379 research outputs found
An alternative to Riemann-Siegel type formulas
Simple unsmoothed formulas to compute the Riemann zeta function, and
Dirichlet -functions to a power-full modulus, are derived by elementary
means (Taylor expansions and the geometric series). The formulas enable
square-root of the analytic conductor complexity, up to logarithmic loss, and
have an explicit remainder term that is easy to control. The formula for zeta
yields a convexity bound of the same strength as that from the Riemann-Siegel
formula, up to a constant factor. Practical parameter choices are discussed.Comment: 16 page
Numerical study of the derivative of the Riemann zeta function at zeros
The derivative of the Riemann zeta function was computed numerically on
several large sets of zeros at large heights. Comparisons to known and
conjectured asymptotics are presented.Comment: 13 pages, 5 figures; minor typos fixe
Uniform asymptotics for the full moment conjecture of the Riemann zeta function
Conrey, Farmer, Keating, Rubinstein, and Snaith, recently conjectured
formulas for the full asymptotics of the moments of -functions. In the case
of the Riemann zeta function, their conjecture states that the -th absolute
moment of zeta on the critical line is asymptotically given by a certain
-fold residue integral. This residue integral can be expressed as a
polynomial of degree , whose coefficients are given in exact form by
elaborate and complicated formulas. In this article, uniform asymptotics for
roughly the first coefficients of the moment polynomial are derived.
Numerical data to support our asymptotic formula are presented. An application
to bounding the maximal size of the zeta function is considered.Comment: 53 pages, 1 figure, 2 table
The zeta function on the critical line: Numerical evidence for moments and random matrix theory models
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
- …