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

TL;DR

This paper explores summatory relations of the Stieltjes constants, connecting them to fundamental constants, prime products, hyperbolic volumes, and providing new series representations and bounds for the Riemann zeta function.

Contribution

It introduces new summatory formulas for Stieltjes constants, relates them to prime products and hyperbolic volumes, and derives novel series and bounds for the Riemann zeta function.

Findings

Summatory relations involving Catalan constant and prime products.

Infinite series of differences of Stieltjes constants evaluate as hyperbolic volumes.

New series representation and bounds for the Riemann zeta function.

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.