbers and one for functions. This is not too surprising since univer-sality is an approximation property. Last but not least we shall make use of theprime number theorem and classical function theory.

It is more convenient to work with series than with products. Therefore, weconsider the logarithms of the functions in question. Since g(s) has no zerosin |s| ≤ r, its logarithm exists and we may define an analytic function f(s) byg(s) = exp f(s) for |s| < r. Conversely, if f(s) is analytic, then g(s) = exp f(s)is analytic and non-vanishing. Now we formulate


and suppose that f(s) is a continuous function onthe disk |s| ≤ r, which is analytic in the interior. Then, for any ǫ > 0,

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max|s|≤r

∣

∣log ζ(

s+ 34

+ iτ)

− f(s)∣

∣ < ǫ

> 0.


Note that the zeros of the zeta-function are negligible since they form a set ofdensity zero by Theorem 2.2 whereas the set of approximating τ has positivelower density. Once Theorem 3.1 is proved, Voronin’s universality theorem 1.3follows. To see this we observe

g(s) − ζ(

s+ 34

+ iτ)

= g(s)(

1 − exp(

log ζ(

s+ 34

+ iτ)

− f(s)))

.

Assume that f is also analytic on the boundary. Then, taking the maximumover all values of s with |s| ≤ r, the desired estimate follows from the expansionexp z − 1 = z + O(|z|2). If f is not analytic on the boundary, we may concludewith a simple continuity argument. Hence, it suffices to prove Theorem 3.1.Its lengthy proof is organized as follows: a rearrangement theorem in a certainHilbert space (Theorem 3.2 in §3.2) allows to approximate the target functionby the logarithms of certain truncated Euler products (Theorem 3.6 in §3.3).The transition to the logarithm of the zeta-function is realized by diophantineapproximation theory (§3.4) and integration in order to obtain a set of the desiredtranslates τ having positive lower density (§3.5).

3.2. Rearrangement of conditionally convergent series. A series∑

n an ofreal numbers an is said to be conditionally convergent, if

∑

n |an| is divergent but∑

n an is convergent for an appropriate rearrangement of the terms an. Riemannproved that any conditionally convergent series can be rearranged such that itssum converges to an arbitrary preassigned real number or infinity. For instance,to any given c ∈ R there exists a permutation σ of N such that

∑

n∈N

(−1)

σ(n)= c.

In some sense, conditionally convergent series are universal with respect to R.

It is the aim of this section to extend Riemann’s rearrangement theorem toHilbert spaces. In what follows let H be a Hilbert space and denote, as usual,its inner product by 〈x, y〉 and its norm by ‖x‖ =

√

〈x, x〉.Theorem 3.2. Assume that a series

∑

n un of vectors in a real Hilbert space Hsatisfies

∞∑

n=1

‖un‖2 <∞,

and for any e ∈ H with ‖e‖ = 1 the series∑

n〈un, e〉 converges conditionally(with some rearrangement). Then for any v ∈ H there is a permutation σ of Nsuch that

∞∑

n=1

uσ(n) = v

in the norm of H.


This theorem is due to Pechersky who proved it on demand of Voronin. Theproof is slightly more complicated than the one for Riemann’s rearrangementtheorem. We start with

Lemma 3.3. Under the assumptions of Theorem 3.2, for any v ∈ H and anyǫ > 0 there exist a positive integer N and numbers ǫ1, . . . , ǫN , equal to 0 or 1,such that

∥

∥

∥s−

N∑

n=1

ǫnun

∥

∥

∥< ǫ.

Proof. We choose an integer m such that∞∑

n=m

‖un‖2 < 19ǫ2.

Denote by Pm the set of all linear combinations

N∑

n=m

λnun with λn ∈ [0, 1] and N = m,m+ 1, m+ 2, . . . .

Obviously, Pm is convex. Let Pm be the closure of Pm with respect to the norm ofH; consequently Pm is a closed convex set. First of all we show that Pm coincideswith H.

The seperation theorem for linear operators states that if X is a normed linearspace and D is a convex subset of X which is closed in the norm of X, then forany s ∈ X \D there exist ǫ > 0 and a linear functional F on X such that

F (x) ≤ F (s) − ǫ for all x ∈ D.

The proof follows from the well-known theorem of Hahn-Banach, which relateslinear functionals to convex sets. A simple consequence is that for any properconvex subset D of real Hilbert space H, which is closed in the norm of H, thereexists a vector e ∈ H with ‖e‖ = 1 such that

supx∈D

〈x, e〉 <∞.

We return to our problem: suppose that Pm 6= H, then, by the above reasoning,there exists e ∈ H with ‖e‖ = 1 such that supx∈Pm

〈x, e〉 <∞. Since, by assump-tion, the series

∑∞n≥m〈un, e〉 converges conditionally with some arrangement of

the terms, the subseries consisting of the positive terms is divergent. Thus, forany C there exist an N and a sequence ǫm, . . . , ǫN , each ǫn being equal to 0 or1, such that

N∑

n=m

ǫn〈un, e〉 > C.

Since∑N

m ǫnun ∈ Pm, it follows that supx∈Pm〈x, e〉 = ∞, giving the contradiction.


So we have shown Pm = H. Consequently, there exist N ≥ m and λm, . . . , λN ∈[0, 1] such that

∥

∥

∥v −

N∑

n=m

λnun

∥

∥

∥< 1

3ǫ.

By induction we can construct ǫm, . . . , ǫN , equal to 0 or 1, such that for any Mwith m ≤ M ≤ N the inequality

∥

∥

∥

M∑

n=m

λnun −M∑

n=m

ǫnun

∥

∥

∥≤

M∑

n=m

‖un‖2

holds. We may set ǫm = 1 and suppose that ǫm, . . . , ǫM have been chosen so thatthe last inequality is fulfilled. With ǫM+1, equal to 0 or 1, satisfying

(λM+1 − ǫM+1)⟨

M∑

n=m

(λn − ǫn)un, uM+1

⟩

≤ 0,

we get

∥

∥

∥

M+1∑

n=m

λnun −M+1∑

n=m

ǫnun

∥

∥

∥

2

≤∥

∥

∥

M∑

n=m

(λn − ǫn)un

∥

∥

∥

2

+ ‖uM+1‖2 ≤M+1∑

n=m

‖un‖2.

Hence, we can find a sequence of numbers ǫm, . . . , ǫN , each being 0 or 1, suchthat

∥

∥

∥

N∑

n=m

λnun −N∑

n=m

ǫnun

∥

∥

∥

2

≤N∑

n=m

‖un‖2 < 19ǫ2.

Thus,

∥

∥

∥v −

N∑

n=m

ǫnun

∥

∥

∥≤∥

∥

∥v −

N∑

n=m

λnun

∥

∥

∥+∥

∥

∥

N∑

n=m

λnun −N∑

n=m

ǫnun

∥

∥

∥< 2

3ǫ,

which proves the lemma. •

The next step is

Lemma 3.4. Under the assumptions of Theorem 3.2, there exists a permutationnk of N such that some subsequence of the partial sums of the series

∑

k unk

converges to v in the norm of H.

Proof. We construct the sequence n1, n2, . . . as follows. First let n1 = 1. Ap-plying Lemma 3.3 to the series

∑

n≥2 un, yields the existence of a finite setT1 ⊂ 2, 3, . . . such that

∥

∥

∥v − u1 −

∑

n∈T1

un

∥

∥

∥< 1

2.

Now write the indices in T1 in an arbitrary order. If 2 6∈ T1, then write also 2.Denote by T2 the set of all indices we have so far, and define N1 = maxn ∈ T2.


Applying Lemma 3.3 to the series∑∞

n=N1+1 un, shows that there exists a finiteset T3 ⊂ N1 + 1, N1 + 2, . . . such that

∥

∥

∥v −

∑

n∈T2

un −∑

n∈T3

un

∥

∥

∥< 1

4.

Now write out the indices of first T2 and then T3, each in arbitrary order, write3 if 3 6∈ T2 ∪ T3. Continuing this process, the assertion of the lemma follows. •.

Further, we have to prove

Lemma 3.5. Let v1, . . . , vN be arbitrary elements in a real Hilbert space H. Thenthere exists a permutation σ of the set 1, . . . , N such that

max1≤m≤N

∥

∥

∥

m∑

k=1

vσ(k)

∥

∥

∥≤(

N∑

n=1

‖vn‖2

)12

+ 2∥

∥

∥

N∑

n=1

vn

∥

∥

∥.

Proof. First, suppose that

N∑

n=1

vn = 0.

Then we shall construct by induction a permutation n1, . . . , nN of 1, . . . , Nsuch that

(3.1) max1≤m≤N

∥

∥

∥

m∑

k=1

vnk

∥

∥

∥≤(

N∑

n=1

‖vn‖2

)12

.

For this aim put n1 = 1 and suppose that n1, . . . , nj with 1 ≤ j ≤ N − 1 havebeen chosen, satisfying

max1≤m≤j

∥

∥

∥

m∑

k=1

vnk

∥

∥

∥

2

≤j∑

n=1

‖vn‖2.

Then we may choose nj+1 from the remaining numbers such that

⟨

j∑

k=1

vnk, vnj+1

⟩

≤ 0.

Such an nj+1 exists since otherwise

∑

i6=nk

⟨

i∑

k=1

vnk, vi

⟩

=⟨

j∑

k=1

vnk,−

j∑

k=1

vnk

⟩

> 0.


Hence,

∥

∥

∥

j+1∑

k=1

vnk

∥

∥

∥

2

=

j∑

k=1

‖vnk‖2 + ‖vnj+1

‖2 + 2

⟨

j∑

k=1

vnk, vnj+1

⟩

≤j+1∑

k=1

‖vnk‖2.

This yields a permutation n1, n2, . . . , nN of 1, 2, . . . , N which satisfies (3.1)

under the assumption∑N

n=1 vn = 0.

For arbitrary v1, . . . , vN define

vN+1 = −N∑

n=1

vn,

and apply the already proved case for v1, . . . , vN , vN+1. This leads to a permu-tation n1, n2, . . . , nN+1 of 1, 2, . . . , N + 1 with

max1≤m≤N+1

∥

∥

∥

m∑

n=1

vσ(n)

∥

∥

∥≤(

N∑

n=1

‖vn‖2

)12

+∥

∥

∥

N∑

n=1

vn

∥

∥

∥.

Removing νN+1 from the set vn1 , . . . , vnN, vnN+1

we get an N -tuple of vectorswhich satisfies the inequality of the lemma. •

Now we are in the position for the

Proof of Theorem 3.2. By Lemma 3.4 we may assume that some subsequenceof the partial sums of the series

∑

k uk converges to v in the norm of H. Wedefine

Un =

n∑

k=1

uk,

and suppose that a sequence of partial sums Unjconverges to v. For each j ∈ N

there is a permutation σ of the set of vectors Unj+1, . . . , Unj+1 in such a way

that the value of

mj := max1≤m≤nj+1−nj

∥

∥

∥

nj+m∑

n=nj+1

uσ(n)

∥

∥

∥

is minimal. By Lemma 3.5 it follows that

mj ≤

∞∑

n=nj+1

‖un‖2

12

+ 2‖Unj+1− Unj

‖,

which tends to zero as j → ∞. Hence, the corresponding series converges to vin the norm of H. Theorem 3.2 is proved. •


In the sequel we shall apply Pechersky’s rearrangement theorem 3.2 to the fol-lowing Hilbert space. Let R be a positive real number, then the so-called Hardyspace HR

2 is the set of functions f(s) which are analytic for |s| < R and for which

‖f‖ := limr→R−

∫∫

|s|<r|f(s)| dσ dt <∞.

We define on HR2 an inner product by

(3.2) 〈f, g〉 = Re

∫∫

|s|≤Rf(s)g(s) dσ dt.

Hence HR2 is a real Hilbert space.

3.3. Finite Euler products. Let Ω denote the set of all sequences of realnumbers indexed by the primes, that are all infinite vectors of the form ω :=(ω2, ω3, . . .) with ωp ∈ R. Then we define for any finite subset M of the set of allprimes, any ω ∈ Ω and complex s

ζM(s, ω) =∏

p∈M

(

1 − exp(−2πiωp)

ps

)−1

.

Obviously, ζM(s, ω) is an analytic function in s without zeros in the half-planeσ > 0. Consequently, its logarithm exists and equals

log ζM(s, ω) = −∑

p∈Mlog

(

1 − exp(−2πiωp)

ps

)

;

here as for log ζ(s) we may take the principal branch of the logarithm on thepositive real axis.

The first step in the proof of Theorem 3.1 is to show


and suppose that f(s) is continuous on |s| ≤ r

and analytic in the interior. Further, let ω0 =(

14, 2

4, 3

4, . . .

)

. Then for any ǫ > 0and any y > 0 there exists a finite set M of prime numbers, containing at leastall primes p ≤ y, such that

max|s|≤r

∣

∣log ζM(

s+ 34, ω0

)

− f(s)∣

∣ < ǫ.

Proof. Since f(s) is continuous for |s| ≤ r, there exists κ > 1 such that κ2r < 14

and

(3.3) max|s|≤r

∣

∣

∣f( s

κ2

)

− f(s)∣

∣

∣<ǫ

2.

The function f(

sκ2

)

is bounded on the disc |s| ≤ κr =: R, and thus belongs to

the Hardy space HR2 .


Denote by pk the kth prime number. We consider the series

∞∑

k=1

uk(s) with uk(s) := log(

1 − exp(−2πiωpk)p

−s− 34

k

)−1

.

First, we shall prove that for every v ∈ HR2 there exists a rearrangement of the

series∑

uk(s) for which∞∑

k=1

ujk(s) = v(s).

In view of the Taylor expansion of the logarithm the series∑

k uk(s) differs from

∞∑

k=1

ηk(s) with ηk(s) := exp(

−2πik4

)

p−s− 3

4k

by an absolutely convergent series. Hence, it suffices to verify the conditions ofthe rearrangement theorem 3.2 for the series

∑

k ηk(s). Since R < 14,

∞∑

k=1

‖ηk(s)‖2 ≪∑

p

1

p32−2R

<∞.

Further, we have to check that for any φ ∈ HR2 with ‖φ‖2 = 1 the series

(3.4)

∞∑

k=1

〈ηk, φ〉

is conditionally convergent for some rearrangement of its terms. By the Cauchy-Schwarz inequality,

∞∑

k=1

〈ηk, φ〉 ≤∥

∥

∥

∞∑

k=1

ηk

∥

∥

∥

12 · ‖φ

∥

∥

∥

12

=∥

∥

∥

∞∑

k=1

ηk

∥

∥

∥

12<∞,

and so it is sufficient to show that there exist two subseries of (3.4), where oneis diverging to +∞ and the other one to −∞.

By (3.2),

(3.5) 〈ηk, φ〉 = Re

exp(

−2πik4

)

∫∫

|s|≤Rp−s− 3

4k φ(s) dσ dt

.

Now define

∆(x) =

∫∫

|s|≤Rexp

(

−x(

s+ 34

))

φ(s) dσ dt,


then the integral appearing on the right of (3.5) equals ∆(log pk). Further, letφ(s) =

∑∞m=0 αms

m. Then we may express ∆(x) in terms of the Taylor coeffi-cients αm as follows:

∆(x) = exp(

−3x4

)

∫∫

|s|≤Rexp(−sx)φ(s) dσ dt

= exp(

−3x4

)

∫∫

|s|≤R

∞∑

n=0

(−sx)nn!

∞∑

m=0

αm sm dσ dt

= exp(

−3x4

)

∞∑

m=0

∞∑

n=0

(−1)nxn

n!αm

∫∫

|s|≤Rsmsn dσ dt.

We compute∫∫

|s|≤Rsmsn dσ dt =

∫ R

0

∫ 2π

0

ρm+n exp(iθ(n−m)) dθ dρ

=

2πR2m+2

2m+2if m = n,

0 if m 6= n.

This yields

(3.6) ∆(x) = πR2 exp

(

−3x

4

) ∞∑

m=0

βmm!

(xR)m,

where

βm = (−1)mαmR

m

m+ 1.

Since ‖φ‖ = 1, we get

1 =

∫∫

|s|≤R|φ(s)|2 dσ dt =

∞∑

m=0

|αm|2∫∫

|s|≤R|s|2m dσ dt

= πR2

∞∑

m=0

|αm|2m+ 1

R2m.

Hence,

(3.7) 0 <

∞∑

m=0

|βm|2 ≪ 1,

which implies that βm is bounded. Consequently, the function F (z), given by

F (z) =

∞∑

m=0

βmm!zm,

defines an entire function in z.


Next we shall show that for any δ > 0 there exists a sequence of positive realnumbers zj, tending to +∞, for which

(3.8) |F (zj)| > exp(−(1 + 2δ)zj).

Suppose the contrary. Then there is some δ ∈ (0, 1) and a constant B such that|F (z)| < B exp(−(1 + 2δ)z) for any z ≥ 0. It follows that

(3.9) | exp((1 + δ)z)F (z)| < B exp(−δ|z|) for z ≥ 0;

since |βm| ≪ 1, this estimate even holds for z < 0 by a suitable change of theconstant B.

Here we shall apply two theorems from Fourier analysis. First, recall the theoremof Paley-Wiener: given an entire function G(z), then the relation

(3.10) G(z) =

∫ α

−αg(ξ) exp(iξz) dξ

holds for some square integrable function g(ξ) if and only if∫ ∞

−∞|G(z)|2 dz <∞,

and G(z) has an analytic continuation throughout the complex plane satisfyingG(z) ≪ exp((α + ǫ)|z|) for any ǫ > 0, where the implicit constant may dependon ǫ (this characterizes all transcendent functions of fixed exponential type ≤ α).Plancherel’s theorem states that for any such G(z) with (3.10) also

g(ξ) =1

2π

∫ ∞

−∞G(z) exp(−iξz) dz

holds almost everywhere in R.

Application of the theorem of Paley-Wiener with G(z) = exp((1+δ)z)F (z) yieldswith regard to (3.9) the representation

exp((1 + δ)z)F (z) =

∫ 3

−3

f(ξ) exp(iξz) dξ,

where f(ξ) is a square integrable function with support on the interval [−3, 3](not to be confused with our target function). Further, Plancherel’s theoremimplies

f(ξ) =1

2π

∫ ∞

−∞F (z) exp((1 + δ)z − iξz) dz

almost everywhere. Hence, f(ξ) is analytic in a strip covering the real axis. Sincethe support of f(ξ) lies inside a compact interval, the integral above has to bezero outside this interval. Hence, F (z) has to vanish identically, contradictingthe existence of a sequence of positive real numbers zj diverging to +∞ with(3.8).


Let xj =zj

R. Then it follows from (3.6) and (3.8) that

|∆(xj)| > πR2 exp(

−34xj)

F (xjR) ≥ πR2 exp(

−xj(

34

+R(1 + 2δ)))

.

Thus, for sufficiently small δ′ > 0 we obtain the existence of a sequence of positivereal numbers xj , tending to +∞, such that

(3.11) |∆(xj)| > exp(−(1 − δ′)xj).

Now we shall approximate F and ∆ by polynomials. Let Nj = [xj ] + 1 andassume that xj − 1 ≤ x ≤ xj + 1. Since |βm| ≪ 1,

∞∑

m=N2j +1

βmm!

(xR)m ≪ (xR)N2j

(N2j )!

∞∑

m=0

(xR)m

m!≪

NN2

j

j exp(Nj)

(N2j )!

≪ exp(−2xj),

by Stirling’s formula. Trivially,∞∑

m=N2j +1

1

m!

(

−3x4

)m ≪ exp(−2xj)

for the same x. Hence,

F (xR) =

N2j

∑

m=0

+

∞∑

m=N2j +1

βmm!

(xR)m = Pj(x) +O(exp(−2xj))

and analogouslyexp

(

−3x4

)

= Pj(x) +O(exp(−2xj)),

where Pj and Pj are polynomials of degree ≤ N2j . This yields in view of (3.6)

∆(x) = Qj(x) + o(exp(−xj)) for xj − 1 ≤ x ≤ xj + 1,

where Qj = PjPj is a polynomial of degree ≤ N4j .

In order to find lower bounds for ∆(x) we have to apply a classical theoremof A.A. Markov which states that if Q is a polynomial of degree N with realcoefficients which satisfies the inequality

max−1≤x≤1

|Q(x)| ≤ 1,

thenmax

−1≤x≤1|Q′(x)| ≤ N2.

For the first, lets assume that Qj is a real polynomial. Choose ξ ∈ [xj−1, xj +1]such that

|Qj(ξ)| = maxxj−1≤x≤xj+1

|Qj(x)|.Then Markov’s inequality implies

maxxj−1≤x≤xj+1

|Q′j(x)| ≤ N8

j |Qj(ξ)|.


For |x− ξ| ≤ δξ2

with sufficiently small δ satisfying 0 < δ < N−8j , it follows that

|Qj(x)| ≥ |Qj(ξ)| − |x− ξ| maxxj−1≤x≤xj+1

|Q′j(x)|

≥ |Qj(ξ)| − O(|x− ξ|ξ2|Qj(ξ)|)≥ 1

2|Qj(ξ)| ≥ 1

2|Qj(xj)|.

Hence, for x ∈ [ξ − δξ2, ξ + δ

ξ2] ∩ [xj − 1, xj + 1]

|∆(x)| ≥ |Qj(x)| + o(exp(−xj))≥ 1

2|Qj(xj)| + o(exp(−xj)) ≥ 1

2|∆(xj)| + o(exp(−xj)).

We have assumed that Qj has real coefficients. If this is not true, then the abovereasoning may be applied to both, the real part and the imaginary part of Qj .Hence, for sufficiently large xj , the intervals [xj − 1, xj + 1] contain intervals[α, α + β] of length ≥ 1

200N−8j all of whose points satisfy at least one of the

inequalities

(3.12) |Re ∆(x)| > 1

200exp(−(1 − δ′)x) , |Im ∆(x)| > 1

200exp(−(1 − δ′)x).

In order to prove the divergence of a subseries of (3.4) we note that one of theinequalities in (3.12) is satisfied infinitely often as x → ∞; we may assume thatit is the one with the real part. By the prime number theorem 2.21, the interval[exp(α), exp(α+ β)] contains

∫ exp(α+β)

exp(α)

du

log u+O

(

exp(

α + β − cα19

))

≫ exp(α)

α

(

exp(β) − 1 +O(

exp(

β − cα19

)))

many primes, where c > 0 is some absolute constant. Under these prime numberspk ∈ [exp(α), exp(α+ β)] we choose those with k ≡ 0 mod 4. Since ωpk

= k4, we

deduce from (3.5) and (3.11)∑

k≡0 mod 4α≤log pk≤α+β

〈ηk, φ〉 =∑

k≡0 mod 4α≤log pk≤α+β

Re ∆(log pk) ≫ exp(12δ′xj),

which diverges with xj → ∞.

Thus, we have shown that the series (3.4) satsifies the conditions of Theorem3.2. Hence, there exists a rearrangement of the series

∑

k uk(s) such that

(3.13)∞∑

k=1

ujk(s) = f( s

κ2

)

,

where f is our target function. Before we can finish the proof of Theorem 3.6 wehave to prove the following easy


Lemma 3.7. Suppose that G(s) is analytic on |s− s0| ≤ R and∫∫

|s−s0|≤r|G(s)|2 dσ dt = M.

Then, for any fixed r satisfying r < R and any s with |s− s0| ≤ r,

|G(s)| ≤ 1

R− r

(

M

π

) 12

.

Proof. By Cauchy’s formula,

G(s)2 =1

2πi

∮

|z−s|=ρ

G(z)2

z − sdz =

1

2π

∫ 2π

0

G2(s + ρ exp(iθ)) dθ

for any ρ < R. Taking the absolute modulus and integrating with respect to ρ,we obtain

|G(s)|2∫ R−r

0

ρ dρ ≤ 1

2π

∫ 2π

0

∫ R−r

0

|G(s+ ρ exp(iθ))|2ρ dρ dθ =M

2π.

This yields the assertion. •

We return to the proof of Theorem 3.6. According to (3.13),

limn→∞

n∑

k=1

ujk(s) = f( s

κ2

)

in the norm of HR2 . This implies

limn→∞

∫∫

|s|≤R

∣

∣

∣

∣

∣

f( s

κ2

)

−n∑

k=1

ujk(s)

∣

∣

∣

∣

∣

2

dσ dt = 0

uniformly on |s| ≤ R. Thus, application of Lemma 3.7 shows that for sufficientlylarge m

max|s|≤R

∣

∣

∣

∣

∣

f( s

κ2

)

−m∑

k=1

ujk(s)

∣

∣

∣

∣

∣

< 12ǫ.

Hence, there exists a finite set M , containing without loss of generality all primesp ≤ y, such that

log ζM(

s+ 34, ω0

)

=m∑

k=1

ujk(s).

approximates g(s). More precisely, in view of (3.3) it follows that

max|s|≤r

∣

∣log ζM(

s + 34, ω0

)

− f(s)∣

∣

≤ max|s|≤r∣

∣log ζM(

s+ 34, ω0

)

− f(

sκ2

)∣

∣+ max|s|≤r∣

∣f(

sκ2

)

− f(s)∣

∣ < ǫ.

This finishes the proof of Theorem 3.6. •


Before we continue with the proof of Voronin’s universality theorem, we needsome arithmetical tools from the theory of diophantine approximation.

3.4. Diophantine approximation. In the theory of diophantine approxima-tions one investigates how good an irrational number can be approximated byrational numbers. This has plenty of applications in various fields of mathematicsand natural sciences.

For abbreviation we denote vectors of RN by x = (x1, . . . , xN ), we define τx =(τx1, . . . , τxN ) for τ ∈ R and x · y = x1y1 + . . . + xNyN . Further, for x ∈ RN

and γ ⊂ RN we write x ∈ γ mod 1 if there exists y ∈ ZN such that x − y ∈ γ.Moreover, we shall introduce the notion of Jordan volume of a region γ ⊂ RN .Therefore, we consider the sets of parallelepipeds γ1 and γ2 with sides parallel tothe axes and of volume Γ1 and Γ2 with γ1 ⊂ γ ⊂ γ2; if there are γ1 and γ2 suchthat lim supγ1 Γ1 coincides with lim infγ2 Γ2, then γ has the Jordan volume

Γ = lim supγ1

Γ1 = lim infγ2

Γ2.

The Jordan sense of volume is more restrictive than the one of Lebesgue, but ifthe Jordan volume exists it is also defined in the sense of Lebesgue and equal toit.

Weyl [69] proved

Theorem 3.8. Let a1, . . . , aN ∈ R be linearly independent over the field ofrational numbers, write a = (a1, . . . , aN ), and let γ be a subregion of the N-dimensional unit cube with Jordan volume Γ. Then

limT→∞

1

Tmeas τ ∈ (0, T ) : τa ∈ γ mod 1 = Γ.

Proof. From the definition of the Jordan volume it follows that for any ǫ > 0there exist two finite sets of open parallelepipeds ∏−

j and ∏+j inside the

unit cube such that

(3.14)⋃

∏−j ⊂ int(γ) ⊂ γ ⊂

⋃

∏+j

and

meas(

⋃

∏+j \⋃∏−

j

)

< ǫ;

here, as usual, M denotes the closure of the set M , and int(M) its interior.Denote by 1± the characteristic function of

⋃∏±j , i.e.

1±(x) =

1 if x ∈ ⋃∏±j ,

0 if x 6∈ ⋃∏±j .

Further, let 1 be the characteristic function of γ mod 1. Consequently,

0 ≤ 1−(x) ≤ 1(x) ≤ 1+(x) ≤ 1,


and∫

[0,1]N(1+(x) − 1−(x)) dx < ǫ,

where the integral is N -dimensional with dx = dx1 · · · dxN . Define

Φ(x) =

0 if |x| ≥ 12,

c exp(

−(

1x+ 1

2

+ 1x− 1

2

))

if |x| < 12,

where c is defined via∫ 1

2

− 12

Φ(x) dx = 1.

Consequently, Φ(x) is an infintely differentiable function, and hence the func-tions, given by

1±(x) = δ−N∫

[0,1]N1±(y)Φ

(

x1 − y1

δ

)

· · ·Φ(

xN − yNδ

)

dy

for 0 < δ < 1, are infinitely differentiable functions too. From (3.14) it followsthat for sufficiently small δ we have

0 ≤ 1−(x) ≤ 1(x) ≤ 1+(x) ≤ 1,

and

(3.15) 0 ≤∫

[0,1]N(1+(x) − 1−(x)) dx < 2ǫ.

We conclude

(3.16)

∫ T

0

1−(τa) dτ ≤ meas τ ∈ (0, T ) : τa ∈ γ mod 1 ≤∫ T

0

1+(τa) dτ

and

0 ≤∫ T

0

1+(τa) dτ −∫ T

0

1−(τa) dτ ≤ 2ǫT.

Both integrands above are infinitely differentiable functions which are 1-periodicin each variable. Thus, we have the Fourier expansion

1±(x) =∑

n∈ZN

c±n

exp(2πin · x),

where

c±n

=

∫

[0,1]N1±(x) exp(−2πin · x) dx.

Note that c±0

is the volume of⋃∏±

j . Integration by parts gives

c±n≪

N∏

j=1

(|nj| + 1)−k for k = 1, 2, . . . ,


where the implicit constant depends only on k. This shows that the Fourierseries converges absolutely, and hence, for every ǫ > 0, there exists a finite setM ⊂ ZN such that

1±(x) =∑

n∈Mc±n

exp(2πin · x) +R(x) with |R(x)| < ǫ.

This yields

1

T

∫ T

0

1±(τa) dτ =1

T

∫ T

0

∑

n∈Mc±n

exp(2πiτ n · a) dτ + θǫ

with some θ satisfying |θ| < 1. Consequently,

1

T

∫ T

0

1±(τa) dτ = c±0

+∑

0 6=n∈Mc±n

1

T

∫ T

0

exp(2πiτ n · a) dτ + θǫ.

Since the an are linearly independent over Q, we have n · a 6= 0 for n 6= 0. Itfollows for such n that

∫ T

0

exp(2πiτ n · a)) dτ ≪ 1.

Since ǫ > 0 is arbitrary, we obtain

limT→∞

1

T

∫ T

0

1±(τa) dτ = c±0.

Thus, we get in (3.16)

c−0− ǫ ≤ lim inf

T→∞

1

Tmeas τ ∈ (0, T ) : τa ∈ γ mod 1

≤ lim supT→∞

1

Tmeas τ ∈ (0, T ) : τa ∈ γ mod 1 ≤ c+

0+ ǫ

for any positive ǫ. From (3.15) it follows that 0 ≤ c+0− c−

0≤ 2ǫ. Now sending

ǫ→ 0, the theorem is proved. •

As an immediate consequence of Theorem 3.8 we get the classical inhomogeneousKronecker approximation theorem:

Corollary 3.9. Let α1, . . . , αN ∈ R be linearly independent over Q, let β1, . . . , βNbe arbitrary real numbers, and let q be a positive number. Then there exists anumber τ > 0 and integers x1, . . . , xN such that

|ταn − βn − xn| <1

qfor 1 ≤ n ≤ N.


We conclude with the notion of uniform distribution modulo 1. Let γ(τ) be acontinuous function with domain of definition [0,∞) and range RN . Then thecurve γ(τ) is said to be uniformly distributed mod 1 in RN if

limT→∞

1

Tmeas τ ∈ (0, T ) : γ(τ) ∈∏ mod 1 =

N∏

j=1

(βj − α1)

for every parallelepiped∏

= [α1, β1] × . . . × [αN , βN ] with 0 ≤ αj < βj ≤ 1 for1 ≤ j ≤ N . In a sense, a curve is uniformly distributed mod 1 if the number ofvalues which lie in any given measurable subset of the unit cube is proportionalto the measure of the subset.

In questions about uniform distribution mod 1 one is interested in the fractionalpart only. Hence, we define for a curve γ(τ) = (γ1(τ), . . . , γN(τ)) in RN

γ(τ) = (γ1(τ) − [γ1(τ)], . . . , γN(τ) − [γN(τ)]);

recall that [x] denotes the integral part of x ∈ R.

Theorem 3.10. Suppose that the curve γ(τ) is uniformly distributed mod 1 inRN . Let D be a closed and Jordan measurable subregion of the unit cube in RN

and let Ω be a family of complex-valued continuous functions defined on D. If Ωis uniformly bounded and equicontinuous, then

limT→∞

1

T

∫ T

0

f(γ(τ))1γD(τ) dτ =

∫

D

f(x) dx

uniformly with respect to f ∈ Ω, where 1γD(τ) is equal to 1 if γ(τ) ∈ D mod 1,

and equal to zero otherwise.

Proof. By the definition of the Riemann integral as a limit of Riemann sums,we have for any Riemann integrable function F on the unit cube in RN

(3.17) limT→∞

1

T

∫ T

0

F (γ(τ)) dτ =

∫

[0,1]NF (x) dx.

By the assumptions on Ω, for any ǫ > 0 there exist f1, . . . , fn ∈ Ω such that forevery f ∈ Ω there is an fj among them satisfying

supx∈D

|f(x) − fj(x)| < ǫ.

By (3.17) there exists T0 such that for any T > T0 and for each function f1, . . . , fnone has

∣

∣

∣

∣

∫

D

fj(x) dx − 1

T

∫ T

0

fj(γ(τ))1γD(τ) dτ

∣

∣

∣

∣

< ǫ.


Now, for any f ∈ Ω,∣

∣

∣

∣

∫

D

f(x) dx − 1

T

∫ T

0

f(γ(τ))1γD(τ) dτ

∣

∣

∣

∣

≤∣

∣

∣

∣

∫

D

fj(x) dx − 1

T

∫ T

0

fj(γ(τ))1γD(τ) dτ

∣

∣

∣

∣

+

∣

∣

∣

∣

∫

D

(f(x) − fj(x)) dx

∣

∣

∣

∣

+1

T

∣

∣

∣

∣

∫ T

0

(fj(γ(τ)) − f(γ(τ)))1γD(τ) dτ

∣

∣

∣

∣

By the appropriate choice of fj it follows from the estimates above that this isbounded by 3ǫ. Since ǫ > 0 is arbitrary, the assertion of the theorem follows. •

Besides, there is also the notion of uniformly distributed sequences, defined in asimilar way. It was proved by Hlawka [28] that the imaginary parts of the zerosof the zeta-function are uniformly distributed modulo 1.

3.5. Approximation in the mean — end of proof. We choose κ > 1 andǫ1 ∈ (0, 1) such that κr < 1

4and

max|s|≤r

∣

∣

∣f( s

κ

)

− f(s)∣

∣

∣< ǫ1.

Let Q = p ≤ z and E = s : −κr < σ ≤ 2,−1 ≤ t ≤ T. We shall estimate

(3.18) I :=

∫ 2T

T

∫∫

E

∣

∣ζ−1Q

(

s+ 34

+ iτ, 0)

ζ(

s + 34

+ iτ)

− 1∣

∣

2dσ dt dτ,

where 0 = (0, 0, . . .). By Theorem 2.10,

ζ(s+ iτ) =∑

n≤T

1

ns+iτ+O(T−σ).

This gives

I =

∫∫

E+ 34

∫ 2T

T

|ζ−1Q (s+ iτ, 0)ζ(s+ iτ) − 1|2 dτ dσ dt

≪∫∫

E+ 34

∫ 2T

T

∣

∣

∣

∣

∣

ζ−1Q (s+ iτ, 0)

∑

n≤T

1

ns+iτ− 1

∣

∣

∣

∣

∣

2

dτ dσ dt

+

∫∫

E+ 34

∫ 2T

T

T−σ|ζ−1Q (s+ iτ, 0)|2 dτ dσ dt,(3.19)

where E + 34

is the set of all s with s− 34∈ E . By definition,

ζ−1Q (s, 0) =

∏

p∈Q

(

1 − 1

ps

)

=∞∑

m=1p|m⇒p∈Q

µ(m)

ms;


recall that µ(m) is Mobius µ-function defined by (2.28). We may bound thesecond term appearing on the right-hand side of (3.19) by

T−2( 34−κr) max

s∈E+ 34

∫ 2T

T

|ζ−1Q (s+ iτ, 0)|2 dτ ≪ T 2κr− 1

2

∣

∣ζ−1Q

(

34− κr, 0

)∣

∣

2.

Furthermore, for T > z a simple computation gives

ζ−1Q (s, 0)

∑

n≤T

1

ns= 1 +

∑

z<k≤zzT

bkks

with bk =∑

m|kp|m⇒p∈Q;k≤mT

1.

By estimate (2.29) for the divisor function, we have

(3.20) |bk| ≤ d(k) ≪ kǫ for any ǫ > 0.

Hence, for T > z

∫ 2T

T

∣

∣

∣

∣

∣

ζ−1Q (s+ iτ, 0)

∑

n≤T

1

ns+iτ− 1

∣

∣

∣

∣

∣

2

dτ =

∫ 2T

T

∣

∣

∣

∣

∣

∑

z<k≤zzT

bkks+iτ

∣

∣

∣

∣

∣

2

dτ

= T∑

z<k≤zzT

|bk|2k2σ

+O

(

∑

0<ℓ<k≤zzT

|bkbℓ|(kℓ)σ

∣

∣

∣

∣

∣

∫ 2T

T

(

k

ℓ

)iτ

dτ

∣

∣

∣

∣

∣

)

.

Using estimate (3.20) with ǫ = ǫ12, the above is bounded by

T∑

k>z

d2(k)

k2σ+

∑

0<ℓ<k≤zzT

d(k)d(ℓ)

(kℓ)σ log kℓ

≪ Tz1−2σ+ǫ1 + (zzT )ǫ1∑

0<ℓ<k≤zzT

1

(kℓ)σ log kℓ

.

The sum on the right can be estimated by ((zzT )2−2σ + 1) log2(zzT ) similarly aswe did in the proof of Theorem 2.13. Thus, we finally arrive at

∫∫

E+ 34

∫ 2T

T

∣

∣

∣

∣

∣

ζ−1Q (s+ iτ, 0)

∑

n≤T

1

ns+iτ− 1

∣

∣

∣

∣

∣

2

dτ dσ dt

≪∫∫

E+ 34

(

Tz1−2σ+ǫ1 + (zzT )ǫ1((zzT )2−2σ + 1) log2(zzT ))

dσ dt

≪ z2κr+ǫ1− 12T.

In view of (3.19) we conclude that for any ǫ2 > 0

(3.21) I ≪ ǫ42T,


provided that z and T are sufficiently large, say z > z0 and T > T0, dependingonly on ǫ2. Define

AT =

τ ∈ [T, 2T ] :

∫∫

E+ 34

|ζ−1Q (s+ iτ, 0)ζ(s+ iτ) − 1|2 dσ dt < ǫ22

.

Then it follows from (3.19) and (3.21) that for sufficiently large z and T

(3.22) measAT > (1 − ǫ2)T,

which is surprisingly large. Application of Lemma 3.7 gives for τ ∈ AT

max|s|≤r

|ζ−1Q (s+ iτ, 0)ζ(s+ iτ) − 1| < Cǫ2,

where C is a positive constant, depending only on κ. For sufficiently small ǫ2 wededuce

max|s|≤r

∣

∣log ζ(

s+ 34

+ iτ)

− log ζQ(

s+ 34

+ iτ, 0)∣

∣ < 2Cǫ2,(3.23)

provided τ ∈ AT .

By Theorem 3.6 there exists a sequence of finite sets of prime numbers M1 ⊂M2 ⊂ . . . such that ∪∞

k=1Mk contains all primes and

limk→∞

max|s|≤κr

∣

∣

∣log ζMk

(

s+ 34, ω0

)

− f( s

κ

)∣

∣

∣= 0.(3.24)

Let ω′ = (ω′2, ω

′3, . . .) and ω = (ω2, ω3, . . .). By the continuity of log ζM

(

s+ 34, ω)

with respect to ω, for any ǫ1 > 0 there exists a positive δ with the property thatwhenever

(3.25) ‖ωp − ω′p‖ < δ for all p ∈Mk,

then

max|s|≤κr

∣

∣log ζMk

(

s+ 34, ω0

)

− log ζMk

(

s+ 34, ω′)∣

∣ < ǫ.(3.26)

Setting

BT =

τ ∈ [T, 2T ] :∥

∥

∥τlog p

2π− ωp

∥

∥

∥< δ

,

we get

1

T

∫

B

∫∫

|s|≤κr

∣

∣log ζQ(

s+ 34

+ iτ, 0)

− log ζMk

(

s+ 34

+ iτ, 0)∣

∣

2dσ dt dτ

=

∫∫

|s|≤κr

1

T

∫

BT

∣

∣log ζQ(

s+ 34

+ iτ, 0)

− log ζMk

(

s+ 34

+ iτ, 0)∣

∣

2dτ dσ dt.

Putting ω(τ) =(

τ log 22π, τ log 3

2π, . . .

)

, we may rewrite the inner integral as∫

BT

∣

∣log ζQ(

s+ 34, ω(τ)

)

− log ζMk

(

s+ 34, ω(τ)

)∣

∣

2dτ.


The logarithms of the prime numbers are linearly independent over Q (this fol-lows easily from the unique prime factorization of the integers). Thus, by Weyl’stheorem 3.8, the curve γ(τ) =

(

τ log 22π, τ log 3

2π, . . . , τ log pN

2π

)

is uniformly distributedmod 1. Application of Theorem 3.10 yields

limT→∞

1

T

∫

BT

∣

∣log ζQ(

s+ 34, ω(τ)

)

− log ζMk

(

s+ 34, ω(τ)

)∣

∣

2dτ

=

∫

D

∣

∣log ζQ(

s+ 34, ω)

− log ζMk

(

s+ 34, ω)∣

∣

2dµ

uniformly in s for |s| ≤ κr, where D is the subregion of the unit cube in RN givenby the inequalities (3.25) and dµ is the Lebesgue measure. By the definition ofζM(s, ω) it follows that for Mk ⊂ Q

ζQ(s, ω) = ζMk(s, ω)ζQ\Mk

(s, ω),

and thus∫

D

∣

∣log ζQ(

s+ 34, ω)

− log ζMk

(

s+ 34, ω)∣

∣

2dµ

=

∫

D

∣

∣log ζQ\Mk

(

s + 34, ω)∣

∣

2dµ = measD ·

∫

[0,1]N

∣

∣log ζQ\Mk

(

s+ 34, ω)∣

∣

2dµ.

Since

log ζQ\Mk

(

s+ 34, ω)

=∑

p∈Q\Mk

∞∑

n=1

exp(−2πinωp)

npn(s+34)

,

we obtain∫

[0,1]N

∣

∣log ζQ\Mk

(

s+ 34, ω)∣

∣

2dµ =

∑

p∈Q\Mk

∞∑

n=1

1

n2p2nσ+ 3n2

.

If Mk contains all primes ≤ yk, then

∑

p∈Q\Mk

∞∑

n=1

1

n2p2nσ+ 3n2

≪ y2κr− 1

2k .

Hence, we finally get

1

T

∫

BT

∫∫

|s|≤κr

∣

∣log ζQ(

s + 34

+ iτ, 0)

− log ζMk

(

s+ 34

+ iτ, 0)∣

∣

2dσ dt dτ

≪ y2κr− 1

2k measD.

As already noticed above, the curve γ(τ) is uniformly distributed mod 1. Hence,application of Theorem 3.8 shows that

limT→∞

1

TmeasBT = measD,


which implies for sufficiently large yk

meas

τ ∈ BT :

∫∫

|s|≤κr

∣

∣log ζQ(

s+ 34

+ iτ, 0)

− log ζMk

(

s+ 34

+ iτ, 0)∣

∣

2dσ dt

< yκr− 1

4k

> 12measD · T.

Now application of Theorem 3.10 yields

meas

τ ∈ BT : max|s|≤κr

∣

∣log ζQ(

s+ 34

+ iτ, 0)

− log ζMk

(

s+ 34

+ iτ, 0)∣

∣

< y15(κr−

14)

k

> 12measD · T.(3.27)

If we now take 0 < ǫ2 <12measD, then (3.22) implies

lim infT→∞

1

Tmeas AT ∩ BT > 0.

Thus, in view of (3.24) we may approximate f(

sκ

)

by log ζMk

(

s + 34, 0)

(inde-

pendent on τ), with (3.26) and (3.27) the latter function by log ζQ(

s+ 34, 0)

,

and finally with regard to (3.23) by log ζ(

s+ 34

+ iτ)

on a set AT ∩BT of τ withpositive measure. This finishes the proof of Theorem 3.1 (as well as Voronin’suniversality theorem 1.3). •

3.6. Reich’s discrete universality theorem and other related results.

Reich [56] and Bagchi [1] improved Voronin’s result significantly in replacingthe disk by an arbitrary compact set in the right half of the critical strip withconnected complement, and by giving a lucid proof in the language of probabilitytheory. The strongest version of Voronin’s theorem has the form:

Theorem 3.11. Suppose that K is a compact subset of the strip 12< σ < 1 with

connected complement, and let g(s) be a non-vanishing continuous function onK which is analytic in the interior of K. Then, for any ǫ > 0,

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : maxs∈K

|ζ(s+ iτ) − g(s)| < ǫ

> 0.

The topological restriction on K is necessary. This follows from basic facts inapproximation theory. Notice that the interior of a compact line segment K isempty and therefore the target function g only needs to be continuous and zero-free for such sets K. The restriction on g to be non-vanishing cannot be removedas we shall show in Section 4.2. The domain in which the uniform approximationof admissible target functions takes place is called the strip of universality. Inthe case of the zeta-function this strip of universality is the open right half ofthe critical strip. It is impossible to extend the universality property of the zeta-function to any region covering the critical line, since there are too many zerosof the zeta-function on the critical line (see also Garunkstis & Steuding [18]).


An interesting variation of Voronin’s theorem is due to Reich. In [56] he intro-duced the concept of discrete universality by restricting the approximating shiftsto arithmetic progressions. Surprisingly, this still leads to a positive lower densityfor the number of solutions to the corresponding approximation problem. Hereis Reich’s theorem in its strongest form:

Theorem 3.12. Suppose that K is a compact subset of the strip 12< σ < 1 with

connected complement, and let g(s) be a non-vanishing continuous function onK which is analytic in the interior of K. Then, for any ∆ > 0 and any ǫ > 0,

lim infN→∞

1

N♯

1 ≤ n ≤ N : maxs∈K

|ζ(s+ in∆) − g(s)| < ǫ

> 0.

Neither does Voronin’s theorem imply Reich’s theorem nor the other way around(by our current knowledge). Nevertheless, his argument follows in the main partsalong the lines of Voronin’s proof. The integral I given by (3.18) has to bereplaced by the sum

1

N

N∑

n=1

|ζ(s+ in∆) − ζQ(s+ in∆, 0)|2.

Using Gallagher’s lemma 2.17 (see also Montgomery [50]), the latter expressioncan be bounded by the corresponding integrals:

lim supN→∞

1

N

N∑

n=1

|ζ(s+ in∆) − ζQ(s+ in∆, 0)|2

≪ lim supT→∞

1

T

∫ T

0

|ζ(s+ iu) − ζQ(s+ iu, 0)|2 du

+

(

∆ lim supT→∞

1

T

∫ T

0

|ζ(s+ iu) − ζQ(s+ iu, 0)|2 du

)

12

×

×(

lim supT→∞

1

T

∫ T

0

|ζ ′(s + iu) − ζ ′Q(s+ iu, 0)|2 du

)

12

,

and the right-hand side can be treated as before. The remaining parts of theproof are very similar.

We shall briefly mention another line of investigation. Recently, Kaczorowski,Laurincikas & Steuding [32] studied shifts of universal Dirichlet series with re-spect to universality and their value-distribution. Assume that K1, . . . ,Kn aredisjoint compact subsets of 1

2< σ < 1 with connected complements. Let g(s) be

any non-vanishing continuous function, defined on⋃nj=1 Kj , which is analytic in

the interior. If now for any ǫ > 0, there exists a real number τ such that

maxs∈∪n

j=1Kj

|ζ(s+ iτ) − g(s)| < ǫ,


then also

max1≤j≤n

maxs∈Kj

|ζ(s+ iτ) − gj(s)| < ǫ,

where the gj(s) are defined as restriction of g(s) on Kj . Equivalently, one canconsider all gj(s) being defined on some compact subset K of 1

2< σ < 1 with

connected complement and study shifts of ζ(s). Let λ1, . . . , λn be complex num-bers, K be any compact set, and define Kj := s + λj : s ∈ K. Then theshifts

ζλj(s) := ζ(s+ λj)

are said to be jointly universal with respect to λ1, . . . , λn if, for every compactK with connected complement and for which the sets Kj are disjoint subsets of12< σ < 1, every family of (non-vanishing) continuous functions gj(s) defined on

K which are analytic in the interior, and for any ǫ > 0, we have

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max1≤j≤n

maxs∈K

: ζλj(s+ iτ) − gj(s)| < ǫ

> 0.

Clearly, the assumption on the Kj to be disjoint is necessary. According to theforegoing remarks we see that ζ(s) and ζ(s + iλ) can approximate uniformlyany pair of suitable target functions on sufficiently small disks simultaneouslyprovided that λ 6= 0. This simultaneous approximation property may be regardedas a first example of a phenomenon which is called joint universality.

A rolling stone gathers no moss! Here are the exercises for this chapter. As an im-mediate consequence of Voronin’s universality theorem one can obtain universality forcertain relatives of ζ(s):

Exercise 10. Show that ζ(s)−1 is universal.

It might be a good exercise to prove all technical details of the lengthy proof ofVoronin’s theorem which we left for the reader:

Exercise 11. Work out all estimates from Section 3.5.

Exercise 12. Show that the logarithm of the prime numbers are linearly independentover Q.

The latter assertion was essential for the application of Weyl’s approximation theorem3.8. Here is another application. Although the zeta-function has no zeros in its half-plane of absolute convergence it assumes arbitrarily small values:

Exercise 13. Use Kronecker’s approximation theorem, Corollary 3.9, to show that

infσ>1

|ζ(σ + it)| = 0.

The following two exercises might be not too easy; help can be found in [56, 61].


Exercise 14. Prove Theorem 3.11. For this aim use Mergelyan’s celebrated approxi-mation theorem.

Exercise 15. Prove Reich’s discrete universality theorem 3.12.

4. Applications, extensions, and open problems

The universality property of the Riemann zeta-function allows several interesting ap-

plications, maybe the most important one is functional independence. Besides we shall

also present Bagchi’s theorem which connects universality with the Riemann hypoth-

esis. Moreover, we discuss several generalizations and open questions.

4.1. Functional independence. We state some consequences of universality.To begin with we extend Bohr’s classical result about the denseness of the setof values taken by ζ(s) on a vertical line σ ∈ (1

2, 1). The following theorem is

essentially Voronin’s theorem 1.2 from the introduction:

Theorem 4.1. Let 12< σ < 1 be fixed, then the sets

(log ζ(σ + it), (log ζ(σ + it))′, . . . , (log ζ(σ + it))(n−1)) : t ∈ Rand

(ζ(σ + it), ζ ′(σ + it), . . . , ζ (n−1)(σ + it)) : t ∈ Rlie everywhere dense in Cn.

Proof. Suppose that we are given a vector (b0, b1, . . . , bn−1) ∈ Cn. Let

r = 14− 1

2min

σ − 12, 1 − σ

and define

f(s) =n−1∑

k=0

bkk!sk.

Obviously, f (k)(0) = bk for k = 0, 1, . . . , n− 1. By Cauchy’s formula, we have forany analytic function g(s) on |s| ≤ ρ

(4.1) g(k)(0) =k!

2πi

∮

|s|=ρ

g(s)

sk+1ds.

By Voronin’s universality theorem 1.3 the function f(s) can be approximated toarbitrary precision on the disk |s| ≤ r by log ζ

(

s+ 34

+ iτ)

for some τ . Hence,taking

g(s) = f(s) − log ζ(

s+ 34

+ iτ)

and ρ < r in (4.1), shows that (log ζ(σ + it), log ζ ′(σ + it), . . . , log ζ(σ+ it)(n−1))with fixed σ ∈ (1

2, 1) lies for some values of t as close to (f(0), f ′(0), . . . , f (n−1)(0)) =

(b0, b1, . . . , bn−1) as we want. This implies the statement for the first set.


We use induction onm to prove that for any (m+1)-tuple (a0, a1, . . . , am) ∈ Cm+1

with a0 6= 0, there exists (b0, b1, . . . , bm) ∈ Cm+1 for which

exp

(

m∑

k=0

bksk

)

≡m∑

k=0

akk!sk mod sm+1;

here the notation ≡ modsm+1 means that the power series expansion of thedifference of both sides of ≡ consists of no terms with sk for k < m + 1. Form = 0 one only has to choose b0 = log a0. By the induction assumption we mayassume that with some α

exp

(

m∑

k=0

bksk

)

≡m∑

k=0

akk!sk + αsm+1 mod sm+2.

Thus,

exp

(

m∑

k=0

bksk + βsm+1

)

≡ (1 + βsm+1)

(

m∑

k=0

akk!sk + αsm+1

)

mod sm+2.

Now, let bm+1 = β be the solution of the equation

βa0 + α =am+1

(m+ 1)!,

which exists by the restriction on a0 to be non-vanishing. This leads to

exp

(

m+1∑

k=0

bksk

)

≡m+1∑

k=0

akk!sk mod sm+2,

and hence the claim.

Finally, let

g(s) := exp

(

n−1∑

k=0

bksk

)

≡n−1∑

k=0

akk!sk mod sn.

By Voronin’s theorem 1.3 there exists a sequence τj, tending with j to infinity,such that

limj→∞

max|s|≤r

∣

∣ζ(

s+ 34

+ iτj)

− g(s)∣

∣ = 0

for some r ∈ (0, 14). In view of (4.1) we obtain

limj→∞

max|s|≤r−ǫ

∣

∣ζ (k)(

s+ 34

+ iτj)

− g(k)(s)∣

∣ = 0

for k = 1, . . . , n − 1 and any ǫ ∈ (0, r). By the same reasoning as above, thisproves the theorem. •

Further, universality implies functional independence:


Theorem 4.2. Let z = (z0, z1, . . . , zn−1) ∈ Cn. If F0(z), F1(z), . . . , FN(z) arecontinuous functions, not all identically zero, then there exists some s ∈ C suchthat

N∑

k=0

skFk(ζ(s), ζ′(s), . . . , ζ (n−1)(s)) 6= 0.

In particular it follows that the zeta-function is hypertranscendental, i.e., ζ(s)does not satisfy any algebraic differential equation. This solves one of Hilbert’sfamous problems which he posed at the International Congress of Mathemati-cians in Paris 1900. The first proof of the hypertranscendence of the zeta-functionwas given by Stadigh and in a more general setting by Ostrowski, both, of course,by a different reasoning (see [54]).

Proof. First, we shall show that if F (z) is a continuous function and

F (ζ(s), ζ ′(s), . . . , ζ (n−1)(s)) = 0

identically in s ∈ C, then F vanishes identically.

Suppose the contrary, i.e. F (z) 6≡ 0. Then there exists a ∈ Cn for whichF (a) 6= 0. Since F is continuous, there exist a neighbourhood U of a and apositive ǫ such that

|F (z)| > ǫ for z ∈ U.

Choosing an arbitrary σ ∈ (12, 1), application of Theorem 4.1 yields the existence

of some t for which

(ζ(σ + it), ζ ′(σ + it), . . . , ζ (n−1)(σ + it)) ∈ U,

giving the desired contradiction. This proves our claim, resp. the assertion ofthe theorem with N = 0.

Without loss of generality we may assume that F0(z) is not identically zero. Asabove there exist an open bounded set U and a positive ǫ such that

|F0(z)| > ǫ for z ∈ U.

Denote by M the maximum of all indices m for which

supz∈U

|Fm(z)| 6= 0.

For M = 0 the assertion of the theorem follows from the special case from above.Otherwise, we may choose an open subset V ⊂ U such that

infz∈V

|FM(z)| > ǫ

for some positive ǫ. By Theorem 4.1, there exists a sequence tj, tending with jto infinity, such that

(ζ(σ + itj), ζ′(σ + itj), . . . , ζ

(n−1)(σ + itj)) ∈ V.


This implies

limj→∞

∣

∣

∣

∣

∣

M∑

k=0

(σ + itj)kFk(ζ(σ + itj), ζ

′(σ + itj), . . . , ζ(n−1)(σ + itj))

∣

∣

∣

∣

∣

= ∞.

This proves the theorem. •

4.2. Self-recurrence and the Riemann hypothesis. It is a natural questionto ask whether the condition on g(s) to be non-vanishing is necessary or is itpossible to approximate uniformly functions having a zero by the zeta-function?The answer is negative.

To see this assume that g(s) is an analytic function on the disk |s| ≤ r with azero ξ in the interior of the disk but no zero on the boundary. An application ofRouche’s theorem shows that whenever the inequality

(4.2) max|s|=r

∣

∣ζ(

s + 34

+ iτ)

− g(s)∣

∣ < min|s|=r

|g(s)|

holds, ζ(

s+ 34

+ iτ)

has a zero inside |s| < r too. The zeros of an analyticfunction lie either discretely distributed or the function vanishes identically, andthus inequality (4.2) holds if the left-hand side is sufficiently small. If now forany ǫ > 0

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max|s|≤r

∣

∣ζ(

s+ 34

+ iτ)

− g(s)∣

∣ < ǫ

> 0,

then we expect ≫ T many τ in the interval [0, T ] each of which corresponds via(4.2) to a complex zero of ζ(s) in the strip 3

4− r < σ < 3

4+ r up to level T (for a

rigorous proof one has to consider the densities of values τ satisfying (4.2) whichcan be done along the lines of the proof of Theorem 4.3 below). This contradictsdensity theorem 2.16 which gives

N(

34− r, T

)

= o(T ).

Thus, a given function with a zero cannot be approximated uniformly by thezeta-function (in the sense of Voronin’s theorem)!

The above reasoning shows that the location of the complex zeros of Riemann’szeta-function is intimately related to the universality property. This observa-tion is essential for the following observation which links universality with theRiemann hypothesis.

Bohr introduced the fruitful notion of almost periodicity to analysis. An analyticfunction f(s), defined on some vertical strip a < σ < b, is called almost periodicif, for any positive ε and any α, β with a < α < β < b, there exists a lengthℓ = ℓ(f, α, β, ε) > 0 such that in every interval (t1, t2) of length ℓ there is anumber τ ∈ (t1, t2) such that

|f(σ + it+ iτ) − f(σ + it)| < ε for any α ≤ σ ≤ β, t ∈ R.


Bohr [7] proved that any Dirichlet series is almost-periodic in its half-plane ofabsolute convergence. Bohr discovered an interesting relation between the Rie-mann hypothesis and almost periodicity; indeed, his aim in introducing the con-cept of almost periodicity might have been Riemann’s hypothesis. His approachfailed for the Riemann zeta-function but he succeeded for some relatives, namelyDirichlet L-functions associated to some arithmetical functions called characters(defined in §4.4). Bohr showed that if χ is a non-principal character, then theanalogue of Riemann’s hypothesis for the Dirichlet L-function L(s, χ) is equiva-lent to the almost periodicity of L(s, χ) in the half-plane σ > 1

2. The condition on

the character looks artificial but is necessary for Bohr’s reasoning. His argumentrelies in the main part on diophantine approximation applied to the coefficientsof the Dirichlet series representation. The Dirichlet series for L(s, χ) with anon-principal character χ converges throughout the critical strip, however theDirichlet series for the zeta-function does not.

More than half a century later Bagchi [1] proved that the Riemann hypothesisis true if and only if for any compact subset K of the strip 1

2< σ < 1 with

connected complement and for any ǫ > 0

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : maxs∈K

|ζ(s+ iτ) − ζ(s)| < ǫ

> 0.

In [2], Bagchi generalized this result in various directions; in particular for Dirich-let L-functions to arbitrary characters. One implication of his proof relies essen-tially on Voronin’s universality theorem which, of course, was unknown to Bohr.Later, Bagchi [3] gave another proof in the language of topological dynamics,independent of universality, and therefore this property, equivalent to Riemann’shypothesis, is called strong recurrence. Following [61] we extend Bagchi’s resultslightly to

Theorem 4.3. Let θ ≥ 12. Then ζ(s) is non-vanishing in the half-plane σ > θ

if and only if, for any ǫ > 0, any z with θ < Re z < 1, and for any 0 < r <minRe z − θ, 1 − Re z,

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max|s−z|≤r

|ζ(s+ iτ) − ζ(s)| < ǫ

> 0.

Proof. If Riemann’s hypothesis is true, we can apply Voronin’s universality the-orem 1.3 with g(s) = ζ(s), which implies the strong recurrence. More generally,the non-vanishing of ζ(s) for σ > θ would allow to approximate ζ(s) by shiftsζ(s+ iτ) uniformly on appropriate subsets of the strip θ < σ < 1. The idea forthe proof of the other implication is that if there is at least one zero to the rightof the line σ = θ, then the strong recurrence property implies the existence ofmany zeros, in fact too many with regard to density theorem 2.16.

Suppose that there exists a zero ξ of ζ(s) with Re ξ > θ. Without loss ofgenerality we may assume that Im ξ > 0. We shall show that there exists a


disk with center ξ and radius r, satisfying the conditions of the theorem, and apositive ǫ such that

(4.3) lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max|s−z|≤r

|ζ(s+ iτ) − ζ(s)| < ǫ

= 0.

Locally, the zeta-function has the expansion

(4.4) ζ(s) = c(s− ξ)m +O(

|s− ξ|m+1|)

with some non-zero c ∈ C and m ∈ N. Now assume that for a neighbourhoodKδ := s ∈ C : |s− ξ| ≤ δ of ξ the relation

(4.5) maxs∈Kδ

|ζ(s+ iτ) − ζ(s)| < ǫ ≤ min|s|=δ

|ζ(s)|

holds; the second inequality is fulfilled for sufficiently small ǫ (by an argumentalready discussed above). Then Rouche’s theorem implies the existence of a zeroρ of ζ(s) in

Kδ + iτ := s ∈ C : |s− iτ − ξ| ≤ δ.We say that the zero ρ of ζ(s) is generated by the zero ξ. With regard to (4.4)and (4.5) the zeros ξ and ρ = β + iγ are intimately related; more precisely,

ǫ > |ζ(ρ) − ζ(ρ− iτ)| = |ζ(ρ− iτ)| ≥ |c| · |ρ− iτ − ξ|m − O(δm+1).

Hence,

|ρ− iτ − ξ| ≤(

ǫ

|c|

)1m

+O(

δ1+ 1m

)

.

In particular,

12< Re ξ − 2

(

ǫ

|c|

)1m

< β < 1,

and

|γ − (τ + Im ξ)| < 2

(

ǫ

|c|

)1m

,

for sufficiently small ǫ and δ = o(ǫm+1). Next we have to count the generatedzeros in terms of τ . Two different shifts τ1 and τ2 can lead to the same zero ρ,but their distance is bounded by

|τ1 − τ2| < 4

(

ǫ

|c|

)1m

.

If we now write

I(T ) :=⋃

j

Ij(T ) :=

τ ∈ [0, T ] : maxs∈Kδ

|ζ(s+ iτ) − ζ(s)| < ǫ

,


where the Ij(T ) are disjoint intervals, it follows that there are

≥[

14

( |c|ǫ

)1m

meas Ij(T )

]

+ 1 > 14

( |c|ǫ

)1m

meas Ij(T )

many distinct zeros according to τ ∈ Ij(T ), generated by ξ. The number ofgenerated zeros is a lower bound for the number of all zeros. For the number ofall zeros having real part > Re ξ − 2( ǫ

|c|)1m up to level T this yields

♯

ρ = β + iγ : β > Re ξ − 2

(

ǫ

|c|

)1m

, 0 < γ < T + Im ξ + 2

(

ǫ

|c|

)1m

≥ 14

( |c|ǫ

)1m

meas I(T ).

This in combination with density theorem 2.16 yields

meas I(T ) = o(T ),

which implies (4.3). The theorem is proved. •

4.3. The effectivity problem. The known proofs of universality theoremsare ineffective, giving neither an estimate for the first approximating shift τ norbounds for the positive lower density. There are attempts by Good, Laurincikas,and Garunkstis which we shall now shortly discuss.

If the Riemann hypothesis is true, then

(4.6) log∣

∣ζ(

12

+ it)∣

∣ = O

(

log t

log log t

)

as t→ ∞. This is a significant improvement of the bound for ζ(s) on the criticalline predicted by the Lindelof hypothesis, but we may ask whether it is thecorrect order? On the contrary, Montgomery [51] proved, for fixed 1

2< σ < 1,

there exists an absolute positive constant C such that

maxT

13 (σ− 1

2 )<t≤T|ζ(σ + it)| ≥ exp

(

C(log t)1−σ

(log log t)σ

)

;(4.7)

the same estimate is valid for σ = 12

under assumption of the truth of theRiemann hypothesis. By a different method, Balasubramanian & Ramachandra[4] obtained the same estimate for σ = 1

2unconditionally. These results were

only slight improvements of earlier results, however, some probabilistic heuristicssuggest these estimates to be best possible, i.e., the quantity in (4.7) describesthe exact order of growth of ζ(s).∗∗

∗∗The recent random matrix model predicts significantly larger values: in analogy to largedeviations for characteristic polynomials one may expect that the estimate in (4.6) gives thetrue order.


The proofs of the universality theorem, neither Voronin’s original one nor Bagchi’sprobabilistic proof and its variations, do not give any information about the ques-tion how soon a given target function is approximated by ζ(s+ iτ) within a givenrange of accuracy, and Montgomery’s approach does not give us any idea of theshape of the set of values of ζ(s) on vertical lines. Good [20] combined Voronin’suniversality theorem with the work of Montgomery on extreme values of thezeta-function. This enabled him to complement Voronin’s qualitative picturewith Montgomery’s quantitative estimates. Recently, Garunkstis [16] proved an-other, more satisfying effective universality theorem along the lines of Voronin’sproof and building on Good’s ideas. In particular, his remarkable result showsthat if f(s) is analytic in |s| ≤ 0.06 with max|s|≤0.06 |f(s)| ≤ 1, then for any0 < ǫ < 1

2there exists a

(4.8) 0 ≤ τ ≤ exp(

exp(

10ǫ−13))

such thatmax

|s|≤0.0001

∣

∣log ζ(

s+ 34

+ iτ)

− f(s)∣

∣ < ǫ,

and further

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max|s|≤0.0001

∣

∣log ζ(

s+ 34

+ iτ)

− f(s)∣

∣ < ǫ

≥ exp(

−ǫ−13)

.(4.9)

The original theorem is too complicated to be given here. Laurincikas foundanother approach which gives conditional effective results subject to certain as-sumptions on the speed of convergence of a related limit distribution. However,the rate of convergence of weakly convergent probability measures in the spaceof analytic functions is not understood very well.

Following [59, 61], we shall investigate the converse problem, namely effectiveupper bounds for the upper density of universality:

Theorem 4.4. Suppose that g(s) is a non-constant, non-vanishing analytic func-tion defined on |s| ≤ r, where r ∈ (0, 1

4). Then, for any sufficiently small ǫ > 0,

d(ǫ, r, g) := lim supT→∞

1

Tmeas

τ ∈ [0, T ] : max|s|≤r

∣

∣ζ(

s+ 34

+ iτ)

− g(s)∣

∣ < ǫ

.

= o(ǫ).

Thus, the decay of d(ǫ, r, g) with ǫ→ 0 is more than linear in ǫ for any suitablefunction g.

Proof. Assume that g(s) is a non-constant, non-vanishing analytic functiondefined on Br := s : |s| ≤ r. Then there exists a complex number c in theinterior of g(Br) (which is not empty since g(s) is not constant) such that

(4.10) g(s) = c+ γ(s− λc) +O(

|s− λc|2)


for some λc of modulus less than r and some γ 6= 0; this means that λc is a c-valueof g(s) of multiplicity one. To see this suppose that for all c in the interior ofg(Br) the local expansion is different from (4.10), i.e., g′(s) vanishes identicallyin the interior. Then g is a constant function, contradicting the assumption ofthe theorem.

Now suppose that

max|s|=r

∣

∣

ζ(

s + 34

+ iτ)

− c

− g(s) − c∣

∣ < min|s|=r

|g(s) − c|.

Then, by Rouche’s theorem, ζ(z) has at least one c-value ρc in z = s+ 34

+ iτ :|s| < r. We rewrite the latter inequality as

(4.11) max|s|≤r

∣

∣ζ(

s+ 34

+ iτ)

− g(s)∣

∣ < ǫ ≤ min|s|=r

|g(s) − c|.

By Voronin’s universality theorem 1.3 the first inequality holds for a set of τwith positive lower density. The second one follows for sufficiently small ǫ fromthe fact that c = g(λc) has positive distance to the boundary of g(Br). Thus, ac-value of g(s) generates many c-values of ζ(z).

Assume that ρc = sj + 34

+ iτj with |sj| < r for j = 1, 2. It follows from (4.11)that

(4.12) |g(sj) − c| = |g(sj) − g(λc)| < ǫ.

Since g′(λc) = γ 6= 0, there exists a neighborhood of c where the inverse functiong−1 exists and is a one-valued continuous function. By continuity, (4.12) implies

(4.13) |sj − λc| < ε = ε(ǫ),

where ε(ǫ) tends with ǫ to zero; since g(s) behaves locally as a linear function by(4.10), we have ε(ǫ) ≍ ǫ. Now (4.13) implies

(4.14) |τ2 − τ1| = |s1 − s2| ≤ |s1 − λc| + |λc − s2| < 2ε.

Denote by Ij(T ) the disjoint intervals in [0, T ] such that (4.11) is valid exactlyfor

τ ∈⋃

j

Ij(T ) =: I(T ).

Inequality (4.14) implies that in every interval Ij(T ) lie at least

1 +

[

1

2εmeas Ij(T )

]

≥ 1

2εmeas Ij(T )

c-values ρc of ζ(s) in the strip 12< σ < 1. Thus, the number Nc(T ) of these

c-values ρc (counting multiplicities) satisfies the estimate

2εNc(T ) ≥ meas I(T ).(4.15)

Next we locate the real parts of these c-values more precisely. Obviously, by(4.13),

Reλc + 34− ε < Re ρc = Re sj + 3

4< Reλc + 3

4+ ε.


Clearly, for sufficiently small ε this range for the c-values lies in the interior ofthe strip of universality. Hence, if we let Nc(σ1, σ2, T ) count all c-values of ζ(s)in the region σ1 < σ < σ2, 0 < t ≤ T (counting multiplicities), then we canrewrite (4.15) as

meas I(T ) ≤ 2εNc

(

Reλc + 34− ε,Reλc + 3

4+ ε, T

)

.(4.16)

In view of the universality theorem 1.3 there exists an increasing sequence (Tk)with limk→∞ Tk = ∞ such that for any δ > 0

meas I(Tk) ≥ ( d(ǫ, r, g) − δ)Tk.

Consequently, this together with (4.16) leads to

( d(ǫ, r, g) − δ)Tk ≤ 2εNc

(


4+ ε, T

)

.

Sending δ → 0, yields

(4.17) d(ǫ, r, g) ≤ lim supT→∞

2ε

TNc

(


4+ ε, T

)

.

Here we shall use a classical theorem of Bohr & Jessen [10]: for any complexc 6= 0,

(4.18) limT→∞

1

TNc

(


4+ ε, T

)

= o(1)

as ǫ → 0. Substituting this in (4.17) implies (4.4) and the assertion of thetheorem follows. •As substitute of the deep result of Bohr & Jessen we can give an alternativeargument at the expense of a slightly weaker estimate as follows. For this purposedefine

ℓ(s) =

11−c(ζ(s) − c) if c 6= 1,

2s−1(ζ(s) − 1) if c = 1.

Then the c-values of ζ(s) correspond one-to-one to the zeros of ℓ(s) (having thesame multiplicity) and

(4.19) ℓ(σ + it) = 1 + λ−σ−it +O(Λ−σ)

with some constants λ,Λ satisfying 1 < λ < Λ, as σ → ∞. Hence, there existsa real number σ2 > 1 such that there are no zeros of ℓ(s) to the right of σ2 − 1.Now let Nc(σ, T ) count the number of c-values of ζ(s) with real-part greater thanσ and imaginary part in (0, T ], resp. the zeros of ℓ(s) (counting multiplicities).Then Littlewood’s lemma 2.14 yields

∫ σ2

σ1

Nc(σ, T ) dσ =1

2πi

∫

Rlog ℓ(s) ds+O(1),(4.20)

where R is the rectangular contour with vertices σ1, σ2, σ1 + iT, σ2 + iT with12< σ1 < 1 < σ2. Here the error term arises from the pole of ζ(s) (to define here

log ℓ(s) we choose the principal branch of the logarithm on the real axis whereasfor other points s the value of the logarithm is obtained by continuous variation).


A standard application of Jensen’s inequality (as in Section 2.4) shows that theright-hand side of (4.20) can be replaced by

1

2π

∫ T

0

log |ζ(σ1 + it)| dt+O(T ) ≤ T

4πlog

(

1

T

∫ T

0

|ζ(σ1 + it)|2 dt

)

+O(T ).

The right-hand side can be estimated by the mean-square theorem 2.13. Thisgives in (4.20)

∑

Re ρc>σ10<Im ρc≤T

(Re ρc − σ1) ≪ T,

as T → ∞; here the sum on the left-hand side is taken over all c-values ρc ofζ(s) (not necessarily generated by λc). Since, for 1

2< σ1 < σ3,

Nc(σ3, T ) ≤ 1

σ3 − σ1

∑

Re ρc>σ10<Im ρc≤T

(Re ρc − σ1),

we may estimate

Nc

(


4+ ε, T

)

≤ Nc(12(1

2+ Reλc + 3

4− ε), T ) ≪ T.

Thus, we deduce from (4.17) the bound d(ǫ, r, g) = O(ǫ), which is slightly weakerthan the bound from the theorem.

We return to the problem of effectivity in the universality theorem for ζ(s).Comparing the lower bound (4.9) of Garunkstis from the beginning with theupper bounds of Theorem 4.4, we may ask which estimate is more close to thetruth. If a given function g(s) is sufficiently nice, i.e., if its logarithm f(s) satisfiesthe condition of Garunkstis’ theorem, then

exp(

−ǫ−13)

≪ d(ǫ, g, r) ≤ d(ǫ, g, r) = o(ǫ).

Given a positive ǫ and a sufficiently small disk K located in the right half ofthe critical strip, in principle, estimate (4.8) allows to find algorithmically anapproximating τ such that

maxs∈K

|ζ(s+ iτ) − g(s)| < ǫ;

unfortunately, we cannot expect a reasonable running time for such an algorithmwhen ǫ is small. Anyway, this idea was considered in a project by Garunkstis,Slezeviciene–Steuding & Steuding. For certain smooth functions g(s) and ratherlarge values for ǫ approximating shifts τ were computed. Quite many of theseτ were found but it is impossible to deduce any information about the densityof universality as long as the running time of the underlying algorithm cannotbe significantly improved. Nevertheless, we shall illustrate this attempt towardeffective universality by some data.


Consider the exponential function on a small disk centered at the origin. Forexample, we have

max|s|≤0.006

∣

∣ζ(

s+ 34

+ 12 963 i)

− exp(s)∣

∣ < 0.05.

The shift τ is a positive integer since the discrete variant of universality, Reich’stheorem 3.12, was used in order to simplify the algorithm.

1 2 3 4 5 6

0.99

0.9925

0.995

0.9975

1.0025

1.005

1 2 3 4 5 6

-0.01

-0.005

0.005

Figure 6. ζ(s+ 34+ 12 963 i) ≈ exp(s) for s = 0.006 exp(iφ) with

0 ≤ φ ≤ 2π. On the left the real parts, on the right the imaginaryparts are plotted; the zeta-function is given in black, exp in grey.

We conclude with another application of discrete universality. The argument inthe proof of Theorem 4.4 which gave us a factor ǫ for the upper bound does notapply if we consider discrete shifts and so, in general, we do not get an upperbound which tends with ǫ to zero. However, via Reich’s discrete universalitytheorem 3.12 and (4.18) one can prove

(4.21) lim supN→∞

1

N♯

1 ≤ n ≤ N : maxs∈K

|ζ (s+ in∆) − g(s)| < ǫ

= o(1)

as ǫ→ 0 for any real ∆ > 0 and any suitable function g on K. This is of interestwith respect to an estimate of Reich concerning small values of Dirichlet serieson arithmetic progressions. Let F (s) be a Dirichlet series, not identically zero,which has a half-plane of absolute convergence σ > σa, an analytic continuationto σ > σm (σm < σa) except for at most a finite number of poles on the lineσ = σa, such that its mean square exists and F (s) is of finite order of growth inany closed strip in σm < σ < σa. Reich [57] proved under these assumptions forany σ > σm, σ 6= σa, any sufficiently small ǫ > 0, and any real ∆, neither beingequal to zero nor of the form 2πℓ log( q

r)−1 with positive integers ℓ, q, r and q 6= r,

that the relation

lim supN→∞

1

N♯ 1 ≤ n ≤ N : |F (σ + in∆)| < ǫ < 1

holds. In particular, it follows that F (σ + i∆n) cannot converge to zero asn → ∞, and hence sn = σ + i∆n cannot be a sequence of zeros of F (s). Itshould be noticed that Reich’s theorem also includes estimates for c-values onarithmetic progressions (since with F (s) also F (s) − c satisfies the conditions).


In the special case of the Riemann zeta-function we note the following improve-ment of Reich’s theorem:

Corollary 4.5. Let c be any constant and σ ∈ (12, 1), and ∆ > 0 be real. Then

limǫ→0

lim supN→∞

1

N♯ 1 ≤ n ≤ N : |ζ(σ + in∆) − c| < ǫ = 0.

In particular, there does not exist an arithmetic progression sn = σ+ i∆n (withσ and ∆ as in the theorem) on which ζ(s) converges to any complex number c.

We sketch the easy proof. Let g(s) be a non-constant, non-vanishing, analyticfunction defined on a small disk centered at σ ∈ (1

2, 1) such that its closure lies

inside the strip of universality for the zeta-function. Further assume that

|g(s) − c| < ǫ;

this choice for g(s) is certainly possible for any given complex number c. By thetriangle inequality,

|ζ(σ + in∆) − c| ≤ |ζ(σ + in∆) − g(s)| + |g(s) − c|for any s. Hence, applying (4.21) yields

lim supN→∞

1

N♯ 1 ≤ n ≤ N : |ζ(σ + in∆) − c| < ǫ = o(1)

as ǫ→ 0. This is the assertion of the corollary. •

There are remarkable results for a related problem. Extending a classical resultof Putnam on the impossibility of an infinite vertical arithmetic progressions ofzeros (or even approximate zeros), van Frankenhuijsen [15] recently proved that

ζ(σ + in∆) = 0 for 0 < |n| < N

with fixed σ,∆ > 0 and N ≥ 2 cannot hold for

N ≥ 60

(

∆

2π

)1σ−1

log ∆.

It is conjectured that there are no arithmetic progressions at all. Moreover, thereare even no zeros known of the form 1

2+ iγ and 1

2+ i2γ. It is conjectured that

the ordinates of the nontrivial zeros of ζ(s) are linearly independent over Q.

4.4. L-functions and joint universality. A special role in number theory isplayed by multiplicative arithmetical functions and their associated generatingfunctions. Multiplicative functions respect the multiplicative structure of N: anarithmetic function f is called multiplicative if f(1) = 1 and

f(m · n) = f(m) · f(n)

for all coprime integers m,n; if the latter identity holds for all integers, f is saidto be completely multiplicative.


Let q be a positive integer. A Dirichlet character χ mod q is a non-vanishinggroup homomorphism from the group (Z/qZ)∗ of prime residue classes modulo qto C. The character, which is identically one, is called principal, and is denotedby χ0. By setting χ(a) = 0 on the non-prime residue classes any Dirichletcharacter extends via χ(n) = χ(a) for n ≡ a mod q to a completely multiplicativearithmetical function. For σ > 1, the Dirichlet L-function attached to a characterχ mod q is given by

L(s, χ) =

∞∑

n=1

χ(n)

ns=∏

p

(

1 − χ(p)

ps

)−1

.

The zeta-function ζ(s) may be regarded as the Dirichlet L-function to the prin-cipal character χ0 mod 1. It is possible that for values of n coprime with q thecharacter χ(n) may have a period less than q. If so, we say that χ is imprimitive,and otherwise primitive; the principal character is not regarded as a primitivecharacter. Every non-principal imprimitive character is induced by a primitivecharacter. Two characters are non-equivalent if they are not induced by the samecharacter. The characters to a common modulus are pairwise non-equivalent. Ifχ mod q is induced by a primitive character χ∗ mod q∗, then

L(s, χ) = L(s, χ∗)∏

p|q

(

1 − χ∗(p)

ps

)

.(4.22)

Being twists of the Riemann zeta-function with multiplicative characters, Dirich-let L-functions share many properties with the zeta-function. For instance, thereis an analytic continuation to the complex plane, only with the difference thatL(s, χ) is regular at s = 1 if and only if χ is non-principal. Furthermore, L-functions to primitive characters satisfy a functional equation of the Riemann-type; namely,

( q

π

)s+δ2

Γ

(

s+ δ

2

)

L(s, χ) =τ(χ)

iδ√q

( q

π

)1+δ−s

2Γ

(

1 + δ − s

2

)

L(1 − s, χ),

where δ := 12(1 − χ(−1)) and

τ(χ) :=∑

a mod q

χ(a) exp

(

2πia

q

)

is the Gauss sum attached to χ. One finds similar zero-free regions (with theexception of hypothetical Siegel zeros on the real line), density theorems, andalso for Dirichlet L-functions it is expected that the analogue of the Riemann hy-pothesis holds; the so-called Generalized Riemann hypothesis states that neitherζ(s) nor any L(s, χ) has a zero in the half-plane σ > 1

2.

Dirichlet L-functions were constructed by Dirichlet to investigate the distributionof primes in arithmetic progressions. The main ingredient in his approach are the


orthogonality relations for characters linking prime residue classes with charactersums:

1

ϕ(q)

∑

a mod q

χ(a) =

1 if χ = χ0,0 otherwise,

(4.23)

and its dual variant

1

ϕ(q)

∑

χ mod q

χ(a) =

1 if a ≡ 1 mod q,0 otherwise,

valid for a coprime with q, where ϕ(q) is Euler’s ϕ-function which counts thenumber of prime residue classes mod q. By the latter relation a suitable linearcombination of characters can be used as indicator function of prime residueclasses modulo q. Using similar techniques as for ζ(s), one can prove a primenumber theorem for arithmetic progressions: if π(x; a mod q) denotes the num-ber of primes p ≤ x in the residue class a mod q, then, for a coprime with q,

π(x; a mod q) ∼ 1

ϕ(q)π(x).(4.24)

This shows that the primes are uniformly distributed in the prime residue classes.One can prove also an asymptotic formula with error term and the theorem ofPage-Siegel-Walfisz gives an asymptotic formula which is uniform in a smallregion of values q. Under assumption of the Generalized Riemann hypothesisone has

π(x; a mod q) =1

ϕ(q)li (x) +O

(

x12 log(qx)

)

for x ≥ 2, q ≥ 1, and a coprime with q, the implicit constant being absolute.There are plenty of results which hold if Riemann’s hypothesis is true. Often onecan replace this assumption by the celebrated theorem of Bombieri-Vinogradovwhich states that, for any A ≥ 1,

∑

q≤Qmax

a mod q(a,q)=1

maxy≤x

∣

∣

∣

∣

π(y; a mod q) − 1

ϕ(q)li(y)

∣

∣

∣

∣

≪ x

(log x)A+Qx

12 (logQx)6.

This shows that the error term in the prime number theorem (4.24) is, on av-

erage over q ≤ x12 (log x)−A−7, of comparable size as predicted by the Riemann

hypothesis. All these results can be found in [33, 62].

We return to universality. Voronin [68] proved that a collection of DirichletL-functions to non-equivalent characters can uniformly approximate simultane-ously non-vanishing analytic functions. This is called joint universality and itsstrongest version is given in:

Theorem 4.6. Let χ1 mod q1, . . . , χℓ mod qℓ be pairwise non-equivalent Dirich-let characters, K1, . . . ,Kℓ be compact subsets of the strip 1

2< σ < 1 with con-

nected complements. Further, for each 1 ≤ j ≤ ℓ, let gj(s) be a continuous


non-vanishing function on Kj which is analytic in the interior of Kj. Then, forany ǫ > 0,

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max1≤j≤ℓ

maxs∈Kj

|L(s+ iτ, χj) − gj(s)| < ε

> 0.

The proof of this joint universality theorem can be found in [61]. The proof usesthe orthogonality relation (4.23). Although this relation is a rather simple fact,the resulting independence is essential for joint universality. Consider a characterχ mod q induced by another character χ∗ mod q∗. Because of (4.22) it followsthat both L(s, χ∗) and L(s, χ) cannot approximate uniformly a given functionjointly.

Another type of universality was discovered by Bagchi. In [1], he proved univer-sality for Dirichlet L-functions with respect to the characters; more precisely, ifK is a compact subset of 1

2< σ < 1 with connected complement and g(s) is a

non-vanishing continuous function on K, which is analytic in the interior, then,for any sufficiently large prime number p and any ε > 0, there exist a Dirichletcharacter χ mod p such that

maxs∈K

∣

∣L(

s+ 34, χ)

− g(s)∣

∣ < ε;

moreover, the latter inequality holds for more than cp characters χ mod p, wherec is a positive constant (recall that there are ϕ(p) = p− 1 characters χ mod p).

Another interesting class of universal L-functions are those built from modularforms. First we recall some basic facts from the theory of automorphic formswhich can be found in the book [31] of Iwaniec. Denote by H the upper half-plane z := x + iy ∈ C : y > 0, and let k and N be positive integers, k beingeven. The subgroup

Γ0(N) :=

(

a b

c d

)

∈ SL2(Z) : c ≡ 0 mod N

of the full modular group SL2(Z) is called Hecke subgroup of level N or congru-ence subgroup modN . A holomorphic function f(z) on H is said to be a cuspform of weight k and level N , if

f

(

az + b

cz + d

)

= (cz + d)kf(z)

for all z ∈ H and all matrices(

a b

c d

)

∈ Γ0(N),

and if f vanishes at all cusps. The vanishing of f at the cusps is equivalent withthe boundedness of the mapping

z := x+ iy 7→ yk|f(z)|2


on H. In this case, f possesses for z ∈ H a Fourier expansion:

(4.25) f(z) =

∞∑

n=1

c(n) exp(2πinz).

The cusp forms on Γ0(N) of weight k form a finite dimensional complex vectorspace, denoted by Sk(Γ0(N)), with the Petersson inner product, defined by

〈f, g〉 =

∫

H/Γ0(N)

f(z)g(z)ykdx dy

y2

for f, g ∈ Sk(Γ0(N)). Suppose that M |N . If f ∈ Sk(Γ0(M)) and dM |N , thenz 7→ f(dz) is a cusp form on Γ0(N) of weight k too. The forms which maybe obtained in this way from divisors M of the level N with M 6= N span asubspace Sold

k (Γ0(N)), called the space of oldforms. Its orthogonal complementwith respect to the Petersson inner product is denoted Snew

k (Γ0(N)). For n ∈ Nwe define the Hecke operator T (n) by

T (n)f =1

n

∑

ad=n

ak∑

0≤b<df

(

az + b

d

)

for f ∈ Sk(Γ0(N)). The operators T (n) are multiplicative, i.e., T (mn) =T (m)T (n) for coprime m,n, and they encode plenty of arithmetic informationabout modular forms. The theory of Hecke operators implies the existence of anorthogonal basis of Snew

k (Γ0(N)) made of eigenfunctions of the operators T (n)for n coprime with N . By the multiplicity-one principle of Atkin & Lehner, theelements f of this basis are in fact eigenfunctions of all T (n), i.e., there existcomplex numbers λf(n) for which

T (n)f = λf(n)f and c(n) = λf(n)c(1) for all n ∈ N.

Furthermore, it follows that the first Fourier coefficient c(1) of such an f is non-zero. Such a simultaneous eigenfunction is said to be an eigenform. A newformis defined to be an eigenform that does not come from a space of lower leveland is normalized to have c(1) = 1. The newforms form a finite set which isan orthogonal basis of the space Snew

k (Γ0(N)). For instance, Ramanujan’s cuspform

(4.26)

∞∑

n=1

τ(n) exp(2πinz) := exp(2πiz)

∞∏

n=1

(1 − exp(2πinz))24

is a normalized eigenform of weight 12 to the full modular group, and hence anewform of level 1. Ramanujan [55] conjectured that the coefficients τ(n) are

multiplicative and satisfy the estimate |τ(p)| ≤ 2p112 for every prime number p.

The multiplicativity was proved by Mordell [52], in particular by the beautifulformula

τ(m)τ(n) =∑

d|(m,n)

d11τ(mn

d2

)

.


The estimate was shown by Deligne. More precisely, Deligne [14] proved for thecoefficients of any newform f of weight k the estimate

|c(n)| ≤ nk−12 d(n).(4.27)

In the 1930s, Hecke [27] started investigations on modular forms and Dirichletseries with a Riemann-type functional equation; his studies were completed byAtkin & Lehner (for newforms). Here we shall focus on newforms. Given anewform f with Fourier expansion (4.25), we define the associated L-function by

L(s, f) =

∞∑

n=1

c(n)

ns.(4.28)

In view of the classical bound d(n) ≪ nǫ it follows from (4.27) that the series(4.28) converges absolutely for σ > k+1

2. By the theory of Hecke operators, it

turns out that the Fourier coefficients of newforms are multiplicative. Hence, inthe half-plane of absolute convergence, there is an Euler product representation:

(4.29) L(s, f) =∏

p|N

(

1 − c(p)

ps

)−1∏

p∤N

(

1 − c(p)

ps+

1

p2s+1−k

)−1

.

Hecke, resp. Atkin & Lehner, proved that L(s, f) has an analytic continuationto an entire function and satisfies the functional equation

Ns2 (2π)−sΓ(s)L(s, f) = ω(−1)

k2N

k−s2 (2π)s−kΓ(k − s)L(k − s, f),

where ω = ±1 is the eigenvalue of the Atkin-Lehner involution (0 −N1 0

) onSk(Γ0(N)). Hecke proved a converse theorem which gives a characterizationof these L-functions by their functional equation; this beautiful result general-izes Hamburger’s theorem for the Riemann zeta-function (see Titchmarsh [63],§2.13).

Laurincikas & Matsumoto obtained a universality theorem for L-functions at-tached to normalized eigenforms of the full modular group. Laurincikas, Mat-sumoto & Steuding [41] extended this result to newforms:

Theorem 4.7. Suppose that f is a newform of weight k and level N . Let K be acompact subset of the strip k

2< σ < k+1

2with connected complement, and let g(s)

be a continuous non-vanishing function on K which is analytic in the interior ofK. Then, for any ǫ > 0,

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : maxs∈K

|L(s+ iτ, f) − g(s)| < ǫ

> 0.

Laurincikas & Matsumoto [40] obtained also a joint universality theorem for L-functions associated with newforms twisted by characters. Let f ∈ Sk(Γ0(N))


be a newform with Fourier expansion (4.25) and let χ be a Dirichlet charactermod q where q is coprime with N . The twisted L-function is defined by

Lχ(s, f) =∞∑

n=1

c(n)

nsχ(n).

As in the non-twisted case (4.28), this Dirichlet series has an Euler product andextends to an entire function.

Theorem 4.8. Let q1, . . . , qn be positive integers coprime with N and let χ1 modq1, . . . , χn mod qn be pairwise non-equivalent character. Further, for 1 ≤ j ≤ n,let gj be a continuous function on Kj which is non-vanishing in the interior,where Kj is a compact subset of the strip s ∈ C : k

2< σ < k+1

2 with connected

complement. Then, for any ǫ > 0,

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : max1≤j≤n

maxs∈Kj

|Lχj(s+ iτ, f) − gj(s)| < ε

> 0.

The proof relies on a joint limit theorem due to Laurincikas and some kind ofprime number theorem for the coefficients of cusp forms with respect to arith-metic progressions, namely

∑

p≤xp≡a mod q

c(p)2p1−k ∼ 1

ϕ(q)

x

log x,(4.30)

where a is coprime with q. The proof of the latter formula uses ideas of Rankin.

By Wiles’ celebrated proof of the Shimura-Taniyama conjecture for semistablemodular forms [70] (which led to the proof of Fermat’s last theorem), and theproof by Breuil et al. [12] of the general case, every L-function attached to anelliptic curve over the rationals is the L-function to some newform of weight 2 forsome congruence subgroup. Consequently, Theorem 4.7 yields the universalityof L-functions associated with elliptic curves. Laurincikas & Steuding [42] usedTheorem 4.8 to give an example of jointly universal L-functions associated withelliptic curves. Here one may choose any finite family of elliptic curves of theform

Em : Y 2 = X3 −m2X with squarefree m ∈ N;

these curves were first studied in Tunnell’s work on the congruent number prob-lem. For this family one can avoid Wiles’ proof of the Shimura-Taniyama-Weilconjecture and show more or less directly that the L-function associated withE1 corresponds to a newform f ∈ S2(Γ0(32)) and that the L-function to Em is atwist of E1 with the Kronecker symbol

(

m.

)

.

4.5. The Linnik-Ibragimov conjecture. Meanwhile universality has beenproved for quite many Dirichlet series. We list some significant examples.


A number field K is a finite algebraic extension of Q. The Dedekind zeta-functionof a number field K is given by

ζK(s) =∑

a

1

N(a)s=∏

p

(

1 − 1

N(p)s

)−1

,

where the sum is taken over all non-zero integral ideals, the product is taken overall prime ideals of the ring of integers of K, and N(a) is the norm of the ideala. The Riemann zeta-function may be regarded as the Dedekind zeta-functionfor Q. Universality for the Dedekind zeta-function was first obtained by Voronin[68] and Gonek [19] for some special cases, and in full generality by Reich [56].Here the strip of universality is restricted to 1 − min1

2, 1d < σ < 1, where d

is the degree of K over Q. This restriction depends on the mean-square half-plane for ζK(s); it is conjectured that for any Dedekind zeta-function the stripof universality can be extended to the open right half of the critical strip. Insome cases more is known, namely, if K is abelian (e.g., a subfield of a cyclotomicfield), then ζK(s) splits into a product of Dirichlet L-functions to pairwise non-equivalent characters. Using the joint universality for these L-functions, it iseasy to deduce the unrestricted universality of ζK(s) in 1

2< σ < 1.

There are other interesting examples which are strongly universal: they can ap-proximate functions with zeros on a set of positive lower density. The first exam-ple is the logarithm of the Riemann zeta-function as we have seen by Theorem3.1 (and of course the same argument applies to all universal Euler products asfor example ζK(s)). Next we present a completely different example.

For 0 < α ≤ 1, λ ∈ R, the Lerch zeta-function is given by

L(λ, α, s) =∞∑

n=0

exp(2πiλn)

(n+ α)s.

This series converges absolutely for σ > 1. The analytic properties of L(λ, α, s)are quite different, if λ ∈ Z or not. If λ 6∈ Z, the series converges for σ > 0 andL(λ, α, s) can be continued analytically to the whole complex plane. For λ ∈ Zthe Lerch zeta-function becomes the Hurwitz zeta-function

ζ(s, α) =

∞∑

m=0

1

(m+ α)s;

this function has an analytic continuation to C except for a simple pole at s = 1with residue 1. Denote by λ the fractional part of a real number λ. Setting

λ+ = 1 − λ and λ− =

1 if λ ∈ Z,λ otherwise,


one can prove the functional equation

L(λ, α, 1 − s) =Γ(s)

(2π)s

(

exp(

2πi(

s4− αλ−

))

L(−α, λ−, s)

+ exp(

2πi(

− s4

+ αλ+))

L(α, λ+, s))

.

Twists with additive characters destroy the point symmetry of Riemann-typefunctional equations. Gonek [19] and Bagchi [1] (independently) obtained stronguniversality for the Hurwitz zeta-function ζ(s, α) if α is transcendental or ra-tional 6= 1

2, 1. Laurincikas [37] extended this result by proving that the Lerch

zeta-function L(λ, α, s) is strongly universal if λ is not an integer and α is tran-scendental. All examples of strongly universal Dirichlet series do not have anEuler product and have many zeros in their region of universality; indeed, theproperty of approximating analytic functions with zeros is intimately related tothe distribution of zeros of the Dirichlet series in question. Euler products forwhich the analogue of Riemann’s hypothesis is expected should not be capableof approximating functions with zeros.

Roughly speaking, there are two methods to prove universality. Firstly, one cantry to mimic Voronin’s proof or Bagchi’s probabilistic approach. This soundsmore simple than it actually is, because one has to assure many analytic andarithmetic properties of the function in question. The second way is to find arepresentation as a linear combination or a product of jointly universal func-tions. All known proofs of universality of the first type depend on a certain kindof independence. For instance, the logarithms of the prime numbers are linearlyindependent over Q (we used this property in the proof of Voronin’s universalitytheorem when we applied Weyl’s refinement of Kronecker’s approximation theo-rem). Another example are the numbers log(n+α) with non-negative integral nwhich are linearly independent over Q if α is transcendental. In order to proveuniversality for the Hurwitz zeta-function, the first type of proof yields the resultaimed at for transcendental α. If α is rational 6= 1

2, 1, one can find a represen-

tation of ζ(s, α) as a linear combination of non-equivalent Dirichlet L-functionsfor which we have the joint universality theorem. In the cases α = 1

2and α = 1

the Hurwitz zeta-function has an Euler product representation and is equal tothe Riemann zeta-function for α = 1, resp., for α = 1

2,

ζ(

s, 12

)

= 2sL(s, χ),

where χ is the unique character mod 2. In both rational cases the Hurwitz zeta-function is universal but does not have the strong universality property. It is aninteresting open problem whether ζ(s, α) is universal or even strongly universalif α is algebraic irrational.

It was conjectured by Linnik and Ibragimov that all reasonable functions givenby Dirichlet series and analytically continuable to the left of the half-plane ofabsolute convergence are universal. Here we need to explain what is meant by


’reasonable’. For example, put a(n) = 1 if n = 2k with k ∈ N0, and a(n) = 0otherwise. Then

∞∑

n=1

a(n)

ns=

∞∑

k=0

1

2ks= (1 − 2−s)−1,

and obviously, this function is far away from being universal. So one has toask for natural conditions needed to prove universality. In [60, 61] a rathergeneral universality theorem for an axiomatically defined set of L-functions wasproved. If one is willing to accept some widely believed conjectures from numbertheory (e.g., the Ramanujan-Petersson conjecture, Langlands conjectures), thenthis class contains all arithmetically relevant L-functions. A satisfying jointuniversality result for this class is not yet found.

No pains - no gains! The following exercises may be used to repeat the whole contentof these notes; for some help we refer to [33, 62]:

Exercise 16. Prove the prime number theorem for arithmetic progressions.

Exercise 17. Prove universality for a Dirichlet L-function L(s, χ).

Exercise 18. Use Reich’s discrete universality theorem 3.12 in order to prove thatthe Riemann hypothesis is true if and only if for any ǫ > 0, any real number ∆ > 0,any z with 1

2 < Re z < 1, and any 0 < r < minRe z − 12 , 1 − Re z,

lim infN→∞

1

N♯

1 ≤ n ≤ N : max|s−z|≤r

|ζ(s + i∆n) − ζ(s)| < ǫ

> 0.

Extend the assertion to Dirichlet L-functions.

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Jorn Steuding E-mail: [email protected]: Institut fur Mathematik, Universitat Wurzburg, Am Hubland, 97 074 Wurzburg,

Germany

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