On the fourth moment of theta functions at their central
point
Amadou Diogo BARRY and St´ ephane R. LOUBOUTIN
Abstract. Let χ be a Dirichlet character of prime conductor p 3. Set A =
(χ(1) χ(−1))/2 {0, 1} and let θ(x, χ) =
P
n≥1
nAχ(n) exp(−πn2x/p)
(x
0) be its associated theta series whose functional equation is used to obtain
the analytic continuation and functional equation of the L-series L(s, χ) =
P
n≥1
χ(n)n−s.
These functional equations depend on some root numbers
W (χ), complex numbers of absolute values equal to one. In particular, if
θ(1, χ) = 0, then numerical approximations to W (χ) = θ(1, χ)/θ(1, χ) can
be efficiently computed, which leads to a fast algorithm for computing class
numbers and relative class numbers of real or imaginary abelian number fields
(see the bibliography). According to numerical computations, it is reasonable
to conjecture that θ(1, χ) = 0 for any primitive Dirichlet character χ. One way
to prove that this conjecture at least holds true for infinitely many primitive
characters is to study the moments
P
χ
|θ(1,
χ)|2k
for k Z≥1, where χ ranges
over all the even or odd primitive Dirichlet characters of conductor p. This
paper is devoted to proving a lower bound on these moments for 2k = 4.
1. Introduction
We restrict ourselves to even characters of prime conductors. Let p 3 be a
prime. Let Xp
+
be the set of the (p 3)/2 primitive even Dirichlet characters of
conductor p. For χ Xp +, set
θ(x, χ) =
n≥1
χ(n)e−πn2x/p
(x 0).
The analytic continuation and the functional equation satisfied by the Dirichlet
L-series L(s, χ) =

n≥1
χ(n)n−s,
(s) 0, stem from the functional equation
satisfied by the associated theta function (see [Dav, Chapter 9]):
(1) θ(x, χ) =
W (χ)
x1/2
θ(1/x, ¯), χ
where W (χ) = τ (χ)/

p (the Artin root number, a complex number of absolute
value equal to 1 ) with τ (χ) =
∑p
k=1
χ(k)e2πik/p (Gauss sum).
1991 Mathematics Subject Classification. 2000 Mathematics Subject Classification. Primary
11R42, 11N37. Secondary 11M06.
Key words and phrases. Dirichlet characters, Artin root numbers, Theta functions, Divisor
function.
1
Contemporary Mathematics
Volume 487, 2009
1
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