The zeta function on the critical line: Numerical evidence for moments and random matrix theory models

TL;DR

This paper presents extensive numerical computations of the Riemann zeta function's moments on the critical line, comparing results with random matrix theory predictions and exploring variability, correlations, and asymptotic behaviors.

Contribution

It provides new large-scale numerical evidence for moments of the zeta function, analyzing variability and correlations, and evaluates computational methods for future research.

Findings

High variability of moments over large intervals

Power law modeling of extreme value decline

Detection of long-range correlations and oscillations

Abstract

Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those and competing predictions. It is shown that for high moments and at large heights, the variability of moment values over adjacent intervals is substantial, even when those intervals are long, as long as a block containing 10^9 zeros near zero number 10^23. More than anything else, the variability illustrates the limits of what one can learn about the zeta function from numerical evidence. It is shown the rate of decline of extreme values of the moments is modelled relatively well by power laws. Also, some long range correlations in the values of the second moment, as well as asymptotic oscillations in the values of the shifted fourth moment, are found. The computations described here relied on several representations of the zeta function. The numerical comparison of their effectiveness that is presented is of independent interest, for future large scale computations.