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

Kenichi Kawagoe; Masato Wakayama; Yoshinori Yamasaki

doi:10.1515/forum.2008.001

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

Kenichi Kawagoe·Masato Wakayama·Yoshinori Yamasaki

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Abstract

A q-analogue of the Riemann zeta function 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 the q-analogue left open in [Kaneko et al. 03]. We also examine a "crystal" limit (i.e. q->0) behavior of the q-analogue. The q-trajectories of the trivial and essential zeros of the Riemann zeta function are investigated numerically when q moves in (0,1]. Moreover, conjectures for the crystal limit behavior of zeros of the q-analogue are given.

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Kenichi Kawagoe, Masato Wakayama, Yoshinori Yamasaki. 2005-01-20. q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function. https://doi.org/10.1515/forum.2008.001

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