Rocky Mountain Journal of Mathematics
- Rocky Mountain J. Math.
- Volume 44, Number 2 (2014), 443-477.
Series representations for the Stieltjes constants
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.
Rocky Mountain J. Math., Volume 44, Number 2 (2014), 443-477.
First available in Project Euclid: 4 August 2014
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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