TY - JOUR

T1 - The zeta function on the critical line

T2 - Numerical evidence for moments and random matrix theory models

AU - Hiary, Ghaith A.

AU - Odlyzko, Andrew M.

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2012

Y1 - 2012

N2 - 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 109 zeros near zero number 1023. 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.

AB - 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 109 zeros near zero number 1023. 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.

KW - Moments

KW - Odlyzko-Schönhage algorithm

KW - Riemann zeta function

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U2 - 10.1090/S0025-5718-2011-02573-1

DO - 10.1090/S0025-5718-2011-02573-1

M3 - Article

AN - SCOPUS:84866760245

VL - 81

SP - 1723

EP - 1752

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 279

ER -