ACTA ARITHMETICA
* (200*)
The second moment of Dirichlet twists of Hecke L-functions
by
Peng Gao (Singapore), Rizwanur Khan (Los Angeles, CA)
and Guillaume Ricotta (Bordeaux)
1. Introduction. Let f be a fixed holomorphic Hecke eigenform of
level 1 and weight k. For =(z) > 0 we have a Fourier expansion of f :
X
f (z) =
af (n)n(k−1)/2 e2πinz ,
n≥1
where the coefficients af (n) are real and satisfy the Ramanujan–Petersson
bound |a(n)| ≤ d(n), and we have normalized so that af (1) = 1. Let χ be a
Dirichlet character of modulus q, and form the Dirichlet series of the twist
of f by χ given by
X af (n)χ(n) Y af (p)χ(p) χ(p2 ) −1
=
1−
+ 2s
L(f × χ, s) =
ns
ps
p
p
n≥1
for <(s) > 1. When χ is primitive this is an L-function, which continues to
an entire function and satisfies the functional equation
s
q
(1.1)
Γ (k/2 + s)L(f × χ, 1/2 + s)
2π
−s
q
= ıχ
Γ (k/2 − s)L(f × χ, 1/2 − s),
2π
where ıχ = ik τ (χ)2 /q and τ (χ) is the Gauss sum associated to χ (thus
|ıχ | = 1); cf. [3, Prop. 14.20].
For a large modulus q, we are interested in the second power moment of
L(f × χ, 1/2) as χ ranges over all primitive characters modulo q. Stefanicki
[7] proved
X∗
(1.2)
|L(f × χ, 1/2)|2 = KPq (1)ψ(q)q log q + O(2ν(q) q(log q)0.935 ),
χ mod q
2010 Mathematics Subject Classification: Primary 11M99.
Key words and phrases: mean values of L-functions.
[1]
2
P. Gao et al.
where ∗ denotes summation only over primitive characters, K is a constant,
Pq (1)ψ(q) is a finite Euler product over the prime divisors of q, and ν(q)
is the number of distinct prime factors of q. A modulus q may have logloglogq q
distinct prime factors, but the main term in (1.2) fails to dominate the error
1
1
term when ν(q) is larger than 10
log log q. The condition ν(q) < 10
log log q
holds for very few integers q, a set of zero density in fact, by a theorem
of Hardy and Ramanujan [1]. We are interested in the following problem:
can we establish an asymptotic for the second moment for a larger class of
integers q; if not all integers, then almost all? This can be considered an
analogue of a problem studied by Soundararajan: in [6], he adapts HeathBrown’s [2] asymptotic formula for the fourth power moment of Dirichlet
L-functions to hold for all moduli. Unfortunately, his methods do not seem
to immediately carry over to this case and we must do something different.
We should also mention that it is an open problem to find an asymptotic
expression with a power saving error term for the left hand side of (1.2),
even for prime q. The corresponding problem for the fourth power moment
of Dirichlet L-functions with prime moduli was recently solved by Young [8].
Let
6(4π)k
K=
kf k2 ,
πΓ (k)
where the norm is the Petersson norm. Define the multiplicative function
ψ(q) by setting ψ(p) = (1 − 2/p) and ψ(pk ) = (1 − 1/p)2 for k ≥ 2, so that
qψ(q) is the number of primitive characters modulo q. Let
Y
af (p)2 − 2
1 2
1
1 −1
Pq (s) =
1− s
1−
+ 2s
1 − 2s
.
p
ps
p
p
p|q
Our main theorem is
Theorem 1.1. For integers q satisfying
X1
(1.3)
≤ (log log q)−10 ,
p
p|q
p>x
log q
where x = exp 200 log
log log log q , we have
X∗
(1.4)
|L(f × χ, 1/2)|2 = KPq (1)ψ(q)q log q(1 + O((log log q)−1 )),
χ mod q
where the implied constant depends on f .
log q
Condition (1.3) holds when for example ν(q) exp 300 log
log log log q , which
is certainly true for almost all integers. We will need the following consequence of this condition.
Second moment of Dirichlet twists
3
Lemma 1.2. If (1.3) holds then
Q
√
X
|µ(d)| p|d (1 + 10/ p)
(1.5)
(log log q)−9 .
d
d|q
d≥(log q)0.05
Proof. Using “Rankin’s trick” we have
Q
√
X
|µ(d)| p|d (1 + 10/ p)
(1.6)
d
d|q
d≥(log q)0.05
p|d⇒p≤x
≤
X
d|q
p|d⇒p≤x
d
(log q)0.05
(log log q)−10
(log x)−1
|µ(d)|
Q
p|d (1
√
+ 10/ p)
d
Y
1+
p≤x
1
p1−(log x)−1
(log log q)−9 .
Thus
(1.7)
|µ(d)|
X
Q
√
+ 10/ p)
d
d|q
d≥(log q)0.05
p|d (1
X
|µ(d)|
Q
p|d (1
√
+ 10/ p)
d
d|q
d≥(log q)0.05
p|d⇒p>x
+ (log log q)−9
Q
√ X X
|µ(d)| p|d (1 + 10/ p)
1
+ (log log q)−9 .
p
d
p|q
p>x
d|q
This
Q completes the proof as the sum over divisors in the last line of (1.7) is
p|q (1 + 1/p) log log q.
Throughout the paper all implied constants may depend implicitly on f .
2. Proof of Theorem 1.1. The following standard results can be found
in Sections 2 and 3.1 respectively of [7].
Lemma 2.1 (Orthogonality). For (nm, q) = 1 we have
X∗
X
(2.1)
χ(n)χ(m) =
µ(d)φ(q/d).
χ mod q
d|q
n≡m mod q/d
4
P. Gao et al.
Lemma 2.2 (Approximate functional equation). If χ is a primitive character modulo q then
X af (n)af (m)χ(n)χ(m) nm 2
√
|L(f × χ, 1/2)| =
V
(2.2)
,
q2
nm
n,m
where
V (x) =
dy
1 G(y)x−y
πi
y
(c)
for any c > 0, and
G(s) =
Γ (k/2 + s)2
(2π)2s Γ (k/2)2
decays rapidly in vertical lines. We have V (x) C min{1, x−C } for any
C > 0, so that the sum above is essentially supported on nm < q 2+ε for any
ε > 0.
Thus we have
X∗
(2.3)
|L(f × χ, 1/2)|2
χ mod q
X X∗ af (n)af (m) nm X∗
√
=
V
χ(n)χ(m)
q2
nm
n,m χ mod q
χ mod q
X
X
af (n)af (m)
nm
√
V
.
=
µ(d)φ(q/d)
q2
nm
d|q
n≡m mod q/d
(mn,q)=1
The main idea is to treat separately the cases d < (log q)0.05 and d ≥
(log q)0.05 . Note that in the main term of the theorem, we have Pq (1)ψ(q) (log log q)−6 , so we need to bound any error terms by q log q(log log q)−7 .
Small divisors
Lemma 2.3. We have, for some θ > 0,
X
af (n)af (m)
nm
√
(2.4)
φ(l)−1 q(log q)1−0.065 + q ε l−θ .
V
q2
nm
n6=m
n≡m mod l
(nm,q)=1
The above lemma is a combination of Lemmas 2, 3 and 7 of [7]. For
n and m both close to l, a shifted convolution problem is solved to get a
power saving bound. For n or m much larger than l, the sum is bounded
absolutely by appealing to Shiu’s [5] estimate for sums of multiplicative
functions in arithmetic progression combined with Rankin’s [4] result that
|af (n)| is bounded by (log n)−0.065 on average. This gives only a log power
Second moment of Dirichlet twists
5
saving. The remaining ranges of n and m are trivially bounded with power
savings. Thus we have
X
X
af (n)af (m)
nm
√
(2.5)
µ(d)φ(q/d)
V
q2
nm
d|q
d<(log q)0.05
n≡m mod q/d
n6=m
(mn,q)=1
X
d|q
d<(log q)0.05
−θ −1
q
q
0.935
ε q
φ
|µ(d)|φ
q(log q)
+q
d
d
d
q(log q)0.985 .
Now we turn to the “diagonal”, terms i.e. those with n = m:
X
2
X
af (n)2
q
n
µ(d)φ
(2.6)
V
.
d
n
q2
n
d|q
d<(log q)0.05
(n,q)=1
First note that by Lemma 1.2 we have
X
X
q
q
(2.7)
=
+ O(q(log log q)−9 )
µ(d)φ
µ(d)φ
d
d
d|q
d<(log q)0.05
d|q
= qψ(q) + O(q(log log q)−9 ).
The last line follows by putting n = m = 1 in Lemma 2.1. The sum
over n can be written as an integral involving the Rankin–Selberg L-function
L(f ⊗ f, s) to get
Lemma 2.4.
(2.8)
X af (n)2 n2 V
n
q2
n
(n,q)=1
= KPq (1) log q + K1 Pq (1) + K2 Pq0 (1) + O(q −1/5 ),
for some constants Ki and
K = residues=1 2L(f ⊗ f, s) =
6(4π)k
kf k2 .
πΓ (k)
This is shown in Section 4 of [7], where it is also observed that Pq0 (1) ν(q)Pq (1). Now as ν(q) logloglogq q , we find that the diagonal terms give the
main term of Theorem 1.1.
6
P. Gao et al.
Large divisors. For d | q we can express the condition n ≡ m mod q/d
using the Dirichlet characters modulo q/d. We have
(2.9)
φ(q/d)
X
n≡m mod q/d
(mn,q)=1
af (n)af (m)
√
V
nm
X
=
X
n,m
χ mod q/d
(mn,q)=1
nm
q2
af (n)af (m)χ(n)χ(m)
√
V
nm
nm
.
q2
Now the last line equals
X
1 L(f × χ, s + 1/2)L(f × χ, s + 1/2)
πi
(c) χ mod q/d
Y
af (p)χ(p) χ(p2 ) Y
af (p)χ(p) χ(p2 )
ds
×
1−
+ 2s+1
1−
+ 2s+1 G(s)q 2s .
s+1/2
s+1/2
p
p
s
p
p
(2.10)
p|d
p|d
Here the extra Euler products arise due to the condition (nm, q) = 1. On
taking c = 1/log q and using Lemma 3.2 from the final section, we observe
that (2.10) is less than a constant multiple of
Y
q
10
4
2s ds (2.11)
log q log log q
1+ √
(s + 1) G(s)q
.
d
p
s
p|d
1/log q
The last integral is easily found to be log log q. Thus we have
X
X
af (n)af (m)
nm
√
(2.12)
V
µ(d)φ(q/d)
q2
nm
d|q
d≥(log q)0.05
n≡m mod q/d
(mn,q)=1
2
q log q(log log q)
X
d|q
d≥(log q)0.05
|µ(d)|
Q
p|d (1
d
√
+ 10/ p)
.
This is q log q(log log q)−7 by Lemma 1.2.
3. An upper bound for the second moment. In this section we will
need a different form of the approximate functional equation.
Lemma 3.1 (Approximate functional equation). For a primitive character χ modulo q and <(s) = 0 we have
Second moment of Dirichlet twists
(3.1)
L(f × χ, 1/2 + s) =
+ ıχ
where
Ws (x) =
n
q
n1/2+s
(2π)2s Γ (k/2 − s) X af (n)χ(n)
X af (n)χ(n)
n
7
Ws
q 2s Γ (k/2 + s)
n
n1/2−s
n
W−s
,
q
1 Γ (k/2 + s + y) y2
dy
e (2πx)−y
2πi
Γ (k/2 + s)
y
(c)
for any c > 0. We have Ws (x) C min{1, |s + 1|C x−C } for any C > 0, so
that the sums above are essentially supported on n < q 1+ε for any ε > 0.
This is a special case of Theorem 5.3 in [3]. The bound on Ws (x) can be
seen by moving the line of integration far to the right when x > 1 and just
to the left of the origin when x ≤ 1.
We show the following.
Lemma 3.2. For <(s) ≥ 0 and integers q and r we have
X
L(f × χ, 1/2 + s)L(f × χ, 1/2 + s)
(3.2)
χ mod q
Y
af (p)χ(p) χ(p2 ) af (p)χ(p) χ(p2 )
×
+ 2s+1
1−
+ 2s+1 1−
p
p
ps+1/2
ps+1/2
p|r
Y
10
4
|s + 1| q log q log log q
1+ √ .
p
p|r
Proof. The statement of the lemma certainly holds for <(s) ≥ 2, so by
the Phragmén–Lindelöf principle (cf. [3, Theorem 5.53]) it is enough to prove
the result for <(s) = 0. Thus throughout this proof, assume that s is purely
imaginary. First we demonstrate the bound
X∗
(3.3)
|L(f × χ, 1/2 + s)|2 |s + 1|4 d log d.
χ mod d
By Lemma 3.1 it is enough to bound
2
X∗ X af (n)χ(n)
n (3.4)
W
s
d n1/2+s
n
χ mod d
2
2
X X af (n)χ(n)
X X af (n)χ(n)
n n +
.
Ws
Ws
1/2+s
1/2+s
d
d n
n
χ mod d n≤d
χ mod d n>d
By a large sieve inequality (cf. [3, Section 7.5]) and the Rankin–Selberg
8
bound
P. Gao et al.
P
2
x, we have
2
X X af (n)χ(n)
X af (n)2
n W
d
d log d.
s
1/2+s
d n
n
χ mod d n≤d
n≤d
n≤x af (n)
The remaining sum in (3.4)
X
φ(d)
equals, by orthogonality,
X
2
a
(n)
n f
Ws
.
1/2+s
d n
1≤r≤d
n>d
(r,d)=1 n≡r mod d
If we drop the condition (r, d) = 1, this is bounded by
X X af (n)af (n + hd)
n
n + hd
W
W
d
s
s
d
d
n1/2+s (n + hd)1/2−s
h≥0 n>d
X X af (n)2 + af (n + hd)2 d d
(1 + h)−3 |1 + s|4 d|1 + s|4 .
n
n1/2 d1/2
h≥0 n>d
This establishes (3.3).
Now, the left hand side of (3.2) is bounded by
X
Y
10
|L(f × χ, 1/2 + s)|2 .
1+ √
p
p|r
χ mod q
We have
X
|L(f × χ, 1/2 + s)|2 = |L(f × χ0 , 1/2 + s)|2
χ mod q
+
Y af (p)χ(p) χ(p2 ) 2
|L(f × χ, 1/2 + s)|
1 − p1/2+s + p2s .
χ mod d
X X∗
d|q
2
p|q/d
Using (3.3) we have
X
X
Y 10
|L(f × χ, 1/2 + s)|2 |s + 1|4
d log d
1+ √
p
χ mod q
d|q
p|q/d
Y
1
4
|s + 1| q log q
1+
p
p|q
4
|s + 1| q log q log log q.
Acknowledgements. We would like to thank Prof. K. Soundararajan
for his encouragement. We also thank the referee for a careful reading and
helpful suggestions on exposition. The first author is supported by a research
fellowship from an Academic Research Fund Tier 1 grant at Nanyang Technological University. The second author is partially supported by the Na-
Second moment of Dirichlet twists
9
tional Science Foundation. The third author is financed by the ANR project
“Aspects Arithmétiques des Matrices Aléatoires et du Chaos Quantique”.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
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593–600.
P. Shiu, A Brun–Titchmarsh theorem for multiplicative functions, J. Reine Angew.
Math. 313 (1980), 161–170.
K. Soundararajan, The fourth moment of Dirichlet L-functions, in: Analytic Number
Theory, Clay Math. Proc. 7, Amer. Math. Soc., Providence, RI, 2007, 239–246.
T. Stefanicki, Non-vanishing of L-functions attached to automorphic representations
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math.NT/0610335.
Division of Mathematical Sciences
Nanyang Technological University
21 Nanyang Link
Singapore 637371
E-mail: penggao@ntu.edu.sg
Department of Mathematics
University of California, Los Angeles
520 Portola Plaza
Los Angeles, CA 90095-1555, U.S.A.
E-mail: rrkhan@math.ucla.edu
Institut de Mathématiques de Bordeaux
Université de Bordeaux 1
Laboratoire A2X, Bâtiment A33, Bureau 301 M
351 cours de la Libération
33405 Talence Cedex, France
E-mail: guillaume.ricotta@math.u-bordeaux1.fr
Received on 13.12.2008
and in revised form on 4.3.2009
(5891)