The zetafunction on the critical line: numerical evidencefor moments and random matrix theory models

Ghaith Hiary, Andrew M Odlyzko

Research output: Contribution to journalArticle (Academic Journal)peer-review

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 zeros near zero number . 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.
Original languageEnglish
Pages (from-to)1723-1752
Number of pages30
JournalMathematics of Computation
Volume81
Issue number279
DOIs
Publication statusPublished - 2012

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