High Precision Computation of Riemann's Zeta Function by the Riemann-Siegel Formula, II

TL;DR

This paper presents a method for high-precision computation of the Riemann zeta function using the Riemann-Siegel formula, including error analysis and implementation in Python's mpmath library.

Contribution

It introduces a detailed approach for computing the zeta function at any complex point with controlled error, and integrates the method into open-source computational tools.

Findings

Achieved high-precision calculations of ζ(s) and Z(t)

Developed error analysis for floating point and inherent errors

Implemented the method in the mpmath Python library

Abstract

(This is only a first preliminary version, any suggestions about it will be welcome.) In this paper it is shown how to compute Riemann's zeta function $\zeta(s)$ (and Riemann-Siegel $Z(t)$) at any point $s\in\mathbf C$ with a prescribed error $\varepsilon$ applying the, Riemann-Siegel formula as described in my paper "High Precision ... I", Math of Comp. 80 (2011) 995--1009. This includes the study of how many terms to compute and to what precision to get the desired result. All possible errors are considered, even those inherent to the use of floating point representation of the numbers. The result has been used to implement the computation. The programs have been included in"mpmath", a public library in Python for the computation of special functions. Hence they are included also in Sage.