TY - JOUR

T1 - Q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function

AU - Kawagoe, Kenichi

AU - Wakayama, Masato

AU - Yamasaki, Yoshinori

AU - Sarnak, Peter

PY - 2008/1/1

Y1 - 2008/1/1

N2 - 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.

AB - 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.

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U2 - https://doi.org/10.1515/FORUM.2008.001

DO - https://doi.org/10.1515/FORUM.2008.001

M3 - Article

SN - 0933-7741

VL - 20

SP - 1

EP - 26

JO - Forum Mathematicum

JF - Forum Mathematicum

IS - 1

ER -