Functional equations for the Stieltjes constants

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.