The Riemann Zeros and Eigenvalue Asymptotics.
Comparison between formulae for the counting functions of the heights tn  of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn  are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl. Many features of Hcl 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 tn have a similar structure to those of the semiclassical En; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues. The Riemann-Siegel 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 Hcl=XP.
log x x (x2 1) x log x xImtn
Retn>0
(see section 1.18 of [7]). Here the numbers tn in the oscillatory contributions are
related to the complex Riemann zeros, dened as follows.
Riemann's zeta function, depending on the complex variable s, is dened as
Y X1
ns (Res>1)
and by analytic continuation elsewhere in the s plane. It is known that the complex
zeros (i.e., those with nonzero imaginary part) of (s) lie in the \critical strip" 0 <
Res<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 tn in (1.4) are dened by

(Retn =0):If the Riemann hypothesis is true, all the (innitely many) tn are real, and are the
heights of the zeros above the real s axis. It is known by computation that the rst
1,500,000,001 complex zeros lie on the line [9], as do more than one-third of all of
them [10].
Each term in the sum in (1.4) describes an oscillatory contribution to the uctuations
of the density of primes, with larger Retn corresponding to higher frequencies.
M. V. BERRY AND J. P. KEATING
Fig. 1 Complex s plane, showing the critical strip (shaded) and the complex Riemann zeros (there
are trivial zeros at
Because of the logarithmic dependence, each oscillation gets slower as x increases.
This slowing-down can be eliminated by the change of variable

Retn>0
If the Riemann hypothesis is true, 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
tn are reminiscent of the decomposition of a musical sound into its constituent
harmonics. Therefore there is a sense in which we can give a one-line 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 dened in (1.7) and then converting it into
an audio signal. In order for the human ear to hear the lowest Riemann zero, with
t1 =14:13 :::; it would be necessary to play N 100 periods of cos(t1u), requiring
primes in the range 0 <x<exp(2N=t1) exp(45) 1019.
On this acoustic analogy, the heights tn (hereinafter referred to simply as \the
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 self-adjoint (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 mechan-
ics, where the eigenvalues are the energies of stationary states (\levels"), rather than
frequencies as in acoustics or optics, and the operator is the hamiltonian. Reecting
this catholicity of context, we will refer to the tn 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 Riemann-quantum analogy, which
is an identication of the periodic orbits in the conjectured dynamics underlying the
Riemann zeros, made by comparing formulae for the counting functions of the tn and
of asymptotic quantum eigenvalues. Section 3 explains the signicance of the long
periodic orbits in giving rise to universal (that is, system-independent) 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 tn
(the Riemann-Siegel formula), with a physical interpretation in terms of resurgence
of long periodic orbits that implies new interpretations of the periodic-orbit 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
2. The Analogy. The basis of the analogy is a formal similarity between representations
for the uctuations of the counting functions for the Riemann zeros tn and
for vibration frequencies associated with a system whose rays are chaotic. For the tn
(assumed real), the counting function is dened for t>0as
where denotes the unit step. Central to our arguments is the fact that N(t) can
be decomposed as follows [11]:
where

and

(The branch of the logarithm is chosen to be continuous, with N(0)=0:)
These two components can be interpreted as the smooth and uctuating parts of
the counting function. Here and hereinafter the notation hidenotes a local average
of a uctuating quantity, over a range large compared with the length scales of the
uctuations but small compared with any secular variation. Implicit in such averaging
is an asymptotic parameter; in the present case this is t, and the averaging range is
large compared with the mean spacing of the zeros but small compared with t itself.
The formula for hNi can be obtained from the functional equation for (s) [7]. It
follows by dierentiating the last member of (2.3) that the asymptotic density of the
zeros is

M. V. BERRY AND J. P. KEATING
Fig. 2 Thick line: Divergent series (2.6) for the counting function uctuations N of the Riemann
zeros, including all values of m and the rst 50 primes p. Thin line: Exact calculation of
N from (2.4).
and therefore that the mean spacing between the zeros decreases logarithmically with
increasing t. Underlying the formula for N are the observations that the phase of a
function jumps by on passing close to a zero, and that
that between the jumps in N this function varies smoothly, implying that its average
value is zero.
Now we substitute into (2.4) the Euler product (1.5), disregarding the fact that
this does not converge in the critical strip, and obtain the divergent but formally
exact expression

This formula gives the uctuations as a series of oscillatory contributions, each labelled
by a prime p and an integer m, corresponding to the prime power pm. Terms with
m>1 are exponentially smaller than those with 1. The oscillation corresponding
to p has a \wavelength" (that is,
log p
In order to discriminate individual zeros, suciently many terms must be included
in the sum for this wavelength to be less than the mean spacing; from (2.5), this gives
p<t=2. When truncated in this way, the sum (2.6) can reproduce the jumps quite
accurately for low-lying zeros, as Figure 2 shows, even though the complete sum
diverges.
Consider now a classical dynamical system [12] in a conguration space with D
freedoms, coordinates fp1;:::;pDg. Trajectories
are generated by a hamiltonian function H(q, p) on the two-dimensional phase space
fq, pg, whose conserved value is the energy E. In quantum physics, q and p are
operators, with commutation relation [q, p]=i~, where ~ h=2 is Planck's con-
stant. Then H(q, p), augmented by boundary conditions, becomes a hermitean wave
operator, whose eigenvalues, discrete if the system is bound, are the quantum energy
levels En. More generally, this formalism applies to any wave system (e.g., water
waves [13]) with coordinates q and wavenumber k, dened by a dispersion relation
!(q, k), the connection between the quantum and wave formalisms being
Familiar wave equations appear when the commutation relations are implemented
with Hamilton's equations are the corresponding ray equations (in
optics these are the rays generated by Snell's law or Fermat's principle). For example,
a locally uniform medium (H independent of q) with impenetrable walls corresponds
to \quantum billiards," where waves are governed by the Helmholtz equation with
Dirichlet boundary conditions, and the (straight) rays are reected specularly at the
walls [14]. Of special interest to us is the asymptotics of the eigenvalues En in the
semiclassical limit ~ ! 0, which from (2.8) is equivalent to the short-wavelength or
high-frequency limit.
Waves, in particular the eigenfunctions of H, usually depend not on individual
trajectories but on families of trajectories, whose global structure is an important
determinant of the energy-level asymptotics. Of interest here is the case where the
trajectories are chaotic [15, 16, 17], that is, where E is the only globally conserved
quantity and neighbouring trajectories diverge exponentially. Then on a given energy
shell (that is, for given E), the usual structure|and the one we will consider
here|is that all initial conditions generate trajectories that explore the (2D 1)-
dimensional energy surface ergodically, except for a set, dense but of zero measure, of
(one-dimensional) isolated unstable periodic orbits.
An important result of modern mathematical physics, central to the Riemann-
quantum analogy, is that these isolated periodic trajectories determine the uctuations
in the counting function N(E) of the energy levels [18, 19, 20, 21]. Using the notation
(2.2), with E replacing t, we can separate N(E) into its smooth and uctuating parts
hN(E)i and N(E). The averaging is over an energy interval large compared with
the mean level spacing but classically small, that is, vanishing with ~. We state the
formula for N(E) and then explain it:

The symbol indicates that the formula applies asymptotically, that is, for small
~. (In the special case of the Selberg trace formula [21], corresponding to waves on
a compact surface of constant negative curvature, the formula is exact.) The index
labels primitive periodic orbits, that is, orbits traversed once. The index m labels
their repetitions. Therefore, the two sums together include all periodic orbits. Sp(E)
is the action of the primitive orbit p, that is,
I
In terms of Sp, the period of the orbit is
The hyperbolic symplectic matrix Mp (the monodromy matrix) describes the exponential
growth of deviations from p of nearby (linearized) trajectories, between successive
M. V. BERRY AND J. P. KEATING
crossings of a Poincare surface of section transverse to p. p is the Maslov phase, determined
[22] by the winding round p of the stable and unstable manifolds containing
the orbit.
Physically, the appearance of periodic orbits is not surprising. The levels En,
counted by N, are associated with stationary states, that is, states or modes that are
time-independent. By the correspondence principle, their asymptotics should depend
on phase space structures unchanged by evolution along rays, that is, the invariant
manifolds with energy E. In the type of chaotic dynamics we are considering, there are
two types of invariant manifold: the whole energy surface, which determines hN(E)i
as we will see, and, decorating this, the tracery of periodic orbits, which determines
the ner details of the spectrum as embodied in the uctuations N(E).
For long orbits, the determinant is dominated by its expanding eigenvalues, and,
for large Tp,

where p is the Liapunov (instability) exponent of the orbit p. Thus, approximately,

Now we can make the formal analogy with the corresponding formula (2.6) for the
counting function uctuations of the Riemann zeros:
Quantum Riemann
Dimensionless
actions
mt log p
~
Periods mTp m log
Stabilities 12 pTp 21 log
Asymptotics
The nonappearance of ~ on the \Riemann" side indicates that the dynamical system
underlying the zeros is scaling, in the sense that the trajectories are the same for all
\energies" t, as in the most familiar scaling system, namely, quantum billiards, where,
for a particle of mass m, energy scales according to the combination
and, for an orbit of length Lp, the analogy, primes acquire a new
signicance, as primitive periodic orbits, whose periods are logp. The index m in (2.6)
then labels their repetitions.
The fact that all orbits have the same instability exponent (unity) indicates that
the Riemann dynamics is homogeneously unstable, that is, uniformly chaotic. More-
over, the dynamics does not possess time-reversal symmetry. If it did, degeneracy
of actions between each orbit and its time-reversed partner would lead to their contributing
coherently to N(t), so that for most orbits (those that are not self-retracing)
the prefactor in (2.6) would be 2= rather than 1=.
An alternative form of the periodic-orbit sum (2.9), which will be useful later, is
in terms of the level density
d N (E)
Denoting primitive and repeated periodic orbits by the common index j (= fp; mg),
we can write
~
where for convenience we have absorbed the Maslov indices into the actions, and the
amplitude Aj is
as ~ ! 0. For the Riemann zeros, the corresponding formula, from (2.6) and (2.14),
has j =0,
log
and is an identity rather than an asymptotic approximation.
There are two discordant features of the analogy [1], to which we will return. First,
the exponential decay of long orbits in the quantum formula (2.13) is an approximation
to the determinant in (2.9), whereas for the Riemann zeros the exponential in (2.6)
is exact. Second, the negative sign in (2.6) indicates that when the Maslov phases
mp=2 are reinstated in (2.13) their value should be for all orbits, but this is hard
to understand because if the index is for a given orbit it should be 2 for the same
orbit traversed twice.
The smooth part hN(E)i of the counting function is, to leading order in ~, the
number of phase space quantum cells (volume hD) in the volume (E) of the energy
For billiards, is proportional to the
spatial volume conning the system (this is Weyl's asymptotics [23]). The mean level
density is thus
In the quantum formula (2.13), each orbit contributes an oscillation to N(E),
with energy \wavelength" (cf. (2.7))
This should be compared with the mean spacing of the eigenvalues, which is the
reciprocal of the mean level density and so (from (2.19)) of order ~D. An important
implication is that the oscillation contributed by a given orbit has, asymptotically, a
wavelength much larger than the mean level spacing. Thus in order to have a chance
of resolving individual levels it is necessary to include at least all those orbits with
periods up to
This evokes the time-energy uncertainty relation, so TH is called the Heisenberg time.
Asymptotically, TH corresponds to very long orbits, or, in the Riemann case, large
primes pH(t)=t=2 (cf. the discussion following (2.7)). In what follows, this emphasis
on long orbits will play a key role.
M. V. BERRY AND J. P. KEATING
3. Long Orbits and Universality. In a classically chaotic system, the periodic
orbits proliferate exponentially as their period increases [24], with density
number of orbits with periods between T and T
dT
exp (T)
as T !1:
Here, is the topological entropy of the system. In the cases we are interested in,
can be identied with a suitable average of the instability exponents of long periodic
orbits (cf. (2.12)). In the Riemann case, where according to (2.14) the periodic orbits
correspond to primes, (3.1) nicely reproduces the prime number theorem (1.2) and
thereby reinforces the analogy (the repetitions, labelled by m, give exponentially
smaller corrections).
From (2.18), the proliferation in (3.1) cancels the decay of the intensities A2 for
long orbits. One way to write this is
(3.2) lim A2j (T Tj)=1:
This is the sum rule of Hannay and Ozorio de Almeida [25]. Its importance is threefold:
rst, it does not contain ~ and so is a classical sum rule. Second, the amplitudes Aj
nevertheless have signicance in quantum (i.e., wave) asymptotics, because they give
the strengths of the contributions to spectral density uctuations. Third, the rule is
universal: (3.2) contains no specic feature of the dynamics|it holds for all systems
that are ergodic. One way to appreciate the naturalness of this universality is to
imagine that a long orbit with energy E, inscribed on the constant-energy surface
forms an intricate tracery that, with the slightest smoothing, could cover
the surface uniformly with respect to the microcanonical (Liouville) measure. This
\phase-space democracy" is the basis of Hannay and Ozorio de Almeida's derivation.
Expressed mathematically, this ergodicity-related sum rule corresponds to an
(associated with the invariant measure) of the Perron{Frobenius
operator that generates the classical ow in phase space. Equivalently [26, 27], it
corresponds to a simple pole at of the dynamical zeta function D(s), dened
(for two-dimensional systems, for example) by
where p is the larger eigenvalue (jpj > 1) of the monodromy matrix M. The rest of
the spectrum of the Perron{Frobenius operator, or equivalently the analytic structure
of D(s) away from determines the rate of approach to ergodicity|that is, it
is related to the system-specic short-time dynamics.
Now recall that according to (2.21) the long orbits determine spectral uctuations
on the scale of the mean level separation. The universality of the classical sum rule
suggests that the spectral uctuations should also show universality on this scale.
And by the Riemann-quantum analogy, we expect this spectral universality to extend
to the Riemann zeros tn.
It is in the statistics of the levels and Riemann zeros that the universality appears.
This is to be expected, since ergodicity is a statistical property of long orbits.
It is important to note that we are here considering individual systems and not
ensembles, so statistics cannot be dened in the usual way, as ensemble averages.
Instead, we rely on the presence of an asymptotic parameter (see the remarks after
(2.4), and before (2.9)): high in the spectrum (or for large t in the Riemann case),
there are many levels (or zeros) in a range where there is no secular variation, and it
is this large number that enables averages to be performed. Universality then emerges
in the limit ~ ! 0 (or t !1) for correlations between xed numbers of levels or
zeros.
A mathematical theory of universal spectral uctuations already exists in the more
conventional context where statistics are dened by averaging over an ensemble. This
is random-matrix theory [28, 29, 30, 31, 32], where the correlations between matrix
eigenvalues are calculated by averaging over ensembles of matrices whose elements
are randomly distributed, in the limit where the dimension of the matrices tends
to innity. Here the relevant ensemble is that of complex hermitean matrices: the
\Gaussian unitary ensemble" (GUE). As will be discussed in the next section, it is
precisely these statistics that apply to high eigenvalues of individual chaotic systems
without time-reversal symmetry, and also to high Riemann zeros, in the sense that
the spectral or Riemann-zero averages described in the previous paragraph coincide
with GUE averages.
First, however, we give a very simple argument [33] showing that the approach to
universality must be nonuniform. The classical sum rule (3.2) applies to long orbits
but not to short ones, because these will reect the specic dynamics of the system
whose spectrum is being considered. Therefore, spectral features that depend on short
orbits can be expected to be nonuniversal. From (2.20), these are uctuations on the
energy scale is the period of the shortest orbit. This scale is
asymptotically small but still large compared with the separation of order hD between
neighbouring eigenvalues. On this basis, we expect universality to be a good approximation
for correlations between eigenvalues separated by up to O(1=h(D1)) mean
spacings, but not for larger separations. For the Riemann zeros,
tion (2.14)), whereas the mean separation between zeros is 2=log(t=2). Therefore
universality for zeros near t should break down beyond about log(t=2)= log 2 mean
spacings. We regard the observation of the breakdown of random-matrix universality
for the Riemann zeros [34], in accordance with this prediction, as giving powerful
support to the analogy with quantum or wave eigenvalues.
4. Periodic-Orbit Theory for Spectral Statistics. In discussing statistics, it will
be simplest to measure intervals between eigenvalues or Riemann zeros in units of the
local mean spacing. We denote such intervals by x, and the corresponding levels or
zeros, referred to a local origin, by xn; in these units, 1. We will mainly
be concerned with statistics that are bilinear in the level density, the simplest being
the pair correlation of the density uctuations, dened in [31], in the sense of a
distribution, as
R (x; y) probability density of separations x of levels or zeros
close to a scaled position y
(In the second member, the sum is over a stretch of N levels near y, with N>1.) R
gives the correlation between levels near E, or, correspondingly, Riemann zeros near
M. V. BERRY AND J. P. KEATING
t; for simplicity of notation, we will henceforth not indicate these base levels (denoted
y in (4.1)).
Closely related to R is the form factor K() (the name comes from crystallogra-
phy), dened as
(R
where the sum is as in (4.1). Here the variable (conjugate to x) is the scaled time
TH
where TH is the Heisenberg time (equation (2.21)). With the denitions given, both
R and K tend to 1 at long range; the term (x)inR ensures that this requirement is
compatible with (4.2).
Other statistics that are bilinear in d can be expressed in terms of K or R.A
useful one is the number variance:
variance of number of levels or zeros in
anDinterval w1here the mean1 number2iEs x
x
correlation function (4.1) is determined by the spectral density uctuations,
for which there is the semiclassical formula (2.16). Our aim in this section is to explain
how to employ this observation to calculate these bilinear statistics, obtaining not only
the universal random-matrix limit but also the corrections to this corresponding to
large eigenvalue or zero separations, or short times. The argument is subtle and has
several levels of renement, of which we start with the simplest [3, 5, 33].
We will calculate K(). The rst step is to substitute (2.16) into (4.1), thereby
obtaining a double sum over periodic orbits. Since all the actions are positive, we can
simplify the averages (over a small interval of eigenvalues or along the critical line)
using
The dimensionless intervals x that we will be considering may be large but must
correspond to classically small energy ranges, so we can approximate the actions
using
x xTj x2
where Sj, Tj, and d are evaluated at E0. Elementary manipulations, and evaluating
the integral in (4.2), give the asymptotic (that is, small-~) form factor as the double
sum
It is convenient now to consider separately the diagonal part Kdiag of the sum
(terms with and the o-diagonal part Ko (terms with k). For Kdiag,wehave

Kdiag ()=
In the limit ~ ! 0, xed, the sum over orbits can be evaluated using the Hannay-
Ozorio sum rule (3.2), giving
This is universal: all details of the specic dynamics have disappeared. Because of the
Riemann-quantum analogy, the same behaviour should hold for the pair correlation
of the Riemann zeros. Here we make contact with the seminal work of Montgomery
[35], who indeed proved (4.9) in that case.
Now we observe that in random-matrix theory the exact form factor of the GUE
is
( is the unit step.) For later reference, the GUE pair distribution function, obtained
from (4.2), issin (x)
x
Evidently the approximation (4.9), based on periodic orbits, captures exactly the
random-matrix behaviour for jj < 1, without invoking any random matrices. This
led Montgomery [35] to conjecture (following a suggestion of Dyson and independently
of any semiclassical argument) that for the Riemann zeros K()=KGUE()inthe
limit t !1.
Clearly, (4.9) does not give the random-matrix result when jj > 1. Indeed
it fails drastically by not satisfying the requirement, necessary for any form factor
representing a discrete set of points (eigenvalues or zeros), that K() ! 1as !1.
This failure reects the importance of Ko, and implies that for large (long orbits)
the o-diagonal terms in the double sum (4.7) cannot vanish through incoherence,
as might naively be thought, but must conspire by destructive coherent interference
to cancel the term from Kdiag and replace it by 1. This is consistent with the
Montgomery conjecture, which implies
in the limit t !1.
One reason why Kdiag alone is inadequate is the proliferation of orbits: for sufciently
long times, there will be many pairs of orbits whose actions dier by less
M. V. BERRY AND J. P. KEATING
than ~, so that they cannot be regarded as incoherent in (4.7). This phenomenon,
that in some appropriate sense the large- limit of the double sum must be 1, is the
semiclassical sum rule. Originally [33] the rule was obtained by a dierent argument,
and was mysterious. Now there is a better understanding of the mechanism by which
the cancellation occurs [36, 37]; we will discuss it later.
Indeed, for the Riemann zeros, (4.12) can be derived [4] using a conjecture of
Hardy and Littlewood [38] concerning the pair distribution of the prime numbers.
These correlations are important because if the logarithms of the primes (primitive
orbit periods) were pairwise uncorrelated, Ko, being the average of a sum of random
phases, would be zero. The Hardy-Littlewood conjecture is that 2(k; X), dened
as the number of primes p X such that p + k is also a prime, has the following
asymptotic form for large X:
log X
with> 0ifk is odd
pjk
where the q-product includes all odd primes, and the p-product includes all odd
prime divisors of k. Pairwise randomness would correspond to C(k)=1. Itcanbe
demonstrated [4] that as K !1
and so on average(4.16) C
for large k. This in turn was shown to imply (4.12) in the limit t !1[4].
We have seen that Kdiag is universal in the limit ~ ! 0; that is, it is independent of
the specic features of the dynamics. These reappear|in a dramatically nonuniform
way|in the approach to the limit. To see this, note rst that it is only for short orbits,
that is, when <1, that universality breaks down. Next, choose a corresponding
to a time much longer than the shortest period T0 and shorter than the Heisenberg
time TH, that is,
We continue to use the Hannay-Ozorio sum rule for >, the limit (4.12) for Ko
ensuring the correct GUE formula (4.10) for >1, but take the contributions from
orbits with period Tj < 2hdi~ directly from (4.8). Thus
Fig. 3 Number variance 2(x) (4.4) of the Riemann zeros tn near n =P1012, calculated from
(with =1=4), (2.14), and (2.18) (thin line), compared with 2(x) computed from
numerically calculated zeros by Odlyzko [39, 40] (thick line); all the zeros are close to
2:677 1011, and their smoothed density is hdi =3:895 :::: Note the resurgence resonances
(cf. (4.23)) associated with the lowest zeros t1, t2, and t3, and that the theory fails to capture
small, fast oscillations in the data.
is a candidate for a semiclassical formula for the form factor. Later we will see that this
is not quite correct: the proper incorporation of the o-diagonal terms in the double
sum introduces a small but important modication near = 1. For the moment, we
continue to discuss (4.18).
This formula for K(), applied to the Riemann zeros, is extremely accurate. When
employed in conjunction with (4.4) to calculate the number variance of the zeros [34],
it reproduces almost perfectly this statistic as computed from numerical values of high
zeros [39, 40].

Figure

3 shows that the agreement extends from the random-matrix
regime (small x) to the far nonuniversal regime. Note however the tiny oscillatory
deviations; we will return to these later.
For the pair correlation, we have
Remarkably, it is possible to calculate the correction Rc explicitly and in closed form
at this level of approximation. The formula was obtained for both the Riemann zeros
and for general systems in [41], and independently in [42] for the Riemann zeros. From
(4.18), (4.2), and (2.14), we get

R (x)
c c 2( hdi)2 m;p pm hdi
Z

M. V. BERRY AND J. P. KEATING
where hdi is given by (2.15). The sum is insensitive to the value of provided this
is not too small, so we set = 1. Next, we write
m) log p:
This corresponds to separating into contributions from primitive orbits (rst term)
c
and repetitions (second term). In the repetitions, the sum over m can be evaluated
explicitly. For the rst term, we use [7]
Some tricky but elementary manipulations now give
c 2( hdi)2 2
(p exp fi log pg1)2
where
x
hdi:
This formula has a very interesting structure, worth discussing in detail. First,
in the limit t !1for any xed x, and so the pole in the zeta function
cancels the singularity 1=2. Second, the prefactor 1=hdi2 ensures that the correction
Rc1 is asymptotically small in comparison with RGUE (equation (4.11)). Third, the
dependence on shows that R1 involves the separation between zeros in the original
c
variable t =Ims (heights of zeros along the critical line), rather than the scaled
separation x; this means that structural features of R1 appear asymptotically at larger
c
x than the oscillations in RGUE, as expected for nonuniversal features of correlations.
Fourth, the contributions from repetitions (the sum over p in (4.23)) are less signicant
than those from primitive orbits (rst two terms), as Figure 4 shows. Fifth, and most
important, the appearance of (1 i) indicates an astonishing resurgence property
of the zeros: in the pair correlation of high Riemann zeros, the low Riemann zeros
appear as resonances. This is illustrated in Figure 5. The resonances also appear as
peaks in the nonuniversal part of the number variance (Figure 3).
For generic dynamical systems without time-reversal symmetry, it can be veried
directly that the analogue of (4.23) is [41, 43]R1 (x)=
where D is the dynamical zeta function dened in (3.3), and now = x=~hdi. Again,
the pole in the zeta function (now at cancels the singularity 1=2. In this
case, the resonances discussed above are caused by singularities of log D(s) away
from that is, by subdominant eigenvalues of the Perron{Frobenius operator.
Fig. 4 Nonuniversal correction to the pair correlation of the Riemann zeros, calculated from (4.23)
as Rc1 scaled () 2(hdi)2 Rc (x). Parts (a) and (c) include repetitions; (b) and (d) omit
repetitions.
Fig. 5 Pair correlation R(x) of the Riemann zeros, calculated \semiclassically" (thick line) from
and (4.23), for zeros near n =105, and random-matrix behaviour RGUE(x) (thin line);
note the rst nonuniversal resurgence resonance near x =21.
Now we return to the tiny oscillatory deviations noticeable in Figure 3, reecting
small errors in (4.19) and (4.23). These are again associated with the approach to the
t !1limit of the form factor, rather than the limit itself: whereas (4.23) captures
the appropriate large-t asymptotics of Kdiag, the GUE-motivated replacement (4.12)
incorporates only the t !1limit of Ko.
For the Riemann zeta function, this can be corrected as follows. We have already
noted above that the formula (4.12) for Ko can be derived using the smoothed
expression (4.16) for the Hardy-Littlewood conjecture. The large-t asymptotics we
M. V. BERRY AND J. P. KEATING
Fig. 6 Number variance 2(x) (4.4) of the Riemann zeros tn near n =109, calculated from (4.19)
and (4.26), including the o-diagonal correction (4.27){(4.28) [70] (full line), compared with
P(x) computed from numerically calculated zeros by Odlyzko [39, 40] (dots); all the zeros
are close to t =3:719 108, and their smoothed density is hdi =2:848 :::.
seek comes from using the original unsmoothed form (4.14) [5, 41]. The result is that
in which R1 is given by (4.23), and
c

c 2( hdi)2 2
where
(p 1)2
is a convergent product over the primes. As with the diagonal term (cf. the discussion
after (4.24)), convergence as ! 0 is ensured by the pole of the zeta function.
This second correction, although small, does incorporate the small oscillations,
through the trigonometric functions with argument 2x. Asymptotically (that is, as
t !1), these oscillations are fast (cf. (4.23){(4.24)) in comparison with the variations
from the resonances of the zeta function. When employed in conjunction with (4.4),
the correction accurately reproduces the oscillatory deviation (Figure 3) in the number
variance of the zeros; this is illustrated in Figure 6.
Unfortunately, this derivation of (4.27) for the Riemann zeros cannot be imitated
for general chaotic dynamical systems because we have no a priori knowledge
of the correlations between the actions of dierent periodic orbits, analogous to the
Hardy-Littlewood conjecture for the primes. It is possible to get some information by
working backwards and, assuming that the GUE expression (4.10) or (4.11) describes
the pair correlation of eigenvalues in generic chaotic systems without time-reversal
symmetry, deriving the universal limiting form of the implied action correlations [37].
(This procedure essentially follows an analogous derivation for the primes themselves,
assuming the Montgomery conjecture [44]). An interesting feature of this approach
is that it leads to predictions about classical trajectories based on the distribution of
quantum energy levels. However, it gives no information about the deviations from
random-matrix universality that are the focus of our concern here.
Recently a theory has been developed that overcomes these diculties [5, 41]. It
is based on two observations.
First, as already noted above, quantum eigenvalues (or Riemann zeros) are re-solved
by the trace formula if the sum (2.9) over periodic orbits is truncated near the
time TH (this will be made more precise in the next section). Hence, if
the trace formula thus truncated generates the approximation
to the counting function, the quantities E~n dened by

should be good semiclassical approximations to the exact eigenvalues. The theory is
based on calculating the correlations in this approximate spectrum.
Second, the diagonal terms Kdiag() are asymptotically dominant in the form
factor for <1, corresponding to times less than TH. This implies that orbits with
periods less than TH make contributions that are eectively uncorrelated; treating
them in this way allows the correlations in the E~n spectrum to be computed exactly.
For chaotic systems without time-reversal symmetry, the result is that when x>>
1 the deviations from the GUE formula can also be represented in the form (4.26),
(4.25), and (4.27), where replaced by D(i) (dened by (3.3)) and, in
(4.28), b() is replaced by

x=~hdi). Here is the residue of the pole at s =0ofD(s), 21 is the q-
hypergeometric function [45], and is the pth element of the product over primitive
orbits in (3.3).
The formal similarity between the results for the Riemann zeros and for the
semiclassical eigenvalues is striking, and reinforced by the fact that the derivation
of (4.31) just outlined leads precisely to (4.28) when applied to the zeros. Indeed,
by Fourier-transforming (4.26) with respect to t, this can be regarded as a heuristic
derivation of the Hardy-Littlewood conjecture. In the same way, Fourier-transforming
the corresponding result for dynamical systems with respect to 1=~ leads to a classical
periodic orbit correlation function corresponding directly to the Hardy-Littlewood
conjecture and reducing to the universal form conjectured in [37] in the long-time
limit. It is a challenge to derive these correlations within classical mechanics.
We nish this section on connections between statistics of the Riemann zeros
and quantum eigenvalues by remarking that the results for pair correlations extend
to correlations of higher order. Thus Montgomery's conjecture for the two-point
correlation of the Riemann zeros generalizes to all n-point correlations. Specically,
the irreducible n-point correlation function
M. V. BERRY AND J. P. KEATING
tends asymptotically to the corresponding GUE expression:
where the elements Sij of the n n matrix S are given by
sin f (xi xj)g
The analogue of Montgomery's theorem for the diagonal contributions to R~n was
proved for and then for all n 2 [47]. The o-diagonal contributions were
calculated using a generalization of the Hardy-Littlewood conjecture for
and then for all n 2 [49]. In all cases the results conrm the conjecture
(4.33) and (4.34). The nonuniversal deviations from the GUE formulae (4.33){(4.34)
were calculated for using the method outlined above, and take
a form (related to the structure of (s)ass ! 1) directly analogous to that already
discussed. As expected, this extends to the higher order correlations of quantum
eigenvalues.
5. Riemann-Siegel Formulae. A powerful stimulus to the development of analogies
between quantum eigenvalues and the Riemann zeros has been the Riemann-Siegel
formula for (s). As explained in [7], this very eective way of computing the zeros
(especially high ones)|employed in most numerical computations nowadays|was
discovered by Siegel in the 1920s among papers left by Riemann after his death
years earlier. We present the formula in an elementary way, chosen to facilitate our
subsequent exploration of its intricate interplay with quantum mechanics. Riemann's
derivation [11, 50] was dierent, and a remarkable achievement, because although it
was one of the rst applications of his method of steepest descent for integrals it was
more sophisticated than most applications today, in that the saddle about which the
integrand is expanded is accompanied by an innite string of poles.
It is a consequence of the functional equation satised by (s) [11] that the following
function Z(t) is even, and real for real t:

:Here (t) is the function appearing in the smoothed counting function (2.3) for the
zeros. Naive substitution of the Dirichlet series (1.5) gives the formal expression
X1 exp fit log ng
This is doubly unsatisfactory. First, it does not converge|a defect shared with its
relative similarly originating in the inadmissibility of
(1.5) in the critical strip. Second, it is not manifestly real as Z(t) must be.
Both defects can be eliminated by truncating the series (5.2) at a nite
and resumming the tail. The truncation n(t) is chosen to be the term whose phase
(t)t log n is stationary with respect to t; the asymptotic formula for (last member
of (2.3)) gives
A crude resummation [1] using the Poisson summation formula leads to a result equivalent
to the \approximate functional equation" [11]:

ng
This is a remarkable example of resurgence: the resummed terms in the tail n>n(t)
are the complex conjugates of the early terms 1 n n(t), so that the series in
(5.4)|called the \main sum" of the Riemann-Siegel expansion|is real, like the exact
Z(t). The zeros generated by the rst term alone (n = 1), that is, cos (t)=0,
have the correct mean density (cf. (2.3)). Higher terms shift the zeros closer to their
true positions, and introduce the random-matrix uctuations. It is worth mentioning
that the zeros obtained by including successive terms in (5.4) cannot be regarded as
the eigenvalues of hermitean operators that approximate the still-unknown Riemann
operator, because these partial sums of the main sum each have zeros for complex t [6].
Unfortunately, the truncation (5.3) introduces another defect: the sum is a discontinuous
function of t, unlike Z(t), which is analytic. The discontinuities can be
eliminated by formally expanding the dierence between (5.2) and the sum in (5.4)
about the truncation limit N(t), to obtain the correction terms in (5.4). This will depend
on the fractional part of (t=2) as well as its integer part n, so it is convenient
to dene
r
(5.5) a (t) n (t)+ (1 z
The expansion is in powers of 1=a (henceforth we do not write the t-dependences
explicitly), and gives
ng (1)n+1 X1 Cr (z)
ar
This procedure was devised in [4], where it was used to calculate the rst correction
term C0(z), and elaborated in [51] in a study of the higher corrections.
The sum over r is the Riemann-Siegel expansion. Its terms Cr(z) are constructed
from derivatives (up to the 3rth) of

cos fzg
with coecients determined by an explicit recurrence relation involving the coecients
(Bernoulli numbers) in the Stirling expansion of (t) for large t. The next few
coecients are
M. V. BERRY AND J. P. KEATING
(superscripts in brackets denote derivatives). Gabcke [50] calculated Cr(z) for r 12.
Later terms get very complicated; for example,
+106489993378346112059965404000022
713214794639C (z)
50407933481C (z)
An elaborate asymptotic analysis [51] shows that the high orders (\asymptotics
of the asymptotics") can be represented compactly as a \decorated factorial series"
whose terms are

;where for large r
sin f(2m +1) rg cos
cos f(2m +1) rg sin odd)Comparison with numerically computed Cr(z) (up to using special techniques
to evaluate the derivatives of C0(z)) shows that these formulae capture the ne details
of the Riemann-Siegel coecients, even for small r.
The factorial in (5.10) means that the sum over r in (5.6) is divergent in the
manner familiar in asymptotics: the terms get smaller and then diverge. Asymptotics
folkore suggests, and Borel summation (implemented analytically and checked numer-
ically) conrms, that optimal accuracy obtainable from the Riemann-Siegel formula
(without further resummation) corresponds to truncating the sum at the least term.
This has
and the resulting error is of order
ng (1)n+1 Xr Cr (z)
ar
The accuracy is very high: even for the lowest Riemann zero,
expft1g1020. Nevertheless, it is possible to do better, as we shall see later.
Now we turn to the quantum analogues of the Riemann-Siegel formula for classically
chaotic systems with D>1, as envisaged in [1], explored in detail in [52],
and derived in [53]. These studies are motivated by the hope that such an eective
method of computing Riemann zeros might lead to a useful way to calculate quantum
eigenvalues.
First, the counterpart of Z(t) in (5.1) is a function with zeros at the quantum
energy levels En; this is the quantum spectral determinant
where H is the hermitean wave operator (section 2) and the real factor A is introduced
to make the product converge. Hermiticity implies that is real for real E; this
\quantum functional equation" is analogous to the functional equation for (s), which
implies that Z(t) is real for real t.
To nd the counterpart of the Dirichlet series (5.2), we note that the quantum
eigenvalue counting function can be written (cf. (2.4)) as(5.15) N (E)= lim ImTrlog f1
"!0
Now the decomposition into smooth and uctuating parts, together with the periodic-
leads to
Y < X1 exp fimSp (E)
where B(E) is real and nonzero for real E and where we have absorbed the Maslov
indices into S.
Expanding the product over primitive orbits p and the exponential of the sum
over repetitions m, we obtain a series of terms that can be labelled by
Here mp represents the number of repetitions of the orbit p. Each term corresponds
to a sum over actions:
The expansions lead to
M. V. BERRY AND J. P. KEATING
with an explicit form for the coecients Dn that we do not give here [52]. As (5.18)
indicates, the terms n correspond to composite orbits, or pseudo-orbits, consisting of
combinations of repetitions of dierent periodic orbits. We label the composite orbits
so that increasing n corresponds to increasing period
with representing no orbit at all, that is, which the coecient
The sum (5.19) is the counterpart of the Dirichlet series (5.2) for Z(t), with
composite orbits n related to primitive orbits p in the same way that the integers n
are related to the primes p (cf. (1.5)). Moreover (5.19) diverges, like the sum (2.9)
from which it was obtained, and it is not manifestly real as the exact (E) must be.
Our interpretation of the Riemann-Siegel formula suggests a similar resummation of
the tail of the series (5.19) after truncation at the term whose phase is stationary with
respect to E. This term|the counterpart of n(t) in (5.3)|represents the composite
orbit dened by
d
The corresponding period T (E)is
where TH(E) is the Heisenberg time (2.20).
Comparison with the Riemann-Siegel main sum in (5.4) suggests that the sum of
the composite orbits with approximately, the complex conjugate of the
sum of the orbits with In fact, this relation can be derived using arguments
based on analytic continuation with respect to E [53]. These arguments also indicate
a more detailed correspondence: between the sums of groups of terms with periods
X. The resulting \Riemann-Siegel lookalike" formula is
(For a dierent derivation, see [54].)
With (5.23) it is possible to reproduce some low-lying quantum eigenvalues, and
of course the fact that the sum is nite is a major advantage over the innite divergent
series (2.9) and (5.19). However, for a chaotic system with D>1 the number of terms
with exponentially large in 1=~, so the Riemann-Siegel lookalike is not as
useful for calculating high quantum eigenvalues as (5.4) is for calculating Riemann
zeros. The origin of the dierence is the exponential proliferation of periodic orbits
(and composite orbits), together with the fact that hdi increases as 1=~D, whereas for
the Riemann zeros, whose classical counterpart appears to be quasi-one-dimensional,
hdi increases as log t. Moreover, (5.23) is discontinuous at the energies of composite
orbits with period T .
No way has yet been found to implement the obvious suggestion of cancelling the
discontinuities in the quantum formula (5.23) by a series of corrections analogous to
the terms involving Cr(z) in the Riemann-Siegel expansion (5.6). However, a dierent
completion of the Riemann-Siegel main sum was discovered ([55], generalizing an idea
in [4]), that does have a quantum analogue.
In this alternative approach to the resummed Dirichlet series, the abrupt truncation
is replaced by a smoothed cuto involving the complementary error function
and an optimization parameter K. An argument involving analytic continuation in t
leads to
with an explicit expression for the correction terms. With K chosen appropriately, this
smoothed sum can reproduce Z(t) to an accuracy equivalent to that of the Riemann-
Siegel main sum together with several correction terms. The corrections in (5.24) form
an explicit asymptotic series enabling Z(t) to be calculated with an accuracy of order
exp(t2); this improvement over the Riemann-Siegel exp(t) is possible because
(5.24) involves the higher transcendental function Erfc, whereas the Riemann-Siegel
expansion involves only elementary functions. Several related representations of Z(t)
are now known [56, 57, 58].
The improved representation (5.24), together with the explicit correction terms,
can readily be adapted to the quantum spectral determinant. The smoothed version
of the Riemann-Siegel lookalike (5.23) is obtained by an argument involving analytic
continuation with respect to 1=~, leading to
where Ni denotes the ith derivative of N with respect to 1=~. A numerical test
of this formula for the hyperbola billiard (a classically chaotic system with D =2)
shows that it can reproduce quantum eigenvalues with high accuracy, even resolving
near-degenerate pairs of levels [59].
Finally, we note an important clue to the Riemann dynamics, hidden in the
asymptotics (5.10), (5.11) of the Riemann-Siegel expansion (5.6). It concerns the
implied small exponential expftg (cf. the error in (5.13)). The same exponential
appears in the asymptotics of the gamma functions in (t) (equation (2.3)). Quantum
mechanics suggests this is the \phase factor" corresponding to a periodic orbit with
imaginary action (an \instanton" in physics jargon). If we write
(remembering for the Riemann zeros), the implied period is
@S @S
So, it seems that as well as the real periodic orbits in (2.14), with periods m log p,
there are complex periodic orbits, with periods that are multiples of i.
6. Spectral Speculations. Although we do not know the conjectured Riemann
operator H whose eigenvalues (all real) are the heights tn of the Riemann zeros, the
analogies presented so far suggest a great deal about it. To summarize:
260 M. V. BERRY AND J. P. KEATING
a. H has a classical counterpart (the \Riemann dynamics"), corresponding to a
hamiltonian ow, or a symplectic transformation, in a phase space.
b. The Riemann dynamics is chaotic, that is, unstable and bounded.
c. The Riemann dynamics does not have time-reversal symmetry. In addition,
we note the recent discovery [60, 61] of modied statistics of the low zeros for the
ensemble of Dirichlet L-functions, associated with a symplectic structure.
d. The Riemann dynamics is homogeneously unstable.
e. The classical periodic orbits of the Riemann dynamics have periods that are
independent of \energy" t, and given by multiples of logarithms of prime numbers.
In terms of symbolic dynamics, the Riemann dynamics is peculiar, and resembles
Chinese: each primitive orbit is labelled by its own symbol (the prime p) in contrast
to the usual situation where periodic orbits can be represented as words made of
letters in a nite alphabet.
f. The Maslov phases associated with the orbits are also peculiar: they are all .
The result appears paradoxical in view of the relation between these phases and the
winding numbers of the stable and unstable manifolds associated with periodic orbits
[22], but nds an explanation in a scheme of Connes [62].
g. The Riemann dynamics possesses complex periodic orbits (instantons) whose
periods are multiples of i.
h. For the Riemann operator, leading-order semiclassical mechanics is exact: as
in the case of the Selberg trace formula [21], (1=2+it) is a product over classical
periodic orbits, without corrections.
i. The Riemann dynamics is quasi-one-dimensional. There are two indications of
this. First, the number of zeros less than t increases as t log t; for a D-dimensional
scaling system, with energy parameter (E) proportional to 1=~, the number of energy
levels increases as (E)D. Second, the presence of the factor pm=2 in the counting
function uctuation formula (2.6), rather than the determinant in the more general
Gutzwiller formula (2.9), suggests that there is a single expanding direction and no
contracting direction.
j. The functional equation for resembles the corresponding relation|a consequence
of hermiticity|for the quantum spectral determinant.
We have speculated [6] that the conjectured Riemann operator H might be some
quantization of the following extraordinarily simple classical hamiltonian function
Hcl(X;P) of a single coordinate X and its conjugate momentum P:
Now we outline the reasons for this tentative association of XP with (s).
At the classical level, (6.1) has a hyperbolic point at the origin in the innite-
phase (X;P) plane, and generates the following equations of motion and trajectories:
Thus classical evolution is uniformly unstable, with stretching in X and contraction in
P. Furthermore, the motion has the desired lack of time-reversal symmetry: velocity
cannot be reversed (X_ is tied to X in (6.2)) and so the orbit cannot be retraced.
At the semiclassical level, we can try to estimate the smoothed counting function
hN(E)i of energy levels En generated by the quantum version of (6.1). For this
it is necessary to specify a value of Planck's constant ~. We choose
choices simply rescale the energies. hN(E)i is the area A under the constant-energy
measured in units of the \Planck cell" area 2~ =2, with a
Fig. 7 Phase space for with cutos for semiclassical regularization.
Maslov index correction given by =4, where is the angle turned through along
the orbit in phase space (this correction gives the \1/2" in the quantization of the
harmonic oscillator). We encounter the immediate diculty that A is innite: motion
generated by is unbounded, and so does not give discrete quantum energies.
As will be clear later, closing the phase space to make the motion bounded is a central
unsolved problem. In the interim, a simple (perhaps the simplest) expedient is to
regularize by truncating in X and P as indicated in Figure 7. The result (unaltered
by representing the Planck cell by a rectangle instead of a square) is that hN(E)i is
precisely the asymptotics of the smoothed counting function for the Riemann zeros
(last member of (2.3)), including the term 7/8, with t replaced by the energy E.
At the quantum level, the simplest formally hermitean operator corresponding to
(6.1) is

The formal eigenfunctions, satisfying
are
A
We note the appearance of the power Xs appearing in the Dirichlet series for
(as integers) and the Euler product (as primes), with the symmetrization (6.3)
placing s on the critical line.
It is evident that XP is simply a canonically rotated version of the inverted
harmonic oscillator P2 X2, which in turn is a complexied version of the usual
X2. Some of these connections have been noted before [63,
64, 65, 66, 67]. The rst-order operator XP is the simplest representative of this
class, with the monomials (6.5) avoiding the complications of the parabolic cylinder
eigenfunctions of P2 X2.
262 M. V. BERRY AND J. P. KEATING
To evaluate the corresponding momentum eigenfunction E(P) (Fourier transform
of (6.5)), it is necessary to specify a continuation across simplest
choice, for a reason to be given later, is to make the wavefunction even in X, that is,
to replace X by jXj. Then

A
It follows that, up to factors that can easily be made symmetrical, the position and
momentum eigenfunctions are each other's time-reverses. Thus we nd a physical
interpretation of the function (t) (dened in (2.3)) at the heart of the functional
equation (cf. (5.1)) for (s).
The major problem remaining is to nd boundary conditions that would convert
XP into a well-dened hermitean operator with discrete eigenvalues. This is equivalent
to specifying the way in which parts of the (X;P) plane are connected so as to
compactify the (quantum and classical) motion. Some hints in this direction follow.
Our observations about the complex periodic orbits of the Riemann dynamics
(see the last paragraph of section 5) suggest that X and X should be identied.
The reason is that the complex orbits of X, obtained by replacing t by i in (6.2),
have period 2i, which becomes the desired i (equation (5.27)) on identifying X.
To proceed further, we consider the symmetries of XP, in the hope (so far un-
realized) of superposing solutions of (6.4) acted on by operations in the symmetry
group, with each solution multiplied by the appropriate group character. An obvious
symmetry is dilation: XP is invariant under
From (6.2), K corresponds to evolution after time log K. This implies that the operator
generates dilations, in the same way that the momentum operator generates
translations, and the following series of transformations makes this obvious:

d
d log X
dX K1=2iH
One possibility is to choose the integer dilations and the characters unity.
Then the superposition of solutions (6.5) does contain (1=2iE) as a factor, but there
seems no reason to impose the condition that this must vanish. Moreover, the set of
integer dilations does not form a group (the inverse multiplications 1=m are missing).
Another possibility, closely related to the ideas of [62], is to use not all integers
but the group of integers under multiplication (mod This would have two
advantages. First, it involves only integer dilations. Second, including the characters
(n) of this group (sets of k complex numbers with unit modulus) opens the possibility
of widening the interpretation as eigenvalues of XP, to include the zeros of Dirichlet
L-functions. These are dened by the series
ns
(The special case = 1 corresponds to (s):) It is conjectured that for all these L-functions
the complex zeros lie on the line Res =1=2. On this interpretation, each L-function
corresponds to a dierent self-adjoint extension of XP under identication of
positions X that are related by dilations in the group of integers under multiplication
(mod k). An analogy is with the quantum mechanics of a particle in a periodic
potential (e.g., an electron in a crystal): from the Bloch-Floquet theorem, solutions
of the underlying dierential equation are all periodic up to a phase factor exp(i);
each choice of is a dierent self-adjoint extension, and generates a discrete spectrum.
The analogy is imperfect, because is continuous, whereas the L-functions cannot be
continuously parameterized. A closer analogy is with quantization on a torus phase
space [69], where for topological reasons the permited phases are discrete.
The dynamics (6.2) suggests that the system might be closed by connecting the
asymptotic positions with the asymptotic momenta. Then particles owing out at
would be reinjected at 1. Related to this is a class of dilations
where K is H-dependent (of course these are still symmetries of H). Specically, the
choice K =2=(XP) yields the canonical transformation
corresponding to exchange of X and P (the more familiar does not
leave XP invariant). A short calculation gives the transformed quantum wavefunction
in terms of the untransformed momentum wavefunction as
1=4
We do not know how to convert this \quantum exchange" into an eective boundary
condition, but note its connection with the following intriguing identity, obtained from
the momentum wavefunction formula (6.6) and the functional equation for (s):

where PX =2 (=
If (only) the minus were a plus, this would be a condition generating the Riemann
zeros.
We can sum up these scattered remarks about XP by returning to the properties
listed at the beginning of this section. XP is consistent with point a, part of b (XP
dynamics is unstable but not bounded), and c, d, g, h, i, and j. Concerning point e,
the appearance of times that are logarithms of integers begins to be plausible in view
of the association between dilation and evolution, but primes do not appear in any
obvious way. We have no explanation of property f.


--R

Riemann's zeta function: A model for quantum chaos?
More than one third of the zeros of Riemann's zeta-function are on =1=2
The theory of the Riemann zeta-function
Mathematical Methods of Classical Mechanics
The Hamiltonian method and its application to water waves
Regularity and chaos in classical mechanics
Regular and irregular motion
An Introduction


Chaos in classical and quantum mechanics
Distribution of eigenfrequencies for the wave equation in a nite domain: III.
Chaos on the pseudosphere
Maslov indices in the Gutzwiller trace formula
Spectra of Finite Systems
Introduction to Ergodic Theory



Random Matrices and the Statistical Theory of Energy Levels
A Brownian-motion model for the eigenvalues of a random matrix
Statistical theory of the energy levels of complex systems IV
Statistical Theories of Spectra Fluctuations
in Mathematical and Computational Methods in Nuclear Physics
Semiclassical theory of spectral rigidity
Semiclassical formula for the number variance of the Riemann zeros
The pair correlation of zeros of the zeta function
The semiclassical sum rule and Riemann's zeta function
Correlations in the actions of periodic orbits derived from quantum chaos
Some problems in
On the distribution of spacings between zeros of the zeta function
The 1020th zero of the Riemann zeta function and 175 million of its neigh- bours
Gutzwiller's trace formula and spectral statistics: Beyond the diagonal approximation
On the Berry Conjecture
random matrix theory
Pair correlation of zeros and primes in short intervals
Cambridge University Press
On the Triple Correlation of Zeros of the Zeta Function
Zeros of principal L-functions and random-matrix theory
Random matrix theory and the Riemann zeros I: Three- and four-point correlations

Neue Herleitung und Explizite Restabscha
The Riemann-Siegel formula for the zeta function: High orders and remainders
A rule for quantizing chaos?

Semiclassical quantization of multidimensional systems
A new approximation for (1
An asymptotic representation for the Riemann zeta function on the critical line
An exponentially-improved Gram-type formula for the Riemann zeta function
An asymptotic representation for (1
Calculation of spectral determinants
Zeros of Zeta Functions
symmetry and zeta functions
Formule de trace en geometrie non-commutative et hypothese de Riemann
Eigenstate structures around a hyperbolic point
Phase of the Riemann zeta function and the inverted harmonic oscillator
The phase of the Riemann zeta function
The Riemann hypothesis and the Hamiltonian of a quantum mechanical system

Introduction to Analytic Number Theory
Quantum boundary conditions for torus maps
Asymptotics of the Pair Correlation of Riemann Zeros
--TR

--CTR
T. M. Dunster, Uniform asymptotic approximations for incomplete Riemann Zeta functions, Journal of Computational and Applied Mathematics, v.190 n.1, p.339-353, 1 June 2006
number theory;spectral asymptotics