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

Fingerprint

Dirichlet L-function
Q-analogue
Riemann zeta function
Crystal
Crystals
Euler
Trivial
Zero
Q-series
Limit Behavior
Riemann hypothesis
Classical Limit
Trajectory
Analogue
Predict
Trajectories

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

Q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function. / Kawagoe, Kenichi; Wakayama, Masato; Yamasaki, Yoshinori; Sarnak, Peter.

In: Forum Mathematicum, Vol. 20, No. 1, 01.01.2008, p. 1-26.

Research output: Contribution to journalArticle

Kawagoe, Kenichi ; Wakayama, Masato ; Yamasaki, Yoshinori ; Sarnak, Peter. / Q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function. In: Forum Mathematicum. 2008 ; Vol. 20, No. 1. pp. 1-26.
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