Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


On $q$-analogues of the Euler constant and Lerch’s limit formula
HTML articles powered by AMS MathViewer

by Nobushige Kurokawa and Masato Wakayama PDF
Proc. Amer. Math. Soc. 132 (2004), 935-943 Request permission


We introduce and study a $q$-analogue $\gamma (q)$ of the Euler constant via a suitably defined $q$-analogue of the Riemann zeta function. We show, in particular, that the value $\gamma (2)$ is irrational. We also present a $q$-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 $\gamma (q)$.
  • Richard Askey, The $q$-gamma and $q$-beta functions, Applicable Anal. 8 (1978/79), no. 2, 125–141. MR 523950, DOI 10.1080/00036817808839221
  • Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
  • 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).
  • F. H. Jackson, On $q$-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193–203.
  • 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.
  • 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.)
  • M. Lerch, Dalši studie v oboru Malmsténovských řad, Rozpravy České Akad. 3 No. 28 (1894), 1–61.
  • Daniel S. Moak, The $q$-gamma function for $q>1$, Aequationes Math. 20 (1980), no. 2-3, 278–285. MR 577493, DOI 10.1007/BF02190519
  • Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11M35, 33D05
  • Retrieve articles in all journals with MSC (2000): 11M35, 33D05
Additional Information
  • Nobushige Kurokawa
  • Affiliation: Department of Mathematics, Tokyo Institute of Technology, Meguro, Tokyo, 152-0033 Japan
  • Email:
  • Masato Wakayama
  • Affiliation: Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka, 812-8581 Japan
  • Email:
  • Received by editor(s): September 3, 2002
  • Published electronically: 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 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 935-943
  • MSC (2000): Primary 11M35, 33D05
  • DOI:
  • MathSciNet review: 2045407