2 AMADOU DIOGO BARRY AND STEPHANE
´
R. LOUBOUTIN
Define the moments of order 2k:
S2k(p) =
χ∈Xp+
|θ(1,
χ)|2k
(k Z≥1).
Proposition 1. S2(p) is asymptotic to
c2p3/2
as p ∞, where c2 =
1
4


.
Proof. Recall that
χ∈Xp
+
χ(a)¯(b) χ =

⎪(p



3)/2 if b ±a (mod p) and gcd(a, p) = gcd(b, p) = 1,
−1 if b ±a (mod p) and gcd(a, p) = gcd(b, p) = 1,
0 otherwise.
It follows that
S2(p) =
p 1
2

a,b
b≡±a (mod p)
e−π(a2+b2)/p


a
e−πa2/p
2
(where starred sums indicate sums over indices not divisible by p). The desired
result follows.
Proposition 2. (See [Lou99]). There exists c 0 such that S4(p)
cp2
log p
for p 3.
Corollary 3. It holds that that θ(1, χ) = 0 for at least p/ log p of the
characters χ Xp +.
Proof. The Cauchy-Schwarz inequality yields that θ(1, χ) = 0 for at least
S2(p)2/S4(p)
of the χ’s in Xp
+.
Remark 4. For such a character χ, we have W (χ) = θ(1, χ)/θ(1, χ), by (1)
(hence numerical approximations to W (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 [Lou98], [Lou02] and [Lou07]).
The aim of this paper is to prove the following new result:
Theorem 5. There exists c 0 such that S4(p)
cp2
log p for p 3.
According to Proposition 2, Corollary 3, Theorem 5 and extended numerical
computations, we conjecture that θ(1, χ) = 0 for any primitive Dirichlet character
χ = 1, and the more precise behavior (see [Bar]):
Conjecture 6. There exists c4 0 such that S4(p) is asymptotic to
c4p2
log p
as p ∞.
As for the second and fourth moments of the values of Dirichlet L-functions
L(s, χ) at their central point s = 1/2, the following asymptotics are known:
1=χ mod p
|L(1/2,
χ)|2
p log p
(see [Rama, Remark 3]) and
1=χ mod p
|L(1/2,
χ)|4

1
2π2
p
log4
p
2
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