Computing Dirichlet character sums to a power-full modulus

The Postnikov character formula is used to express large portions of a Dirichlet character sum in terms of quadratic exponential sums. The quadratic sums are then computed using an analytic algorithm previously derived by the author. This leads to a power-saving if the modulus is smooth enough. As an application, a fast, and potentially practical, method to compute Dirichlet L-functions with complexity exponent 1/3 for smooth enough moduli is derived.Comment: 23 pages, minor adjustment

An explicit hybrid estimate for L(1/2+it,χ)L(1/2+it,\chi)

An explicit hybrid estimate for $L(1/2+it,\chi)$ is derived, where $\chi$ is a Dirichlet character modulo $q$. The estimate applies when $t$ is bounded away from zero, and is most effective when $q$ is powerfull, yielding an explicit Weyl bound in this case. The estimate takes a particularly simple form if $q$ is a sixth power. Several hybrid lemmas of van der Corput--Weyl type are presented.Comment: 25 page

An alternative to Riemann-Siegel type formulas

Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $L$-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

Asymptotics and formulas for cubic exponential sums

Several asymptotic expansions and formulas for cubic exponential sums are derived. The expansions are most useful when the cubic coefficient is in a restricted range. This generalizes previous results in the quadratic case and helps to clarify how to numerically approximate cubic exponential sums and how to obtain upper bounds for them in some cases.Comment: 18 page

New computations of the Riemann zeta function on the critical line

We present highlights of computations of the Riemann zeta function around large values and high zeros. The main new ingredient in these computations is an implementation of the second author's fast algorithm for numerically evaluating quadratic exponential sums. In addition, we use a new simple multi-evaluation method to compute the zeta function in a very small range at little more than the cost of evaluation at a single point.Comment: 26 page

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