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

2017/01/19 by Mark W. Coffey, Coffey, Mark W. · 1 citation
Mathematics · #Analytic Number Theory Research #Advanced Topics in Algebra #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1701.07064

Abstract

The Stieltjes constants γ_k(a) appear in the regular part of the Laurent expansion for the Hurwitz zeta function ζ(s,a). We present summatory results for these constants γ_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 ζ(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 ζ(1/2), ζ(s)=ζ(s,1) being the Riemann zeta function.

Citations

• Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links

Cited by

• Some integrals and series involving the Stieltjes constants

Related

• Series of zeta values, the Stieltjes constants, and a sum S_γ(n)
• Series representations for the Stieltjes constants
• Functional equations for the Stieltjes constants
• On representations and differences of Stieltjes coefficients, and other relations
• Double series expression for the Stieltjes constants