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 |
|---|---|
| Pages (from-to) | 1-26 |
| Number of pages | 26 |
| Journal | Forum Mathematicum |
| Volume | 20 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - Jan 1 2008 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
- Applied Mathematics
Cite this
Q-Analogues of the Riemann zeta, the Dirichlet L-functions, and a crystal zeta function. / Kawagoe, Kenichi; Wakayama, Masato; Yamasaki, Yoshinori; Sarnak, Peter.
In: Forum Mathematicum, Vol. 20, No. 1, 01.01.2008, p. 1-26.Research output: Contribution to journal › Article
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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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UR - http://www.scopus.com/inward/citedby.url?scp=40549141120&partnerID=8YFLogxK
U2 - 10.1515/FORUM.2008.001
DO - 10.1515/FORUM.2008.001
M3 - Article
AN - SCOPUS:40549141120
VL - 20
SP - 1
EP - 26
JO - Forum Mathematicum
JF - Forum Mathematicum
SN - 0933-7741
IS - 1
ER -