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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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: 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
PII: S 0002-9939(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.