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AMO - Advanced Modeling and Optimization, Volume 14, Number 3, 2012
Neural Network prediction of Riemann
zeta zeros
O. Shanker ∗
Abstract
The theory underlying the location of zeroes of the Riemann Zeta
Function is one of the key unsolved problems. Researchers are making
extensive numerical studies to complement the theoretical work. The
evaluation of the zeta function at large heights involves time consuming
calculations. It would be helpful if one could have good predictions for the
possible locations of the zeros, since one could then reduce the number
of function evaluations required for locating the zeros. In this work we
apply neural network regression as a tool to aid the empirical studies of
the locations of the zeros. We use values evaluated at the Gram points
as the input feature set for the predictions. The range of t studied is
267653395648.87 to 267653398472.51.
∗Mountain View, CA 94041, U. S. A. Email: oshanker@gmail.com
*AMO - Advanced Modeling and Optimization. ISSN: 1841-4311
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O. Shanker
Figure 1: Riemann-Siegal function Z(gn) evaluated at 500 Gram points. X-axis
is t−267653395647.
1 Introduction
In the 1850’s Riemann came up with an intriguing hypothesis about the lo-
cation of the roots of the Riemann zeta function, which challenges us to this
day. There is an impressive body of empirical evidence for his hypothesis, but a
formal proof has been elusive. The numerical studies have found regions where
the hypothesis is almost violated, but no counter-example has been found. The
numerical calculations are time-consuming, and would benefit from tools which
give approximate locations for the zeros of the function. We apply neural net-
work regression to aid the empirical studies of the locations of the zeros. The
paper is organised as follows. Section 2 establishes the required notation for
the Riemann Zeta Function and L-functions. We also review breifly the the-
oretical work related to the locations of the zeros to show the importance of
the topic. Section 3 describes the numerical evaluation of the Riemann zeta
function. In section 4 we present results on training a neural network to predict
the zero locations. The feature set used for training the neural network is based
on the values at Gram points. The range of t studied is 267653395648.87 to
267653398472.51, which encompasses the range of zeros from 1012 to 1012 +104.
Section 5 gives a brief summary of the results.
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Neural Network prediction of Riemann zeta zeros
2 Theory of zero distributions of Generalised
zeta functions
In this section we establish the required notation for the Riemann Zeta Function
and L-functions. We also discuss the theory of the zero distributions, which gives
the subject its importance.
The Riemann Zeta function is defined for Re(s)>1 by
ζ(s) =
∞
∑
n=1
n−s=∏
p∈primes
(1−p−s)−1.(1)
Eq. (1) converges for Re(s)>1. It was shown by Riemann [1, 2, 3, 4] that
ζ(s) has a continuation to the complex plane and satisfies a functional equation
ξ(s) := π−s/2Γ(s/2) ζ(s) = ξ(1 −s); (2)
ξ(s) is entire except for simple poles at s= 0 and 1. Riemann multiplied the
definition by s(s−1) to remove the poles. We write the zeroes of ξ(s) as 1/2+iγ.
The Riemann Hypothesis asserts that γis real for the non-trivial zeroes. We
order the γs in increasing order, with
......γ−1<0< γ1≤γ2. . . . (3)
Then γj=−γ−jfor j= 1,2,..., and γ1,γ2,. . . are roughly 14.1347, 21.0220,
. . ..
Asymptotically, for the Riemann zeta function the mean number of zeros
with height less than γ(the smoothed Riemann zeta staircase) is [4]
<NR(γ)>= (γ/2π)(ln(γ /2π)−1) −7
8.(4)
Thus, the mean spacing of the zeros at height γis 2π(ln(γ/2π))−1. For the
range of tvalues studied by us this spacing is essentially constant at 0.2567.
The study of the zeroes of the Riemann zeta function and Generalised Zeta
functions is of interest to mathematicians and physicists. Mathematicians study
the spacings because of its applications to analytic number theory, while physi-
cists study it because of its relation to the theory of the spectra of random
matrix theories (RMT) and the spectra of classically chaotic quantum systems.
Odlyzko [5, 6] has made extensive numerical studies of the zeroes of the Rie-
mann zeta function and their local spacings. He also studied their relation to
the random matrix models of physics. Wigner [7] suggested that the resonance
lines of a heavy nucleus might be modeled by the spectrum of a large random
matrix. Gaudin [9] and Gaudin-Mehta [8] gave results for the local (scaled)
spacing distributions between the eigenvalues of typical members of the ensem-
bles as N→ ∞, based on their study of orthogonal polynomials. Later Dyson
[10] introduced the closely related circular ensembles.
Odlyzko confirmed numerically that the local spacings of the zeroes of the
Riemann Zeta function obey the laws for the (scaled) spacings between the
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O. Shanker
eigenvalues of a typical large unitary matrix. That is, they obey the laws of
the Gaussian Unitary Ensemble (GUE) [7, 8, 9, 10]. Katz and Sarnak [12] state
that at the phenomenological level this may be the most striking discovery about
zeta since Riemann. Odlyzko’s computations thus verified the discoveries and
conjectures of Montgomery [13, 14, 15].
Further evidence for the connection between random matrices and gener-
alised zeta functions comes from calculations of the zero correlation functions
[16, 17, 18, 19], and the study of the low-lying zeros of families of L-functions
[20]. Extensive numerical computations [5, 11] have strengthened the connec-
tion.
Several authors [21, 22, 23, 24, 25] have studied the moments of the Rie-
mann zeta function and families of L-functions, and the relation to characteris-
tic polynomials of random matrices. The autocorrelation functions were studied
in [26, 27]. The relation of the Riemann zeta function to probability laws was
studied in [28]. The author of this work studied the distributions of the zero
spacings using Rescaled Range Analysis [29].
It has been shown that the long-range statistics of the zeroes of the Riemann
zeta function are better described in terms of primes than by the GUE RMT.
Berry [30, 31, 32, 33] has related this to a study of the semiclassical behaviour of
classically chaotic physical systems. The primitive closed orbits of the physical
system are analogous to the primes p. The analogy comes from formulae that
connect zeros of the zeta function and prime numbers [34, 35, 36]
Quantum chaology is defined by Berry [32] as the study of semiclassical, but
non-classical, behaviour characteristic of systems whose classical motion exhibits
chaos. By semiclassical one means the limit as Planck’s constant ℏ→0. The
distribution of eigenvalues of quantum chaotic systems shows universality. The
universality class depends on the symmetries of the system’s Hamiltonian. For
systems without time-reversal invariance the distribution of eigenvalue spacings
approaches that of the GUE. The connection between quantum chaology and
the Riemann zeta function comes about because the Riemann Hypothesis would
follow if the imaginary parts γjof the non-trivial zeros of the Riemann zeta
function are eigenvalues of a self-adjoint operator.
The remarkable properties of the Riemann Zeta Function can be generalised
to a host of other zeta and L-functions. The simplest of the generalisations are
for the Dirichlet L-functions L(s, χ) defined as follows: q≥1 is an integer and
χis a primitive character of the Abelian group formed by all integers smaller
than and relatively prime to q. χis extended to all integer values by making it
periodic, and χ(m) = 0 if m and q have a common factor. Then
L(s, χ) =
∞
∑
n=1
χ(n)n−s=∏
p
(1−χ(p)p−s)−1.(5)
The analogue of the functional equation Eq. (2) is known for the generalised zeta
functions, and they also seem to satisfy the generalised Riemann Hypothesis. q
is called the conductor of the L-function. In this work we perform studies on
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Neural Network prediction of Riemann zeta zeros
Figure 2: Comparision of neural network prediction vs actual value for the
distance from a Gram point to the next zero. The x axis shows 50 Gram points
beginning at index 1000000008584.
the Riemann zeta function. It would be worthwhile to extend the study to the
other zeta functions.
The next section gives the details of the numerical calculations.
3 Empirical Calculations
In this section we discuss the details of the numerical work. The numerical
analysis takes advantage of the functional equation Eq. (2). One defines
θ(t) = arg(πit/2Γ( 1
4+it
2)),(6)
where the argument is defined by continuous variation of tstarting with the
value 0 at t= 0. For large t θ has the aymptotic expansion
θ(t) = t
2ln( t
2π)−t
2−π
8+1
48t−1
5760t3.(7)
A consequence of the zeta functional equation is that the function Z(t) =
exp(iθ(t))γ(1/2 + it), known as the Riemann-Siegel Z-function, is real valued
for real t. Moreover we have |Z(t)|=|ζ(1/2 + it)|. Thus the zeros of Z(t) are
the imaginary part of the zeros of ζ(s) which lie on the critical line. We are
led to finding the change of sign of a real valued function to find zeros on the
critical line. This is a very convenient property in the numerical verification
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O. Shanker
of the Riemann Hypothesis. Another very helpful property is that many of the
zeros are separated by the ”Gram points”. When t≥7, the θfunction Eq.(6)
is monotonic increasing. For n≥1, the n−th Gram point gnis defined as
the unique solution >7 to θ(gn) = nπ. The Gram points are as dense as the
zeros of ζ(s) but are much more regularly distributed. Their locations can be
found without any evaluations of the Riemann-Siegal series Eq.(8). Grams law
is the empirical observation that Z(t) usually changes its sign in each Gram
interval Gn= [gn, gn+1). This law fails infinitely often, but it is true in a large
proportion of cases. The average value of Z(gn) is 2 for even nand −2 for odd
n[3], and hence Z(gn) undergoes an infinite number of sign changes. Figure 1
shows the Riemann-Siegal function Z(gn) evaluated at 500 Gram points start-
ing at n= 99999999999. Given the desirable properties of the Gram points, it
seems natural to use the values at these points as the feature set for the neural
network regression.
The Riemann-Siegel Z-function is evaluated using the Riemann-Siegal series
Z(t) = 2
m
∑
n=1
cos(θ(t)−tln(n))
√n+R(t),(8)
where mis the integer part of √t/(2π), and R(t) is a small remainder term which
can be evaluated to the desired level of accuracy. The most important source
for loss of accuracy at large heights is the cancellation between large numbers
that occur in the arguments of the cos terms in Eq. (8). We use a high precision
module to evaluate the arguments. The rest of the calculation is done using
regular double precision accuracy. The range of t studied is 267653395648.87 to
267653398472.51, which encompasses the range of zeros from 1012 to 1012 +104.
Odlyzko [5] has published very accurate values for the zeros in this range. By
comparing our calculations with the the calculations of Odlyzko, we estimate
that our accuracy for the Zfunction evaluation is better than 10−8.
In the next section we discuss the application of neural network regression
to aid the location of the zeros.
4 Neural Network Regression
This section describes the feature set used in training our neural networks, and
the results of the training. Neural Network Regression (NNR) models have
been applied for forecasting and are included in state of the art techniques in
forecasting methods [37, 38, 39]. Cybenkos [40] and Hornik et al. [41] demon-
strated that specific NNR models can approximate any continuous function if
their hidden layer is sufficiently large. Furthermore, NNR models are equivalent
to several nonlinear nonparametric models, i.e. models where no decisive as-
sumption about the generating process must be made in advance [42]. Kouam
et al. [43] have shown that most forecasting models (ARMA models, bilinear
models, autoregressive models with thresholds, non- parametric models with
kernel regression, etc.) are embedded in NNR. The advantage of NNR mod-
els can be summarised as follows: if, in practice, the best model for a given
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Neural Network prediction of Riemann zeta zeros
Figure 3: Comparision of neural network prediction vs actual value for the
smallest zero difference in the Gram interval (the left hand zero lies in the Gram
interval). The x axis shows 50 Gram points beginning at index 1000000008584.
problem cannot be determined, it is a good idea to use a modelling strategy
which is a generalisation of a large number of models, rather than to impose a
priori a given model specification. This is the reason for the interest in NNR
applications [44, 45, 46, 47, 48, 49].
As explained in Section 3, we use values defined at the Gram points for the
input feature set provided to the neural network. This is because the positions
of the Gram points can be determined to the desired accuracy without extensive
costly calculations. The features we use are the values of the Riemann-Segal
Z-function, the first ten terms in the Riemann-Segal series Eq. (8), and nine
terms with the cos function replaced by the respective sin function, for a pair of
consecutive Gram points. We drop the first sin term because it is always zero
at Gram points. Thus, the size of the input feature set is 40. We use a two
layer neural network with 200 neurons in the hidden layer.
We train the neural network to separately predict two quantities. The first
is the distance from a Gram point to the next zero. The second is the smallest
zero difference in the Gram interval with the left hand zero lying in the Gram
interval. The reason for choosing these output functions is that close pairs of
zeros are interesting. They correspond to cases for which the RH is nearly
false. Calculating the Riemann zeta function in zones where two zeros are very
close is a stringent test of the Riemann Hypothesis (often described as Lehmer’s
phenomenon since Lehmer was the first to observe such situations). We use the
scaled difference:
δj= (γj+1 −γj) ln(γj/2π)/(2π).(9)
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O. Shanker
Next zero Mean Std.Deviation
T raining set
Acutal 0.579 0.41
P redicted 0.579 0.37
V alidation set
Acutal 0.599 0.42
P redicted 0.590 0.38
T est set
Acutal 0.576 0.43
P redicted 0.581 0.39
Table 1: Prediction of the distance from a Gram point to the next zero.
Smallest zero Mean Std.Deviation
difference
T raining set
Acutal 0.965 0.41
P redicted 0.926 0.39
V alidation set
Acutal 0.959 0.41
P redicted 0.926 0.39
T est set
Acutal 0.972 0.43
P redicted 0.934 0.42
Table 2: Prediction of the smallest zero difference in the Gram interval with the
left hand zero lying in the Gram interval.
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Neural Network prediction of Riemann zeta zeros
We divide the 10000 data points into three sets, of sizes 9000, 500 and 500
respectively. The first set of 9000 points is used for training the neural network.
The second set of 500 points is used as a validation set during the training.
Finally, the last set of 500 points is used to check how well the neural network
has learned to predict the output function.
Figure 2 shows the comparision of the neural network prediction vs actual
value for the distance from a Gram point to the next zero for 50 Gram points
beginning at index 1000000008584. Table 1 presents summary results for the
complete data set. The table shows the the mean value and standard deviation
for the actual data set and also for the predicted values. Figure 3 shows the
comparision of the neural network prediction vs actual value for the smallest
zero difference in the Gram interval with the left hand zero lying in the Gram
interval, for the same 50 Gram points. Table 2 presents summary results for the
complete data set. The table shows the the mean value and standard deviation
for the actual data set and also for the predicted values.
In both cases, we see that the neural network gives fairly good predictions.
It follows the peaks and dips of the output function faithfully. It is successful in
reproducing the mean value as well as the variation of the output it is attempting
to predict. Given this encouraging result, it seems worthwhile to consider using
more features in the input layer and more neurons in the hidden layer to increase
the fidelity of the predictions.
In the next section we give a brief summary of the results.
5 Conclusions
We studied the Riemann zeta function for the range of t from 267653395648.87
to 267653398472.51. We trained neural networks to predict the distance from
a Gram point to the next zero, and the smallest zero difference in the Gram
interval with the left hand zero lying in the Gram interval. In both cases, the
neural network gives fairly good predictions. It follows the peaks and dips of the
output function faithfully. It is successful in reproducing the mean value as well
as the variation of the output it is attempting to predict. Given this encouraging
result, it seems worthwhile to consider using more features in the input layer
and more neurons in the hidden layer to increase the fidelity of the predictions.
The use of neural networks should also be extended to the Generalised zeta
functions.
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