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.Comment: 31 pages, 10 figures, 19 table