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

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 $10^{-10}$.