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

We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function ζ(s, a) for s,a∈ℂ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s, a \in \mathbb {C}$\end{document}, 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.