Affiliation:
1. Pure Mathematics, University of Waterloo, 200 University Avenue W, Waterloo, Ontario, Canada N2L3G1
Abstract
Let
q
be an odd prime power, and
denote the set of square-free monic polynomials
D
(
x
)∈
F
q
[
x
] of degree
d
. Katz and Sarnak showed that the moments, over
, of the zeta functions associated to the curves
y
2
=
D
(
x
), evaluated at the central point, tend, as
, to the moments of characteristic polynomials, evaluated at the central point, of matrices in
USp
(2⌊(
d
−1)/2⌋). Using techniques that were originally developed for studying moments of
L
-functions over number fields, Andrade and Keating conjectured an asymptotic formula for the moments for
q
fixed and
. We provide theoretical and numerical evidence in favour of their conjecture. In some cases, we are able to work out exact formulae for the moments and use these to precisely determine the size of the remainder term in the predicted moments.
Subject
General Physics and Astronomy,General Engineering,General Mathematics
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