Series representations for the Stieltjes constants

Details Series representations for the Stieltjes constants Series representations for the Stieltjes constants

2009-05-07

arxiv.org/abs/0905.1111v2

Physics

Mathematical Physics

37 pages, no figures New material added at end, including Corollary 6

Scientific paper

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 series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of all orders of the Stieltjes coefficients are given. Many other results are obtained, including instances of an exponentially fast converging series representation for \gamma_k=\gamma_k(1). Some extensions are briefly described, as well as the relevance to expansions of Dirichlet L functions.

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