Series of zeta values, the Stieltjes constants, and a sum S_\gamma(n)

TL;DR

This paper develops new series and integral representations for the Stieltjes constants and related sums, which are crucial in understanding the properties of the Hurwitz zeta function and the Riemann hypothesis.

Contribution

It introduces novel series and integral formulas for the Stieltjes constants and the sum S_gamma(n), enhancing analytical tools for number theory and the Riemann hypothesis.

Findings

Derived new series representations for Stieltjes constants

Obtained integral representations of the sum S_gamma(n)

Provided insights into the application of S_gamma(n) in the Li criterion

Abstract

We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function zeta(s,a) about s=1. Additionally we obtain series and integral representations of a sum S_\gamma(n) formed as an alternating binomial series from the Stieltjes constants. The slowly varying sum S_\gamma(n)+n is an important subsum in application of the Li criterion for the Riemann hypothesis.