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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 and , with the Riemann zeta function and its alternating form.

- [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Washington, National Bureau of Standards (1964).
**[2]**George E. Andrews, Richard Askey, and Ranjan Roy,*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958****[3]**Tom M. Apostol,*Introduction to analytic number theory*, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. MR**0434929**- [4] M. W. Coffey, Series representations for the Stieltjes constants, arXiv:0905.1111 (2009), to appear in Rocky Mtn. J. Math.
**[5]**Mark W. Coffey,*Addison-type series representation for the Stieltjes constants*, J. Number Theory**130**(2010), no. 9, 2049–2064. MR**2653214**, https://doi.org/10.1016/j.jnt.2010.01.003- [6] M. W. Coffey, Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant, arXiv:1106.5147 (2011).
- [7] M. W. Coffey, Parameterized summation relations for the Stieltjes constants, arXiv:1002.4684 (2010).
- [8] M. W. Coffey, Evaluation of some second moment and other integrals for the Riemann, Hurwitz, and Lerch zeta functions, arXiv:1101.5722 (2011).
- [9]
R. E. Crandall, Unified algorithms for polylogarithm, -series, and zeta variants, preprint (2012).
`http://www.perfscipress.com/papers/universalTOC25.pdf` **[10]**Richard Crandall and Carl Pomerance,*Prime numbers*, Springer-Verlag, New York, 2001. A computational perspective. MR**1821158**- [11] http://dlmf.nist.gov/
**[12]**K. O. Geddes, M. L. Glasser, R. A. Moore, and T. C. Scott,*Evaluation of classes of definite integrals involving elementary functions via differentiation of special functions*, Appl. Algebra Engrg. Comm. Comput.**1**(1990), no. 2, 149–165. MR**1325519**, https://doi.org/10.1007/BF01810298- [13] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York (1980).
**[14]**M. S. Milgram,*The generalized integro-exponential function*, Math. Comp.**44**(1985), no. 170, 443–458. MR**777276**, https://doi.org/10.1090/S0025-5718-1985-0777276-4**[15]**R. B. Paris and D. Kaminski,*Asymptotics and Mellin-Barnes integrals*, Encyclopedia of Mathematics and its Applications, vol. 85, Cambridge University Press, Cambridge, 2001. MR**1854469****[16]**E. T. Whittaker and G. N. Watson,*A course of modern analysis*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR**1424469**

Retrieve articles in *Mathematics of Computation*
with MSC (2010):
11M06,
11M35,
11Y35,
11Y60

Retrieve articles in all journals with MSC (2010): 11M06, 11M35, 11Y35, 11Y60

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