Acta Mathematica Sinica, New Series
1997, Oct., Vol.13, No.4, pp. 443–452
On the Third and Fourth Power Moments of
Fourier Coefficients of Cusp Forms
Cai Yingchun
(Department of Mathematics, Shandong Normal University, Jinan 250014, China)
Abstract The asymptotic formulae for the third and fourth power moments of Fourier coefficients
of cusp forms are proved in this paper.
Keywords Cusp form, Fourier coefficients, The estimation of mean value, Asymptotic formula
1991MR Subject Classification 11F
Chinese Library Classification O156
1
Introduction
Let a(n) be the nth Fourier coefficient of a cusp form of weight ω = 2m(m ≥ 6) for the full
modular group. In 1974 Deligne[1] proved the following deep result:
a(n) n
ω−1
2
d(n)
(1)
where d(n) denotes the divisor function, and the constant in is absolute. In fact this result
is best possible and is one of the crowning achievements of mathematics.
Assume that the a(n)’s are real and let
a(n).
A(x) =
n≤x
Some mathematicians have made contributions to the estimation of A(x).
In 1973 Joris[2] proved:
ω
1
(2)
A(x) = Ω± (x 2 − 4 log log log x).
√
In 1990 Ivic[3] showed that the interval [T, T + C T ] contains two points t1 and t2 such
that
ω
ω
−1
−1
(3)
A(t1 ) > Bt12 4 , A(t2 ) < −Bt22 4
where B > 0, C > 0.
By some similarity between A(x) and (x) (the error term in the Dirichlet divisor problem)
it is conjectured that
ω−1
1
(4)
A(x) = O(x 2 + 4 +ε )
Received February 23, 1995, Revised March 12, 1996, Accepted May 6, 1996
444
Acta Mathematica Sinica, New Series
Vol.13 No.4
should hold.
Jutila[4] proved a truncated formula for A(x), that is,
√
ω
1
ω
1
ω
1
1
π
A(x) = √ x 2 − 4
a(n)n− 2 − 4 cos 4π xn −
+ O(x 2 +ε N − 2 )
4
2π
n≤N
(5)
where 1 N x.
By (1), (5) and a trivial estimation one gets
ω
1
A(x) x 2 − 6 +ε .
(6)
1990 Ivic[5] proved the following square mean value formula for A(x),
X
1
A2 (x)dx = CX ω+ 2 + B(X),
(7)
1
where
C=
∞
1
1
a2 (n)n−ω− 2 ,
(4ω + 2)π 2 n=1
B(X) = O(X ω log5 X),
3
1 (log log log X)
.
B(X) = Ω X ω− 4
log X
Ivic[3] also proved the following upper bound estimation for the eighth power moment of
A(x), that is,
X
A8 (x)dx X 4ω−1+ε .
(8)
1
Thus in a moderate sense the conjecture (4) is true, one can easily derive this result from
(7) or (8).
As a corollary of (7) Ivic[3] proved the following sharpening of (6), that is,
ω
1
A(x) x 2 − 6 log2 x.
By the large value technique, Ivic[3] proved
X
ω−1
1
|A(x)|B dx X 1+B( 2 + 4 )+ε ,
1
(9)
0 ≤ B ≤ 8.
(10)
Inspired by [5], in this paper we shall prove the following third and fourth power moments
for A(x), that is,
Theorem 1
X
3ω
1
3ω
5
A3 (x)dx = c1 X 2 + 4 + O(X 2 + 28 +ε ),
1
√
where
c1 =
Theorem 2
2
π (6ω + 1) √
3
√
√
n+ m= k
1
X
1
ω
a(m)a(n)a(k)(mnk)− 4 − 2 .
1
A4 (x)dx = c2 X 2ω + O(X 2ω− 23 +ε ),
Cai Yingchun
On the Third and Fourth Power Moments of Fourier Coefficients of Cusp Forms
where
c2 =
3
64π 4 ω
√
ω
√
√
√
m+ n= k+ l
445
1
a(m)a(n)a(k)a(l)(mnkl)− 2 − 4 .
As a corollary of Theorem 2 and (7) we get the following sharpening of (10) for 0 ≤ B ≤ 4,
that is,
X
ω−1
1
Corollary
|A(x)|B dx X 1+B( 2 + 4 ) , 0 ≤ B ≤ 4.
1
The proofs of Theorems 1 and 2 are based on the Voronoi-type formula (5) for A(x). In the
proofs we have not made attempt to get the best results for the O terms.
2
Some Lemmas
Lemma 1[1]
Lemma 2[4]
a(n) n
ω−1
2
d(n).
ω
1
√
ω
1
ω
1
x 2 −4 π
A(x) = √
a(n)n− 2 − 4 cos 4π nx −
+ O(x 2 +ε N − 2 ),
4
2π n≤N
(1 N x).
Lemma 3[5]
If m, n, k are natural numbers such that
√
m+
√
√
n = k then
√
√
√
1
3
| m + n − k| ≥
(max(m, n, k))− 2 .
27
Lemma 4[5]
If m, n, k, and l are natural numbers such that
√
√
√
√
√
√
√
√
m + n = k + l or
m + n + k = l,
then
√
√
√
√
2
| m + n − k − l| (max(m, n, k, l))− 7
or
√
√
√
√
2
| m + n + k − l| (max(m, n, k, l))− 7 .
Lemma 5[5] . For any real numbers α = 0, β and 0 < δ < 12 , we have
1
1
1
3
Kδ + |α| 3 K 2 + |α|− 2 K 4 ,
K<k≤2K
√
α k+β<δ
the implied constant in being absolute.
The following Lemma 6 is well known.
Lemma 6 If g(x) and h(x) are continuous real-valued functions of x and g(x) is monotonic then
v
b
g(x)h(x)dx max |g(x)|
max h(x)dx .
a≤x≤b
a≤u<v≤b
a
u
446
Acta Mathematica Sinica, New Series
Vol.13 No.4
The Third Power Moment of A(x)
3
Let
ω
1
√
ω
1
π
x 2 −4 a(n)n− 2 − 4 cos 4π xn −
,
(x, N ) = √
4
2π n≤N
ω
1
R(x, N ) = O(x 2 +ε N − 2 ),
1 N x.
For simplicity, put
ω
1
r = r(m, n, k) = a(m)a(n)a(k)(mnk)− 2 − 4 ,
m, n, k ≤ N
and
r = 0,
otherwise.
9
14
In this section 2 ≤ H ≤ X
2 ,N = H .
By the elementary formula
cos A cos B cos C =
1
(cos(−A + B + C) + cos(A − B + C) + cos(A + B − C) + cos(A + B + C)),
4
we have
3
where
(x, N )
√ √
√
√
3
3 = √3 3 x 2 ω− 4
r cos 4π x( m + n − k) − π
4
8 2π
√
√
√
√
3ω
3
r cos 4π x( m + n + k) − 34 π
+ √1 3 x 2 − 4
8 2π
= 0+ 1+ 2
0
3ω
3
= √3 3 x 2 − 4
8 2π
1
3ω
3
= √3 3 x 2 − 4
8 2π
2
3ω
3
= √1 3 x 2 − 4
8 2π
Thus
2H
√
√
√
m+ n= k
r,
√ √
√
√
π
r cos 4π x( m + n − k) −
,
4
√
√
√
m+ n=
k
√ √
√
√
r cos 4π x( m + n + k) − 34 π .
2H
3ω
3
3 dx = √ 3
x 2 − 4 dx.
r
0
8 2π
H
H
By Lemma 1
or
ω
m,n,
k>N
√
√
√
m+ n= k
1
a(m)a(n)a(k)(mnk)− 2 − 4
√
√
√
m+ n= k
m,n,
k>N
or
3
d(m)d(n)d(k)(mnk)− 4
Cai Yingchun
On the Third and Fourth Power Moments of Fourier Coefficients of Cusp Forms

447



m>N √
√
√
m+ n= k
+
3

 d(m)d(n)d(k)(mnk)− 4 ,
k>N √
√
√
m+ n= k
3
d(m)d(n)d(k)(mnk)− 4
m>N √
√
√
m+ n= k
√
√
√
1
( k = m + n (mn) 4 )
3
(mnk)− 4 +ε
m>N √
√
√
m+ n= k
3
m>N
3
(mn)− 4 +ε (mn)− 8 +ε 3
k>N √
√
√
m+ n= k
r=
2H
3
(mnk)− 4 +ε k>N √
√
√
m+ n= k
3
1
(mnk)− 4 +ε N − 8 +ε .
m> N
√ 10 √
√
m+ n= k
ω
1
1
a(m)a(n)a(k)(mnk)− 2 − 4 + O(N − 8 +ε ),
H
1
m>N
d(m)d(n)d(k)(mnk)− 4 Hence
9
m− 8 +ε N − 8 +ε ,
ω
1
√3 3
a(m)a(n)a(k)(mnk)− 2 − 4
0 dx =
8 2π
+O(H
3ω
1
2 +4
2H
x
3ω
3
2 −4
dx
(3.1)
H
1
N − 8 +ε ).
By Lemma 6
2H
H
Hence
2H
H
2 dx
√
xp cos(A x + B)dx
(A = 0)
√
A
√ cos(A x + B) dx
=
2A x
2 x
H
v
√
1
A
p+ 12 −1
√ cos(A x + B)dx H p+ 2 A−1 .
A
max
H
H≤u<v≤2H u 2 x
2H
−1 p+ 12
H
3ω
1
2 −4
H
3ω
1
2 −4
H
3ω
1
2 −4
(3.2)
r
√
√
√
m+ n+ k
3
(mnk)− 4 +ε
√
√
√
m+ n+ k
m,n,k≤N
11
(mnk)− 12 +ε H
√
√
√
1
1
( m + n + k ≥ (mnk) 6 )
3
3ω
1
2 −4
1
N 4.
m,n,k≤N
For simplicity, let =
√
√
√
3
m + n − k. When | | ≥ H − 7 by (3.2) we have
2H
x
H
3ω
3
2 −4
√
3ω
1
π
cos 4π x −
dx H 2 − 4 −1 ,
4
(3.3)
448
Acta Mathematica Sinica, New Series
Vol.13 No.4
3
and when | | < H − 7 we have the trivial upper bound
2H
x
3ω
3
2 −4
H
√
3ω
1
π
cos 4π x −
dx H 2 + 4 .
4
(3.4)
We dedeuce from (3.3) and (3.4) that
2H
H
H
1 dx
3ω
1
2 −4
r −1 +H
3
||≥H − 7
||<H


H
3ω
1
2 −4


3
1
=H
(M1 + M2 ) + H
r
3
7

3ω
1
 r −1 +H 2 + 4
+
H − 7 <||<m 4
3ω
1
2 −4
3ω
1
2 +4
(3.5)
r
3
1
||<H − 7
||>m 4
3ω
1
2 +4
M3 .
3
for given m√≤ n ≤ N there is at most one k such that | | < H − 7 , since
√ For√M3 and
√
k = m + n − and | N | = o(1). Such k, if exists, must be greater than n. By Lemma
3
3
3
2
4, 0 < | | < H − 7 implies n− 2 H − 7 , that is, n H 7 ; hence
3
3
5
1
(mn)− 4 n− 4 H ε
n− 4 H − 14 +ε .
(3.6)
M3 H ε
m≤n≤N
2
nH 7
In the sum M1 , for m ≤ n ≤ N,
√
√
√
k = ( m + n)2 + O(| | n) > n,
so there are
√
1+|| m
such k. Hence
Hε
M1
√
3
3
(mn)− 4 | |−1 (1 + | m|)m− 4
m≤n≤N
Hε
H
Finally,
M2 3
3
3
n− 4 m− 2 H 7 + H ε
m≤n≤N
3
ε
7 +ε
1
3
n− 4 m−1
(3.7)
m≤n≤N
3
+ H N 4 H 7 +ε .
r| |−1 H ε
3
1
1
(mnk)− 4 m− 4 H ε N 2 .
(3.8)
m,n,k≤N
1
||>m 4
From (3.5), (3.6), (3.7) and (3.8) we deduce that
2H H
1
dx H
3ω
5
2 + 28 +ε
.
(3.9)
Combiming (3.1), (3.2) and (3.9) we get
2H H
3
(x, N )dx = c1 ((2H)
3ω
1
2 +4
−H
3ω
1
2 +4
) + O(H
3ω
5
2 + 28 +ε
).
(3.10)
Cai Yingchun
On the Third and Fourth Power Moments of Fourier Coefficients of Cusp Forms
449
By the same but simpler arguments it can be proved that
2H H
2
1
(x, N )dx H ω+ 2 .
(3.11)
Since
(a + b)3 = a3 + O(a2 |b| + |b|3 ),
it follows from (3.10) and (3.11) that
2H
3
A (x)dx
2H
=
H
H
2H
=
3
(x, N )dx + O H
ω
9
2 − 28 +ε
2H 2
(x, N )dx
+ O(H
3ω
1
2 − 28 +ε
3
(x, N )dx + O(H
3ω
5
2 + 28 +ε
).
H
(3.12)
Lastly it is a routine argument to deduce from (3.10) and (3.12) that
X
1
4
A3 (x)dx = c1 X
3ω
1
2 +4
+ O(x
3ω
5
2 + 28 +ε
).
The Fourth Power Moment of A(x)
For simplicity, put
ω
1
r1 = r1 (m, n, k, l) = a(m)a(n)a(k)a(l)(mnkl)− 2 − 4 ,
r1 = 0,
m, n, k, l ≤ N,
otherwise,
and all the notations are the same as in Section 3 except
27
N = H 46 .
By the elementary formula
cos A cos B cos C cos D
= 81 (cos(−A + B + C + D) + cos(A − B + C + D)
+ cos(A + B − C + D) + cos(A + B + C − D)
+ cos(−A + B + C − D) + cos(A − B + C − D)
+ cos(A + B − C − D) + cos(A + B + C + D)),
we have
4
(x, N )
)
H
√ √
√
√
2ω−1 √
= x 4
r1 cos 4π x( m + n + k − l) − π
2
8π
√
√
2ω−1 √
√
√
r1 cos(4π x( m + n − k − l))
+ 3x 4
32π
√
√
2n−1 √ √
√
+x 4
r1 cos(4π x( m + n + k + l) − π)
32π
= 3+ 4+ 5+ 6
450
Acta Mathematica Sinica, New Series
where
3
4
5
2ω−1
= 3x 4
32π
√
√
√
√
m+ n= k+ l
2ω−1
= 3x 4
32π
√
√
√
√
m+ n= k+ l
r1 ,
√
√
√ √
√
r1 cos(4π x( m + n − k − l)),
√ √
√
√
2ω−1 √
=x 4
r1 cos 4π x( m + n + k − l) − π
2 ,
8π
2ω−1
6
Vol.13 No.4
= −x 4
32π
√
√
√ √
√
r1 cos(4π x( m + n + k + l)).
The following arguments are similar to that of [5] Section 3.√ √
√
√
For natural numbers m, n, k and l, the condition m+ n = k+ l holds iff (m, n) = (k, l)
or m, n, k, l all have the same square-free part, h, say, such that m = α2 h, n = β 2 h, k = θ2 h, l =
ϕ2 h and α + β = θ + ϕ. So in the sum
ω
1
=
a(m)a(n)a(k)a(l)(mnkl)− 2 − 4
√
√
√
√
m+ n= k+ l
m,n,k,l≤N
if let the variables m, n, k and l run over all natural numbers, then by Lemma 1 the error thus
induced is
∞
3
k=1
n>N
N
3
n− 2 k− 2 d2 (n)d2 (k) +
− 12 +ε
+
α
− 32
α,β,θ,ϕ≥1
α2h >N
d(α2 )h−3 d4 (h)
α2 h>N
1
3
(αβθϕ)− 2 h−3 d(α2 )d(β 2 )d(θ2 )d(ϕ2 )d4 (h)
1
1
N − 2 +ε + N − 4 +ε N − 4 +ε .
Thus
2H 3
H
The treatment of
3
32π 4
dx =
√
√
√
√
m+ n= k+ l
is similar to that of
6
2H
H
r1
2
2H
1
x2ω−1 dx + O(H 2ω N − 4 +ε ).
(4.1)
H
in Section 3. Applying (3.2) we have
√
√
√ √
√
x2ω−1 cos(4π x( m + n + k + l))dx
√
√
√
1 √
H 2ω− 2 ( m + n + k + l)−1 .
Hence by Lemma 1
2H
H
6 dx
1
H 2ω− 2
1
H 2ω− 2 +ε
√
√
√
√
r1 ( m + n + k + l)−1
m,n,k,l<N
√
√
√
3 √
(mnkl)− 4 ( m + n + k + l)−1
√
√
√
√
1
( m + n + k + l ≥ 41 (mnkl) 8 )
7
1
1
1
(mnkl)− 8 H 2ω− 2 +ε N 2 .
H 2ω− 2 +ε
m,n,k,l<N
(4.2)
Cai Yingchun
On the Third and Fourth Power Moments of Fourier Coefficients of Cusp Forms
451
√
√
√
√
21 . Similarly to (3.2), when | | ≥ H −c3 we have
m + n + k − l and c3 = 46
1
2H
√ √
√
√
√
1
π
dx H 2ω− 2 | 1 |−1 .
x2ω−1 cos 4π x( m + n + k − l) −
2
H
Let 1 =
When | 1 | < H −c3 we have the trivial bound
2H
√ √
√
√
√
π
x2ω−1 cos 4π x( m + n + k − l) −
dx H 2ω .
2
H
Thus
2H H
5
dx H 2ω
1
r1 + H 2ω− 2
0<|1 |<H −c3
1
r1 | 1 |−1 = H 2ω M4 + H 2ω− 2 M5 , (4.3)
|1 |≥H −c3
say.
First consider M4 . Without loss of generality we may assume m ≤ n ≤ k ≤ N. The condition
| 1 | ≤ H −c3 implies
√
√
√
√
√
l ≤ m + n + k + | 1 | ≤ 4 k,
so that
√
√
√
√
√
√
√
√
√
| m + n + k + l| | m + n + k − l| ≤ 7 kH −c3 ,
that is,
√
√
√
√
|1 − (m + n + k + 2( mn + nk + mk))| ≤ 7 kH −c3 .
Since
(4.4)
√
√
7 kH −c3 ≤ 7 N H −c3 = o(1),
there will be no natural number l for which 0 < | l | < H −c3 unless
√
√
√
√
0 < ||2( mn + nk + mk|| < 7 kH −c3 .
(4.5)
In the latter case, there will be at most one eligible l. In view of (4.5), such an l must also
satisfy k ≤ l ≤ 10k. Hence by Lemma 1,
3
3
3
(mnkl)− 4 N ε N ε
(mn)− 4 k− 2 .
M4 m≤n≤k≤N
m≤n≤k≤N
(4.5)
0<|1 |≤H −c3
√
√
√
√
Applying Lemma 5 with α = 2( m + n), β = 2 mn and δ = 7 2kH −c3 , we have
3
3
3
1
1
1
3
k− 2 K − 2 (K 2 H −c3 + n 6 K 2 + n− 4 K 4 ),
(4.7)
K<k<2K
whence
m≤n≤2K
K≤k≤2K
(4.5)
3
3
(mn)− 4 k− 2
3
1
1
3
(mn)− 4 (H −c3 + n 6 K −1 + n− 4 K − 4 )
m≤n≤2K
1
2
K H −c3 + K
(4.8)
− 13
+K
− 12
1
2
K H −c3 + K
2
− 13
.
3
By Lemma 4, 0 < |1 | < H −c3 implies k− 7 H −c3 , since k ≤ l ≤ 10k. Hence H 7 c3 k N.
We deduce from (4.7) and (4.8) that
1
2
1
M4 N ε (N 2 H −c3 + H − 21 c3 ) H ε− 23 .
(4.9)
452
Acta Mathematica Sinica, New Series
Vol.13 No.4
1
To estimate M5 , we split this sum into two sub-sums according to H −c3 < | 1 | ≤ k 4
1
and | 1 | > k 4 respectively. Following the same arguments for M2 and M3 in Section 3, we
3
find that these sub-sums are bounded by H c3 +ε and N 4 +ε respectively. Hence M5 H c3 +ε .
Combining the estimations for M4 and M5 we get
2H 5
H
1
dx H 2ω− 23 +ε .
(4.10)
By a similar argument, it can be proved that
2H 4
H
1
dx H 2ω− 23 +ε .
(4.11)
Collecting (4.1), (4.2), (4.10) and (4.11) we obtain
2H
4
H
1
(x, N )dx = c2 ((2H)2ω − H 2ω ) + O(H 2ω− 23 +ε ).
(4.12)
By the elementary formula: (a + b)4 = a4 + O(|a|3 |b| + |b|4 ), we get
2H
A4 (x)dx =
H
2H
4
ω
1
(x, N )dx + O(H 2 +ε N − 2 )
H
2H
H
|
(x, N )|3 dx + O(H 2ω+1 N −2 ).
(4.13)
Then by Holder’s inequality, this yields
2H
H
|
(x, N )|3 dx 2H
(x, N )
4
34 dx
H
14
2H
dx
H
3
1
H 2 ω+ 4 .
Finally we obtain
2H
H
1
A4 (x)dx = c2 ((2H 2ω ) − H ω ) + O(H 2ω− 23 +ε ).
(4.14)
Lastly it is a routine argument to deduce from (4.14) that
1
X
1
A4 (x)dx = c2 X 2ω + O(X 2ω− 23 +ε ).
References
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2
3
4
5
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KAI-MAN TSANG. Higher-power moments of (x), E(t) and P (x). Proc London Math Soc, 1992, 61:
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