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

Abstract

A q -analogue ζ q ( s ) of the Riemann zeta function ζ( 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), 175–192] 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 ζ 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), 175–192]. We also examine a “crystal” limit (i.e. q ↓ 0) behavior of ζ q ( s ). The q -trajectories of the trivial and essential zeros of ζ( s ) are investigated numerically when q moves in (0, 1]. Moreover, conjectures for the crystal limit behavior of zeros of ζ q ( s ), which predict an interesting distribution of “trivial zeros” and an analogue of the Riemann hypothesis for a crystal zeta function, are given. 2000 Mathematics Subject Classification: 11M06.