Functional equations for the Stieltjes constants

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 γ 1 (a) and γ 2 (a) at rational arguments, this being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for γ 0 (a), γ 1 (a), and γ 2 (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.

A Bound for Stieltjes Constants

“…where, for non-negative integers n, the coefficients γ n (a) are known as the Stieltjes constants ( [20]), which were generalized to fractional values α ∈ R + by Kreminski [12] in 2003. These constants have several interesting and unexpected applications in the zeta function theory, as was shown recently in [4], [3], [8], and [9]. Moreover, the classical Euler-Maclaurin Summation can be used to prove (see [10]) that, if we set C α (a) = γ α (a) − log α (a)…”

On the Laurent series for the Epstein zeta function

“…It should be pointed out that some of the identities (5.1)-(5.7) have been derived previously by Connon (2009) and Coffey (2011) using different and much more general methods. Recently, Blagouchine (2015) and Coffey (2016)…”

Some integrals of the Dedekind $η$ function

“…and γ k (a) are the Stieltjes constants [8,7,9] to be recollected in section 5. The products in this equation are taken over prime numbers p.…”