Series representations for the Stieltjes constants

TL;DR

This paper introduces new series representations for the Stieltjes constants, including an addition formula and rapidly converging series, enhancing both theoretical understanding and computational methods in analytic number theory.

Contribution

It provides novel series formulas for the Stieltjes constants, including an addition formula and exponentially fast converging series, along with derivative expressions and extensions related to Dirichlet L functions.

Findings

Addition formula for Stieltjes constants

Exponentially fast converging series for rac{1}

Expressions for derivatives of Stieltjes coefficients

Abstract

The Stieltjes constants \gamma_k(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function \zeta(s,a) about s=1. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of all orders of the Stieltjes coefficients are given. Many other results are obtained, including instances of an exponentially fast converging series representation for \gamma_k=\gamma_k(1). Some extensions are briefly described, as well as the relevance to expansions of Dirichlet L functions.