Series representations for the Stieltjes constants

2009/05/07 by Mark W. Coffey, Coffey, Mark W. · 4 citations
Mathematics · #11M06 #11M35 #11Y60 (Primary) 05A10 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.0905.1111

openalex publication_date 2009/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Closed-form formula of Riemann zeta function and eta function for all\n non-zero given complex numbers via sums of powers of complex functions to\n disprove Riemann Hypothesis
• Series representation of the Riemann zeta function and other results:\n Complements to a paper of Crandall

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