q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function
نویسندگان
چکیده
A q-analogue ζq(s) of the Riemann zeta function ζ(s) was studied in [Kaneko et al. 03] 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 ζq(s) left open in [Kaneko et al. 03]. We also examine a “crystal” limit (i.e. q ↓ 0) behavior of ζq(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 ζq(s) are given. 2000 Mathematics Subject Classification : 11M06
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