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

Authors: Ghaith A. Hiary and Andrew M. Odlyzko
Journal: Math. Comp. 81 (2012), 1723-1752
MSC (2010): Primary 11M06, 11Y35, 11M50, 15B52
Published electronically: December 19, 2011
MathSciNet review: 2904600
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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 that the rate of decline of extreme values of the moments is modeled 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.

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Additional Information

Ghaith A. Hiary
Affiliation: Pure Mathematics, University of Waterloo, 200 University Ave West, Waterloo, Ontario, Canada, N2L 3G1.

Andrew M. Odlyzko
Affiliation: Department of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, Minnesota 55455.

Keywords: Riemann zeta function, moments, Odlyzko-Schönhage algorithm
Received by editor(s): August 12, 2010
Received by editor(s) in revised form: May 18, 2011
Published electronically: December 19, 2011
Additional Notes: Preparation of this material was partially supported by the National Science Foundation under agreements No. DMS-0757627 (FRG grant) and DMS-0635607. Computations were carried out at the Minnesota Supercomputing Institute.
Article copyright: © Copyright 2011 American Mathematical Society

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