ON THE FOURTH MOMENT OF THETA FUNCTIONS AT THEIR CENTRAL POINT 3
(see [HB, Corollary (page 26)]). It is also conjectured that for k Z≥1 there exists
a positive constant C(k) such that
1=χ mod p
|L(1/2,
χ)|2k
C(k)p
logk2
p
and it is known that (see [RS]))
1=χ mod p
|L(1/2,
χ)|2k
k
p
logk2
p.
2. The second moment of the restricted divisor function
Our proof of Theorem 5 is based on a new method: the study of the moment
of order 2 of the restricted divisor function (see Proposition 7 below). It is known
that (see [Ten]):
S(x) =
1≤mx d|m
1 = x(log x + 1) + O(

x).
It follows that for c 1 the first moment of the restricted divisor function
Sc(x) =
1≤mx
d|m
1
c

m≤d≤c

m
1 = S(x) 2
1≤mx
d|m
d
1
c

m
1
= S(x) 2
d 1
c

x
x
d

c2d
+ O(1) = 2(log c)x + O(

x)
is asymptotic to 2(log c)x as x ∞. Now,
T (n) =
1≤mx
d|m
1
2
is asymptotic to
1
π2
x
log3
x as x ∞. We give an asymptotic for the second
moment of the restricted divisor function:
Proposition 7. Fix c 1. Then,
Tc(x) =
1≤mx
d|m
1
c

m≤d≤c

m
1
2
=
12
log2
c
π2
x log x + O(x).
It follows that
Λ(n) =
1≤mn d|m
e−π(m/d−d)2/n
2
n log n.
Remark 8. It holds that Λ(n) n log n, by [Lou99, Lemme 2].
Proof. Let us prove the second assertion. Fix c 1. If
1
c

m d c

m,
then (m/d d)2/n (c 1/c)2m/n (c 1/c)2 for 1 m n. It follows that
Λ(n)
e−2π(c−1/c)2
Tc(n).
Let us now prove the first assertion. We have
Tc(x) =
1≤mx
d1|m
1
c

m≤d1≤c

m
d2|m
1
c

m≤d2≤c

m
1.
3
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