Strategies To Evaluate The Riemann Zeta Function

TL;DR

This paper explores new series and integral representations of the Riemann Zeta function, aiming for rapid convergence and error bounds to improve understanding and computation of the function.

Contribution

It generalizes existing identities and introduces novel series and integrals for the zeta function, extending classical methods on the complex plane.

Findings

New converging series for the zeta function

Generalized identities involving zeta

Representations with rapid convergence

Abstract

This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta function. The results originate from attempts to extend the zeta function by classical means on the complex plane. This is particularly of interest for representations which converge rapidly in a given area of the complex plane, or for the purpose to make error bounds available.