New rapidly convergent series representations for $\zeta (2n+1)$

Abstract:

We give three series representations for the values of the Riemann zeta function $\zeta (s)$ at positive odd integers. One representation extends Ewell’s result for $\zeta (3)$ [Amer. Math. Monthly 97 (1990), 219–220] and is considerably simpler than the two generalisations proposed earlier. The second representation is even simpler: \[ \zeta (2n+1)=(-1)^n\frac {4(2\pi )^{2n}}{(2n+1)!}\sum _{k=0}^\infty R_{2n+1,k}\zeta (2k),\qquad n\ge 1,\] where the coefficients $R_{2n+1,k}$ for a fixed $n$ are rational in $k$ and are explicitly given by the finite sum involving the Bernoulli numbers. The third representation is obtained from the second by the Kummer transformation. We demonstrate the rapid convergence of this series using several examples.