Q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function

Kenichi Kawagoe, Masato Wakayama, Yoshinori Yamasaki, Peter Sarnak

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

A q-analogue q(s) of the Riemann zeta function (s) was studied in [Kaneko M., Kurokawa N. and Wakayama M.: A variation of Euler's approach to values of the Riemann zeta function. Kyushu J. Math. 57 (2003), 175192] via a certain q-series of two variables. We introduce in a similar way a q-analogue of the Dirichlet L-functions and make a detailed study of them, including some issues concerning the classical limit of q(s) left open in [Kaneko M., Kurokawa N. and Wakayama M.: A variation of Euler's approach to values of the Riemann zeta function. Kyushu J. Math. 57 (2003), 175192]. We also examine a crystal limit (i.e. q 0) behavior of q(s). The q-trajectories of the trivial and essential zeros of (s) are investigated numerically when q moves in (0, 1]. Moreover, conjectures for the crystal limit behavior of zeros of q(s), which predict an interesting distribution of trivial zeros and an analogue of the Riemann hypothesis for a crystal zeta function, are given.

Original languageEnglish
Pages (from-to)1-26
Number of pages26
JournalForum Mathematicum
Volume20
Issue number1
DOIs
Publication statusPublished - Jan 1 2008

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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