On representations and differences of Stieltjes coefficients, and other relations

TL;DR

This paper provides new exact formulas, series, and integral representations for Stieltjes coefficients, linking various topics in analysis and number theory with applications in theoretical and computational contexts.

Contribution

It introduces novel series, summatory, and integral representations for Stieltjes coefficients, enhancing understanding and computation in analytic number theory.

Findings

Derived new series representations for Stieltjes coefficients

Established integral relations linking coefficients at rational arguments

Connected Stieltjes coefficients to derivatives of zeta and L-functions

Abstract

The Stieltjes coefficients $\gamma_k(a)$ arise in the expansion of the Hurwitz zeta function $\zeta(s,a)$ about its single simple pole at $s=1$ and are of fundamental and long-standing importance in analytic number theory and other disciplines. We present an array of exact results for the Stieltjes coefficients, including series representations and summatory relations. Other integral representations provide the difference of Stieltjes coefficients at rational arguments. The presentation serves to link a variety of topics in analysis and special function and special number theory, including logarithmic series, integrals, and the derivatives of the Hurwitz zeta and Dirichlet $L$-functions at special points. The results have a wide range of application, both theoretical and computational.