A New Series Representation Involving Root Of Unity For The Values Of Riemann Zeta Function At Integer Arguments

Abstract

In this paper we provide a new series representation for the values of Riemann zeta function at integer arguments, namely: $ \zeta(m)=\sum_{n=1}^{\infty}\frac{m(-1)^{n-1}\Gamma(1-\omega_{m}n)...\Gamma(1-\omega_{m}^{m-1}n)}{n!n^m}$, where $n$ is an integer that lager than $1$ and $\omega$ is the $m$-th root of unity. This series converges quite fast. It's derived by some technique of infinite partial fraction decomposition. With this technique we also establish other useful formulas related to gamma function.