Numerical calculation of the Riemann zeta function at odd integer arguments: A direct formula method

TL;DR

This paper presents a recurrence formula and an algorithm for efficiently computing the Riemann zeta function at odd integers using values at even integers, achieving high accuracy especially for large arguments.

Contribution

A new recurrence formula and algorithm for calculating the Riemann zeta function at odd integers from even integer values, with proven error bounds.

Findings

Error bound approximately $O(10^{-n})$

High accuracy for large arguments

Achieves over ten decimal places for small arguments

Abstract

In this article, we introduce a recurrence formula which only involves two adjacent values of the Riemann zeta function at integer arguments. Based on the formula, an algorithm to evaluate $\zeta$-values(i.e. the values of Riemann zeta function) at odd-integers from the two nearest $\zeta$-values at even-integers is posed and proved. The behavior of the error bound is $O(10^{-n})$ approximately where $n$ is the argument. Our method is especially powerful for the calculation of Riemann zeta function at large argument, while for smaller ones it can also reach spectacular accuracies such as more than ten decimal places.