An interplay between a generalized-Euler-constant function and the Hurwitz zeta function

Abstract

For the generalized-Euler-constant function \[ a\mapsto \gamma(a):=\underset{n\rightarrow\infty}{\lim} \left(\sum_{i=0}^{n-1}\frac{1}{a+i}-\ln\frac{a+n-1}{a}\right) \] defined on $\R^+$, the expansion $\gamma(a)=\sum_{j=2}^{\infty}\frac{(-1)^j}{j}\,\zeta(j,a)$, where $\zeta(j,a)$ is the Hurwitz zeta function, is derived and a formula for its numerical computation is presented.