The Zetafast algorithm for computing zeta functions

Abstract

We express the Riemann zeta function $ζ\left(s\right)$ of argument $s=σ+iτ$ with imaginary part $τ$ in terms of three absolutely convergent series. The resulting simple algorithm allows to compute, to arbitrary precision, $ζ\left(s\right)$ and its derivatives using at most $C\left(ε\right)\left|τ\right|^{\frac{1}{2}+ε}$ summands for any $ε>0$, with explicit error bounds. It can be regarded as a quantitative version of the approximate functional equation. The numerical implementation is straightforward. The approach works for any type of zeta function with a similar functional equation such as Dirichlet $L$-functions, or the Davenport-Heilbronn type zeta functions.