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

TL;DR

This paper introduces a rapidly converging series representation for the Riemann zeta function at integer arguments using roots of unity and partial fraction techniques, offering new formulas involving gamma functions.

Contribution

The paper presents a novel series representation for zeta at integers involving roots of unity, derived through infinite partial fraction decomposition, and establishes related gamma function formulas.

Findings

Series converges rapidly for integer arguments

Provides new formulas involving gamma functions

Offers a novel derivation technique using partial fractions

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.