A New Effective Asymptotic Formula for the Stieltjes Constants

We derive a new integral formula for the Stieltjes constants. The new formula permits easy computations as well as an exact approximate asymptotic formula. Both the sign oscillations and the leading order of growth are provided. The formula can also be easily extended to generalized Euler constants

Addison-type series representation for the Stieltjes constants

The Stieltjes constants $\gamma_k(a)$ appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about its only pole at $s=1$. We generalize a technique of Addison for the Euler constant $\gamma=\gamma_0(1)$ to show its application to finding series representations for these constants. Other generalizations of representations of $\gamma$ are given.Comment: 21 pages, no figure