Ann. Henri Poincaré 4, Suppl. 2 (2003) S727 – S752
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/02S727-26
DOI 10.1007/s00023-003-0958-2
Annales Henri Poincaré
Prime Correlations and Fluctuations
P. Leboeuf
Abstract. Based on the expansion of the counting function of the prime numbers
in terms of the zeros of the Riemann zeta function, several aspects of the fluctuations of the primes are reviewed and further developed. In particular, the average
density–density pair correlation, and the moments and autocorrelation of the fluctuations of the counting function are considered. The Riemann hypothesis is assumed
throughout the paper.
1 Introduction
In the semiclassical theory of bounded quantum systems two closely connected
“spectra” are of central importance, namely the set of quantum eigenvalues and the
set of classical periodic orbits. Trace formulae express the quantum density of states
in terms of the classical periodic orbits, both for integrable and chaotic systems
[1, 2]. Properties of one set have been identified to corresponding properties of the
other set, like for example the classical Hannay–Ozorio de Almeida sum rule with
the quantum linear growth of the form factor close to the origin [3].
In the realm of analytic number theory, since the work of Riemann two distinct sets are attached to the study of prime numbers, namely the primes themselves and the zeros (and pole) of the Riemann zeta function. Their relationship
with the theory of complex quantum systems received a decisive impulse after
Montgomery’s work on the pair correlation of the zeros. It establishes the validity of the GUE ensemble of random matrices [4], thus opening the road to their
interpretation as the eigenvalues of some “chaotic” hermitian operator. In this dynamical analogy, the prime numbers play the role of the classical periodic orbits of
some unknown chaotic system. This interpretation is based on the formula expressing the density of Riemann zeros as a sum over prime numbers, whose structure
is very similar to Gutzwiller’s trace formula for chaotic systems [5].
The primes and zeros are, however, “doubly–connected”, since the density
of prime numbers can in turn be expressed as a sum over the Riemann zeros
(and pole). Although on large scales the properties of prime numbers are very
regular, as described for example by the prime number theorem, on local scales
comparable to their mean spacing they show erratic fluctuations that make their
prediction quite difficult. These fluctuations are controlled by the zeros, and it
is therefore quite natural to relate as much as possible the properties of both
sets. It is our purpose here to review and further develop some of the statistical
properties of prime numbers, keeping in mind the applications and potential use
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of the results in the context of dynamical systems. Two main quantities are to be
considered, the average pair correlation of the prime numbers and the distribution
and autocorrelation of the counting function π(x) (the number of primes ≤ x).
After introducing the basic formulae in §2, the results for the pair correlation
of primes are presented in §3. It is shown that the average density R2 (y) of prime
pairs separated by a distance y and located around x (with x → ∞) is, locally,
given by the formula




Si 2πx log x+y/2
x−y/2
Si(2πy)
ρ2 (x) 1 −
R2 (y) = ρ2 (x) 1 −
, (1)


πy
πx log x+y/2
x−y/2
x
where Si(x) = 0 sin u/u du is the sine integral function and ρ(x) = log−1 x is the
asymptotic average density of primes. The last approximation on the r.h.s. holds
when y x. If, moreover, y 1, Si(2πy) → π/2 and R2 (y) ≈ ρ2 (1 − 1/2y), a
result in agreement with the average asymptotic pair correlation conjectured by
Hardy and Littlewood [6] (cf §3 for more details). Eq.(1) expresses the presence
of anti-correlations, i.e. on average the prime numbers repel each other on local
distances. However, since the average distance between primes increases with x,
the effective action of these interactions diminishes as x → ∞.
Equation (1) is the analog for prime numbers of the well known random
matrix formula
sin2 (π ρζ (t) )
,
(2)
R2ζ () = ρ2ζ (t) −
π 2 2
that describes the density of Riemann zeros separated by a distance and located
around a point t → ∞ on the critical axis (ρζ (t) = (2π)−1 log(t/2π) is the asymptotic average density of the zeros). This behavior has been conjectured to be valid
generically for quantum chaotic systems [7].
In the same way as Eq.(2) is known to hold only on a limited range in (with non–universal structures appearing at large ’s), Eq.(1) also has a limited
range of validity. This is a further consequence of the properties of the Riemann
zeros on the prime correlations. Denoting t1 = 14.1347 . . . the imaginary part of
the lowest non–trivial zero of the Riemann zeta function, in §3 it is shown that at
y ∼ x/t1 a transition takes place, and that for y >
∼ x/t1 Eq.(1) does not hold any
more. Instead, a different behavior, which includes positive correlations, occurs.
This new regime is also marked by the appearance of sharp peaks of negative
correlation at positions determined by the lowest prime numbers. Again, this is
similar (although less complex) than the hierarchical structure of anticorrelation
peaks observed in the non–universal fluctuations of R2ζ () [8].
§3 contains also a detailed analysis of the Fourier transform of the pair correlation function, the form factor of the primes.
Similar results to those presented here for the pair correlation of primes were
previously obtained in Refs.[9, 10, 11]. In this respect the material of §3 should
be considered to a large extent as a review. The Hardy–Littlewood average tail
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Prime Correlations and Fluctuations
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has been discussed in most of the above mentioned articles. Some aspects of the
long–range fluctuations (i.e., the presence of anticorrelation peaks) can be found in
[11]. Eq.(1) is not in the literature, but could in fact be derived from the inversion
arguments of Ref.[10]. Though similar in nature with respect to other works, the
techniques used here are distinct and provide a systematic, global and alternative
perspective of the correlations acting among primes.
In §4 another aspect of the fluctuations of prime numbers is considered.
Here the central object is the counting function π(x), and more specifically the
fluctuations of that function around its mean value. From a statistical point of
view, the basic problem is to compute the distribution of the fluctuations that
occur in a window centered around a point x. In the asymptotic limit x → ∞ this
problem was solved recently. Under the hypothesis of the linear independence of
the complex part of the Riemann zeros, a closed form for the Fourier transform of
the distribution was obtained in [12]. The distribution was found to be symmetric
with respect to its average value, and different from a Gaussian law.
Here we concentrate on the computation of the moments of the distribution, and on the finite–x corrections to the asymptotic law. The moments can be
expressed in terms of sums over the Riemann zeros. It is shown that the dominant contributions to these sums come from the low–lying zeros. Assuming, as in
Ref.[12], their linear independence, explicit expressions are obtained for the leading order behavior as x → ∞. We find that for large but finite x√the distribution is
asymmetric, with a negative third moment which is of order 1/ x with respect to
the leading behavior of the distribution. This is in agreement with numerical observations. Finite-x corrections to the even moments are also computed, including
those arising from the correlations acting among the zeros.
The autocorrelation of the fluctuating part of π(x) is also computed. This
provides a complementary information with respect to the distribution, it gives
information on how the value of this function depends on neighboring points. An
explicit formula is obtained that shows that the correlations never decay. The
structure of those correlations is also remarkable, a series of parabolae joined by
discontinuities of the first derivative. The origin of this structure is explained by
making a connection with the fluctuations of the sum of prime numbers ≤ x. We
also provide a numerical verification of these oscillations and discontinuities, which
incidentally also demonstrate the existence of the anticorrelation peaks in the pair
correlation of the primes.
§5 summarizes the results and adds some concluding remarks.
2 Prime number density and counting function
The density of prime numbers is defined as
ρ(x) =
p
δ(x − p) ,
(3)
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where the sum extends over all prime numbers p ≥ 2, and δ(x) is Dirac’s distribution. The work of Riemann provides an explicit formula for this function [13],
∞
ρ(x) =
1 µ(r) mω xω/r .
x log x r=1 r ω
(4)
The ω’s label the location in the complex s–plane of the zeros and pole of the
Riemann zeta function ζ(s). The index mω = +1 for the pole, and mω = −1 for
the zeros. For a fixed x, the series in r is finite, and restricted to values satisfying
the condition x1/r ≥ 2. µ(r) is the Möbius function defined as

if r has one or more repeated prime factors
 0
(−1)k if r is a product of k distinct primes
µ(r) =

1
if r = 1 .
The structure of Eq.(4) is better displayed when ρ(x) is split into a smooth
and an oscillatory part, ρ(x) = ρ + ρ. The smooth part ρ is a monotonic function
of x and is obtained by keeping in Eq.(4) only the contributions of the real ω’s
∞
1 µ(r)
ρ(x) =
x log x r=1 r
x1/r −
1
x2/r − 1
.
(5)
The first term inside the parentheses is the contribution of the pole located at ω =
1, while the second comes from the trivial zeros located at ω = −2l, l = 1, 2, 3, . . ..
In the limit x → ∞ the contributions of the latter are negligible.
In contrast to the pole and real zeros, the complex zeros of ζ(s) located at
ω = 1/2 ± itα , α = 1, 2, 3, . . ., introduce oscillatory terms whose interference is
responsible for the singular structure of ρ(x),
∞
∞
2 µ(r) 1/2r tα
ρ(x) = −
x
log x .
cos
x log x r=1 r
r
α=1
(6)
We will assume Riemann hypothesis (i.e., the tα ’s are real) and concentrate on
asymptotic properties of prime numbers as x → ∞. Most of the results that follow
are valid to leading order in x. In general, we do not explicitly write down any
correction terms, unless explicitly stated. The leading order behavior of ρ, obtained
by keeping only the r = 1 term in Eq.(5), provides the well known result
ρ(x) =
1
,
log x
(7)
while the leading r = 1 contribution to the oscillating part is [14]
∞
2
ρ(x) = − √
cos(tα log x) .
x log x α=1
(8)
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In a quantum mechanical context, loosely speaking it is possible to give to
Eq.(8) a “semiclassical” interpretation, as the oscillating part of the trace formula
for the spectrum of an Hermitian operator whose eigenvalues are the prime numbers. In this analogy, the complex zeros of ζ(s) are playing the role of periodic
orbits. If x is regarded as the energy, the action of the orbits is Sα = tα log x and
their period, defined as dSα /dx, is
τα = tα /x .
(9)
Since, as we are going to see in more detail in §3, the primes asymptotically tend
to behave as an uncorrelated sequence, then this suggests that the underlying
classical dynamics should be integrable (i.e., a non-chaotic system).
From Eq.(8) it follows that each complex zero contributes to the prime number density with an oscillatory term whose period of oscillation, in the neighborhood of some point x, is given by
Xα =
2πx
2π
=
.
tα
τα
(10)
The structure of ρ(x) is thus governed by several scales. The first one is
simply x, the neighborhood around which the properties of primes are evaluated.
Asymptotically x → ∞, and this is the large parameter in the problem. The
second, usually intermediate in dynamical systems but here of order x, is given
by the complex zero with the lowest imaginary part, t1 , that produces in ρ(x)
oscillations of wavelength X1 = 2πx/t1 . The third relevant scale is the mean
distance δ between consecutive primes. According to Eq.(7), asymptotically it is
given by
(11)
δ = ρ−1 = log x .
To resolve structures in a scale of order δ, Riemann zeros with imaginary part of
order
2πx
2πx
=
(12)
tδ =
δ
log x
should be included in the sums (6) and (8).
There are expressions analogous to Eqs.(4)–(8) for the counting function of
the prime numbers,
π(x) =
Θ(x − p) ,
(13)
p
related to the density by ρ(x) = dπ(x)/dx. Θ(x) is the unit step Heaviside function.
π(x) increases by one at each prime p. The exact “trace formula” obtained by
Riemann for the counting function reads [13],
π(x) =
∞
µ(r) r=1
r
ω
mω Li xω/r ,
(14)
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P. Leboeuf
Ann. Henri Poincaré
z
where Li(z) = 0 dt/ log t is the logarithmic integral function. As for the density,
for a fixed x the series in r is finite, and restricted to values satisfying the condition
x1/r ≥ 2.
As the density, the counting function can be decomposed into two parts
(smooth and oscillatory), corresponding to the contributions of the real and imaginary ω’s, respectively
π(x) = π(x) + π
(x) .
The smooth part is given by
π(x) =
∞
µ(r)
r=1
r
∞
1/r
−2k/r
−
Li x
Li x
,
(15)
k=1
The first term inside the square brackets is the contribution of the pole at ω = 1,
while the second one, a sum, originates from the trivial zeros. In the limit x → ∞
the contribution of the latter are negligible.
The complex zeros of ζ(s) give oscillatory terms,
π
(x) = −2 Re
∞
∞
µ(r) r=1
r
Li x(1/2+itα )/r .
(16)
α=1
The same three scales x, X1 and δ mentioned before are also relevant for the
counting function. In the limit x → ∞ the dominant terms of the two contributions
are
π(x) = Li(x) ,
(17)
and
√ ∞
1
1
2 x
cos(tα log x) + tα sin(tα log x) .
π
(x) = −
log x α=1 14 + t2α 2
(18)
When only the term r = 1 in Eq.(14) is considered and π(x) is approximated by
(17) and(18), it reproduces in fact the asymptotic behavior of the function [13]
J(x) = r=1 π(x1/r )/r. J(x) increases by 1 at primes, by 1/2 at prime squares,
etc. Our asymptotic results of the forthcoming sections describing the behavior of
π(x) are therefore also valid for J(x).
The functions ρ(x) and π(x) are thus described in its simplest approximation
by a smooth part associated to the pole of ζ(s). Superimposed to this monotonic
behavior are oscillations controlled by the complex zeros. The singular structure of
these functions arises from subtle interferences between the zeros, and imaginary
parts of order tδ are necessary in order to resolve the spectrum on a scale δ.
Aside from these small–scale structures, there are long–range fluctuations in the
density and counting function on scales of order X1 , determined by the lowest
complex zero. The terms with r > 1 in Eqs.(6) and (16) produce oscillations in
even larger scales, but these are asymptotically negligible compared to the leading
contributions.
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3 Pair Correlations
The fluctuations of the prime numbers are quite different from that of the Riemann
zeros. In a simple model proposed by Cramér [15] and later on developed in [16],
the correlations are ignored and, locally, the primes are considered as random
independent numbers. If that were true, the density of prime pairs separated by a
distance k, and located around a point x → ∞, would be simply proportional to
the square of the probability to find one prime number in that region
R2 (k) =
1
.
log2 x
(19)
Cramer’s model provides accurate predictions for some properties concerning the
primes. However, correlations among primes are believed to exist. In a celebrated
paper [6], Hardy and Littlewood conjectured that the density of prime pairs is
given by
σ(k)
R2HL (k) =
.
(20)
log2 x
For k even, σ(k) is expressed in terms of products extending over all odd primes p
p−1
1
σ(k) = 2C2
, where C2 =
1−
= 0.660161 . . . ,
p−2
(p − 1)2
p>2
p|k
p>2
(21)
whereas σ(k) = 0 if k is odd.
σ(k) is an erratic function of k, and definite trends are difficult to extract.
However, it is known [9] that when averaged by a suitable function, and for k 1,
it takes the simple form
1
.
(22)
σ(k) = 1 −
2k
This relation expresses the presence of anti-correlations, i.e. on average the prime
numbers repel each other at large distances (see also [17, 18] for some related
results).
Our aim is to investigate the correlations between the primes by focusing our
attention on the average behavior of σ(k). The approach used is based on Eq.(8),
that expresses the fluctuating properties of the prime numbers in terms of the
imaginary part of the non–trivial zeros of the Riemann zeta function.
3.1
Prime Correlations: smooth part
The Hardy-Littlewood conjecture Eq.(20) describes a departure from Cramér’s
model. As a function of the distance between primes, R2HL (k) predicts highly
irregular oscillations while, on average, it leads to a slow decay of correlations
according to Eq.(22).
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P. Leboeuf
Ann. Henri Poincaré
Analytic methods like Riemann’s expansion of ρ(x) are not well adapted
to describe the highly irregular number–theoretic oscillations of the correlations,
since these are encoded in subtle interferences acting among the tα ’s. In contrast,
they are well adapted to describe coarse–grained properties because appropriate
averages are natural in a wave–oscillatory description. We thus concentrate on
average or mean–values of the correlations, that provide less accurate but more
simple and elegant results.
Our purpose is to compute the asymptotic x → ∞ behavior of the local
average density of pairs of primes separated by a distance y, defined as
R2 (y) = ρ(x − y/2) ρ(x + y/2)∆x − ρ δ(y) .
(23)
In contrast to the variable k used for the argument of R2 in Eqs.(19)–(22), y is
here a continuous variable. The angular brackets denote a local average over the
position of the reference point x,
x+∆x/2
1
f (x)∆x =
f (u)du .
∆x x−∆x/2
The window size ∆x must satisfy two conditions. It must contain a sufficiently
large number of oscillations of f (u) to guarantee the convergence of the average.
This means that it should be large compared to X1 . Moreover, it should be small
compared to x to avoid variations of the smooth part. Although the latter condition
is not strictly satisfied due to the proximity of both scales, in the following we will
neglect the variations of the prefactors in Eq.(8), which introduce
lower–order
x
corrections in x. Alternatively, the full average f (x) = x−1 1 f (u)du could be
considered; it produces equivalent results. For simplicity, from now on we omit the
subindex ∆x in the angular brackets.
The asymptotic behavior of R2 (y) is obtained by approximating in Eq.(23)
ρ = ρ + ρ by Eqs.(7) and (8). The product of the smooth parts gives ρ2 (x). The
average product of the cross terms ρ
ρ is zero, since all oscillations occur on a
scale smaller than ∆x. Then
1
1
(24)
R2 (y) =
+ Y2 (y) =
+ T2 (y) − ρ δ(y) ,
log2 x
log2 x
with
T2 (y) = ρ(x − y/2) ρ(x + y/2) =
4
cos [tα log(x + y/2)] cos [tα log(x − y/2)] . (25)
2 x log x α,α
Using cos a cos b = [cos(a + b) + cos(a − b)]/2, the average over the cosine of the
sum of the arguments vanishes. Then
2
T2 (y) =
cos [tα log(x + y/2) − tα log(x − y/2)] .
(26)
2 x log x α,α
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Prime Correlations and Fluctuations
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There are two distinct contributions to this double sum. The first, diagonal one,
is considered in this subsection. The second is computed in §3.2.
The diagonal terms α = α give
x + y/2
2
diag
cos tα log
T2 (y) =
.
(27)
x − y/2
x log2 x α
To evaluate the sum, we simply approximate it by an integral (to simplify the
notation we omit the brackets),
∞
x + y/2
2
diag
dt ρζ (t) cos t log
T2 (y) =
,
(28)
x − y/2
x log2 x 0
where we have introduced the smooth density of the complex Riemann zeros, ρζ (t).
We only need the asymptotic behavior of ρζ , given by
ρζ (t) =
1
log
2π
t
2π
.
(29)
It is convenient to re-write the integral in Eq.(28) in terms of the variable τ = t/x
(cf Eq.(9))
∞
x + y/2
dτ ρζ (xτ ) cos xτ log
T2diag (y) = 2 ρ2
.
x − y/2
0
Using Eq.(29), we have ρζ (xτ ) = ρζ (τ ) + (2πρ)−1 . Then the diagonal part of T2 (y)
is
∞
x + y/2
T2diag (y) = ρ δ(y) + 2 ρ2
dτ ρζ (τ ) cos xτ log
.
(30)
x − y/2
0
t
∞
Now we evaluate the integral. When it is written as 0 = limt∗ →∞ 0 ∗ , the highly
oscillatory terms arising from the upper limit t∗ are washed out by the average.
Taking into account the definition (24), the result is,
Y2diag (y) = −
1
2
2x log x log
x+y/2
x−y/2
−
1
1
.
2
log x 2y
(31)
The last approximation holds when y x.
This is the first result concerning the two–point correlation function of the
prime numbers, that coincides with Eq.(22) (see also [11]). It confirms, in agreement with the Hardy–Littlewood conjecture, the presence of (anti)correlations. It is
a smooth function valid, in the approximation used to obtain Eq.(31), for arbitrary
distances. However, it is accurate only for large values of y. This is because the
correlations between Riemann zeros can only be ignored for small tα ’s (compared
to the scale associated to the mean spacing). This corresponds, through a Fourier
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P. Leboeuf
Ann. Henri Poincaré
transform, to y log x (cf §3.3 for details). Therefore, Eq.(31) can only be trusted
in that range (in §3.3 we will show that it holds in the range log x y x/t1 ).
The derivation of Eq.(31) depends only on a global property of the complex Riemann zeros, namely their average density. It does not require any further
knowledge, and assumes the Riemann hypothesis. In contrast, the computation of
the off–diagonal terms (to be done in §3.2) requires additional information.
It is important to stress an elementary point here. Since the average distance
between primes increases as x increases, in Eq.(31) it is natural to re-scale all
distances by the mean spacing between consecutive primes. Defining s = y/δ =
y/ log x, the rescaled two–point function R2 (s) = R2 (y = sδ)/ρ2 is given by
R2 (s) = 1 −
1
.
2s log x
(32)
For a fixed (and finite) value of s, clearly R2 (s) → 1 as x → ∞. Therefore, asymptotically the prime numbers become uncorrelated, in agreement with Cramér’s
model. Eq.(32) also means that asymptotically the only relevant part of the average prime pair correlation function is its tail, described by Eq.(31) (and further
elaborated in §3.3).
The prime correlations are thus a finite–x effect, that vanishes asymptotically.
Further finite–x corrections to the Hardy–Littlewood tail may be computed from
the higher orders in the expansion of the logarithm in Eq.(31) .
3.2
Prime correlations: oscillations
In §3.1 we have computed the diagonal contributions to Eq.(26). Here we improve
the calculation by considering the off-diagonal terms α = α ,
T2of f (y) =
2
cos [tα log(x + y/2) − tα log(x − y/2)] .
2 x log x α=α
(33)
We set
tα = tα + ,
i.e., is the (positive or negative) distance between two different Riemann zeros.
For positive values, it varies in the interval (min , ∞) (where min ≥ 0 is the distance to the nearest neighbor), while for negative values it varies in the range
(−min , −(tα − t1 )). Large values of produce highly oscillatory frequencies in
Eq.(33), and these are eliminated by the averaging procedure. We are therefore
allowed to extend the negative range to (−min , −∞) with negligible error. Moreover, Y2 should be an even function of . Re-arranging the sum in terms of positive
values of only, Eq.(33) becomes
T2of f (y) =
∞
4
x log2 x α=1
∞
=min
x + y/2
cos tα log
cos( log x) .
x − y/2
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Similarly to the diagonal terms, in the simplest approximation the double sum
is replaced by integrals. The substitution requires the average density of pairs of
Riemann zeros separated by a distance , R2ζ () = ρ2ζ + Y2ζ (). Because min can
be arbitrarily small we end up with
∞
∞
x + y/2
4
of f
dt cos t log
d Y2ζ () cos( log x) .
T2 (y) =
2
x
−
y/2
x log x 0
0
(34)
ζ
We now need an explicit expression for Y2 (). Following Montgomery’s conjecture
[4], we make use of the pair correlation function of the Gaussian Unitary Ensemble
of random matrices [19],
sin2 (πρζ )
,
(35)
Y2ζ () = −
π 2 2
Making the change of variable θ = πρζ , the second integral in Eq.(34) becomes
0
∞
d Y2ζ () cos( log x) =
ρζ (t) ∞
ρζ (t)
sin2 θ
[|a − 2| − (a − 2)] ,
dθ
cos(a θ) = −
−
2
π
θ
8
0
where a = log x/πρζ (t) = [π ρ(x) ρζ (t)]−1 > 0. The value of the integral is non–
zero only if a ≤ 2. According to the definition of a, this implies t ≥ 2πx. Then
∞
1
1
d Y2ζ () cos( log x) = −
ρζ (t) −
Θ(t − 2πx) .
2
2π ρ(x)
0
Inserting this result in Eq.(34) and changing to the variable τ = t/x it follows that
∞
x + y/2
dτ ρζ (τ ) cos xτ log
Θ(τ − 2π) .
T2of f (y) = − 2 ρ2
x − y/2
0
When this expression is combined with the contribution coming from the
diagonal terms, Eq.(30), we remark that the integrand exactly cancels for values
of τ ≥ 2π, leading to the following expression for T2 = T2diag + T2of f ,
2π
x + y/2
T2 (y) = ρ δ(y) + 2 ρ2
dτ ρζ (τ ) cos xτ log
.
(36)
x − y/2
0
Using Eq.(29) the last integral is straightforward and leads, by the definition (24),
to
Si 2πx log x+y/2
x−y/2
1 Si(2πy)
−
.
(37)
Y2 (y) = −
x+y/2
2
log2 x πy
πx log x log x−y/2
The last approximation holds when y x.
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This is the second result concerning the two–point correlation function between the prime numbers, given by Eq.(1) in the Introduction. It confirms the
presence of (anti)correlations, even for small values of y. Contrary to the diagonal
part, the computation of the off–diagonal terms requires the pair–correlation function between different non–trivial Riemann zeros. Their effect is to superimpose to
the smooth part Eq.(31) some oscillations, and make moreover the function finite
at small values of y, where the behavior is
Y2 (y) 1
log2 x
−2 +
4π 2 y 2
9
,
y→0.
However, in the relevant range y ≥ 2 the function −Si(2πy)/πy does not significantly deviate from its smooth behavior −1/2y, to which it clearly tends when
y 1.
An alternative view of the prime pair–correlations is offered by the Fourier
transform of T2 , called the form factor. It is defined as
K(τ ) = 4π
∞
dy cos(yτ ) T2 (y) ,
0
(38)
whose inverse relation is
T2 (y) =
1
2π 2
0
∞
dτ cos(yτ ) K(τ ) .
(39)
Comparing Eqs.(36) and (39) we deduce that the form factor of the prime numbers
is, in this approximation,
τ K(τ ) = 2πρ + 2πρ2 log
Θ(2π − τ ) .
(40)
2π
For τ ≤ 2π this function behaves as K(τ ) = 2πρ2 log(xτ /2π). In the limit τ → 0,
K(τ ) → −∞. This reflects the fact that the area under the function Y2 (y) is infinite
(cf. Eq.(38)). At τ = 2π, K(τ ) has a discontinuity (of the first derivative), and
saturates at the value K(τ ) = 2πρ = 2π/ log x, which is proportional to the inverse
of the mean spacing between prime numbers. The discontinuity of K(τ ) does not
take place for values of τ of order 2πρ (as it happens for example for the form factor
of GUE random matrices, see e.g. [19]), but on the much larger scale τ = 2π 2πρ.
In contrast, as it does happen in the GUE case, the diagonal approximation is
“exact” until the saturation takes place. In the limit x → ∞ and for a fixed τ ,
K(τ ) = 2πρ2 log(xτ /2π) → 2πρ, and the whole function tends to a constant.
This behavior corresponds to Cramér’s model. The function K(τ ) is represented
in Fig.1, using the value x = e2π . The behavior close to the origin is displayed in
the inset. The function in fact vanishes when 0 ≤ τ ≤ t1 /x = 2.6 × 10−2 . This is
explained in the next subsection.
Vol. 4, 2003
Prime Correlations and Fluctuations
S739
1.2
K
1.0
0.8
0.4
0.6
0.3
0.4
0.2
0.1
0.2
0.0
0.00
0.05
0.10
0.0
0
5
10
15
τ
20
Figure 1: Form factor of the prime numbers (at x = e2π ). Inset: behavior close to
the origin.
3.3
Long–Range Deviations
The form factor and the two–point function calculated in the previous sections
were obtained by a straightforward approximation of the sums by integrals. It is
possible to go beyond that approximation and make more accurate evaluations. In
particular, we will show that for large values of y there are deviations with respect
to the smooth tail predicted by Eqs.(31).
A better description of the form factor is obtained when it is written in terms
of the complex part of the Riemann zeros. Inserting, when x → ∞, the relation
(26) in Eq.(38), and taking the limit y x, we get
tα + tα
4π 2
cos [(tα − tα ) log x] δ τ −
.
(41)
K(τ ) =
2x
x log2 x α,α
This is a very useful relation, that expresses the form factor as the average of
a series of delta peaks located at τ = (tα + tα )/2x. It is the analog for prime
numbers of the semiclassical expression of the spectral form factor [3]. For small
values of τ (of the order of τ1 = t1 /x) only the lowest Riemann zeros contribute
to K(τ ). Assuming incommensurability between the tα ’s, the non-diagonal terms
involving these low–lying zeros are washed-out by the average and the form factor
is approximated by the diagonal contribution. The latter takes the simple form of
S740
P. Leboeuf
a sum over delta peaks located at the rescaled tα ’s
4π 2 tα
K(τ ) KD (τ ) =
δ τ−
,
x
x log2 x α
Ann. Henri Poincaré
τ →0.
When the sum is approximated by an integral, the previous result,
xτ ,
KD (τ ) = 2πρ2 log
2π
(42)
(43)
is recovered. It holds in fact not only in the limit τ → 0 but in the whole range
τ ≤ 2π. However, Eq.(43) is not an accurate description for τ ∼ τ1 . In particular,
it ignores a basic property of K(τ ), present in Eq.(42) but not in (43), namely
K(τ ) ≡ 0 for τ < τ1 . With this property incorporated, K(τ ) takes the form
0
if τ < t1 /x ,
τ K(τ ) =
(44)
Θ(2π − τ )
if τ ≥ t1 /x .
2πρ + 2πρ2 log 2π
This expression justifies the behavior of K(τ ) close to the origin represented in
Fig.1
It is easy to find out the modifications induced in Y2 (y) by the vanishing of
K(τ ) for τ < τ1 . This simply amounts to impose a lower bound, at t = t1 , in the
integral in Eq.(28). Then Y2diag (y) picks up some additional terms related to the
lower bound, and instead of Eq.(37) we now get, in the limit y x,
Y2 (y) =
1
−Si(2πy) + Si(t1 y/x) − log t1 sin(t1 y/x)
.
2
πy
log x
(45)
In the range t1 y/x 1 the contribution of the correction terms tends to
−t1 (log t1 − 1)/(πx log2 x), i.e. they add a (small) constant which is of order 1/x
with respect to the value of −Si(2πy)/πy. However, when t1 y/x ∼ 1, the decay of
the two Si functions cancel out and the sine function dominates, giving
Y2 (y) = −
1
log t1 sin(t1 y/x)
,
2
πy
log x
y
>
∼
x/t1 .
(46)
Thus, the smooth tail −1/2y predicted by Eq.(37) holds only for separations up to
y ∼ x/t1 . For larger values of y the behavior is modified. The exponent of the tail
is unchanged, but now Y2 (y) oscillates with mean value zero according to Eq.(46).
It implies the presence of regions of positive correlation, where the primes tend to
attract instead of repel each other.
Equations (45) and (46) arise from the existence of a lower limit in the integral
in Eq.(28), but otherwise make use of the average density of the Riemann zeros.
Such an approximation ignores the subtle interference mechanisms between the
tα ’s that could be at work in Eq.(27). And indeed such processes exist. In order to
see them, we remark that the structure of Eq.(27) resembles that of the oscillatory
Vol. 4, 2003
Prime Correlations and Fluctuations
S741
part of the density of states, Eq.(8). At some fixed height x, the comparison with
Eq.(8) suggests that a constructive interference is going to occur at values of y
defined by (x + y/2)/(x − y/2) = p, with p some prime number (in the asymptotic
approximation employed here constructive interference would also happen, though
with a lower weight, at integer powers of primes, see [14]). This gives
y = 2x
p−1
,
p+1
p = 2, 3, 5, . . .
(47)
In contrast to ρ(x), in Y2 (y) the sum will not produce delta peaks at these values
because the sum in Eq.(27) is truncated by the average, and only zeros satisfying
2
∼ 2πx /y∆x contribute significantly. Instead of delta peaks, some broaden
tα <
and finite negative peaks of anti–correlation will be located at the positions (47).
These are superimposed to the oscillations described by Eq.(46). The peaks were
also found in Ref.[11]. An indirect numerical test of their existence is given in
§4.2. The structure found is reminiscent of the complex zeros of the Riemann
zeta function [8], where low–lying Riemann zeros create a hierarchical structure
in the tail of their two–point correlation function, although in the present case,
and contrary to what happens for the Riemann zeros, the shape of the correlation
peaks depends on the averaging window size.
4 Fluctuations of π(x)
The fluctuations of π(x) around its mean value are now considered. Many (and
old) problems of number theory are directly related to them. As mentioned in
the introduction, our purpose is to compute the moments of the distribution of
the fluctuations of π(x), as well as their autocorrelation. Before that, we briefly
considered a related problem.
There is a long–standing question about the relative value of π(x) with respect to its leading asymptotic average term, Li(x), and in particular about the
first time π(x) becomes greater than Li(x). From Eq.(14), and to leading order in
x, the asymptotic difference is
√ √
1
x
x
1
−2
cos(tα log x) + tα sin(tα log x) .
π(x) − Li(x) = −
log x
log x α 14 + t2α 2
(48)
In this approximation, the first crossing between π(x) and Li(x) is therefore defined
as the lowest value of x for which
π
N (x) > 1 .
(49)
We have introduced the normalized asymptotic function (cf Eq.(18))
∞
1
1
π
(x)
1
= −2
cos(tα log x) + tα sin(tα log x) . (50)
π
N (x) = √
x/ log x
+ t2α 2
α=1 4
S742
P. Leboeuf
Ann. Henri Poincaré
1.02
1.00
0.98
0.96
0.94
1.3980
1.3982
x (10**316)
1.3984
Figure 2: π
N (x) as a function of x in the neighborhood of the first crossing found
It is possible to investigate numerically Eq.(49) for increasing values of x,
by first using a relatively large step in x (of about 105 points in each interval
10j → 10j+1 ) and then reducing the step whenever π
N (x) is greater than some
arbitrary fixed value. By this procedure, the first crossing we could find was located
at
(51)
x∗ ≈ 1.39815 × 10316 .
A plot of π
N (x) in this neighborhood is shown in Fig.2. π
N (x) has been computed
truncating the sum to the first 2 × 106 Riemann zeros, which were provided by A.
Odlyzko [20].
The value of x∗ is in agreement with the lowest bound known up to date [21],
which considerably reduces the previous bounds found by Lehman [22] (1.65 ×
101165 ) and by te Riele [23] (6.650 × 10370 ).
4.1
The moments
Our aim is to compute the moments of the asymptotic, x → ∞, distribution of
the fluctuations π
(x) = π(x) − π(x). To leading order, it is useful to consider the
normalized function (50), but lower–order terms will be taken into account as well
later on. To characterize the distribution of the fluctuations, we shall compute the
average of the k–th moment over a window ∆x,
k
Mk = π
N
(x) ,
k = 1, 2, 3, . . . .
(52)
The angular brackets denote an average over the position of the reference point x,
as defined in §3.
Vol. 4, 2003
Prime Correlations and Fluctuations
S743
Because of the average, any oscillatory behavior on a scale equal or smaller
than X1 has zero average. Therefore, the first moment of the fluctuating part
vanishes, as it should
M1 = πN (x) = 0 .
The second moment is written
M2 = 2 1
α1 ,α2
1
+ tα1 tα2
4
4
+ t2α1
1
1
4
+ t2α2
cos [(tα1 − tα2 ) log x] +
(tα1 − tα2 )
sin [(tα1 − tα2 )) log x] .
2
(53)
This equation is obtained by reducing the products to single trigonometric functions of the sum and differences of the arguments, and by further ignoring the
terms involving the sum of the arguments because their average vanishes.
In Eq.(53) only the terms satisfying tα1 − tα2 X1−1 = t1 /(2πx) have a
non–zero average and therefore contribute to the double sum. In the large–x limit
this difference is small, and it is legitimate to ignore the difference between tα1
and tα2 in the prefactors. Then the terms involving the sine do not contribute.
Using the asymptotic definition of the form factor of the prime numbers, given by
Eq.(41), the second moment takes the simple and compact form
M2 =
x log2 x
2π 2
0
∞
K(τ )
.
2 2
4 +x τ
dτ 1
(54)
Different degrees of precision are obtained for M2 according to the approximation used for K(τ ). The simplest one is the smooth approximation discussed
in §3.2, and given by Eq.(44). The latter, together with Eq.(54), allow to make a
simple qualitative estimate of the second moment. We notice that since K(τ ) saturates to a constant for large values of τ , in that limit the integral is convergent.
The main contributions come therefore from the lower limit τ → 0, where the
divergence is stopped by the short–time cutoff of K(τ ) at τ = t1 /x. When K(τ ) is
replaced by (44), the integral in Eq.(54) is somewhat cumbersome. To simplify the
result, we only give the leading order of an expansion where x and t1 are assumed
to be large parameters. The result is
t1
1
1
1
M2 =
(55)
1 + log
− 2 0.04078 − 2 .
πt1
2π
2π x
2π x
In the last equality we have used the value t1 = 14.1347 . . . for the imaginary part
of the lowest zero. In Eq.(55) the first term in the r.h.s comes from the diagonal
smooth approximation of the form factor, Eq.(43). A more accurate estimate is
obtained when, instead, a direct evaluation of the diagonal sum α1 = α2 is made
S744
P. Leboeuf
in Eq.(53) ,
∞
Ann. Henri Poincaré
1
1
1
− 2 0.04619 − 2 ,
(56)
2
2π x
2π x
+ tα
α=1 4
1
√
2
where we have used the well known result [13] ∞
α=1 1/ 4 + tα = − log(2 π) +
1 + γ/2; (γ is Euler’s constant).
In contrast to the diagonal contributions, the term −(2π 2 x)−1 arises from
off-diagonal terms in Eq.(53), and contain therefore information about the correlations acting among the Riemann zeros. This information is embodied through
the discontinuity of the form factor of the prime numbers at τ = 2π, and gives rise
to a lower–order correction to the asymptotic second moment. This, however, is
not the leading order correction to M2 . If in the expansion of the Li(x) in Eq.(16)
the term of order x/ log2 x is kept, it leads to a “diagonal” correction to M2 given
−2
. The specificity of the corrections arising from correlaby log−1 x α 14 + t2α
tions, compared to contributions from lower–order diagonal contributions, is the
negative sign.
The same principle can be extended to compute the higher–order moments.
The important point is that, as for the second moment, the main contributions
come in all cases from the lowest terms in the sums. These are given by a generalization of the diagonal approximation that has been used for M2 . Off-diagonal
corrections provide asymptotically negligible contributions. In the following we
ignore those corrections and concentrate on the main terms.
M2 = 2
1
The third moment. When the third power of π
N (x) is computed from Eq.(50), the
product of three trigonometric functions is decomposed as a sum of trigonometric
functions involving the sum (which has zero average) and differences of the arguments, which are of the type tα1 + tα2 − tα3 (plus permutations). The generalized
diagonal approximation consists in finding the terms for which the argument of
the trigonometric functions vanishes. By assuming incommensurability of the tα ’s,
there is no way to cancel linear combinations as those just mentioned. We then
conclude that,
x→∞.
(57)
M3 = 0 ,
The same conclusion can be reached for all odd moments. The distribution of the
values of π(x) is therefore asymptotically an even function, as was also found in
[12].
As we now show, at a finite x the odd moments are non zero. To capture the
non–vanishing terms the repetitions in Riemann’s formula (16) should be kept.
Instead of Eq.(50), we now have
∞ ∞
tα
tα
2 µ(r)x1/2r 1
1
cos
log
x
log
x
+
t
. (58)
sin
π
N (x) = − √
α
x α=1 r=1 4 + t2α
2
r
r
The main qualitative difference introduced by the terms r = 1 are the 1/r factors in the argument of the trigonometric functions, that introduce asymmetries
Vol. 4, 2003
Prime Correlations and Fluctuations
S745
in the behavior of π
N (x). When the third power of π
N (x) is computed, the arguments of the trigonometric functions have, up to a permutation of indices, the
form tα1 /r1 + tα2 /r2 − tα3 /r3 . The cancellation of the argument of the oscillatory
functions imposes the condition
tα1
tα
tα
+ 2 = 3 .
r1
r2
r3
For incommensurate tα ’s, the only possible solutions are α1 = α2 = α3 , (r1 , r2 )
two given (positive) integers and
r3 =
r1 r2
,
r1 + r2
which should also be an integer. The leading asymptotic solution that leads to an
integer value of r3 is r1 = r2 = 2, with r3 = 1. All other solutions are of lower
order in the limit x → ∞. Then the asymptotic non-vanishing expression for the
third moment becomes
∞
1
3 1.113 × 10−4
√
M3 = − √
.
1
2 ≈ −
2
x α=1
x
4 + tα
(59)
1
2 2
To evaluate the r.h.s., we made use of [12, 24] ∞
= γ − log(4π) +
α=1 1/ 4 + tα
2γ1 + γ 2 + 3 − π 2 /8 = 3.71 . . . × 10−5 ; (γ1 = −0.072816 . . .).
To conclude this section, we give the result for the fourth moment. It is
obtained by similar arguments as for the lower moments (repetitions are not necessary). The leading order contribution is found to be
∞
1
1
M4 = 12
+ t2α
α=1 4
2
−6
∞
1
2 =
+ t2α
∞
3M22 − 6
α=1
1
4
α=1
1
4
1
+
2
t2α
6.178 × 10−3 . (60)
−2
The value of the fourth moment differs, by a factor −6 α 14 + t2α
−2.23 ×
10−4 , from the value required by a Gaussian distribution. Similar results are found
for the higher moments.
To summarize, the distribution of π(x)− π(x) is asymptotically an even function whose momenta differ from the Gaussian law, as illustrated by Eq.(60). At
finite values of x the distribution looses its parity and becomes an asymmetric
function, with the third moment given by Eq.(59). Correlations between zeros introduce corrections, that were explicitly computed for the second moment, Eq.(56).
To check these results we have numerically computed the distribution of the
fluctuating part of π(x). This was done by subtracting the smooth part Eq.(15)
to the exact value of π(x) (obtained with Mathematica), and then computing
S746
P. Leboeuf
Ann. Henri Poincaré
2.0
1.5
1.0
0.5
0.0
−1.0
−0.5
0.0
0.5
1.0
Figure 3: Normalized asymptotic distribution of the fluctuations of π(x) (full line),
compared to a Gaussian law having the same variance.
√
the distribution of the re-scaled quantity (π(x) − π(x))/( x/ log x). The result,
for 63000 points computed in the interval 103 → 1010 , is shown in Fig.3. We
have numerically computed the moments of that distribution. We obtained M2 =
4.577 × 10−2 , M3 = −3.123 × 10−4 , and M4 = 5.778 × 10−3 . The second and
fourth moments are in good agreement with the asymptotic predictions M2 =
4.620 × 10−2, and M4 = 6.180 × 10−3. The third moment is too large compared to
the average value expected from Eq.(59).
4.2
The autocorrelation
The function π(x) wildly oscillates around its mean value as a function of x. In
the previous section we have studied the distribution of those oscillations. Another important aspect of this process is how the oscillations are correlated at
different points x. These “memory effects” may be characterized by computing
the autocorrelation function of the fluctuating part of π(x), defined as
N (x + y/2) .
C(y) = πN (x − y/2) π
(61)
The average window satisfy the same conditions as those defined for the moments.
C(y) is asymptotically evaluated by using the expression (50) for π
N (x).
Because of the same reasons as for the second moment, the resulting double sum
is asymptotically dominated by the diagonal terms (the off-diagonal terms add
Vol. 4, 2003
Prime Correlations and Fluctuations
S747
(1+s/2)/(1−s/2)
2
3
4
5
6 7 89
1.0
C(s)
0.5
0.0
−0.5
0.0
0.5
1.0
1.5
s = y/x
Figure 4: Autocorrelation of the normalized oscillatory part of π(x), as a function
of s (lower scale) and (1 + s/2)/(1 − s/2) (upper scale). Dots: numerical values.
Solid line: Eq.(62).
some asymptotically vanishing corrections). This approximation leads to
C(s) = 2
α
1
4
1 + s/2
1
cos tα log
,
1 − s/2
+ t2α
(62)
which depends only on the rescaled variable s = y/x. To test this result, the
exact values of the fluctuating part π
(x) = π(x) − π(x) were computed using
Mathematica for some 60000 values of x in the range 104 –1010. Then the average
autocorrelation function was computed for several windows in that range. The
result is represented, as a function of the rescaled variable s = y/x, in Fig.4. It
is compared to the prediction Eq.(62). Notice that, because of the large values of
x used in the numerical evaluation, the relative distance y takes also very large
values and therefore correlations between points separated by large distances exist.
The first change of sign of C(s) occurs at s ≈ π/(2t1 ) = 0.11 . . .. As seen in the
figure, C(s) is a non–decaying oscillatory function. This behavior is due to the
dominance of the lowest Riemann zeros in the sum (62).
The structure of C(s) as a function of s, which shows a series of (inverted)
parabolae connected by points where the first derivative of the function is discontinuous (edge points) has its origin in the following connection.
S748
P. Leboeuf
Ann. Henri Poincaré
Consider the sum of the prime numbers smaller than x, measured with respect
to x,
U (x) =
(x − p)Θ(x − p) .
(63)
p
From a physical point of view, U (x) represents (up to a sign) the grand–potential at
zero temperature of a non-interacting Fermi gas confined to an external potential
whose single–particle energy levels are given by the prime numbers. In this analogy,
the variable x represents the chemical potential. An analogous quantity for the
zeros of the Riemann zeta function has been studied in detail in [25].
x
The previous sum can be alternatively written as U (x) = 0 π(y)dy. We are
interested in the fluctuations
of this function with respect to its average, defined
(x) = U (x) − x π(y)dy, or
as U
0
x
U(x)
=
π
(y)dy .
0
The qualitative behavior of this function is clear from this expression. Since π
(x)
has a sawtooth shape (with teeth of negative slope and vertical discontinuities),
its integral consists of a series of parabolae (each corresponding to a tooth of π
(x))
connected by discontinuities of the first derivative of U(x),
which occur when x is
equal to a prime number. This behavior coincides, qualitatively, with the behavior
of C(s) observed in the figure.
(x) (appropriately normalWe now show that the asymptotic behavior of U
ized and with a rescaled argument) coincides with Eq.(62). To see this we first
notice
that, according to Eq.(16), π
(x) ∼ Li(xω/r ), with ω = 1/2 + itα . Since
dx Li(xω/r ) = −Li(x1+ω/r ) + x Li(xω/r ), keeping in the limit x → ∞ the approximation Li(x) ∼ x/ log x, we obtain
3/2
(x) ≈ − 2 x
U
log x
∞
α=1
1
4
1
+ t2α 94 + t2α
3
− t2α cos(tα log x) + 2 tα sin(tα log x) . (64)
4
Since in each bracket t2α is much larger than its companion constant, we can neglect
those constants without introducing an important error. By the same reason, the
term containing the sine is neglected compared to the one with the cosine. Then
∞
2 x3/2 1
U(x)
≈
cos(tα log x) .
log x α=1 t2α
Up to normalization of this function by the factor x3/2 / log x and by the transfor (x) therefore coincides
mation x → (1 + s/2)/(1 − s/2), the oscillatory part of U
with C(s), given by Eq.(62) (when the 1/4 in the denominator is neglected). To
Vol. 4, 2003
Prime Correlations and Fluctuations
S749
explicitly show that in C(s) the discontinuities of the derivative joining the parabolae are, as predicted by this connection, related to prime integers, in the upper
part of Fig.4 we have displayed the variable (1 + s/2)/(1 − s/2) for the values of
s indicated in the lower axis. The discontinuities observed at powers of integers
in the plot of Eq.(62) are spurious and due to the asymptotic approximation used
(cf [14]).
Finally, we also notice that the diagonal expression of the autocorrelation of
π(x), Eq.(62), and the diagonal part of the density of prime pairs, Eq.(27), satisfy,
to a good approximation, the relation
T2diag (y = xs)
∂ 2 C(s)
2
≈
−x
log
x
.
∂s2
(s2 /4 − 1)2
It follows from this that the structure of C(s) discussed above and displayed in
Fig.(4) is related to the long–range deviations of Y2 (y) considered in §3.3. In particular, the abrupt changes of slope observed in C(s) at values (1+s/2)/(1−s/2) = p,
p = 2, 3, 5, 7, . . ., provide a numerical test of the presence of anti–correlation peaks
(located also at positions determined by the lowest prime numbers) in the asymptotic density of prime pairs.
5 Summary and concluding remarks
The properties of prime numbers and those of the complex Riemann zeros are
intricately interrelated. The duality and close relationships that operate between
these two sets – one behaving as eigenvalues of random matrices, the other resembling a random uncorrelated sequence – is one of the more striking features
of number theory. The present investigation further illustrates and clarifies this
connection.
Our first task has been to revisit one of the most basic aspects of the Hardy–
Littlewood conjecture, namely that for large distances y 1 the average prime
pair correlation behaves as R2 (y) = ρ2 (1 − 1/2y). This follows from the diagonal
contributions to R2 (y). It is derived under Riemann hypothesis, and requires the
average density of the Riemann zeros as a further ingredient. Some rigorous results
related to this asymptotic behavior can be found in [17]; it was also related to the
discreteness of the sequence of prime numbers in [11].
Montgomery’s random matrix conjecture concerning the correlation function
of the zeros allows to improve the formula for the average behavior of the pair
correlation of the primes. To the smooth decay of R2 (y) mentioned above, now
is superimposed an oscillatory structure described by a sine integral function (cf
Eq.(37)). This provides a more accurate description of the average properties of
the correlations. The corresponding form factor has been explicitly computed. We
ignore if this result can be directly obtained from the pair correlation function
conjectured by Hardy and Littlewood, Eq.(21).
S750
P. Leboeuf
Ann. Henri Poincaré
Finally, we argued that the previous expressions of R2 (y) are only valid in
a finite domain of distances. As a general rule, the correlations between primes
described above hold only for distances y <
∼ x/t1 , where t1 is the imaginary part
of the lowest Riemann zero. In the limit x → ∞, this scale is much larger than the
typical distance, of order log x, between consecutive prime numbers. For values
of y ∼ x/t1 , a transition takes place and a different behavior begins. In the latter regime, regions of positive correlation exist, and sharp anti-correlation peaks
appear at distances determined by the lowest prime numbers.
A related problem is that of the fluctuations of π(x) around its mean value.
The moments of the distribution were explicitly computed in terms of convergent
sums over Riemann zeros. Linear independence of the latter has been assumed.
Asymptotically the distribution is symmetric and non–Gaussian, in agreement
with the findings of Ref.[12]. However, at finite x the odd moments are non–zero,
and arise from terms with r ≥ 2 in Riemann’s formula. Also, the correlations
between zeros add finite–x corrections to the moments, which can be efficiently
computed using the form factor of the prime numbers. Illustrating a different
aspect of the fluctuations, a numerical search made for finding the first crossing
between π(x) and Li(x) gave a value that is in agreement with the best bound
known today.
In the context of dynamical systems, the moments and distribution of the
fluctuations of the counting function of the quantum eigenvalues have been investigated in recent years. The general conclusion [26] is that asymptotically the
distribution is universal (and obeys a Gaussian law) when the corresponding classical dynamics is fully chaotic, while it is system–dependent, non–universal for an
integrable dynamics. The two basic “spectra” encountered in number theory obey
to that general rule. The fluctuations of the counting function N (t) of the Riemann
zeros has been shown to be asymptotically Gaussian [27], in agreement with the
behavior of a chaotic quantum spectrum. In contrast, the fluctuations of π(x) (the
counting function of prime numbers) are not Gaussian. This is consistent with the
fact that, asymptotically, the prime numbers behave as an uncorrelated sequence,
and can therefore be assimilated to the spectrum of an integrable system.
The autocorrelation of the counting function has a peculiar structure. First,
it is a non–decaying function, showing from a different perspective the existence
of long–range √
correlations. Second, the distribution of the fluctuations give the
estimate ∼ 0.2 x/ log x for the typical size of the oscillations with respect to the
mean. The autocorrelation obtained then implies that those oscillations have a
typical size ∼ 0.1x in the x direction. Third, the peculiar structure of the autocorrelation was explained by making a connection with the fluctuations of the sum of
the prime numbers ≤ x, U (x). Their numerical verification (cf Fig.4) also demonstrates the existence of the anticorrelation peaks in the density of prime pairs at
large distances mentioned above.
Vol. 4, 2003
Prime Correlations and Fluctuations
S751
Acknowledgements
We are grateful to O. Bohigas, J. P. Keating and P. Sarnak for useful comments
and remarks, and to A. Odlyzko for providing the tables of Riemann zeros.
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P. Leboeuf
Laboratoire de Physique Théorique et Modèles Statistiques1
Bât. 100
F-91405 Orsay Cedex
France
1 Unité
de recherche de l’Université de Paris XI associée au CNRS