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 ${\mathcal{L}}_q(s)={\sum }_{n=1}^{\infty }{n}^s{q}^n/(1-q^n)$, $s=-(4k\pm 1)$. 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 $10^{-10}$.

16 pages. Submitted for publication