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On  -analogues of the Euler constant and Lerch's limit formula
Author(s):
Nobushige
Kurokawa;
Masato
Wakayama
Journal:
Proc. Amer. Math. Soc.
132
(2004),
935-943.
MSC (2000):
Primary 11M35, 33D05
Posted:
November 13, 2003
MathSciNet review:
2045407
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Abstract:
We introduce and study a  -analogue  of the Euler constant via a suitably defined  -analogue of the Riemann zeta function. We show, in particular, that the value  is irrational. We also present a  -analogue of the Hurwitz zeta function and establish an analogue of the limit formula of Lerch in 1894 for the gamma function. This limit formula can be regarded as a natural generalization of the formula of  .
References:
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- P. Askey, The
 -gamma and  -beta functions, Applicable Analysis 8 (1978), 125-141. MR 80h:33003 - [E]
- P. Erdös, On arithmetical properties of Lambert series, J. Indian Math. Soc. 12 (1948), 63-66. MR 10:594c
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- Y. Hashimoto, Y. Iijima, N. Kurokawa and M. Wakayama, Euler's constants for the Selberg and the Dedekind zeta functions, Bulletin of the Belgian Mathematical Society Simon Stevin (to appear).
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- F. H. Jackson, On
 -definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193-203. - [KKW]
- M. Kaneko, N. Kurokawa and M. Wakayama, A variation of Euler's approach to values of the Riemann zeta function, Kyushu Math. J. 57 (2003), 175-192.
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- N. Kurokawa and M. Wakayama, A comparison between the sum over Selberg's zeroes and Riemann's zeroes, J. Ramanujan Math. Soc. 18 (2003), 221-236. (Errata will also appear.)
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Additional Information:
Nobushige
Kurokawa
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, Meguro, Tokyo, 152-0033 Japan
Email:
kurokawa@math.titech.ac.jp
Masato
Wakayama
Affiliation:
Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka, 812-8581 Japan
Email:
wakayama@math.kyushu-u.ac.jp
DOI:
10.1090/S0002-9939-03-07025-4
PII:
S 0002-9939(03)07025-4
Keywords:
Euler's constant,
Riemann's zeta function,
Hurwitz's zeta function,
$q$-gamma function,
Lerch's limit formula
Received by editor(s):
September 3, 2002
Posted:
November 13, 2003
Additional Notes:
Work in part supported by Grant-in-Aid for Scientific Research (B) No. 11440010, and by Grant-in-Aid for Exploratory Research No. 13874004, Japan Society for the Promotion of Science
Communicated by:
Wen-Ching Winnie Li
Copyright of article:
Copyright
2003,
American Mathematical Society
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