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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.'
author:
- |
    Mark W. Coffey\
    Department of Physics\
    Colorado School of Mines\
    Golden, CO 80401\
    (Received $\mbox{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2009}$)
date: 'February 24, 2009'
title: 'Series of zeta values, the Stieltjes constants, and a sum $S_\gamma(n)$'
---

=25 pt

**Key words and phrases**

Hurwitz zeta function, Riemann zeta function, digamma function, polygamma function, series representation, integral representation, Stieltjes constants, polylogarithms, Laguerre polynomials, Dirichlet $L$ function

**2000 AMS subject classification**

11M06, 11M35, 33B15

**Introduction**

The Hurwitz zeta function, defined by $\zeta(s,a)=\sum_{n=0}^\infty (n+a)^{-s}$ for Re $s>1$ and Re $a>0$, extends to a meromorphic function in the entire complex $s$-plane. This analytic continuation to C has a simple pole of residue one. This is reflected in the Laurent expansion $$\zeta(s,a)={1 \over {s-1}}+ \sum_{n=0}^\infty {{(-1)^n} \over {n!}}\gamma_n(a)
(s-1)^n, \eqno(1)$$ wherein $\gamma_k(a)$ are designated 