Series representations for the Stieltjes constants

Abstract

The Stieltjes constants γ_k(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(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 γ_k=γ_k(1). Some extensions are briefly described, as well as the relevance to expansions of Dirichlet L functions.

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

References in corpus (1)

• New results on the Stieltjes constants: Asymptotic and exact evaluation

Cited by in corpus (10)

• Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant
• Certain logarithmic integrals, including solution of Monthly problem #tbd, zeta values, and expressions for the Stieltjes constants
• Epsilon expansion of Appell and Kampé de Fériet functions
• Summatory relations and prime products for the Stieltjes constants, and other related results
• Rapid computation of special values of Dirichlet $L$-functions
• Closed-form formula of Riemann zeta function and eta function for all non-zero given complex numbers via sums of powers of complex functions to disprove Riemann Hypothesis
• Expressions for two generalized Furdui series
• Series representation of the Riemann zeta function and other results: Complements to a paper of Crandall
• The signs of the Stieltjes constants associated with the Dedekind zeta function
• On some new formulae involving the Stieltjes constants