Summatory relations and prime products for the Stieltjes constants, and other related results

Abstract

The Stieltjes constants $\gamma_k(a)$ appear in the regular part of the Laurent expansion for the Hurwitz zeta function $\zeta(s,a)$. We present summatory results for these constants $\gamma_k(a)$ in terms of fundamental mathematical constants such as the Catalan constant, and further relate them to products of rational functions of prime numbers. We provide examples of infinite series of differences of Stieltjes constants evaluating as volumes in hyperbolic $3$-space. We present a new series representation for the difference of the first Stieltjes constant at rational arguments. We obtain expressions for $\zeta(1/2)L_{-p}(1/2)$, where for primes $p>7$, $L_{-p}(s)$ are certain $L$-series, and remarkably tight bounds for the value $\zeta(1/2)$, $\zeta(s)=\zeta(s,1)$ being the Riemann zeta function.