Rocky Mountain Journal of Mathematics

Series representations for the Stieltjes constants

Mark W. Coffey

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Abstract

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.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 2 (2014), 443-477.

Dates
First available in Project Euclid: 4 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1407154909

Digital Object Identifier
doi:10.1216/RMJ-2014-44-2-443

Mathematical Reviews number (MathSciNet)
MR3240509

Zentralblatt MATH identifier
1320.11085

Subjects
Primary: 11M35: Hurwitz and Lerch zeta functions 11M06: $\zeta (s)$ and $L(s, \chi)$ 11Y60: Evaluation of constants
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx]

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

Citation

Coffey, Mark W. Series representations for the Stieltjes constants. Rocky Mountain J. Math. 44 (2014), no. 2, 443--477. doi:10.1216/RMJ-2014-44-2-443. https://projecteuclid.org/euclid.rmjm/1407154909


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