Deﬁne the moments of order 2k:
(k ∈ Z≥1).
Proposition 1. S2(p) is asymptotic to
as p → ∞, where c2 =
Proof. Recall that
χ(a)¯(b) χ =
− 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,
It follows that
p − 1
b≡±a (mod p)
(where starred sums indicate sums over indices not divisible by p). The desired
Proposition 2. (See [Lou99]). There exists c 0 such that S4(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
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) ≥
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
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
∼ p log p
(see [Rama, Remark 3]) and
1=χ mod p