Riemann zeta value distribution

Relations for Riemann Zeta value distribution on critical axis

For list of low lying zeros of Riemann zeta function, see https://sites.google.com/site/riemannzetazeros/Home 

O. Shanker, Mountain View, CA 94041, U. S. A.           

The behavior of the Riemann zeta function, through its connections with the spectra of random matrix theories and the spectra of classically chaotic quantum systems, is of enduring interest to statistical physics and condensed matter physics. Motivated by this unexpected and deep connection, we investigated the value distribution of the Riemann zeta function on the critical axis. We present a remarkable and striking property for the value distribution, i.e., the distributions for different Generalized Gram points can be expressed in terms of three functions which are independent of the angle characterizing the Generalized Gram point. The observed coefficients in the relation imply symmetry and anti-symmetry relations for the value distribution. We provide empirical verification of the relations for a range of heights along the critical axis covering 16 orders of magnitude. We discuss how random matrix models may shed further light on this fundamental relation. Another promising area to explore is the behavior of a-points along Julia lines of the ζ function.

See:

Random Matrix Theory explanation for Riemann Zeta Value Distribution Symmetry 

Riemann zeta value distribution has a discontinuity at value = 0. 

Keywords: Riemann zeta, Value Distribution, Universality, Symmetry, Random Matrix Theory

1 Introduction

Theoretical physicists studying statistical physics and condensed matter physics have a deep and enduring interest in the behavior of the Riemann zeta function on the critical axis [1]. This is because of its close and unusual connection to the theory of the spectra of random matrix theories (RMT) [2–15] and the spectra of classically chaotic quantum systems in physics [16–19]. Schumayer and Hutchinson [20] give a comprehensive review of the role of the Riemann Hypothesis in several areas of physics, including condensed matter and statistical physics. To get some insight into this, we look at the symmetry and universality properties of the Riemann zeta function value on the critical axis. This provides a complementary way to look at the problem. The universality relations and symmetries exhibited by a system are fundamental aspects of the system. We discuss how random matrix models can be used to gain further insight into the fundamental universal and symmetry properties.

Ref. [21] studied empirically the distribution of Z(t) values at Gram points and showed that the distribution for even Gram points was the negative of the distribution for odd Gram points. Further work [22]) extended the study to Generalized Gram points. The new results in this work are as follows. Our new relation (Eq. 3.3) expresses the value distributions at different Generalized Gram points, in terms of three universal functions. By universal we mean that the functions are independent of the angle φ characterizing the Generalized Gram point. We show that the value distribution of the Hardy Z function at discrete points is anti-symmetrical for reflections around the mid-points of the Gram intervals (Eq. 3.4) and symmetrical for reflections around the Gram points(Eq. 3.5).

Section 2 establishes the required notation for the Riemann Zeta Function. Section 3 presents the conjectured relations and provides evidence for the newly observed relations. We discuss the application of random matrix theory to understand these fundamental properties, and possible explanation in terms of the behavior at Julia lines. In Section 4 we discuss the evidence for the symmetry relations by studying the quantiles of the distributions. Section 5 presents the conclusions.

2. Notation for the Riemann zeta function

In this section we establish the required notation for the Riemann Zeta Function. For Re(s) > 1 the Riemann Zeta function is defined as

ζ(s) can be continued to the complex plane. Riemann’s hypothesis, that the non-trivial zeros of ζ(s) lie on the critical axis 1/2 + it, is probably the most famous unsolved problem in mathematics. The mean spacing δ of the zeros at large height T is δ = 2π(ln(T/2π))−1. For numerical studies of the Riemann hypothesis one defines Hardy’s function

              Z(t) = exp(iθ(t))ζ(1/2 + it)                                                      (2.2)

where

Table 1: Values of the universal functions A(z), B(z) and C(z) for z in the range −3.0 to 3.0, and the R2 from the fit to actual values.

Table 1 gives the fitted values of the universal functions. Fig 1 and Fig. 2, and Table 2, show the comparison of the predictions from the universality relation to the actual probability densities, for some values of the argument z. 

Figure 1: Test of universality. Comparison of probability density prediction from universality with actual values, for z = 0.5. The y axis is the probability density. The x axis is the angle φ characterizing the Generalized Gram point.

Figure 2: Test of universality. Comparison of probability density prediction from universality with actual values, for z = 0.0.

Table 2: Comparison of actual and predicted probability density for some φ

The figures, and Table 2 show the excellent agreement of the actual probability densities with the universality relations prediction. From the symmetry relations Eq. 3.5 and Eq. 3.5 we find that the function A(z) has to be anti-symmetric in z, and B(z) and C(z) have to be symmetric in z. Table 1 confirms these properties. The mean value of z is known from theory to be 2 cos(φ) [22]. Eq. 3.3 gives an understanding of why the mean value is proportional to cos(φ).

Since the A(z), B(z) and C(z) show large variation close to z = 0.0, we plot these functions for this region. Fig 3, Fig 4 and Fig 5 show the variation.

3.1 Random matrix models

A possible fruitful avenue of future investigation may be to relate the distributions to the results of Random Matrix Theory (RMT). Keating and Snaith [9] introduced the characteristic polynomial of a unitary matrix as a RMT model for the Riemann zeta function. Let U(N) be the group of all N × N unitary matrices. If A ∈ U(N), the characteristic polynomial P(θ) is defined as

P(θ)=det(IN −Aexp(iθ)),                       (3.6)

where det is the determinant, and IN is the unit matrix. Hanga and Hughes [28] define Gram points for U(N) and for SU(N), the group of all N × N special unitary matrices. They show that the SU(N) group gives a better approximation to the distribution of zeros in Gram intervals, for large but finite N. It will be useful to push these results further, and maybe define generalized Gram points for these groups.

Another very promising avenue to explore for a possible explanation is the work of Steuding and Suriajaya [23] and Kalpokas [24] on the mean value results for a-points.

Ivi ́c’s monograph [29] has a survey of the closely related Hardy’s function [30]. Selberg in unpublished work showed that at large t log(ζ(1 + it)) is approximately normally distributed with a standard deviation of order 2√log log t (see Ref. [31]). He showed a similar result [32,33] for log(|Z(t)|). Laurincikas [34] used probabilistic number theory to prove various results about the distribution of the Riemann zeta function.

Regarding the value distribution at specific points along the Gram interval, Titchmarsh [35] and Kalpokas and Steuding [24] present results pertaining to the mean value of the Riemann zeta function. Lester’s [36] Ph. D. thesis also considers the distribution of log(|ζ(1 + it)|) for specific points along the Gram interval. The question of the values of Hardy’s Z-function at a discrete sequence of points on the critical axis is quite interesting. The above references give some results in this direction that can be proved rigorously. At the same time, the present state of Riemann zeta function theory gives limited information about the value distribution of Z(t) at discrete sequences of points. Possibly, these analogues could differ significantly from the continuous case. Our numerical studies of the value distribution of Hardy’s Z-function at discrete points helps fill the gaps. 

3.2 Numerical evaluation

We evaluate Hardy’s function Z(t) using the Riemann−Siegel series 

where m is the integer part of √􏰅t/(2π). R(t) is a small remainder term which can be evaluated to the desired level of accuracy. We used the techniques in Refs. [37–39] to efficiently evaluate the zeta function at large t. To evaluate Z(t) at several points in the Gram interval, we have to use band limited function interpolation [40]. We evaluate the coefficients in the series for band limited function interpolation at the Gram points, and use the series to evaluate Z(t) at other points in the Gram interval. 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. (3.7). We use a high precision module to evaluate the arguments. The rest of the calculation is done using regular double precision accuracy. The zeros from Ref [41] were used to check the accuracy of our zeta function calculations. Our evaluations of Z(t) at T = 1012 are accurate to better than 10−6.

4 Evidence for the relations from quantiles

The human mind is attracted by symmetry. Many objects in nature exhibit symmetry. Even more remarkable are the symmetries exhibited in the basic laws of the universe. In this section we present numerical evidence for the symmetry properties Eq. 3.5 and Eq. 3.4. We present the distribution for T = 1012, followed by the distribution for T = 1028. We are interested in how the symmetry properties behave when we span a large range of height T. We will present the symmetry properties in terms of the quantiles [42] for the sample space. This is because for any sample space the quantiles are well defined and always exist. We recall that in statistics, given a fraction f between 0 and 1, the quantile qf is the value such that fraction f of the sample population is at or below qf . For example, q0.5 is the median of the sample. For f very close to 0 or 1, in the limiting case qf will diverge. However we will assume that there is a range of f for which well-defined limits exist. If necessary we can normalize the Z values by some power of ln(T/2π) without affecting the symmetry properties. Since the standard deviation is known to be sqrt(ln(T/2π)) [22], normalizing Z by this factor will make qf finite for finite f. Certainly, for our two samples separated by several orders of magnitude, we find that the quantile distribution changes very little.

In terms of quantiles the anti-symmetry relation Eq. 3.4 can be written

qf (φ) = −q1−f (π − φ).                                      (4.1)

Eq. 4.1 is a generalization of the result in Ref. [21] for the anti-symmetry of the distribution of Z(t) at even and odd Gram points. The symmetry condition Eq. 3.5 implies

qf (φ) = qf (2π − φ).                                          (4.2)

4.1 Quantiles for Z(t) at T = 1012

Table 3 shows the mean value for Z(T) and the standard deviation at 12 equally spaced values of φ at T = 1012. The mean value is known from theory to be 2cos(φ), and the standard deviation is known to be sqrt(ln(T/2π)) [22]. The table verifies this result to better than one part in a thousand. Table 4 shows the quantiles for Z(T). Eq. 4.1 can be verified, for example, by noting that qf in Table 4 for φ = π/6 and f = 0.25 is 0.017, while it is −0.016 for φ = 5π/6 and f = 0.75. Eq. 4.2 can be verified, for example, by noting that qf is 0.018 for φ = 11π/6 and f = 0.25.

We found that there is a linear dependence of the quantiles on cos(φ). Table 5 shows the intercept and slope when qf for any f is fitted to 2cos(φ)). The observation of this relation itself verifies the symmetry condition Eq. 4.2 (i.e., there is no dependence on sin(φ)). The anti-symmetry condition Eq. 4.1 implies that the intercept is anti-symmetric for reflection around f = 0.5, while the slope is symmetric for such a reflection. This can be verified from the table. Table 5 makes the symmetry and anti-symmetry relations stand out clearly. It must be pointed out that Eq. 4.1 and Eq. 4.2 do not require the existence of the linear relationship of qf to cos(φ). However, the linear relationship is consistent with the symmetry relations. The linear dependence and the symmetry properties imply that the quartile range q1−f − qf is independent of φ.

4.2 Quantiles for Z(t) at T = 1028

To perform a rigorous test of the symmetry relations, we repeat the study for another T separated from the first sample by 16 orders of magnitude. AtT = 1028 we used the zeros from Ref [41] to get the coefficients in the series for band limited function interpolation. We could then use this series to evaluate Z(t) at several points in the Gram interval. 

Table 6 shows the mean value for Z(T) and the standard deviation at 12 equally spaced values of φ at T = 1028. The table verifies that the values are 2cos(φ), to a few parts in a thousand. Table 7 shows the quantiles for Z(T). Eq. 4.1 can be verified, for example, by noting that qf in Table 7 for φ = π/6 and f = 0.25 is −0.135, while it is 0.134 for φ = 5π/6 and f = 0.75. Eq. 4.2 can be verified, for example, by noting that qf is −0.133 for φ = 11π/6 and f = 0.25. 

Table 8 verifies the linear relationship of qf to cos(φ). The anti-symmetry condition Eq. 4.1 and the symmetry condition Eq. 4.2 are also verified in the table.

5 Conclusions

The most exciting new result is the discovery of a hitherto unknown relation for the value distribution of the Riemann zeta function at discrete points. Eq. 3.3 states the relations in terms of three functions independent of the angle φ characterizing the Generalized Gram points. Table 1 gives the fitted values of the universal functions. Fig 1 and Fig. 2 show the comparison of the predictions from the new relation to the actual probability densities, for some values of the argument z. The figures, and Table 2 show the excellent agreement of the actual probability densities with the prediction. The evidence for the anti-symmetry and symmetry properties in terms of quantiles is illuminating. It is noteworthy that the quantile distributions change very little when the height T spans 16 orders of magnitude. Another intriguing observation is the linear dependence of the quantiles on cos(φ). The linear dependence and the symmetry properties imply that the quartile range q1−f − qf is independent of φ.

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