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 and an explanation of . 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.
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. 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: 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/. Other interesting references: www.cerfacs.fr/algor/reports/2010/TR_PA_10_49.ps.gz, http://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 |