A Series Representation for Riemann's Zeta Function and some Interesting Identities that Follow

TL;DR

This paper introduces a new series representation for Riemann's zeta function using complex analysis, leading to novel identities, evaluations of complex integrals, and relationships among special functions and number sequences.

Contribution

It presents a new series representation for ta(s) and ta(s) based on Cauchy's Integral Theorem, enabling the derivation of new identities and integral evaluations.

Findings

Derived a new series for ta(s) and ta(s)

Established identities linking Euler, Bernoulli, and Harmonic numbers

Demonstrated methods to evaluate complex integrals using known functions

Abstract

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions $\zeta^{\pm}(s)$ and $\eta^{\pm}(s)$ are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints.