Functional equations for the Stieltjes constants

TL;DR

This paper derives new functional equations and evaluation formulas for the Stieltjes constants at rational arguments, providing tools for theoretical and computational number theory.

Contribution

It introduces new evaluation methods, multiplication formulas, and asymptotic behaviors for the Stieltjes constants, expanding understanding of their properties at rational points.

Findings

Evaluation formulas for γ₁(a) and γ₂(a) at rational a

Multiplication formulas for γ₀(a), γ₁(a), γ₂(a)

Asymptotic behavior of γ_k(a) as a→0

Abstract

The Stieltjes constants $\gamma_k(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about $s=1$. We present the evaluation of $\gamma_1(a)$ and $\gamma_2(a)$ at rational argument, being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for $\gamma_0(a)$, $\gamma_1(a)$, and $\gamma_2(a)$, 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 $\gamma_k(a)$ as $a \to 0$ as well as a novel technique for evaluating integrals with integrands with $\ln(-\ln x)$ and rational factors.