"The Riemann Zeros and Eigenvalue Asymptotics"
M.V.Berry
and J.P.Keating
[Review article from
SIAM Review, 41, No.2 (1999)
236266]
Abstract
Comparison between formulae for the
counting functions of the heights t_{n}
of the Riemann zeros
and of semiclassical quantum eigenvalues E_{n}
suggests that the t_{n} are
eigenvalues of an (unknown) hermitean operator H,
obtained by quantizing a classical dynamical system with
hamiltonian H_{cl}. Many features of H_{cl}
are provided by the analogy;
for example, the "Riemann dynamics" should be
chaotic and have periodic orbits whose periods are
multiples of logarithms of prime numbers. Statistics of
the t_{n} have a similar structure
to those of the semiclassical E_{n};
in particular, they display randommatrix universality at
short range, and nonuniversal behaviour over longer
ranges. Very refined features of the statistics of the t_{n}
can be computed accurately from formulae with quantum
analogues. The RiemannSiegel formula for the zeta function
is described in detail. Its interpretation as a relation
between long and short periodic orbits gives further
insights into the quantum spectral fluctuations. We
speculate that the Riemann dynamics is related to the
trajectories generated by the classical hamiltonian H_{cl}
= XP.
 Introduction
Our purpose is to report on the
development of an analogy, in which three areas of
mathematics and physics, usually regarded as separate,
are intimately connected. The analogy is tentative and
tantalizing, but nevertheless fruitful.. The three areas
are eigenvalue asymptotics in wave (and particularly
quantum) physics, dynamical chaos, and prime number
theory. At the heart of the analogy is a speculation
concerning the zeros of the Riemann zeta function (an infinite sequence of numbers encoding the primes): the Riemann zeros are related to the
eigenvalues (vibration frequencies, or quantum energies)
of some wave system, underlying which is a dynamical
system whose rays or trajectories are chaotic.
Identification of this dynamical
system would lead directly to a proof of the celebrated Riemann hypothesis. We do not know what the system is, but we
do know many of its properties, and this knowledge has brought insights in
both directions: from mathematics to physics, by
stimulating the development of new spectral asymptotics, and from physics to mathematics, by
indicating previously unsuspected correlations between
the Riemann zeros. We have review some of this material
before [1,2,3,4,5,6], but these accounts do not include
several recent developments to be described here,
especially those in the last part of section 4 and all of
sections 5 and 6.
To motivate the approach from
physics, we begin with the counting function for the
primes, pi(x), defined as the number of
primes less than x (thus pi(3.5)=2); this
is a staircase function, with unit steps at the primes p.
The density of primes is the distribution
At the roughest level of
description, and with the distribution appropriately
smoothed,
(as implied by the prime number theorem: pi(x) ~ x/logx).
One of Riemann's great achievements
[7,8] was to give exact formula for pi'(x),
constructed as follows. First pi'(x) is
expressed in terms of a function J(x) [7, Chap.1]
that has jumps at prime powers:
In this formula,
are the Mobius
numbers (1,1,1,0,1,1,…) [7]. Each of the partial
densities J' is the sum of a smooth part (dominated by
(1.2)) and an infinite series of oscillations:
(see section 1.18 of [7]). Here the
numbers t_{n} in the oscillatory
contributions are related to the complex Riemann zeros,
defined as follows.
Riemann's zeta function, depending on the complex variable s,
is defined as
and by analytic continuation
elsewhere in the s plane. It is known that the
complex zeros (i.e, those with nonzero imaginary part) of
zeta(s) lie in the "critical
strip" [Re s between 0 and 1], and the Riemann hypothesis states that in fact all these zeros lie on
the "critical line" Res = 1/2 (see
Figure 1). The numbers t_{n} in
(1.4) are defined by
0 If the Riemann hypothesis is true,
all the (infinitely many) t_{n} are
real, and are the heights of the zeros above the real s
axis. It is known by computation that the first
1,500,000,001 complex zeros lie on the line [9], as do
more than onethird of all of them [10].
Each term in the sum (1.4)
describes an oscillatory contribution to the fluctuations
of the density of primes, with larger Ret_{n}
corresponding to higher frequencies.
FIGURE 1
Because of the logarithmic
dependence, each oscillation gets slower as x
increases. This slowingdown can be eliminated by the
change of variable u = log x; thus
If the Riemann hypothesis is true,
Im t_{n} = 0 for all n, and
the function f(u), constructed from the
primes, has a discrete spectrum; that is, the support of
its Fourier transform is discrete. If the Riemann
hypothesis is false, this is not the case. The
frequencies t_{n} are reminiscent
of the decomposition of a musical sound into its
constituent harmonics. Therefore there is a sense in
which we can give a oneline nontechnical statement of
the Riemann hypothesis: "The primes have music in
them."
However, readers are cautioned
against thinking that it would be easy to hear this prime
music by constructing f(u) as defined in
(1.7) and then converting it into an audio signal. In
order for the human ear to hear the lowest Riemann zero,
with t_{1} = 14.13…, it would be
necessary to play N ~ 100 periods of cos(t_{1}u),
requiring primes in the range [ x between 0 and
exp(2pi N/t_{1}) ~ exp(45) ~ 10^{19}].
On this acoustic analogy, the
heights t_{n} (hereinafter referred
to simply as "the zeros") are frequencies. This
raises the compelling question: frequencies of what? A
natural answer would be: frequencies of some vibrating
system. Mathematically, such frequencies  real numbers 
are discrete eigenvalues of a selfadjoint (hermitean)
operator. That the search for such an operator might be a
fruitful route to proving the Riemann hypothesis is an
old idea, going back at least to Hilbert and
Polya [7]; what is new is the physical interpretation of
this operator and the detailed information now available
about it.
The mathematics of almost all
eigenvalue problems encountered in wave physics is
essentially the same, but the richest source of such
problems is quantum mechanics, where the eigenvalues are
the energies of stationary states ("levels"),
rather than frequencies as in acoustics or optics, and
the operator is the hamiltonian. Reflecting this
catholicity of context, we will refer to the t_{n}
interchangeably as energies or frequencies, and the
operator as H (Hilbert, Hermite, Hamilton…).
To help readers navigate through
this review, here is a brief description of the sections.
In section 2 we describe the basis of the Riemannquantum
analogy, which is an identification of the periodic
orbits in the conjectured dynamics underlying the Riemann
zeros, made by comparing formulae for the counting
functions of the t_{n} and
of asymptotic quantum eigenvalues. Section 3 explains the
significance of the long periodic orbits in giving rise
to universal (that is, systemindependent) behaviour in
classical and semiclassical mechanics and, by analogy,
the Riemann zeros. The application of these ideas to the
statistics of the zeros and quantum eigenvalues is taken
up in section 4. Section 5 is a description of a powerful
method for calculating the t_{n }(the
RiemannSiegel formula), with a physical interpretation
in terms of resurgence of long periodic orbits that
implies new interpretations of the periodicorbit sum for
quantum spectra. The properties of the conjectured
dynamical system are listed in section 6, where it is
speculated that the zeros are eigenvalues of some
quantization of the dynamics generated by the hamiltonian
H_{cl} = XP.
2. The Analogy
3. Long Orbits and Universality
4. PeriodicOrbit Theory for
Spectral Statistics
5. RiemannSiegel Formulae
6. Spectral Speculations
References
[1] M.V.Berry, Riemann's zeta
function: A model for quantum chaos, in Quantum Chaos
and Statistical Nuclear Physics, T.H. Seligman and H.
Nishioka, eds., Lecture Notes in Phys. 263,
SpringerVerlag, New York, 1986, pp. 117.
[2] M.V.Berry, Quantum chaology
(the Bakerian lecture), Proc. Roy. Soc. Lond. Ser. A,
413 (1987) pp. 183198.
[3] M.V. Berry, Some
quantumtoclassical asymptotics, in Chaos and
Quantum Physics, Les Houches Lecture Series 52, M.J.
Giannoni, A. Voros, and J. ZinnJustin, eds.,
NorthHolland, Amsterdam, 1991, pp.251304.
[4] J.P.Keating, The Riemann
zetafunction and quantum chaology, in Quantum Chaos,
G. Casati, I. Guarneri, and V. Smilansky, eds.
NorthHolland, Amsterdam, 1993, pp. 145185.
[5] J.P.Keating, Periodic orbits,
spectral statistics, and the Riemann zeros, in
Supersymmetry and Trace Formulae: Chaos and Disorder,
J.P. Keating, D.E.Khmelnitskii, and I.V.Lerner, eds.,
Plenum, New York, 1998, pp. 115.
[6] M.V.Berry and J.P.Keating, H
= xp and the Riemann zeros, in Supersymmetry and
Trace Formulae: Chaos and Disorder, J.P. Keating,
D.E.Khmelnitskii, and I.V.Lerner, eds., Plenum, New York,
1998, pp.355367
[7] H.M.Edwards, Riemann's Zeta
Function, Academic Press, New York, London, 1974
[8] D.
Zagier, The first 50
million prime numbers, Math. Intelligencer, 0 (1977),
pp. 719.
[9] J. Van de Lune, H.J.J.
te Riele, and D.T.Winter, On
the zeros of the Riemann zeta function in the critical
strip. IV, Math. Comp., 46 (1986), pp. 667681.
[10] N. Levinson, More than one
third of the zeros of Riemann's zetafunction are on
rho = 1/2, Adv. Math., 13 (1974), pp. 383436.
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