On the Taylor Coefficients of the Hurwitz Zeta Function

TL;DR

This paper derives new representations for the Maclaurin coefficients of the Hurwitz and Riemann zeta functions using series involving Bernoulli polynomials and Stirling numbers, enhancing understanding of their structure.

Contribution

It introduces a novel series transformation formula to express the coefficients of Hurwitz and Lerch zeta functions in terms of semi-convergent series.

Findings

Representation of Hurwitz zeta coefficients via semi-convergent series

Representation of Riemann zeta coefficients using Bernoulli polynomials and Stirling numbers

Extension of results to Lerch zeta function coefficients

Abstract

We find a representation for the Maclaurin coefficients of the Hurwitz zeta-function in terms of semi-convergent series involving the Bernoulli polynomials and the Stirling numbers of the first kind. In particular, this gives a representation for the coefficients of the Riemann zeta function. Our main instrument is a certain series transformation formula. A similar result is proved also for the Maclaurin coefficients of the Lerch zeta function.