Rapidly converging formulae for $\zeta(4k\pm 1)$

TL;DR

This paper introduces rapidly converging series formulas for the Riemann zeta function at odd integers, significantly improving computational efficiency with explicit error bounds and examples.

Contribution

It presents new Lambert series-based formulas for (4kb1 1) that converge exponentially fast, enabling precise calculations of zeta values.

Findings

Convergence rate of about e^{-\u221a{15}} per term for (4k-1)

Convergence rate of about e^{-4} per term for (4k+1)

First order approximation of (3) with error  10^{-10}

Abstract

We provide rapidly converging formulae for the Riemann zeta function at odd integers using the Lambert series $\mathscr{L}_q(s) = \sum_{n=1}^\infty n^{s} q^{n}/(1-q^n)$, $s=-(4k\pm 1)$. Our main formula for $\zeta(4k-1)$ converges at rate of about $e^{-\sqrt{15}\pi}$ per term, and the formula for $\zeta(4k+1)$, at the rate of $e^{-4\pi}$ per term. For example, the first order approximation yields $\zeta(3)\approx\frac{\pi ^3 \sqrt{15}}{100} +e^{-\sqrt{15} \pi }\left[\frac{9}{4}+\frac{4}{\sqrt{15}}\sinh (\frac{\sqrt{15} \pi }{2})\right]$ which has an error only of order $10^{-10}$.