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Mathematics of Computation

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Series representation of the Riemann zeta function and other results: Complements to a paper of Crandall


Author: Mark W. Coffey
Journal: Math. Comp. 83 (2014), 1383-1395
MSC (2010): Primary 11M06, 11M35, 11Y35, 11Y60
DOI: https://doi.org/10.1090/S0025-5718-2013-02755-X
Published electronically: July 29, 2013
MathSciNet review: 3167463
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Abstract: We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation throughout the whole complex plane. Additionally, we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including $ \int _{\mbox {\tiny {Re}} s=c} {{\zeta (s)} \over s} ds$ and $ \int _{\mbox {\tiny {Re}} s=c} {{\eta (s)} \over s} ds$, with $ \zeta $ the Riemann zeta function and $ \eta $ its alternating form.


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Additional Information

Mark W. Coffey
Affiliation: Department of Physics, Colorado School of Mines, Golden, Colorado 80401

DOI: https://doi.org/10.1090/S0025-5718-2013-02755-X
Keywords: Riemann zeta function, Hurwitz zeta function, Stieltjes constants, Euler polynomials, Bernoulli polynomials, incomplete Gamma function, confluent hypergeometric function, generalized hypergeometric function
Received by editor(s): April 2, 2012
Received by editor(s) in revised form: July 18, 2012, and August 27, 2012
Published electronically: July 29, 2013
Article copyright: © Copyright 2013 retained by the author