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

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.