Rapidly Converging Series for ζ(2n+1) from Fourier Series

Abstract

Ever since Euler first evaluated $\zeta (2)$ and $\zeta (2m)$, numerous interesting solutions of the problem of evaluating the $\zeta (2m)(m\in \mathbb{N})$ have appeared in the mathematical literature. Until now no simple formula analogous to the evaluation of $\zeta (2m)$ $(m\in \mathbb{N})$ is known for $\zeta (2m+1)(m\in \mathbb{N})$ or even for any special case such as $\zeta (3)$. Instead, various rapidly converging series for $\zeta (2m+1)$ have been developed by many authors. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for $\zeta (2m+1)$. In addition, using Fourier series and recalling some indefinite integral formulas, we also give recurrence formulas for evaluations of $\beta (2m+1)$ and $\zeta (2m)(m\in \mathbb{N})$, which have been treated in earlier works.