Distribution of zeroes of Riemann zeta function and Dirichlet L-functions

(O. Shanker)

One hundred and fifty years ago, in 1859, Riemann came up with a statement about the location of the zeros of the Riemann zeta function, a statement which to this day mystifies and attracts the best mathematical brains in the world. Riemann stated that the zeros of the Riemann zeta function all have their real part equal to one-half (apart from zeros on the negative real axis, which are called trivial zeros since it is easy to find their locations). We are still asking the question: "What stern taskmaster is causing the zeros to line up so perfectly, like well-behaved school kids?" On this page we describe a different property of the statistics of zeros of the Riemann zeta function and other L functions, the cause for which is not understood. The statistics of the Riemann zeta zeros are a topic of interest to mathematicians because of their connection to big problems like the Riemann Hypothesis, distribution of prime numbers, etc. Through connections with random matrix theory and quantum chaos, the appeal is even broader. This page gives a brief description and links to my work on the Riemann zeta function and Dirichlet L-functions (see also Generalised Zeta Functions and Self-Similarity of Zero Distributions), and the fractal nature of the distribution of their zeroes.

The zeta function values at Gram Points have some remarkable properties. Typical data for the value of the zeta function at Gram points is available at Riemann zeta function values at Gram points 10^12 - 1 to 10^12 + 10054. (For that range, the mean value for zeta is -1.96 for the odd gram points, and 2.05 for the even gram points. The asymptotic values are -2 and +2). A report on the application of Machine Learning to the study of the zeros is available at Advanced Modeling and Optimization, Volume 14, Number 3,  pp 717-728, 2012 (see also Neural Network Prediction of Riemann Zeta Zeros). We also present the Entropy of the Riemann zeta zero sequence and an explanation of Lehmer's Phenomenon.

Fractal Structure

On a quiet Christmas day in 2005, I was watching the beautiful Southern California beaches from across the Laguna Niguel dunes, and trying to come up with some interesting stuff to investigate. Probably inspired by the idyllic setting, I decided to study the fractal structure of the Riemann zeta zeros using Rescaled Range Analysis. The results were interesting, to put it mildly! The self-similarity of the zero distributions is quite remarkable, and is characterized by a large fractal dimension of 1.9 (equivalently, a Hurst Exponent of 0.1). The differences of the zeros are shown in the figure below. Not only is the fractal dimension unusually high, it is also surprisingly constant, even when calculated over fifteen orders of magnitude for the Riemann function.

Fractal dimension for L-functions of order 1 and 2

The very striking behaviour for the zeros of the Riemann zeta function is also shared by other L functions. The table shows the calculation for the L-functions. In the table, r indicates an index to which of the group character representations is being considered for the L-function.

Order of
Hurst Fractal
largest zero
exponent Dimension
Riemann Zeta
0.091 1.909
0.093 1.907
0.094 1.906
0.100 1.900
Degree 1 L-function, Conductor 3
0.092 1.908
Degree 1 L-function, Conductor 4
0.092 1.908
Degree 1 L-function, Conductor 9

Dirichlet Character

10000000 r=2 conjugate pair, negative roots 0.094 1.906
10000000 r=2 conjugate pair, positive roots 0.097 1.903
10000000 r=3 conjugate pair, negative roots 0.114 1.886
10000000 r=3 conjugate pair, positive roots 0.084 1.916
Degree 1 L-function, Conductor 19

Dirichlet Character

1000000 r=2 conjugate pair, negative roots 0.105 1.895
1000000 r=2 conjugate pair, positive roots 0.116 1.884
1000000 r=3 conjugate pair, negative roots 0.123 1.877
1000000 r=3 conjugate pair, positive roots 0.103 1.897
1000000 r=4 conjugate pair, negative roots 0.096 1.904
1000000 r=4 conjugate pair, positive roots 0.112 1.888
1000000 r=5 conjugate pair, negative roots 0.094 1.906
1000000 r=5 conjugate pair, positive roots 0.105 1.895
1000000 r=6 conjugate pair, negative roots 0.125 1.875
1000000 r=6 conjugate pair, positive roots 0.099 1.901
1000000 r=7 conjugate pair, negative roots 0.100 1.900
1000000 r=7 conjugate pair, positive roots 0.106 1.894
1000000 r=8 conjugate pair, negative roots 0.116 1.884
1000000 r=8 conjugate pair, positive roots 0.104 1.896
1000000 r=9 conjugate pair, negative roots 0.098 1.902
1000000 r=9 conjugate pair, positive roots 0.121 1.879
1000000 r=10 real representation, positive roots 0.089 1.911
Degree 2 Elliptic curve L-function, Conductor 19 isogeny class A
0.091 1.909
Degree 2 L-function, Ramanujan tau (associated cusp form of weight 12, level 1)
0.108 1.892
As far as I have been able to make out, this property of a low Hurst exponent comes because the zeros vary with height N almost linearly (slope varying logarithmically), with a
superposed random term with a normal distribution of mean zero and variance apparently varying as log(log(N)).

Comparision with theory

We compared the behaviour of the Riemann zeros with that of the Random Matrix Theories, which explain many properties of the Riemann zeroes. For the Hurst exponents the Random Matrix results seem to differ from the Riemann zero results. However, this statement has to be treated with caution, since the sample sizes considered for the two systems differ significantly. The low Hurst exponent seems to be connected with the relation between the Riemann zeroes and the prime numbers, as explained in my paper. The role of the primes in statistics of the zeta zeros is closely related to the behaviour of quantum chaotic systems. Berry has several introductory articles on quantum chaos, including applications to the Riemann zeta function.
It is well-known that there are certain statistics connected with the Riemann zeta function that are "universal" if one studies the zeta function, or its zeros, high on the critical line. That is, in the limit of large height up the critical line, these "local" statistics follow predictions from random matrix theory. Then there are other statistics, such as the distribution of values of the zeta function itself, that depend on zeros over much larger ranges, and therefore take into account longer-range correlations between the zeros. These do not show universal behaviour. In fact, this type of statistic has crucial dependence on the prime numbers, exactly as we find for the Hurst exponent. In many cases very precise expressions can be written down showing explicitly the contributions from the primes, so in fact we know rather a lot about the role primes play in zero statistics.

Distribution of Primes

Since the Riemann zeta zeros are related to the distribution of prime numbers, we study the distribution. The distribution for the differences of the prime numbers is shown in the figure below for the fiftieth million set of primes. The horizontal axis shows the difference between consecutive primes, and the vertical axis shows the count for the number of times the difference occurs in the fiftieth million set of primes. The structure in the histogram is interesting, e.g., the peaks when the differences are multiples of 6. When a prime number is divided by 6, the remainder is either 1 or 5. An analysis of the peaks gives information on the correlation between the probability of the remainder being 1 or 5 and the remainder for the previous prime number. From the histogram, we get the following probabilities:
Prob(diff=6k+2) = Prob(diff=6k+4) = .276;
Prob(diff=6k) = 0.448;
If there were no correlations between neighbouring primes, then the values would be closer to
Prob(diff=6k+2) = Prob(diff=6k+4) = .25;
Prob(diff=6k) = 0.50;
Are we seeing signs of a correlation between neighbouring primes here? If so, it would be very exciting. A closer look at the prime number statistics seems warranted. See Distribution of Primes and Rescaled Range Analysis of L-function zeros and Prime Number distribution.

Links to relevant sites

I have used the zeroes from http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html and http://pmmac03.math.uwaterloo.ca/~mrubinst/L_function_public/ZEROS/.
For the serious investigator into number theory, the Number Theory Web gives a good number of links. You may also wish to visit the interesting sites of Watkins on fractality in number theory and Number Theory and Physics. The sites provide many links to work related to what I have done, and I encourage you to look into the papers.
Here are a couple of sites which explain the Hurst Exponent and Rescaled Range analysis. Zeta functions also arise in a variety of other contexts. For example, one can define graph zeta functions which may be applied to study the dimension of a complex network (large network of nodes connected by edges). 

Other interesting references: www.cerfacs.fr/algor/reports/2010/TR_PA_10_49.ps.gzhttp://www.blueberry-brain.org/winterchaos/Sabelli%20Rieman%20paper%20final.htm

https://people.math.osu.edu/hiary.1/amortized.html gives sample data for Z(t) using 10^7 zeros near each height T for T ~ 1e12 to 1e28.

Here are links to: my home page, and O. Shanker publication list.

O. Shanker Email: oshanker.AT.gmail.com

Subpages (1): Programs
O Shanker,
Apr 22, 2009, 6:13 PM