Mathematics of Computation

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ISSN 1088-6842(e) ISSN 0025-5718(p)

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
Posted: December 19, 2011
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Abstract | References | Similar Articles | Additional Information

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 $10^{23}$ . 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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