Integral and Series Representations of Riemann's Zeta function, Dirichelet's Eta Function and a Medley of Related Results

TL;DR

This paper presents contour integral representations for Riemann's Zeta and Dirichlet's Eta functions, deriving known and new series and integral formulas through classical methods not commonly documented.

Contribution

It introduces a unified approach using contour integrals from historical methods to derive both known and novel representations of the Zeta and Eta functions.

Findings

New contour integral representations for Zeta and Eta functions

Alternative derivations of classical series and integral formulas

Identification of previously unreported results

Abstract

Contour integral representations for Riemann's Zeta function and Dirichelet's Eta (alternating Zeta) function are presented and investigated. These representations flow naturally from methods developed in the 1800's, but somehow they do not appear in the standard reference summaries, textbooks or literature. Using these representations as a basis, alternate derivations of known series and integral representations for the Zeta and Eta function are obtained on a unified basis that differs from the textbook approach, and results are developed that appear to be new.