On the values of a class of Dirichlet series at rational arguments

Authors:
K. Chakraborty, S. Kanemitsu and H.-L. Li

Journal:
Proc. Amer. Math. Soc. **138** (2010), 1223-1230

MSC (2010):
Primary 11M35; Secondary 33B15

Published electronically:
December 4, 2009

MathSciNet review:
2578516

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we shall prove that the combination of the general distribution property and the functional equation for the Lipschitz-Lerch transcendent capture the whole spectrum of deeper results on the relations between the values at rational arguments of functions of a class of zeta-functions. By Theorem 1 and its corollaries, we can cover all the previous results in a rather simple and lucid way. By considering the limiting cases, we can also deduce new striking identities for Milnor's gamma functions, among which is the Gauss second formula for the digamma function.

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Additional Information

**K. Chakraborty**

Affiliation:
Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211 019, India

Email:
kalyan@hri.res.in

**S. Kanemitsu**

Affiliation:
Graduate School of Advanced Technology, Kinki University, Iizuka, Fukuoka 820-8555, Japan

Email:
kanemitu@fuk.kindai.ac.jp

**H.-L. Li**

Affiliation:
Department of Mathematics, Weinan Teachers College, Weinan, Shaanxi, 714000, People’s Republic of China

Email:
lihailong@wntc.edu.cn

DOI:
http://dx.doi.org/10.1090/S0002-9939-09-10171-5

Keywords:
Lipschitz-Lerch transcendent,
Hurwitz zeta function,
polylogarithm function,
Gauss' second formula,
Milnor's gamma function.

Received by editor(s):
December 28, 2008

Received by editor(s) in revised form:
August 18, 2009

Published electronically:
December 4, 2009

Dedicated:
Dedicated to Professor Eiichi Bannai on his sixtieth birthday, with great respect and friendship

Communicated by:
Ken Ono

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.