## 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 language | English |
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Pages (from-to) | 1-26 |

Number of pages | 26 |

Journal | Forum Mathematicum |

Volume | 20 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2008 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics