Open Access
2014 Series representations for the Stieltjes constants
Mark W. Coffey
Rocky Mountain J. Math. 44(2): 443-477 (2014). DOI: 10.1216/RMJ-2014-44-2-443

Abstract

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

Citation

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Mark W. Coffey. "Series representations for the Stieltjes constants." Rocky Mountain J. Math. 44 (2) 443 - 477, 2014. https://doi.org/10.1216/RMJ-2014-44-2-443

Information

Published: 2014
First available in Project Euclid: 4 August 2014

zbMATH: 1320.11085
MathSciNet: MR3240509
Digital Object Identifier: 10.1216/RMJ-2014-44-2-443

Subjects:
Primary: 11M06 , 11M35 , 11Y60
Secondary: 05A10

Keywords: Dirichlet L functions , Hurwitz zeta function , Laurent expansion , Lerch zeta function , Riemann zeta function , Stieltjes constants , Stirling numbers of the first kind

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

Vol.44 • No. 2 • 2014
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