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Let ρ j (q) be the q-trajectory of ρ j = ρ j (1) satisfying ζ q (ρ j (q)) = 0. From the numerical calculation developed in the previous subsection, we state here some conjectures -
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Conjecture The function ρ j (q) is continuous for 0 < q ≤ 1. Further, the limit value ρ j (0) := lim q↓0 ρ j (q) exists and satisfies ρ j (0) ∈ Z ≤0 -
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Half zeta functions
- Publisher
- de Gruyter
- Copyright
- © Walter de Gruyter
- ISSN
- 0933-7741
- eISSN
- 1435-5337
- DOI
- 10.1515/FORUM.2008.001
- Publisher site
- See Article on Publisher Site
Abstract
A q -analogue ζ q ( 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 ζ 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), 175–192. We also examine a “crystal” limit (i.e. q ↓ 0) behavior of ζ q ( 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 ζ q ( 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.
Journal
Forum Mathematicum – de Gruyter
Published: Jan 1, 2008