Some Rapidly Converging Series for $\zeta (2n+1)$ from Abstract Operators

Abstract

The author derives new family of series representations for the values of the Riemann Zeta function $\zeta (s)$ at positive odd integers. For $n\in \mathbb{N}$, each of these series representing $\zeta (2n+1)$ converges remarkably rapidly with its general term having the order estimate: $$O(m^{-2k}\cdot k^{-2n+1})(k\to \infty ;m=3,4,6).$$ The method is based on the mapping relationships between analytic functions and periodic functions using the abstract operators $\cos (h{\partial }_x)$ and $\sin (h{\partial }_x)$, including the mapping relationships between power series and trigonometric series, if each coefficient of a power series is respectively equal to that of a trigonometric series. Thus we obtain a general method to find the sum of the Dirichlet series of integer variables. By defining the Zeta function in an abstract operators form, we have further generalized these results on the whole complex plane.

19 pages. arXiv admin note: text overlap with arXiv:1008.5046