q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function
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Abstract
A q-analogue ζq(s) of the Riemann zeta function ζ(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), 175–192] 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 M., Kurokawa N. and Wakayama M.: A variation of Euler's approach to values of the Riemann zeta function. Kyushu J. Math. 57 (2003), 175–192]. 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), which predict an interesting distribution of “trivial zeros” and an analogue of the Riemann hypothesis for a crystal zeta function, are given.
2000 Mathematics Subject Classification: 11M06.
© Walter de Gruyter
Articles in the same Issue
- q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function
- Antifinitary linear groups
- Comparing GLn-representations by characteristic-free isomorphisms between generalized Schur algebras
- Stable equivalences of adjoint type
- A partial solution of the isoperimetric problem for the Heisenberg group
- Brill-Noether problems in higher dimensions
- Second main theorem with truncated counting function in several complex variables for moving targets
- Semifields with free automorphism groups
Articles in the same Issue
- q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function
- Antifinitary linear groups
- Comparing GLn-representations by characteristic-free isomorphisms between generalized Schur algebras
- Stable equivalences of adjoint type
- A partial solution of the isoperimetric problem for the Heisenberg group
- Brill-Noether problems in higher dimensions
- Second main theorem with truncated counting function in several complex variables for moving targets
- Semifields with free automorphism groups
