On A Rapidly Converging Series For The Riemann Zeta Function

TL;DR

This paper introduces a new rapidly converging series for the Riemann zeta function, which converges faster than existing series and is useful for computational evaluations and theoretical investigations related to the Riemann hypothesis.

Contribution

A novel proof of a rapidly converging series for the Riemann zeta function, applicable across the entire complex plane, with detailed convergence analysis and polynomial connections.

Findings

Series converges faster than comparable representations

Evaluation of coefficients is straightforward

Series reduction relates to properties of polynomials

Abstract

To evaluate Riemann's zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special case, a new proof of a rapidly converging series for the Riemann zeta function. The series converges in the entire complex plane, its rate of convergence being significantly faster than comparable representations, and so is a useful basis for evaluation algorithms. The evaluation of corresponding coefficients is not problematic, and precise convergence rates are elaborated in detail. The globally converging series obtained allow to reduce Riemann's hypothesis to similar properties on polynomials. And interestingly, Laguerre's polynomials form a kind of leitmotif through all sections.