Functional equations for the Stieltjes constants

2014/02/16 by Mark W. Coffey, Coffey, Mark W.
Mathematics · #11M06 #11M35 #11Y60 #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT) #Secondary: 05A10 #math.CV #math.NT #msc:05A10 #msc:11M06 #msc:11M35 #msc:11Y60

paper · pdf · doi:10.48550/arxiv.1402.3746

30 pages, no figures

arxiv created 2014/05/17 · arxiv updated 2014/05/20

Abstract

The Stieltjes constants γ_k(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about s=1. We present the evaluation of γ₁(a) and γ₂(a) at rational argument, being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for γ₀(a), γ₁(a), and γ₂(a), and point out that these formulas are cases of an addition formula previously presented. We present certain integral evaluations generalizing Gauss' formula for the digamma function at rational argument. In addition, we give the asymptotic form of γ_k(a) as a → 0 as well as a novel technique for evaluating integrals with integrands with ln(-ln x) and rational factors.

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