Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

Abstract

We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function $\zeta (s,a)$ for $s,a\in \mathbb{C}$, along with an arbitrary number of derivatives with respect to $s$, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.

15 pages, 2 figures

Cited by in corpus (2)

• Analysis of a Complex approximation to the Li-Keiper coefficients around the K Function
• Rapid computation of special values of Dirichlet $L$-functions