physics of the riemann hypothesis

Colloquium: Physics of the Riemann hypothesis

Daniel Schumayer* and David A.W. Hutchinson

Jack Dodd Centre for Quantum Technology, Department of Physics, University of Otago,Dunedin, New Zealand 9016

(Received 10 July 2008; published 29 April 2011; publisher error corrected 4 May 2011)

Physicists become acquainted with special functions early in their studies. Consider our perennial

model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in

quantum mechanics. Here a particular number-theoretical function is chosen, the Riemann zeta

function, and its influence on the realm of physics is examined and also how physics may be

suggestive for the resolution of one of mathematics’ most famous unconfirmed conjectures, the

Riemann hypothesis. Does physics hold an essential key to the solution for this more than 100-year-

old problem? In this work numerous models from different branches of physics are examined, from

classical mechanics to statistical physics, where this function plays an integral role. This function is

also shown to be related to quantum chaos and how its pole structure encodes when particles can

undergo Bose-Einstein condensation at low temperature. Throughout these examinations light is

shed on how the Riemann hypothesis can highlight physics. Naturally, the aim is not to be

comprehensive, but rather focusing on the major models and aim to give an informed starting point

for the interested reader.

DOI: 10.1103/RevModPhys.83.307 PACS numbers: 02.10.De, 02.30.Gp, 02.70.Hm

CONTENTS

I. Introduction 307

II. Historical Background and ‘‘Mathematical Necessities’’ 307

III. Connections to Physics 310

A. Classical mechanics 310

B. Quantum mechanics 312

1. Scattering state models 314

2. Bound state models 317

C. Nuclear physics 319

D. Condensed matter physics 320

E. Statistical physics 322

IV. Conclusion 326

I. INTRODUCTION

Counting, in the broadest sense, is probably the oldest

mathematical activity and not even uniquely ours. Evenanimals can distinguish one, two, and three, maybe just byrecognizing a pattern, but only humans have developed anabstract language, mathematics, or more specifically numbertheory, which accurately describes the properties of numbers.

In the following we focus on the border between physicsand number theory, and, more precisely, how the Riemann

zeta function �ðsÞ appears in quite different areas of physics.This Colloquium does not intend to be comprehensive, butrather we offer a panoramic view and give a feeling as to whymany physicists find beauty in the structure of this seeminglyrandom function and what one might learn from it. We collect

examples from diverse realms of physics, from classicalmechanics to condensed matter physics, where the Riemannzeta function or its ‘‘descendants’’ play a significantrole. Because of space limitations we do not aspire to be

mathematically precise in our derivations, but we give physi-cal arguments to support results and also direct the interestedreader to relevant sources.

II. HISTORICAL BACKGROUND AND ‘‘MATHEMATICAL

NECESSITIES’’

Natural numbers form the basis of our arithmetic, withvarious operations defined among these numbers. All of uslearn to use four basic operations: addition, subtraction,multiplication, and division. The latter, division, hides oneof the most enigmatic internal structures of the set of thenatural numbers, namely, that there are special numbers, theprimes, among the natural numbers which cannot be dividedby any other natural number, other than unity and themselves,without a remainder. Euclid of Alexandria proved that thereare infinitely many such numbers. Later, Eratosthenes ofCyrene gave a theoretical algorithm, a sieve, for finding theseprimes among the natural numbers. Despite all efforts in thelast 2000 years, the efficient determination as to whether agiven number is prime or not still proves to be a remarkablechallenge.

It is not hard to understand why the distribution of primescould captivate the imagination of many mathematiciansand physicists. These numbers seem to obey two contra-dictory principles. First, they seem to appear randomlyamong composite numbers, but second, they also appear toobey strict rules governing their distribution.

Apart from Euclid’s, numerous proofs exist for the infin-itude of the prime numbers (Ribenboim, 1991). Euler, at theearly age of 30, proved a stronger statement (Euler, 1737),

Xp prime

1

p¼ 1: (1)

This formula clearly proves Euclid’s statement but it alsodemonstrates the frequent occurrence of prime numbers*[email protected]

REVIEWS OF MODERN PHYSICS, VOLUME 83, APRIL–JUNE 2011

0034-6861=2011=83(2)=307(24) 307 � 2011 American Physical Society

http://dx.doi.org/10.1103/RevModPhys.83.307

among composite numbers. A natural continuation of hiswork is the analysis of the arithmetic properties of the series,P

n�k. Substituting k ¼ 1 into this expression we recover thewell-known, divergent harmonic series. Conversely, if k > 1the summation converges. Euler (1737) also showed—usingthe fundamental theorem of arithmetic—that this series canbe written as an infinite product over the prime numbers, p,such that

�ðkÞ ¼ X1n¼1

1

nk¼ Y

p

�1� 1

pk

��1: (2)

One may interpret through this relationship that the primenumbers construct the �ðkÞ function. Since p denotes a primenumber and k > 1, none of the factors in this product can bezero. Therefore, we can conclude that �ðkÞ does not have anyzeros if k > 1.

Bernhard Riemann, who was the first to apply the tools ofcomplex analysis to this function, proved that the functiondefined by the infinite summation (Riemann, 1859),

�ðsÞ ¼ X1n¼1

1

ns; (3)

can be analytically continued over the complex s plane,except for s ¼ 1. This analytic continuation of the functionis called the Riemann zeta function. Here we follow thetraditional notation, with s denoting a complex number s ¼�þ it, where � and t are real numbers and i is the usualimaginary unit.

Riemann also derived a functional equation, containing the�ðsÞ function, which is valid for all complex s and exhibitsmirror symmetry around the � ¼ 1=2 vertical line, called thecritical line, such that

��s=2�

�s

2

��ðsÞ ¼ ��ð1�sÞ=2�

�1� s

2

��ð1� sÞ: (4)

One should note that the zeta function stands on both sides,on the left-hand side with argument s, while on the right-handside with argument 1� s. This relationship between �ðsÞ and�ð1� sÞ provides some insight regarding the location of thezeros of this function. We examine the half-line for which�< 0, and t ¼ 0. The products on either side can be zero if atleast one of the factors is zero. On the right-hand side of (4)all the prefactors of the zeta function are non-negative and donot have any zeros. On the other side, however, the �ð�=2Þfunction has simple poles at all even negative integers. Theequation can hold only if �ð�Þ has simple zeros at the samelocations. These zeros are called trivial, because their loca-tions are inherited from the � function. The same argumentalso shows that all other zeros of the �ðsÞ function have to liein the 0 � � � 1 region, called the critical strip. The zeroslocated in this strip are the nontrivial zeros of the Riemannzeta function. It can also be shown that the nontrivial zeros �are arranged symmetrically, with respect to both the criticalline and the t ¼ 0 axis. Figure 1 shows the pole and zerostructure of �ðsÞ on the complex s plane including the pos-sible zeros off the critical line.

Thus far the statements about the zeros of �ðsÞ and theirlocations on the complex plain have been simple. However,the distribution of the nontrivial zeros holds one of the most

intriguing and enigmatic mathematical mysteries of the lastcentury and a half. It is embarrassingly easy to poseRiemann’s conjecture: All nontrivial zeros of �ðsÞ have theform � ¼ 1=2þ it, where t is a real number. In other words,all nontrivial zeros lie on the critical line. In 1900 Hilbertnominated the Riemann hypothesis as the eighth problem onhis list of compelling problems in mathematics (Hilbert,1902). Since then not just professional mathematicians butmathematical soldiers of fortune tried, and still try, to verifyits validity. The stakes are high. Whoever proves or disprovesthis hypothesis engraves his name on the tablets of the historyof mathematics, and may also receive financial incentive fromthe Clay Mathematics Institute.1

During the last century, the Riemann hypothesis was recastinto many equivalent mathematical statements. A few of themare purely number theoretical in origin, such as the Mertensconjecture, which we will later discuss in the context of aspecial Brownian motion, but other redefinitions are verymuch cross disciplinary. A more advanced mathematicalintroduction to the history of the Riemann hypothesis andits equivalent statements can be found in an excellent mono-graph and compendium (Borwein et al., 2008), which isreadable not just at the expert but also at the undergraduatelevel.

The distribution of the �ðsÞ zeros, with real part equal to1=2, has thus attracted significant interest. One of the mathe-matics’ giants has proven that infinitely many zeros do lie

s

σ

t

− 2− 4− 6

trivial zeros

1

pole

CRITICALSTRIP

CRITICALLINE

NON-TRIVIALZEROS

complex continuation original domain

FIG. 1. The ‘‘anatomy’’ of the Riemann zeta function on the

complex s plane. The black dots represent the zeros of �ðsÞ,including possible zeros which do not lie on the critical line.

1See http://www.claymath.org/millennium/Riemann_Hypothesis.

308 Daniel Schumayer and David A.W. Hutchinson: Colloquium: Physics of the Riemann hypothesis

Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011

on the critical line (Hardy, 1914); however, Riemann’sconjecture is much stronger, requiring all the zeros to be onthe critical line. In 1942 Selberg proved

N0ðTÞ>CT lnðTÞ ðC> 0 and T � T0Þ; (5)

i.e., the number of zeros of the form s ¼ 1=2þ it (0 �t � T), denoted by N0ðTÞ, grows as T lnðTÞ at least for largeT. Three decades later, Levinson (1974) showed that at leastone-third of the nontrivial zeros are on the critical line whichwas later incrementally improved to two-fifths (Conrey,1989). This small step over a period of 20 years is indicativeof the difficulty of the Riemann hypothesis.

We now return to the linkage between the �ðsÞ zeros andprime numbers. Equation (2) shows the strong connectionbetween the �ðsÞ function and the prime numbers. Thisrelationship can be made even more explicit if one examineshow the number of primes below a given threshold behavesas this threshold is increased. Based on empirical evidence,many mathematicians, e.g., Legendre, Gauss, and Chebyshev(Dickson, 2005), have conjectured that the prime-countingfunction �ðxÞ ¼ jfpjp is prime and p � xgj asymptoticallybehaves as the logarithmic integral LiðxÞ. This conjecture isknown nowadays as the prime number theorem afterHadamard (1896) and de la Vallee-Poussin (1896) indepen-dently gave rigorous proofs of this statement. Interestingly,this theorem has a geometrical interpretation: The primenumber theorem is equivalent to the assertion that no zerosof �ðsÞ lie on the � ¼ 1 boundary of the critical strip.

Riemann (1859) published, although von Mangoldt (1895)provided the rigorous proof, the following explicit formulafor the prime-counting function �ðxÞ:

�ðxÞ ¼ X1n¼1

�ðnÞn

Jðx1=nÞ; (6)

where

JðxÞ ¼ LiðxÞ � limT!1

� Xj�j�T

Ei½� logðxÞ��

þZ 1

x

dt

ðt2 � 1Þt logðtÞ � logð2Þ:

Here �ðnÞ is the Mobius function,2 � denotes the nontrivialzeros of the Riemann �ðsÞ function, and LiðxÞ and EiðxÞ standfor the logarithmic and exponential integrals,3 respectively.Therefore, whoever knows the distribution of the nontrivial

zeros of �ðsÞ will also know the distribution of the primenumbers.

Selecting only the first terms of the summand in Eq. (6)exactly reproduces the prime number theorem, i.e.,

�ðxÞ ffi LiðxÞ ffi x

lnðxÞ : (7)

This observation leads us to conclude that LiðxÞ gives themain contribution to �ðxÞ, while the other terms representcorrections, similar to a perturbative calculation in physics—an analogy to which we will return. Figure 2 shows the prime-counting function �ðxÞ and its various approximations. Onemay notice that the leading term LiðxÞ captures the tendencyof �ðxÞ well, and the appearance of the oscillations showshow the zeros �n influence and refine the agreement. Asx ! 1 the curves of LiðxÞ and �ðxÞ will practically coincideon a similar plot.

One may define a density for the complex, nontrivialRiemann zeta zeros as

dðNÞ ¼ Xk

�ðN � �kÞ; (8)

where � is the Dirac-delta distribution. Following Sir MichaelBerry (1985), the spectral density can be separated into asmooth and an oscillatory part, dðNÞ ¼ �dðTÞ þ doscðTÞ, as

�dðTÞ ¼ 1

2�ln

�T

2�

�þ 1� 1

2�þOðT�1Þ; (9a)

doscðTÞ ¼ � 1

�

Xp

X1r¼1

lnðpÞ cos½rT lnðpÞ�ffiffiffiffiffipr

p ; (9b)

where the external summation of doscðTÞ runs over the primenumbers p. The oscillatory part, therefore, gives the fluctua-tions as individual contributions from each prime number p,labeled by an integer r, corresponding to the prime power pr.Based on the smooth density of Riemann zeros one may

0

10

20

30

40

50

0 50 100 150 200

Prim

e co

untin

g fu

nctio

n, π

(x),

and

its

appr

oxim

atio

n, π

10(x

)

x

π10(x)π(x)Li(x)

1

2

3

4

3 5 7

π30(x)π(x)

FIG. 2. The approximation (6) for the prime-counting function

�ðxÞ (dashed line) using only the first term LiðxÞ (dash-dotted line)

and using the first ten nontrivial pairs of zeros of the Riemann �ðsÞfunction (solid line). In the inset we restricted the range to [2, 10]

and used the first 30 nontrivial zeros.

2The Mobius function is defined as follows: �ð1Þ ¼ 1, �ðnÞ ¼ 0if n has a square divisor, and �ðp1p2 � � �pkÞ ¼ ð�1Þk if all pi’s are

different. Thus �ð2Þ ¼ �1 and �ð12Þ ¼ 0, and �ð21Þ ¼ 1.3The notation for the logarithmic integral is ambiguous in the

literature. There are two definitions

I1ðxÞ ¼Z x

0

dt

lnðxÞ and I2ðxÞ ¼Z x

2

dt

lnðxÞ ;where I1 is interpreted as a Cauchy principal value. These integrals

differ only by a constant number. Depending on the book the reader

may consult, either I1ðxÞ or I2ðxÞ is denoted with LiðxÞ. Here, weprefer the former.

Daniel Schumayer and David A.W. Hutchinson: Colloquium: Physics of the Riemann hypothesis 309


derive the number of positive, nontrivial zeros up to a fixedvalue of T0:

Nð� < T0Þ ¼Z T0

0

�dðTÞdT ¼ T0

2�ln

�T0

2�

�� T0

2�: (10)

Changing the variable to T ¼ lnðT0=2�Þ and recasting ourresult using T , we obtain

NðT Þ / eT ; (11)

i.e., the number of �ðsÞ zeros below T increases exponen-tially. Although at this point this change of variable seemssomewhat arbitrary, we will see later that it further strength-ens the similarity between the zeros of �ðsÞ and the periodicorbits of a chaotic system, where the number of periodicorbits also increases exponentially.

Finally, we note the fruitful and diverse area of extensionsof the Riemann zeta function. These generalized zeta func-tions also occur throughout physics,4 primarily in modernquantum field theories. This topic, however, is far beyond thescope of this Colloquium and we suggest Elizalde’s mono-graph (Elizalde, 1995) as an introduction and Lapidus’s book(Lapidus, 2008) for a more authoritative study.

III. CONNECTIONS TO PHYSICS

A. Classical mechanics

In this section we discuss those models of classical me-chanics, such as billiards, which lead to the introduction ofthe notion of integrability and chaos. This development ofideas gave birth to a new paradigm, since it provided aninsight into how the spectrum of quantized analogs of clas-sical systems is connected to classical paths.

Classical mechanics, in its Lagrangian and Hamiltonianforms, is the exemplar for physics in the modern sense. Themajor theories, e.g., statistical and quantum mechanics, arefirst expressed in the language of analytical mechanics withthe development traced to the enlightenment. Although a fewanalytically solvable models, e.g., the Kepler two-body prob-lem and the harmonic oscillator, gave confidence in themachinery of mechanics, it was soon realized that there areimportant cases, e.g., the three-body problem, where one notjust cannot solve the equations of motion analytically, but themotion is proven to be chaotic (Celletti and Perozzi, 2007).This behavior is very peculiar and at first sight seems puz-zling, since the governing equations are deterministic, yet theactual motion seems to behave randomly. The celestial rele-vance of this three-body problem was so fundamental andenticing that King Oscar II of Sweden and Norway offered aprize for the person who could solve the following problem(Barrow-Green, 1994):

For an arbitrary system of mass points which attract eachother according to Newton’s law, assuming that no two points

ever collide, give the coordinates of the individual points forall time as a sum of a uniformly convergent series whoseterms are made up of known functions.

Although this problem had not been solved, Poincare wasawarded this illustrious prize for his impressive contribution.His work revolutionized the analysis of such chaoticallybehaving systems, although one had to wait nearly 100 yearsfor this revolution to really happen.

In classical mechanics we distinguish a special class ofsystems, the integrable dynamical system, which possesses asmany independent integrals of motion In (action variables) asdegrees of freedom N. For these systems the Hamiltoniancan be expressed as a function of these action variables,namely, H ¼ H ðI1; . . . ; INÞ, and the equations of motion(n ¼ 0; 1; . . . ; N)

d’n

dt¼ � @H

@Inand

dIndt

¼ @H@’n

(12)

are easy to solve: In ¼ const and ’n ¼ ’n;0 þ!nt. A theo-

rem of topology then guarantees that these N constants ofmotion, provided they are independent of each other, definean N-dimensional torus and each trajectory with constantenergy lies on that torus. Therefore, as a specific case, thedynamics described by a one-dimensional time-independentHamiltonian is necessarily integrable. In order to considerchaotic dynamics one has to either introduce a time-dependent Hamiltonian or increase the degrees of freedomto 2 or higher.

One of the ‘‘simplest’’ generic models with 2 or moredegrees of freedom is that of classical billiards. These aredynamical systems, where a particle has constant energy andmoves in a finite volume, which may contain impenetrableobstacles. Whenever the particle reaches the boundary itsuffers specular reflection. Depending on the shape of thebilliard, the motion can be integrable or chaotic. The analysisof a circular billiard (see Fig. 3) is straightforward due to therotational symmetry. The incident angle remains the same ateach bounce and each impact can be calculated from theprevious one by rotating the circle twice that angle.Therefore, if the incident angle is a rational multiple of �,i.e., m�=n, the trajectory is periodic with period n and

FIG. 3. A circular billiard and a Bunimovich stadium, which is a

rectangle smoothly joined by semicircles. Two different types of

trajectories, periodic orbits (1) and nonperiodic trajectories (2), are

also depicted.

4The interested reader can find an invaluable source of number

theoretical papers having some connection to physics on the Web

site of Matthew R. Watkins: http://empslocal.ex.ac.uk/people/staff/

mrwatkin//zeta/physics.htm



therefore finite, otherwise it is infinite. In this latter case, the

points where the ball hits the wall will be uniformly distrib-

uted along the circumference of the circle. It was also proven

by Jacobi that in the latter case every interval of the circle

contains points of the trajectory.Before we step beyond billiards and generalize the idea of

periodic orbits, the origin of trace formulas, we take a short

detour around a recent result (Bunimovich and Dettmann,

2005) regarding the circular billiard (see Fig. 4). As dis-

cussed, due to rotational symmetry, or in other words, the

conservation of angular momentum, this billiard model is

integrable and the trajectory is fully described by two angles

� and c , the angle around the circumference measured from

a predetermined point and the incident angle of the trajectory

at the boundary, respectively. With these variables the dy-

namics is governed by the mapping ð�; c Þ � ð�þ ��2c ; c Þ, where all angles are taken modulo 2� and the ball

travels with unit velocity. The phase space of this system can

be described by Birkhoff’s coordinates constructed from two

angles: the arc-length coordinate q ¼ � (measured in radians

and modulo 2�), and the tangential momentum coordinate

defined as p ¼ sinðc Þ. By convenient normalization, the arc

length of the billiard is unity and the velocity of the ball is

also unity, the phase space is restricted to 0 � q < 2�, and�1< p< 1. This choice also introduces a natural unit time

step, the time elapsed between consecutive bounces, �t ¼2 cosðc Þ. The movement of the ball can, therefore, be repre-

sented by a possibly infinite series of points inside this phase-

space area. Despite the rather artificial appearance of this

model, the electromagnetic field in optical or microwave

cavities can be modeled by such billiards (Stockmann and

Stein, 1990; Nockel and Stone, 1997; Alt et al., 1998;

Harayama et al., 2001). Since these experimental billiards

are not ideal, it is interesting to examine what happens to the

dynamics of this system if we cut a small window(s) along the

reflective boundary, thereby, naturally introducing dissipation

or ‘‘leakage.’’ It is natural to ask the questions: What is the

probability PðnÞ of a ball leaving the billiard after n bounces,

what is the mean number of bounces hni before the ball

escapes, or, similarly, what is the probability PðtÞ that escapetakes at least time t?

For strongly chaotic billiards the latter probability decaysexponentially, while for integrable billiards, such as thecircular one, it softens to only power-law decay (Bauer andBertsch, 1990) and can be qualitatively understood using asimple geometrical argument. The probability p that the ballescapes in a bounce is proportional to the size of the gap tothat of the boundary, p ¼ �=L. Moreover, the probability thatthe ball survives the first (n� 1) bounces and escapes only atthe nth bounce is ð1� pÞðn�1Þp. Therefore, the mean numberof bounces occurring until escape is

hnescapei ¼X1k¼1

kð1� pÞðk�1Þp ¼ 1

p/ 1

�: (13)

We now cut two (possibly overlapping) holes, with sizes �,on the boundary and examine the nonescaping periodic or-bits. Based on the geometrical argument used above, weexpect the probability to be �2=�, if the two holes do notoverlap. However, in systems where the trajectories do notdiverge strongly, i.e., the Lyapunov exponent is close to zero,only a small fraction of the trajectories will eventually hit theopening on the boundary, and the mean escape time will beproportional to �.

If the initial incident angle is taken to be c m;n ¼ �=2�m�=n, where m< n are integers and relative primes to eachother, then the trajectory is closed and its period is n. We nowexamine only those initial conditions for which the escapetime is at least t, or, in other words, the number of bounces isat least N ¼ b2�=�c. To fulfill this requirement one mighttake the initial value of c ¼ c m;n þ , where 0 � � �and � can be restricted to the following range:

�002

��þ t

cosðc m;nÞ;0�[�

0 þ�þ t

cosðc m;nÞ;2�

n

�:

The prime indicates that angles are taken modulo 2�=n. Theprobability can, therefore, be calculated if one sums up allpossible values of ðm; nÞ pairs. This is the point where numbertheory enters into this physical problem; we have to guaranteethat m and n are relative primes. Integrating over the permit-ted region of �0 one may find

Pðt; �; Þ � 1

t

XNn¼1

nF ðnÞXm

�1� cos

�2m�

n

��; (14)

where the exact form of F ðnÞ can be found by Bunimovichand Dettmann (2005). Surprisingly, the sum over m can beexplicitly determined. The first, unit term simply counts howmany numbers are relative prime to n and, therefore, it can beformally expressed using a special function of number theory,Euler’s totient function.5 The second term in Eq. (14) is also aspecial expression. If the summation were over all the integernumbers smaller than n, one could connect it to the Fourierseries. However, here one uses only those m’s which are

FIG. 4. A circular billiard with two small openings. From

Bunimovich and Dettmann, 2005.

5Euler’s totient function, �ðnÞ, gives the number of positive

integers smaller than n, which are relative prime to n, e.g., forany prime number �ðpÞ ¼ p� 1, since all integers smaller than pare relative prime to p.



relative primes to n. Converting the cosine term to complexexponentials and using Ramanujan’s identity6 for the sum ofexponentials, the contribution of the cosine term turns out tobe another special function of number theory which we havealready met, the Mobius function �ðnÞ. Therefore, the proba-bility of nonescaping orbits is

P1¼ limt!1ðtPðt;�;ÞÞ�

X1n¼1

n½�ðnÞ��ðnÞ�F ðnÞ: (15)

The leading-order behavior of P1 as a function of � can bedetermined by calculating its Mellin transform

~PðsÞ ¼Z 1

0P1ð�; Þ�s�1d� (16)

and examining the residues of ~PðsÞ on the complex s plane.Bunimovich and Detteman showed that for the two-holeproblem, where these holes are separated by 0, 60, 90,120, and 180, the probability ~PðsÞ is uniquely determinedby the Riemann zeta function �ðsÞ, i.e., by its pole andnontrivial zeros. The first corrections to the leading-orderterm are given by the nontrivial zeros of �ð1þ sÞ, whichare of the order of

ffiffiffi�

plnð�Þm�1, provided for all zeros of �ðsÞ,

ReðsÞ ¼ � � 1=2 with multiplicity m. The Riemann hy-pothesis is then shown to be equivalent to different asymp-totic estimates on the number of zeros (Titchmarsh andHeath-Brown, 2003). Therefore, if the nontrivial zeros pro-vide the second-order corrections to the probability, it isinstructive to examine the deviation of these probabilitiesexperimentally from the leading-order geometric terms,namely,

lim�!0

limt!1

��1=2

�tP1ðtÞ � 2

�

��¼ 0; (17a)

lim�!0

limt!1 ð��1=2½tP1ðtÞ � 2tP2ðtÞ�Þ ¼ 0; (17b)

where P1 and P2 belong to the one- and two-hole problems,respectively. If it is (experimentally) found that for every� > 0 these equations are fulfilled, then it proves the validityof the asymptotic formulas, thus the validity of the Riemannhypothesis. The numerical results by Bunimovich andDetteman do not contradict these equations. Although theirresult has not proven the Riemann hypothesis, it provides aphysically realizable system where actual measurements cansubstantiate, but not prove, the conjecture.

We now turn our attention now to the dynamics of a moregeneral billiard system. If one smoothly deforms the bound-ary of this circle and creates a stadiumlike shape, the analysisis far less straightforward (Bunimovich, 1979). However,qualitatively we may see that the trajectories can be classifiedsimilarly and one may distinguish periodic orbits, i.e.,fqnðt1Þ; pnðt1Þg ¼ fqnðt2Þ; pnðt2Þg for t1 < t2, and nonperiodictrajectories. It is tempting to think that periodic orbits are

exceptional and quite rare among all orbits, since for inte-grable systems the number of periodic orbits grows polyno-mially, and one may expect that violating integrability woulddecrease the number of periodic orbits. In fact, the opposite istrue. These special orbits proliferate among the possibleorbits and their number, for a general Hamiltonian dynamics,grows exponentially with the length of the periodic orbits�eh‘=h‘, where h is called the topological entropy and ‘denotes the length of a given class of periodic orbits(Gutzwiller, 1991; Stockmann, 1999). This is a striking dif-ference between integrable and chaotic systems. It is evenmore surprising that the knowledge of these periodic orbitsserves as a powerful analytical tool for investigating chaoticsystems, and, moreover, they provide the pathway, throughtrace formulas, from classical to quantum mechanics.

We now examine the time evolution of a Hamiltonianflow in general. We denote the trajectory starting its timeevolution from the initial point r0 with rðtÞ ¼ Fðr0; tÞ. Wefurther introduce the evolution operator with the followingdefinition:

Lðt; r0; rÞ ¼ �ðr0 � rðtÞÞ ¼ �ðr0 � Fðr; tÞÞ: (18)

It can be shown rigorously for a generic classical chaoticsystem (Cvitanovic and Eckhardt, 1991), that

TrðLðt; r0; rÞÞ ¼ Xp

Tp

X1r¼1

�ðt� rTpÞj detð1� JrpÞj ; (19)

where the first summation runs over the periodic orbitslabeled by p, while the second takes into account all repeti-tions r. Jp is the Jacobian matrix of F localized around the

periodic orbit, also called the monodromy matrix.Here we make an important observation: Although this

equation looks cumbersome, it does relate the spectrum of theevolution operator to a global behavior of periodic orbits.Therefore, these two sets of abstract objects are intimatelyrelated to one another. The connection of this trace formulaand its quantum mechanical counterpart to the Riemann zetafunction will become clear in the next section.

B. Quantum mechanics

Below we expound on the Polya-Hilbert conjecture. Weenumerate the one-dimensional Hamiltonians proposed forwhich the distribution of energy eigenvalues mimic the non-trivial zeros of the Riemann zeta function and analyze theirrelationship with the Gutzwiller trace formula. We also ex-amine the possible symmetries of a ‘‘Riemann operator,’’since it partially encouraged the development of quantummechanics with only CT or PT symmetry.

In the dawn of the 20th century Bohr postulated a series ofrules for describing the spectrum of the hydrogen atom wellbefore the birth of Schrodinger’s and Heisenberg’s quantummechanics. In these early days ‘‘quantization’’ meant torestrict the possible values of action variables of the classicalsystem (Bohr-Sommerfeld, Wentzel-Kramers-Brillouin, etc.)and the rules worked well, up to an additive constant.However, this description cannot be satisfactory in general,since for the majority of classical systems the only constant ofmotion is the energy, and therefore a method of quantization

6Ramanujan’s sum is defined as

cnðmÞ ¼ Xm

e2�im=n;

where the summation is over those values of m, which are relative

prime to n. Using Mobius inversion for this sum one can prove that

cnðmÞ ¼ �ðnÞ (Hardy and Wright, 1960).



relying on the existence of action-angle variables could not beapplied (Einstein, 1917). On the other hand, we know thatclassical mechanics works well for large systems, and there-fore quantum mechanics must give the same predictions for alarge system as classical mechanics (Bohr’s correspondenceprinciple). This unproven principle ties these two theoriesfirmly together and the same principle inspired the use of theRiemann � function in investigating the relationship of clas-sical to quantum mechanics.

The basic question is: How can we quantize a classicalmechanical system? Could we state anything about thespectrum of a quantum system, at least qualitatively, withoutsolving the corresponding Schrodinger equation?

We cannot expect to be able to infer the complete spectrumof a generic system, but asking only for the average density ofstates may prove feasible. One can give a crude, althoughremarkably precise, estimate: Each quantum state occupiesapproximately ℏf phase space, where ℏ is the Planck constantdivided by 2� and f is the number of degrees of freedom.This result is rather general and the individual quantumsystems differ only in the ‘‘fluctuations’’ around this average.It turns out that the type of fluctuation depends on thebehavior of the classical counterpart; classically regular andchaotic systems are quite different. For example, a classicalrectangular billiard is integrable (regular), whereas its quan-tum analog exhibits a chaotic spectrum, with the spacing ofthe quantum levels s ¼ �n � �n�1 following an exponentialdistribution PðsÞ � e�s (see Sec. III.C). However, a stadiumbilliard is classically chaotic, but the spectrum of its quantumcounterpart is regular, meaning that PðsÞ is small for smallvalues of s and sharply peaked at a finite value indicating aregular distribution of energy levels. We will see that byinterpreting the �ðsÞ zeros as energy levels, their distributionis similar to that of a quantum systems. This has inspiredphysicists to examine whether one could associate a dynami-cal system with the Riemann zeta function.

The advantage of this approach is that the large number of�ðsÞ zeros are known and quick numerical algorithms havealso been developed to find further zeros, thus solving theSchrodinger equation for large energies would be unneces-sary. The Riemann zeta function could play the same role inthe examination of chaotic quantum systems as the harmonicoscillator does for integrable quantum systems. This is thepoint where the examination of the Riemann zeta functionmay help to understand physics or, vice versa, the physicsmay lead us to the solution of this so far intractable mathe-matical problem.

In order to establish a strong formal connection between ageneric chaotic quantum system and the distribution of theRiemann �ðsÞ zeros, we have to elucidate a new description ofquantum systems, the Gutzwiller’s trace formula. This traceformula is the analog of Eqs. (9a) and (9b) for physicalsystems.

We therefore return to a classically integrable system, forwhich the HamiltonianH can be given in terms of conservedquantities H ¼ H ðI1; . . . ;INÞ. Using Bohr’s semiclassicalquantization rules, these action variables take not arbitrary,but fixed values

Ik ¼ ℏ�nk þ�k

4

�ðk ¼ 1; 2; . . . ; NÞ; (20)

where the�k are integers and called Maslov indices (Arnol’d,1997). The density of states therefore becomes

dðEÞ ¼ Xn

�ðE�H ðIÞÞ; (21)

which can be recast as the sum of a smooth and an oscillatoryterm. The former originates from the Thomas-Fermi semi-classical approximation

dTFðEÞ ¼Z

�ðE�H ðp;qÞÞ dpdq

ð2�ℏÞf ; (22)

while the latter is obtained by expanding the effective actionto quadratic order around the classical periodic orbits (Berryand Tabor, 1976; Emile et al., 2006):

doscðEÞ ¼XN

�2�

ℏTp

�ðf�3Þ=2 1

ℏ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidetðNQi;jNÞ

q exp

�iSpℏ

� i�

4N�þ i

�

4�

�; (23)

where Qi;j ¼ detðH Þ H�1i;j is the comatrix of H i;j ¼

@Ii@IjH , while � is related to the signature of H i;j.

We see, as in classical mechanics, one can also express thedensity of states as a sum of a smooth function dTFðEÞ and anoscillatory function which is defined on the periodic orbits ofthe semiclassical system. Because of the correspondenceprinciple, we expect the Thomas-Fermi density of states toremain valid and only the oscillatory part to vary compared tothe semiclassical derivation.

For nonintegrable systems, however, the orbits no longerlie on invariant tori and a different method is needed for theevaluation of the trace

dðEÞ ¼ � 1

�Tr½ImðGEðr; rÞÞ�; (24)

whereGEðr; rÞ is the Green’s function associated with a givenHamiltonian H . This new approach, based on the Green’sfunction, was developed by Gutzwiller (1970, 1971). Here weshall not follow the details of the derivation, but present onlythe final, fully quantum mechanical expression for the densityof states

doscðEÞ ¼Xp:p:o:

Tp

�ℏ

X1n¼1

cosfn½ðSp=ℏÞ � ð�=2Þ�p�gj detðMn

p � 1Þj1=2 ;

(25)

where the summation runs over all primitive periodic orbits(p.p.o.), andMp is the monodromy matrix for these primitive

periodic orbits. Using this new method one can derive asemiclassical expression for the spectrum of a quantumsystem whose classical analog is chaotic, when the usualBohr-Sommerfeld quantization rules cannot be applied.Gutzwiller’s result above, therefore, can be viewed asa bridge between the classical and quantum behavior of asystem and can provide a rule as to how to quantize such asystem. In this interpretation, Gutzwiller’s approach is similarto Feynman’s path integral description, where the quantumsystem is described in terms of an infinite sum over classicalpaths. For the interested reader we suggest Gutzwiller’s bookon classical and quantum chaos (Gutzwiller, 1991) and Brack



and Bhaduri’s monograph giving an overview of semiclassi-

cal physics (Brack and Bhaduri, 2003). In order to help the

reader to visualize the emergence of periodic orbits in a

quantum mechanical system, we reproduce here a few quan-

tum ‘‘scars’’ from Heller’s numerical study. Figure 5 showsthe probability distribution for three quantum eigenstates

of the Bunimovich billiard. It is apparent how the isolated,

unstable classical periodic orbits manifest themselves as

paths along which the probability distribution is greatly

enhanced. Gutzwiller’s idea to extract eigenvalues of achaotic system via the periodic orbits, therefore, seems

most plausible.Based on analogy between the oscillatory part of the

semiclassical density of states (25) and the similar expression

of (9b), one can set up a dictionary (Berry and Keating,1999b; Bohigas, 2005) that maps the Riemann zeta function

onto a so-far unknown chaotic quantum mechanical system.Although the Hamiltonian H which would describe the

chaotic quantum system corresponding to the Riemann zeta

function is still missing, the mutual resemblance of (9b) and(25) reveals some possible properties of H . A thorough and

concise summary of these properties can be found in Berry

and Keating (1999b) from which we cite just a few for later

use:(1) H has a classical counterpart, since the absence of any

analog of ℏ from (9b) indicates the scaling of the

dynamics, namely, the trajectories are the same at all

energy scales.(2) The Riemann dynamics is chaotic and unstable.(3) The dynamics lacks time-reversal symmetry.(4) The dynamics is quasi-one-dimensional, because for a

generic d-dimensional scaling system the number of

energy eigenvalues increases as �Ed, while for �ðsÞthe number of zeros T < NðTÞ � T lnðTÞ< T2.

Moreover, the appearance offfiffiffiffiffipr

pin the denominator

implies one expanding direction and no contracting

one.

Below we pursue the proposed dynamics related to the

Riemann zeta function.In the early days of quantum mechanics, Hilbert thought of

the possibility of verifying Riemann’s hypothesis using

physical arguments according to Polya’s recollection:

I spent two years in Gottingen ending around the

beginning of 1914. I tried to learn analytic number

theory from Landau. He asked me one day: ‘‘You

know some physics. Do you know a physical reason

that the Riemann hypothesis should be true.’’ This

would be the case, I answered, if the nontrivial

zeros of the xi function [G. Polya refers here to

the Riemann �ðsÞ function.] were so connected withthe physical problem that the Riemann hypothesis

would be equivalent to the fact that all the eigen-

values of the physical problem are real.

I never published this remark, but somehow it

became known and it is still remembered. (Private

letter to Odlyzko. See the scanned pages on

Odlyzko’s personal website: http://www.dtc.um-

n.edu/~odlyzko/polya/.)

The zeros of �ðsÞ can be the spectrum of an operator,R ¼ð1=2ÞI þ iH , where H is self-adjoint. This operator Hmight have an interpretation as a Hamiltonian of a physical

system and, therefore, the key to the proof of the Riemann

hypothesis may have been coded in physics. Since the first

occurrence of this conjecture, a number of models have been

promoted (Rosu, 2003). Below we separate the models de-

pending on whether they relate the zeros to the positive

energy spectrum, i.e., the scattering states of a physical

system, or to the negative energy spectrum, i.e., to the boundstates of a quantum system.

1. Scattering state models

We first consider the possibility that the Riemann zeta

function is associated with a quantum scattering problem.A few decades after Riemann created a new geometry with

his revolutionary work (Riemann, 1867, 1873a, 1873b),

Hadamard (1898) examined the geodesics, the trajectories

of freely moving bodies, on surfaces with negative curvature

in detail and noticed the occurrence of families of geodesics

FIG. 5. Three eigenstates of the quantum stadium billiard are

shown together with the major contributing unstable periodic orbits

of the classical counterpart as thick solid lines. In the middle figure

the guiding straight line for the ^ shaped periodic orbit is omitted.

From Heller, 1984.



whose cross section exhibits a fractal-like structure, as wewould call it nowadays. These geodesics diverge exponen-tially; thus the distance between two trajectories �ðtÞ, how-ever small initially, will grow exponentially, �ðtÞ � e�t�ð0Þ,where � is a positive number, called the Lyapunov exponent.This sensitivity of the system to the initial conditions, how-ever, would not necessarily result in chaotic behavior, pro-vided the space for the trajectories is infinite. However, if thesurface is compact, the trajectories cannot escape to infinity,but rather mix on this surface. If one wishes to visualize aparticular example, consider a donut with two holes. On thissurface the trajectories remain bounded on the surface with-out the length of a geodesics being limited (Balazs and Voros,1986; Gutzwiller, 1991; Bogomolny et al., 1995). These twoproperties, exponential sensitivity of the initial conditions andmixing, are the main requirements for chaotic motion(Cvitanovic et al., 2010). The relative simplicity of thedescription of such surfaces with negative curvature, andthe presence of completely chaotic classical motion moti-vated several in the mid-1980s (Gutzwiller, 1983; Balazs andVoros, 1986; Berry, 1987) to examine how such a system canbe quantized, i.e., what properties do the solutions and ei-genvalues of the equation H� ¼ �� possess.

More precisely, for free motion, one seeks the solution of

��n ¼ �n�n; (26)

where �n are required to be square integrable and the appro-priate boundary conditions are also provided. Over a compactdomain Eq. (26) has only discrete eigenvalues. On a surfacewith negative curvature, the non-Euclidean Green’s theoremshows that the eigenvalues must have the form �n ¼ 1=2þi�n (� is real) (Gelfand and Pjatezkii-Shapiro, 1959). Thisresemblance immediately suggests a connection with thezeros of the Riemann �ðsÞ. It is also proven that, for acompact surface, the set of n’s is finite, but for a noncompactsurface, a continuous part of the spectrum can also appear. Inthe latter case the scattering (continuous spectrum) is non-conventional, because it is the result of the geometry (curva-ture, compactedness) and not the physical interactionbetween particles.

In order to express the eigenvalue density the Green’sfunction is needed. Interestingly, on a surface with negativecurvature the Green’s function can be explicitly written as asum of individual Green’s functions corresponding to theperiodic orbits. It is also a fact that all periodic orbits areunstable and their action is SðEÞ ¼ k‘, where k is the mo-mentum related to the energy by 2mE=ℏ2 ¼ k2 þ 1=4, and ‘defines the length of a closed geodesic belonging to a givenconjugacy class. In this geometry the density of states isexpressed by the Selberg trace formula (Selberg, 1949)

��ðkÞ ¼ A

2�k tanhðk�Þ þ 1

2�

X½p�

X1n¼1

‘p cosðnk‘pÞsinhðn‘p=2Þ ; (27)

where A is the area of the surface and the first summation runsover conjugacy classes of primitive elements p, the second,their repetitions. It is important to note that the Selberg traceformula holds exactly, in contrast to other trace formulas,because no semiclassical approximation has been applied,

although its convergence property is similar to the Gutzwillerform: For large k the Selberg and Gutzwiller trace formulasconverge, since the metric is locally Euclidean and waveswith short wavelength lose their sensitivity to the localcurvature of the metric. In this system, the transient scatteringstates were examined by Pavlov and Fadeev (1975) whorelated the nontrivial zeros of the zeta function to the complexpoles of the scattering matrix:

SðkÞ ¼ ��2ik �½ð1=2Þ þ ik��ð1þ 2ikÞ�½ð1=2Þ � ik��ð1� 2ikÞ : (28)

Despite this natural occurrence of the Riemann zeta functionand its nontrivial zeros, no further insight into the zeros hasbeen gained via this route. Detailed discussion of the Selbergtrace formula has been given by Hejhal (1976, 1983) or morephysics oriented approaches byWardlaw and Jaworski (1989)and Stockmann (1999) and in the context of the Casimireffect by Elizalde (1993), Kurokawa and Wakayama(2002), and Schaden (2006).

We now return to the scattering formalism in the standardEuclidean space. Joffily, motivated by Pavlov and Fadeev(1975), examined the scattering states of a nonrelativistic,spinless particle under the influence of a spherically symmet-ric, local and finite potential. He examined the Jost solutionsof this scattering problem (Joffily, 2003), which differ fromthe physical solution of the Schrodinger equation in theirasymptotics7 (Alfaro and Regge, 1965; Newton, 1982).In standard nonrelativistic scattering theory the S matrix isgiven by

SðkÞ ¼ e2i�ðkÞ ¼ f�ðkÞfþðkÞ ; (29)

where �ðkÞ is the phase shift and f�ðkÞ are the Jost solutionsdefined by their boundary conditions limr!1½f�ðkÞe ikr� ¼ 1(Alfaro and Regge, 1965). Provided the potential has a finiterange and decreases sufficiently rapidly, the Jost solutionfþðkÞ is proven to have infinitely many zeros, correspondingto the solutions of the Schrodinger equation as outgoing orincoming waves. Resonances (i.e., states with finite lifetime)occur if SðkÞ has poles on the complex k plane with negativeimaginary parts: k2n ¼ �n � i�n=2, where �n and �n representthe energy and inverse lifetime associated with the nth state,respectively. Joffily introduced amapping between these zerosof fþðkÞ onto the critical line and showed they coincide withthe nontrivial zeros of the Riemann zeta function. He associ-ated this artificial system with a vacuum and the zeros areinterpreted as an infinity of virtual resonances and thus reflect

7The Jost functions are the solutions of the Schrodinger equation

with the following asymptotic behavior:

limx!1½e

ikxfð�; k; xÞ� ¼ 1;

where � ¼ ‘þ ð1=2Þ is the shifted angular momentum, k� ffiffiffiffiE

pand

x 2 ð�1;1Þ. This choice of the boundary condition is motivated

by our physical picture, i.e., the particle should be represented by

free plane waves far from the local potential. The real physical

solution of the Schrodinger equation can be expressed as a linear

combination of the two Jost functions.



the chaotic nature of the vacuum (Joffily, 2003, 2004). Thisinterpretation has also been extended using relativistic scat-tering (Joffily, 2007).

In another scattering based approach, Chadan and Musette(1993) analyzed the so-called ‘‘coupling constant spectrum’’of a radially symmetric three-dimensional Hamiltonian,where the potential is chosen from a singular family offunctions

H CM ¼ � d2

dr2� ‘ð‘þ 1Þ

r2þ 1

r2fCM; (30)

where fCM has logarithmic singularities at r ¼ 0. They ar-gued that the coupling constant spectrum coincides ‘‘approxi-mately’’ with the nontrivial Riemann zeros if the problemis restricted to a finite, closed interval r 2 ½0; e�4�=3�.Mathematically rigorous detailed analysis and the extensionof the potential family was carried out by Khuri (2002).Furthermore, the existence of a three-dimensional potentialURðrÞwas derived whose s-wave scattering amplitude has thecomplex zeros of the Riemann zeta function as ‘‘redundantpoles.’’ Examination of �ðsÞ using a quantum scatteringapproach is further motivated if one compares the plot ofthe phase of �ðsÞ on the complex plane with the usual Arganddiagram8 of the scattering amplitude corresponding to acollision; see Fig. 6.

As a specific example, for completely elastic collisions, thescattering amplitude should be a perfect circle on the complexplane with unit radius centered on (0,1). For inelastic colli-sions this circle deforms. The phase of �ðsÞ, after interchang-ing the roles of the real and imaginary axes, qualitativelyresembles the Argand diagram of a scattering amplitude. Thisgeometric similarity suggests an analysis of �ðsÞ as if itrepresented the scattering amplitude of a real collision ofparticles. This analogy, however, is not perfect since �ðsÞ doesbecome negative while the Argand diagram of the scatteringamplitude corresponding to a realistic collision does not.Bhaduri advocated neglecting these small differences whichdo not affect their most important result, namely, the phaseðtÞ of the Riemann zeta function along the critical line�ð1=2þ itÞ ¼ ZðtÞe�iðtÞ is intimately connected to the quan-tum scattering of a particle on a saddlelike surface (Bhaduriet al., 1995, 1997).

To illustrate this, consider a nonrelativistic particle movingin an inverted harmonic oscillator potential along the half-line(x � 0). The Schrodinger equation reads as

� ℏ2

2m

d2

dx2�ðxÞ � 1

2m!2x2�ðxÞ ¼ E�ðxÞ; (31)

where we require that �ðx ¼ 0Þ ¼ 0. This problem can bemapped onto a repulsive Coulomb problem of which thephase shift �ðtÞ can be exactly expressed (Flugge, 1974).The oscillatory part of the phase shift is given by

�ðtÞ ¼ �smoothðtÞ þ Im

�ln

��

�1

4þ i

t

2

��

� ln

��

�1

4� i

t

2

��; (32)

which is exactly the phase of the Riemann zeta function. Twoyears after their first result, Bhaduri et al. extended this one-dimensional model to a two-dimensional one, where in onedirection the potential is a traditional confining parabolicpotential, and in the perpendicular direction (y) they keptthe inverted harmonic oscillator (Bhaduri et al., 1997). Thischoice was motivated by the analysis of the Gutzwiller traceformula on the � ¼ 1 border of the critical line and also bythe form of the electrostatic potential at the bottleneck of aquantum contact in a mesoscopic structure (Buttiker, 1990).

While the inverted oscillator reproduced the oscillatingpart of the �ðsÞ phase in Bhaduri’s work, Berry and Keating(1999a) showed that a regularization of a surprisingly simpleone-dimensional classical Hamiltonian H ¼ xp reproducesthe smooth counting function of the zeros. We note here thatthis choice ofH is a canonically rotated form of the invertedoscillator Hamiltonian �ðp2 � x2Þ. Moreover, the quantummechanical model of the corresponding symmetrizedHamiltonian H ¼ ðxpþ pxÞ=2 has also been investigatedand exactly solved preserving the self-adjoint property of theHamiltonian (Twamley and Milburn, 2006; Sierra, 2007). Thebeauty of the xp, or inverted oscillator model, is that itsatisfies most of the properties listed earlier (see Sec. III.B):it is valid as a classical mechanical model, the dynamics isone dimensional and uniformly unstable since the solution ofthe Hamiltonian equations are exponentially decaying ordiverging, and it lacks time-reversal symmetry. However,the trajectories are not bounded causing significant hardshipin the semiclassical quantization. As in the hyperbolic case,the boundary conditions or the way the phase space is regu-larized and/or compactified become decisive. Berry andKeating (1999a) suggested a simple regularization by intro-ducing a cutoff in both position and momentum. This processresults in a finite area, which can be filled up with Planck cells

-3

-2

-1

0

1

2

3

-2 -1 0 1 2 3 4

Im(ζ

(1/2

+ it

))

Re(ζ(1/2 + it))

FIG. 6. An Argand diagram of the Riemann zeta function on the

critical line �ð1=2þ itÞ, where t ¼ 0–49:77, the latter of which is

� �10.

8Argand diagrams can be thought of as a parametric plot of the

inherently complex scattering amplitude on the complex plane, and

the collision energy plays the role of the parameter [see, for

example, Bhaduri (1988) or Bohm and Loewe (2001)].



of size h, thus counting the number of available quantumstates. Another approach is available if one notices the dila-tion symmetry of the Hamiltonian (x � �x, and p � p=�).This symmetry manifests itself in the transformation of thewave function as

c ð�xÞ ¼ 1

�1=2�iEc ðxÞ; (33)

and one might suggest restricting ourselves to � being apositive integer. This could be an attractive suggestion, be-cause the wave packet, generated by the uniform superposi-tions of all these transformed wave functions, is

�ðxÞ ¼ X1�¼1

c ð�xÞ ¼ �

�1

2� iE

�c ðxÞ: (34)

However, there is no physical motivation that would requirethis � prefactor to vanish. Furthermore, this integer-baseddilation symmetry does not form a group, because the multi-plicative inverse element (which would be � ¼ 1=m) ismissing.

Berry and Keating (1999b) also established a peculiarcanonical transformation (X ¼ 2�=p, P ¼ xp2=2�) for thisHamiltonian, which exchanges and mixes the roles of thephysical position and momentum, but was uncertain ‘‘how toconvert this ’quantum exchange’ into an effective boundarycondition.’’ Aneva (1999, 2001a, 2001b) also analyzed thisboundary condition for a hyperbolic dynamical system withconformal geometry and showed how this exchange trans-formation arises as a result of boundary conditions.

Later Sierra (2008) generalized Berry’s model in twodifferent ways, first by incorporating the fluctuation termsdoscðEÞ [see Eq. (9b)] via changed boundary conditions.Together with Townsend, they also considered the motionof a charged particle (electron) moving on the [x-y] plane in aconstant uniform perpendicular magnetic field, and in anelectric potential described by the following Hamiltonian:

H ¼ 1

2�

�p2x þ

�py þ eB

cx

�2�þ e�xy: (35)

In this model, the number of semiclassical quantum stateswith energy less than E has the same functional form as thecounting function of the �ðsÞ zeros, Eq. (10), i.e., the smoothpart of the Riemann zeros is reconstructed by the lowest-lyingLandau level of the charged particle. The fluctuation term (asthey speculate) might be explained by the contribution ofhigher Landau levels. This surmise, however, is only sup-ported by estimating the order of magnitude of this highercontribution and comparing it to that of the Riemann �ðsÞ.This model has the additional attraction of being potentiallyaccessible to experimentalists, including in lower spatialdimensions (Toet et al., 1991; Li and Andrei, 2007;Park et al., 2009).

Exploiting the x $ p exchange symmetry of this modeland using the Riemann-Siegel formula for the �ðsÞ function,Sierra created a new model in which the Jost solutions aredirectly proportional to the Riemann zeta function, and thenontrivial zeros become the energies of the bound states. Thisachievement does not, however, prove the Riemann hypothe-sis, as Sierra explicitly states ‘‘we cannot exclude the exis-tence of zeros outside the critical line.’’

In summary, we first introduced, motivated by Gutzwiller’strace formula, a quantum mechanical model on a surface withnegative curvature, which leads us to the mathematicallyexact Selberg trace formula. The importance of this resultis at least twofold. First it reassures us that describing chaoticsystems via the periodic orbits is likely to be feasible, andsecond it demonstrates the role of periodic orbits in a genericsystem in determining the smooth and fluctuating parts of thedensity of states. We further elaborated on another non-Euclidean model, proposed by Pavlov and Fadeev, in whichthe Riemann zeta function determines the S matrix over thecomplex energy plane.

Converting these results into the usual Euclidean space,however, seems challenging. Although a few models havesuccessfully reproduced the smooth part of the density ofquantum states, the fluctuation terms of these models differfrom those of the Riemann zeta function.

2. Bound state models

From the 1950s, a new approach, the random matrixtheory, emerged from the study of the spectrum of heavynuclei. The same statistical apparatus had also been used toanalyze the statistical properties of the seemingly randomRiemann zeta zeros and led to the conjecture that the �ðsÞzeros belong to one particular universality class (Bohigaset al., 1984a, 1986), the so-called Gaussian unitary ensemble(see Sec. III.C). This result suggested property 3 on Berry’slist. However, Wu and Sprung (1993) generated a one-dimensional, therefore integrable, quantum mechanicalmodel which can possess the Riemann zeta zeros as energyeigenvalues and showed the same level repulsion as thatobserved in quantum chaos. This was a contradictory resultsince, on the one hand, the �ðsÞ zeros follow a statisticsspecific for systems violating time-reversal symmetry; onthe other hand, Wu and Sprung’s model, by definition, wasinvariant under time reversal. However, the proposed modelwas not lacking in irregularity, since the potential reproduc-ing the Riemann zeta zeros appeared to be a fractal, a self-similar mathematical object. Nevertheless, they derived forthe first time a smooth, semiclassical potential which gener-ated the smooth part of NðEÞ,9 through

NðEÞ ¼ 1

h

ZZH�E

dxdp ¼ 2

�

Z xmax

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE� VðxÞ

pdx (36)

with 2m=ℏ2 set to unity. Solving this Abel-type integralequation, one may derive the following implicit expressionfor VðxÞ:

xðVÞ ¼ 1

�

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV � V0

pln

�V0

2�

�

þ ffiffiffiffiV

pln

� ffiffiffiffiV

p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV � V0

pffiffiffiffiV

p � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV � V0

p��

; (37)

where V0 has to be chosen such that the potential is notmultivalued, i.e., V0 � 2�. The choice of V0 affects thepotential at its bottom (x � 0), but for large x it does nothave a significant impact and for x � 1

9In the mathematical literature the argument is usually denoted by

T, as in Sec. II. Motivated by physics, we use here E.



xðVÞ ¼ffiffiffiffiV

p�

ln

�2V

�e2

�(38)

[see Fig. 1 in Wu and Sprung (1993)]. We note here thatMussardo (1997), using similar semiclassical arguments asWu and Sprung, recently also gave a simple expression for asmooth potential supporting the prime numbers as energyeigenvalues. Furthermore, Mussardo also proposed a hypo-thetical resonance experiment to carry out primality testing;this theoretically infinite potential could be truncated at somehigh energy, thus transforming it into a finite well. If anincident wave radiated onto this well has energy E ¼ nℏ!,where n is a prime number, then it should cause a sharpresonance peak in the transmission spectrum arguedMussardo.

Turning back to the smooth potential studied by Wu andSprung, which is able to ‘‘roughly’’ reproduce NðEÞ, it is thenmodified to have the low-lying �ðsÞ zeros exactly. In order toachieve this goal Wu and Sprung set up a least-square mini-mization routine, to minimize the difference between theactual energy eigenvalues and the exact zeros. The resultwas surprising, since the potential curve became coarse andresembled a random potential. They analyzed this curve usingthe standard box-counting technique and measured a d ¼ 1:5fractal dimension for the potential reconstructing theRiemann zeta zeros.

Ramani et al. (1995) pointed out that the apparent contra-diction between Berry’s conjecture and Wu and Sprung’smodel, i.e., whether or not the physical system exhibitstime-reversal symmetry, is caused by the coarse curve ofthe potential, since any smooth one-dimensional potentialwould lead to locally evenly spread energy levels, which isnot the case for Wu’s potential. They also provided a veryefficient algorithm, the ‘‘dressing transformation,’’ withwhich one can build up the quantum potential from individualenergy eigenvalues. However, they standardized the spectrumusing the ‘‘spectrum unfolding’’ technique which eventuallyled them to the following conclusion: The fractal dimensionof the potential supporting the Riemann zeta zeros has d ! 2rather than that measured by Wu and Sprung. Wu and Sprung(1995) pointed out that this difference in fractal dimension isputatively caused by the alternative choice of the spectrum.As they argued, Ramani’s spectrum does not have the sameaverage density, long-range correlation and nearest-levelspacing distribution as the Riemann zeta function, and there-fore one cannot draw valuable conclusions regarding thepotential.

Nearly a decade after Wu and Sprung’s original article,van Zyl and Hutchinson (2003) attempted to clarify thequestions raised by the two previous works. They showedthat for the same set of energy levels, different potentialgenerating techniques (the variational approach used by Wuand Sprung, the dressing transformation used by Ramaniet al.) led to the same potential, shown in Fig. 7. This resulthad been further strengthened by Schumayer et al. (2008)who used the inverse scattering transformation as a thirdtechnique obtaining the same potentials as in the earlierworks. It is noteworthy to mention that the inverse scatteringtransform guarantees the uniqueness of the potential in onedimension. This analysis, therefore, elucidated that the dif-

ference in measured fractal dimension cannot originate fromthe method of inversion. Moreover, they confirmed d ¼ 1:5for the Riemann zeta potential. These works all demonstrated

the importance of long-range correlations in determining thefractal dimension of the potential.

In a manner similar to that of the Riemann �ðsÞ zeros,

the prime numbers can also be considered as an energyspectrum; thus a potential can be associated with them and

it also proves to be fractal, but with a larger fractal dimension,

d ¼ 1:8. This result is somewhat puzzling. The two sets,those of the zeta zeros and the prime numbers, can be mapped

onto each other via Eq. (6), but the nearest-neighbor spacing

distribution of prime numbers is known to be Poisson like(almost uncorrelated random distribution), while that of the

Riemann zeros is rooted in the Gaussian unitary ensemble

and exhibits the corresponding correlations [see Eq. (40) inSec. III.C]. One may, therefore, conclude that Riemann’s

formulas convert two very different random distributions

into each other, or as Sakhr et al. (2003) stated: ‘‘it is possibleto generate the almost uncorrelated sequence of the primes

from the interference of the highly correlated Riemann

zeros.’’Regarding the fractal nature, Schumayer et al. also estab-

lished that the potentials associated either with the zeros of

�ðsÞ or with the prime numbers are multifractals, i.e., thesepotential curves cannot be characterized by one number d, buta range of dimension is necessary to describe their properties

[for definition see Schumayer et al. (2008)].Finally, at the end of this section devoted to the quantum

mechanical models of the Riemann zeta function, we briefly

refer to another alternative spectral interpretation of the zerosproposed by Connes (1999). During the comparison of

Gutzwiller’s trace formula for quantum mechanical systems

and that of the �ðsÞ function we noticed the overall signdifference in dosc [see the negative sign in Eq. (9b) in front

of the summation], i.e., the contribution of the periodic orbits

should be subtracted and not added to the smooth density ofstates, �dðTÞ (Berry, 1986). This sign difference led Connes to

interpret the zeros as gaps, missing lines from the otherwise

0

50

100

150

200

250

300

350

400

0 5 10 15 20

Inve

rsio

n po

tent

ial V

(r)

Distance, x

Inversion potential with N=200Semiclassical approximation

-30

-20

-10

0

10

20

30

0 5 10 15 20

V0(

r) -

V(r

)

FIG. 7. The semiclassical potential (dashed line), and the fractal

potential (solid line) supporting the first 200 zeros of �ðsÞ as energyeigenvalues. The inset depicts the difference of these potentials.

From Schumayer et al. 2008.



continuous energy spectrum rather than discrete energy

levels.

C. Nuclear physics

Random matrix theory (RMT) has been successfully ap-

plied to predict ensemble averages of observables for heavynuclei. Even though the Riemann zeros are distributed ran-

domly, some of their statistical quantities correspond to thoseof the Gauss unitary ensemble. We discuss the RMT briefly

for historical reasons. The reason for brevity is due to two

recent Colloquia devoted to RMT (Papenbrock andWeidenmuller, 2007; Weidenmuller and Mitchell, 2009).

Unfortunately, the degrees of freedom of even a moder-

ately large nucleus are still far beyond our computational

capability, be it analytical or numerical. Similar problems,although the number of components are on a different scale,

have occurred before in physics and engendered the develop-

ment of a new branch of physics, statistical mechanics. Thisis exactly what Wigner (1951) had in mind when he sug-

gested a statistical description of nuclei. He suggested that

nuclei can be statistically described using random matricescarefully chosen from predetermined ensembles. The new

description emerging from this examination is the random

matrix theory.Although random matrix theory emerged from the statis-

tical description of nuclei, it has already infiltrated many

different areas of physics. Recent developments of thisbranch of physics have been reviewed by Bohigas (1989),

Forrester et al. (2003), and Weidenmuller and Mitchell

(2009). Moreover, we can suggest the monograph by one ofthe leading figures of random matrix theory (Mehta, 2004).

But how to choose the ensemble of random matrices

suitable for a certain system, or for the Riemann �ðsÞ func-tion? Throughout classical mechanics, symmetry plays a

decisive role in determining the dynamics of different sys-

tems. If a physical system has a symmetry it implies, viaNoether’s theorem, the existence of a conserved quantity, e.g.,

the translational invariance in time, dictates energy conser-

vation; continuous rotational invariance requires that theangular momentum remain constant. These symmetries limit

the possible forms of the Hamiltonian describing the given

system. Therefore, if one wants to approximate a Hamiltonianwith a large but finite dimensional matrix, these symmetries

will determine the type and structure of the matrix, whether it

is real or complex, symmetric or Hermitian (Dyson, 1962a,1962b, 1962c, 1962d, 1962e).

In the case of an integrable system, the conserved quanti-

ties are all known. Therefore, the Hamiltonian can be diago-nalized, with each eigenvalue forming its own symmetry

class. This leads to the assumption that these eigenvalues

are completely uncorrelated. We also assume that the averagespacing between eigenvalues is unity in the overall sequence

of eigenvalues. If pðsÞ denotes the probability distribution of

nearest neighbor spacings, i.e., if �1 and �2 are eigenvalues ofthe given system, then �1 � �2 ¼ s, and one can express

(Stockmann, 1999) the probability of finding two eigenvalues

in a distance between s and sþ ds with no other eigenvaluesin between. Dividing the distance s into N equal intervals, the

probability is

pðsÞds ¼ limN!1

��1� s

N

�N�ds: (39)

In the N ! 1 limit the right-hand side becomes the expo-nential function. Therefore the probability distributionpðsÞ ¼ expð�sÞ is the Poisson distribution with parameterequal to 1. This is quite a general result for integrable systemssuch as Berry and Tabor (1977) have demonstrated. Similarly,one can deduce similar probability distributions for universal-ity classes of random matrices, e.g., the Gaussian unitaryensemble, the Gaussian orthogonal ensemble, etc. The clas-sification refers to the universality conjecture: If the classicaldynamics is integrable, then pðsÞ corresponds to the Poissonensemble, while in the chaotic case, pðsÞ coincides with thecorresponding quantity for the eigenvalues of a suitableensemble of random matrices (Bohigas et al., 1984a).Furthermore, the local statistics of the eigenvalues convergeas the order of the matrix increases.

How is this connected to the Riemann zeta zeros? Thezeros can be treated as eigenvalues of a fictitious physicalsystem, just as Hilbert and Polya suggested, and their statis-tical properties examined. Montgomery (1973) showed thatthe pair-correlation function of the zeros is

r2ðxÞ ¼ 1��sinð�xÞ�x

�2; (40)

provided the Riemann hypothesis is true. Freeman J. Dyson,during an informal discussion (Cipra, 1999), pointed out toMontgomery that this is exactly the same result as obtainedfor random matrices picked from the Gaussian unitary en-semble. However, this statement is made in the asymptoticlimit, i.e., as one goes to infinity on the critical line, 1=2þ iE,or in the RMT language as the size of the matrices N tends toinfinity. At finite height E or dimensionality N discrepanciesmay occur compared to Eq. (40). Interestingly, it was shownusing heuristic arguments that the nearest-neighbor spacingdistribution of the zeta zeros and that of unitary randommatrices of finite dimension are the same (Bogomolny andKeating, 1995). Moreover, the same authors extended theirstudy of correlation functions (Bogomolny and Keating,1996) rn of order n (n � 2) and proved, in the appropriateasymptotic limit, rn of the Riemann zeta zeros are equivalentto the corresponding GUE result. This result was complimen-tary to Montgomery’s second order (Montgomery, 1973),Hejhal’s third order (Hejhal, 1994), and Rudnick andSarnak’s general result for the nth order correlation function(Rudnick and Sarnak, 1996).

What does this result demand from a model of theRiemann zeros? The striking similarity between the pair-correlation function of the �ðsÞ zeros and the eigenvalues ofrandom matrices from the GUE ensemble only holds forshort-range statistics. Odlyzko (1987) carried out an empiri-cal test, by calculating the statistics for substantial numbers ofzeros, and confirmed Berry’s predictions (Berry, 1985) aboutthe discrepancies between the GUE theory and computedbehavior of the �ðsÞ zeros. The long-range correlation andthe small spacing statistics of the �ðsÞ zeros noticeablydeviate from the GUE prediction. This is expected (Berry,1985, 1988), since long-range correlations are dominated bythe short periodic orbits, which are system specific and there-fore not universal. For �ðsÞ the mean separation between



zeros is lnðE=2�Þ, while the smallest period is � lnð2Þ (seeTable I). Conclusively, the GUE-predicted universal correla-

tion for zeros near E should fail beyond lnðE=2�Þ= lnð2Þ(Berry and Keating, 1999b). Despite the deviation explained

above, the statistics of the �ðsÞ zeros asymptotically coincide

with those of the GUE ensemble, and consequently the

corresponding quantum system ought to violate time-reversal

symmetry (Berry and Keating, 1999a, 1999b). This may have

motivated Berry and Keating’s choice of a �ðxpþ pxÞ as aHamiltonian.

Finally, we mention an unexpected spin-off result of ran-

dom matrix theory related to the Polya-Hilbert conjecture.

Crehan (1995) asserted that for any bounded sequence there

are infinitely many classically integrable Hamiltonians for

which the corresponding quantum spectrum coincides with

this sequence. Furthermore, as an example for his theorem, he

showed that infinitely many classically integrable nonlinear

oscillators are capable of exactly reproducing the Riemann

zeta zeros when they are quantized. Unfortunately, the theo-

rem is an existence theorem and not a constructive one. If

such a system could be created, whether physically or just

theoretically, that would be aesthetically pleasing: It would

connect the most studied physical model (oscillator) with the

basis of our arithmetic (prime numbers). Crehan’s result is

promising and is also supported by the relationship between

the Riemann �ðsÞ zeros and the Painleve Vequation, the latter

of which plays a central role in the theory of completely

integrable dynamical systems (Ablowitz and Clarkson, 1991).Finally, in this section we briefly mention the notion of

quantum ergodicity, which attracted substantial attention in

the last three decades in the search for links between classical

and quantum ergodicity, i.e., what ‘‘fingerprint’’ the classical

chaos leaves in the physical properties if we quantize the

system, especially in the long-time behavior. Only few rig-

orous results (Schnirelman, 1974; de Verdiere, 1985; Zelditch

and Zworski, 1996; Nonnenmacher and Voros, 1998) are

known, and one of them says that the expectation value of

operators over individual eigenstates is almost always the

ergodic, microcanonical average of the classical version of

the operator. However, the theoretically rigorous understand-

ing of quantum ergodicity is still in its infancy. Numerical

simulations suggest though that quantum chaotic systems

exhibit universal behavior at a particular length scale, andat this scale the statistics of the eigenvalues resemble that of

large random matrices chosen from specific ensembles

(Berry, 1977; Bohigas et al., 1984a, 1984b; Heller, 1984;Gutzwiller, 1991; Agam et al., 1995). It is unfortunate that

this length scale is so minute that it substantially hinders thenumerical simulations. Nevertheless, it has also been shown

theoretically (Tomsovic and Heller, 1991; Kaplan and Heller,

1996) that quantum eigenstates must deviate from the RMTpredictions. These corrections may stand out from the spread

out background of RMT, just as the unstable periodic orbitsdo as eigenstates with enhanced amplitudes as shown in

Fig. 5. Although further numerical simulations (Backer

et al., 1998; Kaplan and Heller, 1999) provided some evi-dence regarding the connection between RMT and quantum

ergodicity, its interpretation and strength remain open

questions.

D. Condensed matter physics

In condensed matter physics the fundamental structure isthe crystal lattice. Below we examine the connection of the

lattice with the generalized Riemann hypothesis. We also

show how the specific heat capacity of a solid restricts thelocation of the �ðsÞ zeros.

One of the fundamental bases of modern condensed matter

physics is the geometrical structure of solids: the lattice. Theexamination of this mathematical structure is necessary to

understand even the basic properties of matter. The regular

structure of a perfect lattice is suitable for immediate com-parison with regularities among the natural numbers, and

therefore it is not a surprise that many number-theoreticalfunctions arise in crystallography; e.g., Ninham et al. (1992)

presented a witty review on the Mobius function. For those

mathematically inclined, see Waldschmidt et al. (1995).Moreover, not only the perfect regularity of a lattice, but

also the lack of this regularity can be related to the Riemann

zeta function, as Dyson indicated recently (Dyson, 2009):‘‘A fourth joke of nature is a similarity in behavior between

quasicrystals and the zeros of the Riemann zeta function.’’ Inthe following, we examine why a solid state constituted by

ions should even exist, what binds these ions to each other?Ions arrange themselves into a structure which maximizes

the attractive interaction between unlike and minimizes the

repulsive interaction between like charges. In an ionic crystal,

such as NaCl, the main contribution to the binding energy hasan electrostatic origin with the van der Waals term only a few

percent of the former. The electrostatic term is called the

Madelung energy, and the energy of one ion in the solid iscalled the Madelung constant.

For simplicity, first imagine a one-dimensional infinitely

long ionic lattice. Cations and anions are located next to eachother at a distance a, in a simplified NaCl structure; see Fig. 8.

If simply two unit charges q were positioned at the same

distance a, the electric potential energy of one of the chargeswould beU ¼ q2=4��0a. In a solid each ion is in the field ofall the remaining charges, both positive and negative. Thetotal electrostatic potential energy of one ion at position i inthe lattice is therefore

TABLE I. Dictionary for translating the ‘‘Riemann dynamics’’onto a chaotic quantum dynamics. Based on Berry and Keating(1999b), Brack and Bhaduri (2003), and Bohigas (2005).

Generic chaoticsystem

Riemann zetafunction

Periodic orbit labels Integers PrimesDimensionless action Sp=ℏ T lnðpÞPeriods Tp lnðpÞStability factor

a detðMnp � 1Þ pr

Maslov indexa �p 2b

Asymptotic limit ℏ ! 0 Tp ! 1aDepending on how one maps the oscillatory part of the zetazeros density (9b) onto Gutzwiller’s trace formula (25), thedefinition of the stability factor and the Maslov index canbe different. Here we followed Brack and Bhaduri (2003), whileanother mapping has been given by Berry and Keating (1999b).bTherefore the Maslov phase is �, but this is not unique and onecould also choose 3�, 5�, etc.



Ui ¼Xj�i

1

4��0

ð�1Þji�jjq2

ji� jja ¼ 1

4��0

q2

a

X1k�0

ð�1Þkk

;

(41)

where j runs over all lattice sites except i in the first summa-tion, and in the second equation we have changed the runningvariable to k ¼ ji� jj. In a finite lattice we have 2N ions, butin Eq. (41) each term belongs to two ions; therefore the totalelectrostatic potential energy of the finite lattice is

Utotal ¼ 1

22NUi ¼ N

1

4��0

q2

a

XNk�0

ð�1Þkk

: (42)

This form of Utotal can be divided into three terms: N whichguarantees the extensive nature of the energy, an energyfactor q2=4��0a, and also a numerical factor dependingonly on the lattice structure. One sees directly that theinfluence of the lattice on the total electrostatic energy iscomprised of an infinite sum. Since this energy term has to benegative in order to describe binding, we incorporate this signinto the Madelung constant 1D as

1D ¼ 2X1k¼1

ð�1Þkþ1

k; (43)

where the factor 2 appears because of the mirror symmetryaround the ith ion. The total energy can be written asUtotal ¼� 1DNq2=4��0a, which is negative if 1D > 0.

Generalizing the NaCl structure we examined above for therealistic three-dimensional case, one can write the Madelungconstant for this lattice as

3D ¼ Xði;k;lÞ�ð0;0;0Þ

ð�1Þiþjþkþ1

ði2 þ j2 þ k2Þ1=2 : (44)

Although it is tempting to evaluate this summation by ap-proximating the terms on concentric spheres centered at thereference ion (i ¼ j ¼ k ¼ 0) and utilizing the symmetry, the

resulting series 6� 12=ffiffiffi2

p þ 8=ffiffiffi3

p � � � � is divergent whichis physically unsatisfactory. The convergence properties ofsuch sums have been extensively investigated (Chaba andPathria, 1975, 1976a, 1976b, 1977; Borwein et al., 1985).The sum (44) is an alternating and conditionally convergentsum. The denominator of the summand is a quadratic form,and therefore the Madelung constant for a simple cubicstructure can be formally written as �EPð1=2; �m;nÞ,where �EP is the Epstein zeta function (see below), and m,n ¼ 1, 2, 3. The second argument �m;n is determined by the

type of the lattice, and in crystallography it is a quadraticform P given by the Gram matrix pmn ¼ emen, where em isthe mth lattice vector. Therefore, for example, the Madelungconstant for the body-centered cubic structure can be for-mally written as

bcc3D ¼ �EP

241=2;

2 1 11 2 11 1 2

0@

1A35 ¼ 1:762 675: (45)

Here we have dealt only with the pure Coulomb interaction,but this treatment can be extended to screened electrostaticinteractions as well (Kanemitsu et al., 2005).

The infinite sum in (43) strongly resembles the Riemannzeta function, except each term is weighted by a factorð�1Þkþ1, and its numerical value is 1D ¼ 2 lnð2Þ �1:3863. Although in two and three dimensions the summationcan be written explicitly, obtaining a precise numerical valueis far from easy and the Epstein zeta function is required. Thisfunction can be thought of as a generalized zeta function(Ivic, 2003; Shanker, 2006) which is defined by

�EPðs;P Þ ¼ XP�0

1

P s ; (46)

where P is a quadratic form defined on a d-dimensionallattice. All lattice points for which P � 0 are excludedfrom the summation. This function can be analytically con-tinued to the same domain as the Riemann zeta function and

also has its only pole at s ¼ 1 with residue �=ffiffiffiffi�

p. The

similarity goes further since �EPðs;P Þ also satisfies a func-tional equation expressing mirror symmetry. Thus, there isan inclination to generalize Riemann’s conjecture: All non-trivial zeros of �EPðs;P Þ have a real part of one-half. Thetemptation to do so is strengthened if one chooses specificquadratic forms, e.g., �EPðs; Id1Þ ¼ 2�ð2sÞ, or �EPðs; Id4Þ ��ðsÞ�ðs� 1Þ, where Idn is the n-dimensional identity matrix.Indeed, it was shown 80 years ago that for binary quadraticforms (two-dimensional lattice), infinitely many zeros of �EPlie on the critical line (Potter and Titchmarsh, 1935) in asimilar manner to Hardy (1914) for the Riemann zeta func-tion. Remarkably, however, it has also been shown that in anydimension one can construct such a P that the generalizedhypothesis does not hold (Terras, 1980). This, admittedlynegative, result shows the intriguing connection betweencrystallography and this generalized Riemann hypothesis,but we now depart from the abstract and static crystal struc-ture of solids and examine the dynamics of this system.

The lattice vibrations, phonons, are bosonic quasiparticles.Therefore, if one knows their energy spectrum ℏ!k, then thetotal energy of the phonon gas is simply the sum over allmodes of the crystal

U ¼ Xk

ℏ!k

eℏ!k=kBT � 1: (47)

Since the number of possible modes is large, 3N, where N isthe number of atoms in the lattice, one might convert thisexpression into an integral by introducing the phonon densityof states gð!Þ normalized as

Rgð!Þd! ¼ 3N. Using stan-

dard methods to calculate the specific heat of the solid, adirectly measurable quantity, the following expression can beobtained:

cV ¼Z 1

0

�ℏ!kBT

�2 eℏ!=kBT

ðeℏ!=kBT � 1Þ2 gð!Þd!: (48)

FIG. 8. Schematic structure of a fictitious one-dimensional solid

built up by cations and anions, positioned in an alternating pattern.



The only sample-specific quantity here is gð!Þ. Surprisingly,the number-theoretical Mobius function and the relatedMobius inversion provide a transformation to express gð!Þas a function of the measured specific heat

gð!Þ ¼ 1

kB!2

X1n¼1

�ðnÞL�1

�cVðh=kBuÞ

u2

�; (49)

where the inverse Laplace transform L�1 converts the spaceof u ¼ h=kBT to !=n (Chen, 1990). Around the same timeanother inversion technique appeared in the literature (Xianxiet al., 1990) and was proven to be equivalent to the onediscussed above (Ming et al., 2003). In the latter formulationanother special function, the Riemann zeta function, wasused, but in order to avoid the dependence on the unprovenRiemann hypothesis a free ‘‘regularization’’ parameter s wasalso introduced. The density of states in this formalism is

gð!Þ ¼ 1

2�!

Z 1

�1!ikþsQðkÞ

�ðikþ sþ 2Þ�ðikþ sþ 1Þ dk;(50)

where QðkÞ ¼ R10 uikþs�1cVð1=uÞdu. Physically the density

of states gð!Þ should be independent of the regularizationparameter, although the existence ofQðkÞ requires that smustfall into the 0 � s1 < s < s2 range, where s1 and s2 are theexponents of the specific heat asymptotes at high and lowtemperatures, respectively. Because of the Dulong-Petit law,at high temperature the specific heat is independent of thetemperature; therefore s1 � 0. On the other end of the tem-perature scale the specific heat of phonons vanishes as Td in ddimensions. Therefore, �ðsÞ in the denominator of the inte-grand (50) sweeps through the [1, 1þ d] strip and ensuresthat no zeros of �ðsÞ can occur there. Summarizing, theasymptotes of the specific heat contribution of lattice vibra-tions in a solid provide an experimentally determined zero-free region of �ðsÞ on the complex s plane. Although thisoffers no further restriction than that which is already knownfrom mathematics, it is an example where physics placesindependent bounds upon the location of the zeros.

E. Statistical physics

The description of both bosons and fermions relies on themathematical properties of the Riemann zeta function. Weshow how the problem of the ‘‘grand-canonical catastrophe’’of number fluctuation in an ideal Bose-Einstein condensate isconnected to number theory. We introduce the concept of theprimon gas and also consider number-theoretical models ofBrownian motion.

Although statistical physics, the physics of systems with alarge number of degrees of freedom, relied heavily uponcombinatorics well before the birth of quantum mechanics,probably the first appearance of the Riemann zeta function instatistical physics occurred in Planck’s momentous work onblackbody radiation, the dawn of the quantum era. From thenon, the Riemann zeta function occurs in numerous differentbranches of statistical physics, from Brownian motion tolattice gas models.

Since the topic of ultracold quantum gases has expandedrapidly in the past decade, we interpret the implications of the

distribution of the Riemann zeta zeros in this area first. Westart with the nonrelativistic, noninteracting, spin zero Bosegas and treat the spatial dimension D as a free parameter. It isa standard textbook derivation (Huang, 2001) to show thatthis system undergoes a phase transition at low temperatures,where the de Broglie wavelength � of the particles becomescomparable to the interparticle distance, and thus the quan-tum nature of the constituents becomes decisive. Since theparticles are free, their spectrum is continuous, simply equalto the kinetic energy � ¼ p2=2m. The total number of parti-cles N is the sum of particles in each quantum state

N ¼ 1

ð2�ℏÞDZ

fBEð�ðpÞÞdDqdDp

¼ V

ð2�ℏÞDZ dDp

e½�ðpÞ��=kT � 1: (51)

Changing the integration from momentum to energy leadsdirectly to

N /Z 1

0

�D=2�1

eð��Þ=kT � 1d� / �

�D

2

�: (52)

In the last step we used the fact that the chemical potentialapproaches the energy of the lowest-lying state, i.e., � ¼ 0.

This result shows that the Bose-Einstein condensationphase transition cannot occur in homogeneous noninteractingsystems in dimensions lower than 3. The total number ofatoms is a positive number and fixed for our system. In onespatial dimension, since �ð1=2Þ< 0, the positivity of Ncannot be fulfilled. For two dimensions the right-hand sideof Eq. (52) is divergent due to the pole of the Riemann zetafunction �ðsÞ at s ¼ 1; therefore N appears to be infinite. Theposition of this pole can be interpreted as the manifestation ofthe Mermin-Wagner-Hohenberg theorem, which guaranteesthat a homogeneous two-dimensional system, provided theinteraction is sufficiently weak, cannot undergo a phasetransition. One may thus see that the pole structure of theRiemann zeta function determines whether our system ofinterest can undergo a phase transition or not. We note herethat this phase transition can occur in lower dimensions forinhomogeneous systems (Widom, 1968; Bagnato andKleppner, 1991; Dai and Xie, 2003).

We now turn to another fundamental question of statisticalmechanics: the equivalency of different statistical ensembles.The difference between the predictions for the ‘‘Riemanngas’’ (see below) based on microcanonical, canonical, andgrand-canonical ensembles has been investigated by Tran andBhaduri (2003). However, the motivation for the analysis isrooted in the so-called grand-canonical catastrophe of anideal Bose gas (Ziff et al., 1977). The number fluctuationof an ideal boson gas is

ð�NÞ2 ¼ X1k¼0

hnkiðhnki þ 1Þ; (53)

where hnki denotes the ensemble average of the occupationnumber of the kth energy eigenstate. According to Eq. (53), inthe presence of a macroscopically occupied ground state, thenumber fluctuation is proportional to the total number ofparticles �N0 � N, which, in the thermodynamical limit(N ! 1), leads to divergence.



Grossmann and Holthaus examined the illustrative modelsystem of an ideal Bose gas trapped in a d-dimensionalpotential with a power-law energy spectrum ��i

� ℏ!��i ,

where �i labels the energy eigenstates (Grossmann andHolthaus, 1997b; Weiss and Wilkens, 1997). Later Eckhardtextended the analysis to the mean density of states and thelevel spacing distribution for ideal quantum gases (Eckhardt,1999). Grossmann et al. showed how the dimensionality and�, which, in some sense, measures the strength of the poten-tial, depress or enhance the number fluctuation of the groundstate as a function of the rescaled temperature t ¼ kBT=ℏ!:

ð�N0Þ2 �8<:Ctd=� ð0< d=�< 2Þ;t2 lnðtÞ ðd=� ¼ 2Þ;�ðd=�� 1Þt2 ð2< d=�Þ;

where C is calculated from a d-dimensional Epstein zetafunction (Holthaus et al., 2001), although here its valuedoes not play a significant role. Therefore, in a given spatialdimension the potential can enhance the fluctuationwhile dimensionality depresses it. They also examined thebehavior of the heat capacity around the critical temperaturet0 and proved that the heat capacity changes continuouslyat t0 if 1< d=� � 2, but if d=� > 2 it undergoes a jumpgiven by

C< � C>

NkB

��t0

¼�d

�

�2 �ðd=�Þ�ðd=�� 1Þ ; (54)

where C< and C> denote the asymptotic values of the heatcapacity at t ! t0 from below and above, respectively. It isworthwhile to note that ð�N0Þ2 in the canonical ensemblecould be expressed as the following integral over the complexplane:

ð�N0Þ2 ¼ 1

2�i

Z �þi1

��i1�ðtÞ�ð�; tÞ�ðt� 1Þ; (55)

where �ð�; tÞ ¼ Pð��nÞ�t is the spectral zeta function of agiven spectrum �n, and � is chosen so all the poles of theintegrand lie on the left of the path of integration. Therefore,all the results shown above are determined by the polestructure of the spectral and the Riemann zeta functions�ð�; tÞ and �ðsÞ, respectively. The large-system behavior isextracted from the leading pole, while the finite-size correc-tions are encoded in the next-to-leading poles.

Equations (54) and (55) did not just clarify an importantphysical question, namely, number fluctuation properties of ad-dimensional boson gas below the critical temperature, butalso had valuable number-theoretical consequences. Theproblem solved above is a purely combinatorial one(Grossmann and Holthaus, 1997a; Holthaus et al., 2001;Weiss and Holthaus, 2002; Weiss et al., 2003): How manyways can one distribute n excitation quanta over N particles?This question, for general n andN, is quite difficult. However,in the low-temperature limit the number of excitations n ismuch smaller than the number of particles N. This problemthus becomes tractable and one could obtain the resultsmentioned above. Calculating the number fluctuation of aboson gas in a one-dimensional (d ¼ 1) harmonic potential(� ¼ 1) provides ð�N0Þ2 � t. But t is simply proportional tothe number of energy quanta ‘‘stored’’ in the excited states,t ¼ ðkBT=ℏ!Þ ¼ n, and therefore ð�N0Þ2 � n. A mathema-

tician (according to Grossmann and Holthaus) can now

interpret this formula:If one considers all unrestricted partitions of the integer n

into positive, integer summands, and asks for the root-mean-

square fluctuation of the number of summands, then the

answer is (asymptotically) justffiffiffin

p.

An intriguing consequence of this analysis is that a Bose-

Einstein condensate could be used (in theory at least) to

factorize numbers (Weiss et al., 2004), which could be

treated as a quantum computer calculating the prime factors.Furthermore, using their physical insight, Weiss and

collaborators derived the following nontrivial number-

theoretical result. Let �ðn;MÞ denote the number of parti-

tions of n into M summands regardless of their order [e.g.,

�ð5; 2Þ ¼ 2 while �ð5; 4Þ ¼ 1], and �ðnÞ represent the totalnumber of different partitions, i.e., �ðnÞ ¼ P

nm¼1 �ðn;mÞ.

It is a natural step to introduce the ‘‘probability’’ of

having exactly M terms in a random partition by pðn;MÞ ¼�ðn;MÞ=�ðnÞ. It was then shown that this probability distri-

bution does not become Gaussian, and it adopts its limiting

distribution shape if n > 1010, which itself is a remarkable

fact.Here we mention only that the same combinatorial prob-

lem arises in many different branches of mathematical phys-

ics, such as lattice animals in statistical physics (Wu et al.,

1996; Lima and de Menezes, 2001), numerical analysis on

combinatorial optimization (Mertens, 1998; Majumdar and

Krapivsky, 2002; Andreas and Beichl, 2003; Bauke et al.,

2003), and also in the description of the low-energy excita-

tions of a one-dimensional fermion system such as bosonic

degrees of freedom (bosonization) (Schonhammer and

Meden, 1996).Tran and Bhaduri’s and then Holthaus and Weiss’ works

further underline the fact that the irregular behavior of the

canonical ensemble lies in the combinatorics of partitioning

integers and the microcanonical and canonical ensembles

prognosticate dramatically different ground state number

fluctuations �n0. This is an important example which un-

equivocally shows that the standard statistical ensembles

cannot always be regarded as equivalent.These examples, while not directly related to any attempt

to prove the Riemann hypothesis, but rather just the zeta

function, do illustrate that results in physics can have pro-

found implications for mathematics in general and number

theory, in particular.The interpretation of prime numbers or the Riemann zeta

zeros as energy eigenvalues of particles appears not just in

quantum mechanics but also in statistical mechanics. Below

we review two concepts: the Riemann gas, sometimes called

the primon gas, and the Riemann liquid, although their

definitions vary slightly.Julia (1990) proposed the idea of a fictitious, noninteract-

ing boson gas, where a single particle may have discrete

energy equal to �0; �1; . . . , where �n ¼ �0 lnðpnÞ (n � 1)and pn represents the nth prime number. This is why the

constituents are called primons. Since the particles are

not interacting, a many-body state, in the second quantized

formalism, can be represented by an integer number n.This natural number has a unique factorization, n ¼pm1

1 pm2

2 � � �pmk

k which tells us that m1 particles are in the



jp1i state, m2 particles are in the jp2i state, etc. Because ofthis uniqueness, each many-body state is enumerated onceand only once. Therefore, the total energy of the system, inthe state jni, is En ¼ m1�0 lnðp1Þ þm2�0 lnðp2Þ þ � � � þmk�0 lnðpkÞ ¼ �0 lnðpm1

1 pm2

2 � � �pmk

k Þ ¼ �0 lnðnÞ. In order to

describe this gas we have to construct the partition functionfrom this spectrum

ZB ¼ X1n¼1

exp

�� En

kBT

�¼ X1

n¼1

1

ns¼ �ðsÞ; (56)

where s ¼ �0=kBT ¼ ��0 and � ¼ ðkBTÞ�1 is the inversetemperature. The partition function for the primon gas is thusthe Riemann zeta function �ðsÞ and hence the alternativenomenclature. It is apparent, by looking at the domain of�ðsÞ, that ZB is well behaving for s > 1, i.e., at low tempera-tures, while s � 1 is physically unacceptable. The boundarys ¼ 1 represents a critical temperature, called the Hagedorntemperature (Hagedorn, 1965) above which the system can-not be heated up, since its energy becomes infinite

hEi ¼ � @

@�lnðZBÞ ¼ � �0

�ð��0Þ@�ð��0Þ

@�� 0

s� 1:

(57)

A similar treatment can be built up for fermions rather thanbosons, but here the Pauli exclusion principle has to be takeninto account, i.e., two primons cannot occupy the same singleparticle state. Therefore mi can be 0 or 1 for all i. As aconsequence, the many-body states are labeled not by thenatural numbers, but by the square-free numbers. Thesenumbers are sieved from the natural numbers by theMobius function. The calculation is a bit more complex,but the partition function for a noninteracting fermion primongas reduces to the relatively simple form

ZF ¼ �ðsÞ�ð2sÞ : (58)

The canonical ensemble is of course not the only ensembleused in statistical physics. Julia (1994) entended the study tothe grand-canonical ensemble by introducing a chemicalpotential �, therefore replacing the primes p with new‘‘primes’’ pe��. This generalization of the Riemann gas iscalled the Beurling gas, after the Swedish mathematicianBeurling who generalized the notion of prime numbers.Examining a boson primon gas with fugacity �1 showsthat its partition function is Z0

B ¼ �ð2sÞ=�ðsÞ.This last result has an astonishing interpretation. We know

that for a system, formed by two subsystems not interactingwith each other, the overall partition function is simply theproduct of the individual partition functions of the subsys-tems. Equation (58) has precisely this structure. There are twodecoupled systems: a fermionic ‘‘ghost’’ Riemann gas at zerochemical potential and a boson Riemann gas with energylevels En ¼ 2�0 lnðpnÞ.

Julia (1994) also calculated the appropriate Hagedorntemperatures and analyzed how the partition functions oftwo different number-theoretical gases, the Riemann gasand the ‘‘log gas,’’ behave around the Hagedorn temperature.Although the divergence of the partition function signals thebreakdown of the canonical ensemble, Julia also claimed that

the continuation across or around this critical temperature canhelp understand certain phase transitions in string theory(Deo et al., 1989) or in the study of quark confinement(Julia, 1994). The Riemann gas, as a mathematically tractablemodel, has been followed with much attention because theasymptotic density of states grows exponentially dðEÞ � eE

just as in string theory. Moreover, using arithmetic functions,it is not extremely hard to define a transition between bosonsand fermions by introducing an extra parameter �, whichdefines an imaginary particle, the noninteracting parafer-mions of order �. This extra parameter counts how manyparafermions can occupy the same state, i.e., the occupationnumber of any state falls into the [0, �� 1] range; thus � ¼ 2belongs to normal fermions, while � ! 1 represents normalbosons. The partition function of a free, noninteracting�-parafermion gas can be given as (Bakas and Bowick,1991)

Z�ðsÞ ¼ �ðsÞ�ð�sÞ : (59)

Bakas further demonstrated, using the Dirichlet convolution( ? ), how one can introduce free mixing of parafermions withdifferent orders which do not interact with each other

f ? g ¼ Xdjn

fðdÞg�n

d

�; (60)

where the shorthand notation djn means d is a divisor of n.This operation preserves the multiplicative property of theclassically defined partition functions Z�1?�2

¼ Z�1Z�2

. It is

even more intriguing how the interaction can be incorporatedinto the mixing by modifying the Dirichlet convolution with akernel function or twisting factor

f � g ¼ Xdjn

fðdÞg�n

d

�Kðn; dÞ: (61)

Using the unitary convolution, Bakas established a pedagogi-cally illuminating case, the mixing of two identical bosonRiemann gases. He showed that

ðZ1 Z1Þ ¼ �2ðsÞ�ð2sÞ ¼

�ðsÞ�ð2sÞ �ðsÞ ¼ Z2Z1: (62)

Thus mixing two identical boson Riemann gases interactingwith each other through the unitary twisting is equivalent tomixing a fermion Riemann gas with a boson Riemann gaswhich do not interact with each other. This leads to theinterpretation that one of the original boson componentssuffers a transmutation into a fermion gas. It is worth notingthat the Mobius function, which is the identity function withrespect to the ? operation (i.e., free mixing), reappears insupersymmetric quantum field theories as a possible repre-sentation of the ð�1ÞF operator, where F is the fermionnumber operator (Spector, 1989, 1990, 1998). In this context,the fact that �ðnÞ ¼ 0 for square-free numbers is the mani-festation of the Pauli exclusion principle.

It is interesting that what initiated as rather academicstudies to investigate potential attacks on the Riemann hy-pothesis may lead to advances in physics. But lwe return tothe hypothesis through a slightly different definition of theRiemann gas. Here the energy of the ground state is taken to



be zero and the energy spectrum of the excited state is �n ¼lnðpnÞ, where pn (n ¼ 2; 3; 5; . . . ) runs over the prime num-bers. Let N and E denote the number of particles in theground state and the total energy of the system, respectively.As demonstrated, the fundamental theorem of arithmeticallows only one excited state configuration for a given E ¼lnðnÞ (n is an integer). It immediately means that this gaspreserves its quantum nature at any temperature, since onlyone quantum state is permitted to be occupied. The numberfluctuation of any state (the ground state included) istherefore zero. In contrast, �n0 predicted by the canonicalensemble is a smooth nonvanishing function of the tempera-ture, while the grand-canonical ensemble still exhibits adivergence. This discrepancy between the microcanonical(combinatorial) and the other two ensembles remains evenin the thermodynamic limit.

One may argue that the Riemann gas is fictitious and itsspectrum is unrealizable. However, the spectrum �n ¼ lnðnÞdoes not increase with n more rapidly than n2, and thereforethe existence of a quantum mechanical potential supportingthis spectrum is possible (cf. the inverse scattering transformused in Sec. III.B). The potential has been given byWeiss et al. (2004)

VðxÞ ¼ V0 ln

�jxjL

�; (63)

where V0 and L are positive constants. Within the semiclas-sical approximation the spectrum of this potential is

�n ¼ V0 lnð2nþ 1Þ þ V0 ln

0@ ℏ2L

ffiffiffiffiffiffiffiffiffiffiffiffi�

2mV0

s 1A; (64)

where n ¼ 0; 1; . . . , and the second term represents only aconstant energy shift.

Recently, LeClair published two works (LeClair, 2007,2008), developing and applying a finite-temperature fieldtheoretical formalism for both boson and fermion gases inlow spatial dimensions in which he efficiently disentangleszero temperature dynamics and quantum statistical sums forboth the relativistic and nonrelativistic cases. His alternativeapproach is based on an S-matrix formulation of statisticalmechanics (Dashen et al., 1969), which redefines the quan-tum statistical mechanics directly in terms of dynamical fill-ing fractions fðkÞ. Assuming the two-body scattering kernelK is constant (i.e., constant scattering length) he derives, aspedagogical examples, the well-known results for the boson

Tc ��

n

�ðd=2Þ�2=d

(65)

and also for the fermion gas

�F ��

�dþ 2

2

�n

�2=d

: (66)

In two dimensions the critical temperature for the boson gasvanishes because of the �ðsÞ divergence at s ¼ 1; thereforethis dimension needs further consideration. Because of thisinstability, LeClair extended the examination for energy-dependent two-body kernels, K ¼ �Reð��k

2��1Þ (� is acomplex number and �� is constant), for a one-dimensionalfermion gas and explicitly constructed a quasiperiodic

potential, VðxÞ � cosð logðxÞÞ=x2�, in the real space which

reproduces the given kernel K in the two-body scattering

approximation. Furthermore, the thermodynamic variables,

such as density and pressure, are also shown to be physically

valid (i.e., positive and have finite value) provided 1=2<Reð�Þ< 3=2. This fully covers the right-hand side of the

critical strip divided by the critical line, and due to the

symmetry of �ðsÞ this half-strip can be extended to the whole

critical strip. His argumentation is based on both the non-

vanishing, nondivergent nature of the physical quantities and

on the assumption that an interaction necessarily modifies the

thermodynamical quantities. If �ð�Þ would be zero some-

where in the critical strip, but off the critical line, then the

leading-order contribution to the thermodynamical quantities

would not be zero contradicting the original assumption,

LeClair argued. This contradiction led him to conclude that

�ð�Þ must be nonzero in the 1=2< �< 3=2 strip, which can

automatically be extended to the whole critical strip by using

the symmetries of the Riemann zeta function. LeClair, there-

fore, claimed �ð�Þ can have no zeros in the given range, so

consequently the Riemann hypothesis must be true. The basis

for this conclusion, however, is itself an assumption and so

does not constitute a proof of the Riemann hypothesis, but

does provide another point of attack.Examination of a similar fictitious, fermionic, many-body

system has also been considered by Leboeuf et al. (2001) and

led to the conclusion that ‘‘time-periodic dynamical evolu-

tions have to be considered as serious candidates (for the

Hilbert-Polya Hamiltonian)’’.At the end of this section, we mention an interesting

interlocking area of statistical physics and number theory.

A few have focused on the connection between number-

theoretical functions and Brownian motion (Good and

Churchhouse, 1968; Billingsley, 1973; Shlesinger, 1986;

Wolf, 1998; Evangelou and Katsanos, 2005) or percolation

(Vardi, 1998). The connection seems to be suggestive, espe-

cially if one defines the random motion through the Mobius

function �ðnÞ, i.e., if �ðnÞ ¼ �1, the particle moves up or

down, and if �ðnÞ ¼ 0, it does not move. Therefore the

distance of the particle from the origin after n steps isMðnÞ ¼Pnk �ðkÞ. The importance of this kind of Brownian motion

lies in the so-called Mertens conjecture. This states if

jMðnÞj � ffiffiffin

p, then the Riemann hypothesis is true.

Figure 9 shows the path of the particle for the first 1 106

steps. Although it is tempting to conclude the following: the

cumulative sum of �ðnÞ remains bounded by � ffiffiffin

p, this

conjecture would actually be wrong as te Riele and

Odlyzko indirectly proved (Titchmarsh and Heath-Brown,

2003). There is no explicit counterexample known, but we

have a loose interval [1014;�3:6 101040] in which there

exists an n such that MðnÞ= ffiffiffin

p> 1 (Kotnik and Riele,

2006). Nevertheless, the Mertens conjecture is a sufficiency

condition for the Riemann hypothesis to be true, but not a

necessary one. Its falsity therefore cannot invalidate the

Riemann hypothesis. The failure of the Mertens conjecture

at such a high n value, however, does give cause for concern

regarding numerical evidence for the validity of the Riemann

hypothesis.This is not the only possibility to define a random walk,

however, either on the �ðsÞ zeros or on the prime numbers. In



the early 1970s Billingsley defined a random, but finite, walk(Billingsley, 1973) based on the fundamental theorem ofarithmetic.

Let fðnÞ denote the number of prime factors of nnot counting their multiplicity, e.g., fð40Þ ¼ 2, since 40 ¼23 5. It can be shown that on average, numbers below Nhave lnð lnðNÞÞ factors, a result which on its own is a surprise.For example, numbers below ee

10 � 109566 have only tendistinct factors on average. Based on the factorization onecan define the following random walk: choose an integer n 2½0; N�; starting from the origin we go up by a unit if2 divides n and down if it does not, and continue the testwith 3, 4,. . .. Although this construction does not seem to beas random as a coin-tossing random walk and has few flows(e.g., it is biased), Billingsley suggested a remedy to theseproblems and showed how the similarity to Brownian motionleads to an Erdos-Kac central limit theorem for fðnÞ

P

� � fðnÞ � lnð lnðNÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lnð lnðNÞÞp � �

�! 1ffiffiffiffiffiffiffi

2�p

Z �

e�u2=2du:

(67)

Therefore, the probability of fðnÞ not deviating from theexpected value lnð lnðNÞÞmore than or � times the standarddeviation can be estimated by a Gaussian integral. Therefore,the mapping of the number-theoretical problem onto aBrownian motion helps one to derive a limit theorem forthe number-theoretical function fðnÞ. As an example, if we

choose ¼ �1 and � ¼ 1 for N ¼ 109566 gives Pð�1 �ðfðnÞ � 10Þ= ffiffiffiffiffiffi

10p � 1Þ � 0:68, thus approximately 70% of

the numbers below the chosen N have from 6 to 13 distinctprime factors.

Wolf (1998) defined random walks in a different way andexamined the distribution and correlation of twin primes(where p and pþ 2 are both primes) and also of cousinprimes (p and pþ 4 are both primes). He also suggestednew random number generators with a theoretically infiniteperiod based on this kind of random walk, contrary to thewidely used random number generators (Press et al., 2007).

He also argued and with computations demonstrated themultifractal nature of a subset of prime numbers (Wolf,1989).

IV. CONCLUSION

‘‘All results of the profoundest mathematical investigationmust ultimately be expressible in the simple form of proper-ties of the integers’’ (Leopold Kronecker). Since this reviewis a summary itself in some sense, here we only attempt toconclude with some general remarks.

In many respects the history of the Riemann hypothesis issimilar to that of Fermat’s last theorem, which was stated inthe 17th century and solved 358 years later (Aczel, 1997;Ribenboim, 1999). Along the path toward the final proof itinspired and gave birth to new areas of mathematics, such asthe theory of elliptic curves. Although the Riemann hypothe-sis has not been proven or disproven, it has already stimulatedand influenced many areas of mathematics, e.g., L functions,which can be thought of as generalized zeta functions, forwhich a generalized Riemann hypothesis may hold.Interestingly, for L functions defined over functional spacerather than the number field, a similar hypothesis is rigorouslyproven.

Further evidence also suggests the validity of the Riemannhypothesis. We think of Levinson’s theorem guaranteeing thatat least one-third of the zeros are on the critical line. However,we cannot exclude the possibility of the existence of acounterexample to the Riemann hypothesis, i.e., a very highlying zero s ¼ �þ it for which � � 1=2. Similarly to theMertens conjecture, the counterexample may occur so highon the critical line that we have no machinery to evencalculate zeros at that elevation. The immediate impact ofsuch a collapse of the Riemann hypothesis would be immensesince there exist numerous ‘‘proofs’’ that are contingent uponit (Titchmarsh and Heath-Brown, 2003).

That being said, we cannot miss out in this Colloquium onecomputational masterpiece. Not long after World War II, inwhich mechanical and electrical ‘‘computers’’ were oftenused for encrypting messages (Enigma) and also for research(ENIAC), van der Pol (1947) constructed an electromechani-cal machine that could calculate the first few zeros of theRiemann zeta function. This construction, despite its limitedachievement, deserves to be treated as a gem in the history ofthe natural sciences. Several decades later, on the other end ofthe spectrum, a state-of-the-art application of numerical tech-niques carried out by Brent (1979, 1982), van de Lune andte Riele (1983), and van de Lune et al. (1986) calculated thefirst 1:5 109 zeros. Meanwhile, Odlyzko (1987) exploredthe zeros located around t� 1020 and showed that all zeros(millions of them) he found do lie exactly on the critical line.Here we note that these numerical checkings are, of their ownright, significant achievements, and also have influenced thedevelopment of fast numerical techniques used in physics[see, e.g., Draghicescu (1994) and Greengard (1994)].

In this Colloquium we collected a few examples fromdifferent areas of mathematical physics, starting with classi-cal mechanics and finishing with statistical mechanics, wherethe Riemann zeta function �ðsÞ, especially its zero and polestructure, has a highly influential role.

−1000

−800

−600

−400

−200

0

200

400

600

800

1000

0 1 2 3 4 5 6 7 8 9 10

Cum

ulat

ive

sum

of M

öbiu

s fu

nctio

n, µ

(n)

n × 105

FIG. 9. Function MðnÞ, the cumulative sum of the Mobius func-

tion, is shown with the mean displacement of a random walk, � ffiffiffin

p,

for comparison.



In the section devoted to classical mechanics, we showed

how the Riemann hypothesis can arise in a simple mechanical

system, a ball bouncing on a rigid wall. We also argued how

these billiard systems lead to a revolutionary new way of

describing the dynamics of a chaotic system by introducing

the evolutionary operator. Here we also sketched the connec-

tion, a trace formula, between the dynamics of a chaotic

system and the periodic orbits of the same system.This new descriptive language of dynamics through the

trace formulas of the Green’s function is suitable to develop a

new quantization technique for chaotic quantum systems

which otherwise was impossible using the standard Bohr

quantization rules. Gutzwiller’s trace formula has been ex-

plicitly mentioned, because the Riemann zeta function obeys

a very similar expression. Therefore, we could compare the

two Eqs. (9b) and (25), and imagine what properties a

quantum system might have if its spectrum mimicked the

zeros of the Riemann zeta function.We also surveyed two other attempts to find a quantum

system which has a connection to the Riemann zeta function.

Both of these directions try to associate �ðsÞ with the spec-

trum of the system. The difference between these approaches

is that one of them relates �ðsÞ to the positive energy spec-

trum, i.e., scattering states, while the other, based on the

Hilbert-Polya conjecture, proposes systems where the nega-

tive energies, thus the bound states of the system, coincide

with the zeros of �ðsÞ. This latter case naturally guides us to

condensed matter physics and statistical mechanics, where

one has to evaluate physical observables on the lattice points,

or derive all thermodynamical properties of a given particle-

system provided the spectrum is given.In the sections concentrating on condensed matter physics,

we first showed how the Riemann zeta function, or one of its

ancillary functions, arose when we calculated the binding

energy of a given structure of solid matter. Finally, we showed

how physical requirements for the specific heat of a solid can

provide zero-free regions for the Riemann zeta function.

Research in this direction eventually may offer narrower

zero-free regions and complement the approach in pure

mathematics.In the last section, we discussed three main areas of

statistical physics, where the Riemann zeta function and its

number-theoretical aspects influence the behavior of a physi-

cal system. First, we considered the low-temperature phase

transition of bosons and showed that the pole structure of �ðsÞprohibits Bose-Einstein condensation in one- and two-

dimensional uniform systems. We also reviewed the grand-

canonical catastrophe of an ideal Bose gas, where the

predictions of two ensembles widely used in statistical phys-

ics contradict each other, showing, therefore, that these en-

sembles cannot be equivalent to one another. Finally, we

examined a possible Brownian motion model for the

number-theoretical Mobius function.It would not be without precedent if a completely new

theory or a new mathematical language is needed in which

the Riemann hypothesis can be ‘‘worded’’ naturally for the

hypothesis to be finally proven. As has happened earlier with

mathematics, natural science, and, in particular, physics, can

give impetus and motivate new directions perhaps leading to

the final proof. It is amazing and captivating to see that a

purely number-theoretical function has so many direct linksto classical and modern physics.

Nowadays we are not surprised by Galileo’s famous key-note: ’’[Nature] is written in the language of mathematics,and its characters are triangles, circles, and other geometricfigures without which it is humanly impossible to understanda single word of it; without these, one wanders about in a darklabyrinth’’ (Drake, 1957). Probably we are not meandering ina labyrinth, but we are definitely puzzled by the overwhelm-ing difficulty of proving the Riemann hypothesis. We simplydo not know as yet whether physics will ultimately help inunderstanding such an elegant mathematical statement as theRiemann hypothesis, but we are definitely witnessing theintertwining and invigoration of both disciplines.

ACKNOWLEDGMENTS

D. S. expresses his gratitude to Brandon P. van Zyl forcreating an inspiring and welcoming environment and partialfinancial support through a grant from the National Scienceand Engineering Research Council of Canada. We also thankall the referees of this manuscript. Their comments andsuggestions were highly valuable and led to a significantincrease in the final quality of this paper. This work hasbeen financially supported by the University of Otago andthe Government of New Zealand through the Foundation forResearch, Science and Technology under New EconomyResearch Fund Contract No. NERF-UOOX0703.

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