RiemannZetaValueDistribution

Relations for Riemann Zeta value distribution on critical axis

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

https://sites.google.com/site/riemannzetazeros/ 

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 con- nection, we investigated the value distribution of the Riemann zeta func- tion 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 in- dependent 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.
Random Matrix Theory explanation for Riemann Zeta Value Distribution Symmetry 

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

The argument in Eq. 2.3 is defined by continuous variation of t starting with the value 0 at t = 0. Z(t) is real valued for real t, and 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 present here the functional equation for the ζ(s):

ζ(s) = ∆(s)ζ(1 − s).                                                                               (2.4)

Because of the remarkable results from Steuding and Suriajaya [23] and Kalpokas [24], this seems to be a promising avenue to explore further. For an arbitrary complex number a != 0, Steuding and Suriajaya consider the distribution of values of the Riemann zeta-function ζ at the a-points of the function ∆(s) which appears in the functional equation Eq. 2.4. These a-points δa are clustered around the critical line 1/2 + iR. This line is a Julia line for the essential singularity of ζ at infinity. They observe a remarkable average behavior for the sequence of values ζ(δa).

Gram points [25] play an important role in the the theory because many of the zeros are separated by them. When t ≥ 7, the θ function Eq.(2.3) is monotonic increasing. For n ≥ −1, the n-th Gram point gn is defined as the unique solution > 7 to θ(gn) = nπ. A Gram interval is the interval Gn = [gn,gn+1). In analogy with Gram points, we can associate an angle φ with a point t on the critical axis as follows:

Definition 2.1. For t ≥ 7, t is said to be a generalized Gram point with value φ if θ(t) = 2kπ + φ, where 0 ≤ φ < 2π.

Korolev [26,27] gives a review of some properties of Gram points.

3 Universality and symmetry relations

In this section we present the probability distribution function for the Riemann zeta function values at Generalized Gram points. We present a universality relation satisfied by these distributions.

The sample space for our study is the interval along the critical axis specified by (T1,T2). While empirical studies necessarily use large but finite T1,T2, we are interested in the limit T1 →∞,T2 →∞, T2−T1 →∞,  however

T2 − T1 ≪ T1.                                                               (3.1)

Because of Equation 3.1, we can consider ln(t) to be effectively constant over the interval. The latter condition is not essential but is convenient, in that it simplifies the numerical work. The notation ln(t) stands for the natural logarithm of t. We study the probability distribution function for Z(t) at generalized Gram points, pφ(y):

Definition 3.1.

pφ(z) = Limb->a Count(a,b)/(b-a),                                (3.2)

Count(a,b): = count of Z values with a < Z(t) < b in the sample space. pφ(z) is the probability that a < Z(t) < b when we consider the values of Z(t) for a large number of generalized Gram points in the sample space.

The probability density pφ(z) depends on the sample space (i.e., on the height t and on the size of the sample space). In practice the densities are not sensitive to the choice of the sample space as long as the height t is large enough and the length of the interval from which the sample is collected is large enough (but not too large on log scale). The study of Kalpokas and Steuding [24] implies that the standard deviation of pφ(z) is independent of φ and is proportional to T/2π . The probability distribution will have a finite standard deviation if the argument is normalized by sqrt(T/2π). 􏰆 The new relation for the value distributions is

pφ(z) = A(z) cos(φ) + B(z) cos(2φ) + C(z),                (3.3)

where A(z), B(z) and C(z) are universal functions, i.e., they do not depend on φ. The anti-symmetry relation is

pφ(z) = pφ+π(−z).                                       (3.4) 

The symmetry relation is

pφ(z) = p2π−φ(z).                                    (3.5)

We estimated A(z), B(z) and C(z) by fitting Eq. 3.3 to the actual probability densities for several values of z and φ. 

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.

References

• • [1]  O. Shanker, “Random Matrices, Generalised Zeta Functions and Self- Similarity of Zero Distributions” J. Phys. A: Math. Gen., A 39, 13983- 13997, (2006).

  • [2]  E. Wigner, “Random Matrices in Physics,” Siam Review, 9, 1-23, (1967).

  • [3]  M. Gaudin, M. Mehta, “On the Density of Eigenvalues of a Random Matrix,” Nucl. Phys., 18, 420-427, (1960).

  • [4]  M. Gaudin, “Sur la loi Limite de L’espacement de Valuers Propres D’une Matrics Aleatiore,” Nucl. Phys., 25, 447-458, (1961).

  • [5]  F. Dyson, “Statistical Theory of Energy Levels III,” J. Math. Phys., 3, 166-175, (1962).

  • [6]  E.B. Bogomolny and J.P. Keating, Random matrix theory and the Riemann zeros I: three- and four-point correlations, Nonlinearity, 8:1115–1131, 1995.

  • [7]  E.B. Bogomolny and J.P. Keating, Random matrix theory and the Riemann zeros II:n-point correlations, Nonlinearity, 9:911–935, 1996.

  • [8]  Nicholas M. Katz, Peter Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy, AMS, Providence, Rhode Island, 1999.

  • [9]  J.P. Keating and N.C. Snaith, Random matrix theory and ζ(1/2 + it), Commun. Math. Phys., 214:57–89, 2000.

  • [10]  J.P. Keating and N.C. Snaith, Random matrix theory and L-functions at s = 1/2, Commun. Math. Phys, 214:91–110, 2000.

  • [11]  J.B. Conrey and D.W. Farmer, Mean values of L-functions and symmetry, Int. Math. Res. Notices, 17:883–908, 2000, arXiv:math.nt/9912107.

  • [12]  C.P. Hughes, J.P. Keating, and N. O’Connell, Random matrix theory and the derivative of the Riemann zeta function, Proc. R. Soc. Lond. A, 456:2611–2627, 2000.

  • [13]  C.P. Hughes, J.P. Keating, and N. O’Connell, On the characteristic polyno- mial of a random unitary matrix, Commun. Math. Phys., 220(2):429–451, 2001.

  • [14]  J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein, and N.C. Snaith, Integral moments of zeta- and L-functions, preprint, 2002, arXiv:math.nt/0206018.

  • [15]  J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein, and N.C. Snaith, Autocorrelation of Random Matrix Polynomials Commun. Math. Phys , 237:365-395, 2003

  • [16]  M. V. Berry, “Semiclassical theory of spectral rigidity,” Proc. R. Soc., A 400 , 229-251, (1985).

  • [17]  M. V. Berry, “Riemann’s zeta function: a model for quantum chaos?,” Quantum chaos and statistical nuclear physics (Springer Lecture Notes in Physics), 263, 1-17, (1986).

  • [18]  M. V. Berry, “Quantum Chaology,” Proc. R. Soc. , A 413 , 183-198, (1987).

  • [19]  M. V. Berry, ‘Number variance of the Riemann zeros‘,” NonLinearity , 1, 399-407 , (1988).

  • [20]  Daniel Schumayer and David A.W. Hutchinson, ”Physics of the Riemann Hypothesis” Rev.Mod.Phys. 83, 307-330, (2011)

  • [21]  O. Shanker, “Good to Bad Gram Point Ratio For Riemann Zeta Function”,

    • Experimental Mathematics doi:10.1080/10586458.2018.1492474(2018)

  • [22]  O. Shanker, “Symmetry properties of distribution of Riemann Zeta Func- tion values on critical axis” Advanced Modeling and Optimization, 20, 435- 445, (2018).

  • [23]  J. Steuding and A. I. Suriajaya, “Value-distribution of the Riemann zeta- function Along Its Julia Lines”, Comput. Methods Funct. Theory (2020), https://doi.org/10.1007/s40315-020-00316-x

  • [24]  J. Kalpokas and J. Steuding, “On the value-distribution of the Riemann zeta-function on the critical line”, Moscow Journal of Combinatorics and Number Theory 1(2011), 26-42

  • [25]  J. P. Gram, “Sur les Zeros de la Fonction ζ(s) de Riemann”, Acta Math. 27(1903), 289-304

  • [26]  M. A. Korolev, “Gram’s Law In The Theory Of The Riemann Zeta- Function. Part 1”, Proceedings Of The Steklov Institute Of Mathematics 292(2016)S1-S146

  • [27]  M. A. Korolev, “Gram’s Law In The Theory Of The Riemann Zeta- Function. Part 2”, Proceedings Of The Steklov Institute Of Mathematics 294(2016)S1-S78

  • [28]  Catalin Hanga and Christopher Hughes, ”PROBABILISTIC MODELS FOR GRAM’S LAW”, https://arxiv.org/pdf/1911.03190.pdf, Ph. D. thesis https://etheses.whiterose.ac.uk/27858/ 

  • [29]  Aleksandar Ivi ́c, “The Theory of Hardy’s Z-Function,” Cambridge Univer- sity Press, (2013)

  • [30]  G. H. Hardy and J. E. Littlewood, “Contributions to the theory of the Rie- mann zeta-function and the distribution of primes”, Acta Math. 41(1918), 119-196

  • [31]  D. A. Hejhal, “On a result of Selberg concerning zeros of linear combina- tions of L-functions”, Int. Math. Res. Not. 11(2000), 551-577

  • [32]  A. Selberg, “Selected Papers, Vol. I,” Springer Verlag, (1989)

  • [33]  A. Selberg, “Selected Papers, Vol. II,” Springer Verlag, (1991)

  • [34]  A. Laurincikas, “Limit Theorems for the Riemann Zeta-Function”, Kluwer, Dordrecht, 1996

  • [35]  E. C. Titchmarsh, “On van der Corput’s method and the zeta-function of Riemann (IV)”, Quart. J. Math. Oxford Ser. 5(1934), 98-105

  • [36]  Stephen J. Lester, “The Distribution of Values of the Riemann Zeta- Function,” Ph. D. dissertation, University of Rochester, (2013)

  • [37]  A. Odlyzko, “The 1020-th Zero of the Riemann Zeta Function and 175 Million of its Neighbors”, report, http://www.dtc.umn.edu/~odlyzko/ unpublished/zeta.10to20.1992.pdf, (1992)

  • [38]  G. A. Hiary, “Fast methods to compute the Riemann zeta function”, Annals of Mathematics 174(2011), 891-946

  • [39]  Xavier Gourdon, “The 1013 first zeros of the Riemann Zeta function, and zeros computation at very large height”, report, http://numbers.computation.free.fr/Constants/Miscellaneous/ zetazeros1e13-1e24.pdf, (2004)

  • [40]  A. J. Jerri, “The Shannon sampling theorem - its various extensions and applications”, Proc IEEE 65(1977), 1565-1596

  • [41]  G. A. Hiary, “An amortized-complexity method to compute the Riemann zeta function”, Mathematics of Computation 80(2011), 1785-1796

  • [42]  William Feller, “An Introduction to Probability Theory and Its Applica- tions, Volume II (2 ed.)”, John Wiley and Sons, New York, 1971