Quadratic forms connected with Fourier coefficients of Maass cusp forms Liqun HU

Department of Mathematics, Shandong University, Jinan 250100, China
Department of Mathematics, Nanchang University, Nanchang 330031, China
c
Higher Education Press and Springer-Verlag Berlin Heidelberg 2015
Abstract For the normalized Fourier coefficients of Maass cusp forms λ(n)
and the normalized Fourier coefficients of holomorphic cusp forms a(n), we
give the bound of

m21
+m22
+m23
x λ(m21
+ m22
+ m23
)Λ(m21
+ m22
+ m23
) and

m21
+m22
+m23
x a(m21
+ m22
+ m23
)Λ(m21
+ m22
+ m23
).
Keywords Circle method, Fourier coefficients of Maass cusp forms, quadratic
form, exponential sum
MSC 11F30, 11P55
1 Introduction
The ternary quadratic form m21
+ m22
+ m23
is significant in the number theory
and draws the attention of many researchers. Suppose that x is a large positive
real number. Vinogradov [10] and Chen [2] independently studied the wellknown
sphere problem about the number of lattice points in the 3-dimensional
ball u21
+ u22
+ u23
x and proved the asymptotic formula

m21
+m22
+m23
x
mj
∈Z (j=1,2,3)
1 =
4
3 πx3/2 + O(x2/3).
The exponent 2/3 was improved to 29/44 by Chamizo and Iwaniec [1], and to
21/32 by Heath-Brown [6].
Recently, several authors studied some problems connected with the ternary
quadratic form by different methods. Friedlander and Iwaniec [3] studied the
number of prime vectors among integer lattice points in the 3-dimensional ball.
Received March 14, 2014; accepted July 11, 2014
E-mail: huliqun@ncu.edu.cn
1102 Liqun HU
They proved that the number π3(x) of integer points
(m1,m2,m3) ∈ Z3
with
m21
+ m22
+ m23
= p x
satisfies
π3(x) ∼ 4π
3
x3/2
log x
,
which can be viewed as a generalization of the prime number theorem.
Let Λ(n) stands for the function of von Mangoldt. Guo and Zhai [4] studied

m21
+m22
+m23
x
Λ(m21
+ m22
+ m23
) = 8C3I3x3/2 + O(x3/2 log−A x), (1.1)
where A > 0 is a fixed constant and
C3 =
+∞
q=1
1
q3ϕ(q)

0 a<q
(a,q)=1
G3(a, 0, q)Cq(−a),
I3 =
+∞
−∞
H3(z)dz, H3(z) =
1
0
e(u2z)du
3 1
0
e(−uz)du.
Let λ(n) and a(n) denote the normalized Fourier coefficients of Maass cusp
forms and holomorphic cusp forms, respectively. We can easily obtain

m21
+m22
+m23
x
a(m21
+m22
+ m23
)Λ(m21
+ m22
+ m23
) = O(x
3
2+ε) (1.2)
by the Ramanujan conjecture and the result of (1.1). But for the λ(n), the
bound of
m21
+m22
+m23
x
λ(m21
+ m22
+ m23
)Λ(m21
+ m22
+ m23
) (1.3)
ia not trivial. In this paper, we will use the classical circle method to study
(1.3) and improve the result of (1.2). Our main results are the following.
Theorem 1 Define
πλ,Λ(x) :=

m21
+m22
+m23
x
λ(m21
+ m22
+ m23
)Λ(m21
+ m22
+ m23
).
We have
πλ,Λ(x) = O(x3/2 logc x),
where c > 0 is a fixed constant.
Quadratic forms connected with Fourier coefficients of Maass cusp forms 1103
Theorem 2 Define
πa,Λ(x) :=

m21
+m22
+m23
x
a(m21
+ m22
+ m23
)Λ(m21
+m22
+ m23
).
We have
πa,Λ(x) = O(x3/2 logc
x),
where c > 0 is a fixed constant.
Notation Throughout the paper, the letter ε denotes a positive constant
which is arbitrarily small but may not the same at different occurrences. For
any real number t, [t] denotes its integer part, {t} its fractional part, and
ψ1(t) = t − [t] − 1
2, t = min({t}, 1 − {t}).
The functions ψj(u) (j 2) are defined by
⎧⎪⎪⎨
⎪⎪⎩
ψj+1(u) − ψj+1(0) =
u
0
ψj(t)dt,
1
0
ψj+1(u)du = 0,
j 1.
Finally, G(a, b, q) denotes the quadratic Guass sum
G(a, b, q) =
q
r=1
e
ar2 + br
q

,
where e(z) = exp(2πiz).
2 Outline of method
For any α ∈ R and y > 1, define
S1(α; y) :=

1 m y
e(m2α), S2(α; y) :=

|m| y
e(m2α), (2.1)
T(α; y) :=

1 n y
λ(n)Λ(n)e(nα).
It is easily seen that
S2(α; y) = 2S1(α; y) + 1. (2.2)
By (2.2) and the well-known identity
1
0
e(uα)dα =


1, u= 0,
0, 0 = u ∈ Z,
1104 Liqun HU
we have
πλ,Λ(x) =
1
0
S3
2 (α;
√
x )T(−α; x)dα
= 8
1
0
S3
1(α;
√
x )T(−α; x)dα + 12
1
0
S2
1 (α;
√
x )T(−α; x)dα
+6
1
0
S1(α;
√
x )T(−α; x)dα +
1
0
T(−α; x)dα
= 8
1
0
S3
1(α;
√
x )T(−α; x)dα + O(x log x)
= 8
1− 1
Q
−1/Q
S3
1(α;
√
x )T(−α; x)dα + O(x log x). (2.3)
In order to apply the circle method, we set
P = log5 x, Q = x log−6 x. (2.4)
By Dirichlet’s lemma on rational approximation, each α ∈ [−1/Q, 1 − 1
Q]
may be written in the form
α = a
q
+ z, |z|
1
qQ
,
for some integers a, q with 0 a < q P and (a, q) = 1. We define the major
arc M and minor arc C (M) as follows:
M(a, q) :=
a
q
− 1
PQ
,
a
q
+
1
PQ

,
M :=

1 q P

0 a<q
(a,q)=1
M(a, q), C (M) :=

− 1
Q
, 1 − 1
Q

\M.
Then for Q >0,
πλ,Λ(x) = 8S1(x) + 8S2(x) + O(x log x), (2.5)
where
S1(x) =

M
S3
1 (α;
√
x )T(−α; x)dα, S2(x) =

C (M)
S3
1(α;
√
x )T(−α; x)dα.
3 Some lemmas
Before working on the substantive parts of the proofs of Theorem 1 and 2, we
need present a number of estimates which we will use in the following sections.
Lemmas 3.1 and 3.4 are well known.
Quadratic forms connected with Fourier coefficients of Maass cusp forms 1105
Lemma 3.1 Suppose that bn ∈ C (n 1) are such that their summation
function B(u) can be written as
B(u) :=

n u
bn = M(u) + E(u),
where M(u) is continuously differentiable on (0,+∞). Suppose that f(u) is
continuously differentiable on [u1, u2], where u1 0. Then we have

u1<n<u2
bnf(n) =
u2
u1
f(u)M
(u)du +
u2
u1
f(u)dE(u).
Lemma 3.2 [5] For any H 2, we have
ψ1(u) =

1 |h| H
e(hu)
2πh
+ O

min

1,
1
H u

,
min

1,
1
H u

=
+∞
h=−∞
f(h)e(hu),
f(0) logH
H
, f(h) min
1
|h| ,
H
h2

.
Lemma 3.3 [8] For fixed l 1, we have
ψ2l(u) = (−1)l−1
+∞
h=1
2
(2hπ)2l cos(2πu),
ψ2l+1(u) = (−1)l−1
+∞
h=1
2
(2hπ)2l+1 sin(2πu),
where ψj(u) (j 2) are defined in Section 1.
Lemma 3.4 Suppose q, r ∈ N. Then we have
q
h=1
e
hr
q

=


q, q | r,
0, otherwise.
Lemma 3.5 [7] Suppose q ∈ N, a,b ∈ Z, q 3, (a, q) = 1. Then the Guass
sum satisfies
G(a, b, q) :=
q
h=1
e
ah2 + bh
q


√
q.
Lemma 3.6 [8] Suppose α ∈ R. Then we have

1 n y
e(nα) min

y,
1
α

.
1106 Liqun HU
Lemma 3.7 [8] Suppose α = a
q + z with (a, q) = 1, q 3, and |z| 1/q2.
Then we have

n N
min

y,
1
αn

(y + q log q)

1 + N
q

.
4 Estimation of S1(α;
√
x )
The exponential sum S1(α;
√
x ) is of importance in the proof of Theorem 1.
We will give the asymptotic formula of S1(α;
√
x) on the major arc and the
estimate on the minor arc.
4.1 Estimate of S1(α;
√
x ) on major arc
For α = a
q + z ∈ M with |z| 1
qQ, (a, q) = 1, we have
S1(α;
√
x) =

1 n
√
x
e

n2
a
q
+ z

=
q
r=1
e
r2a
q

1 n
√
x
n≡r (mod q)
e(n2z). (4.1)
Since
n u

n≡r (mod q)
1 = 1+
u − r
q

=
1
2
+ u − r
q
− ψ1
u − r
q

,
using Lemma 3.1 and taking
M(u) = u − r
q
+
1
2, E(u) = −ψ1
u − r
q

,
we have

1 n
√
x
n≡r (mod q)
e(n2z) =
1
q
√
x
0
e(u2z)du −
√
x
0
e(u2z)dψ1
u − r
q

=:

1
−

2
. (4.2)
By the change of u =
√
xv, we get
q
r=1
e
r2a
q

1
= G(a, 0, q)
q
√
x
1
0
e(xzv2)dv. (4.3)
Let
g(u) = e(u2z),
and let l 1 be a fixed positive integer to be determined later. Repeated
partial integration gives
Quadratic forms connected with Fourier coefficients of Maass cusp forms 1107

2
= g(u)ψ1
u − r
q

√
x
0
−
√
x
0
ψ1
u − r
q

g
(u)du
= g(u)ψ1
u − r
q

√
x
0
− q
√
x
0
g
(u)dψ2
u − r
q

= g(u)ψ1
u − r
q

√
x
0
− qg
(u)ψ2
u − r
q

√
x
0
+q
√
x
0
ψ2
u − r
q

g
(u)du
= · · ·
=
l
j=0
(−1)jqjg(j)(u)ψj+1
u − r
q

√
x
0
+(−1)l+1ql
√
x
0
ψl+1
u − r
q

g(l+1)(u)du. (4.4)
Thus, we have
q
r=1
e
r2a
q

2
=
l
j=0
(−1)jqjg(j)(u)
q
r=1
e
r2a
q

ψj+1
u − r
q

√
x
0
+(−1)l+1ql
√
x
0
ψl+1
u − r
q
q
r=1
e
r2a
q

g(l+1)(u)du. (4.5)
We now show for any fixed j 0 and uniformly for 0 u
√
x that
q
r=1
e
r2a
q

ψj+1
u − r
q



√
q log(q + 1), j= 0,
√
q, j 1.
(4.6)
Estimate (4.6) is trivial for q = 1, 2. Suppose now q 3 and j = 0. Taking
H = q2 in Lemma 3.2, we can write
q
r=1
e
r2a
q

ψ1
u − r
q

=:

1
+

2
, (4.7)
where

1
=
q
r=1
e
r2a
q

1 |h| q2
1
2πih
e
h(u − r)
q

,

2
=
q
r=1
O

min

1,
1
q2 (u − r)/q

.
By Lemma 3.5, we get

1
=

1 |h| q2
e(hu/q)
2πih
G(a,−h, q)
√
q log q. (4.8)
1108 Liqun HU
For

2, by the second expression in Lemma 3.2 and Lemma 3.4, we get

2


h∈Z
f(h)e
hu
q
q
r=1
e

− hr
q

q

h∈Z
q|h
|f(h)| log q. (4.9)
From (4.7)–(4.9), we see that (4.6) holds for j = 0. Similarly, the case j 1
follows from Lemmas 3.3 and 3.5.
Now, we bound g(u) for 0 u
√
x . We shall show that for any fixed
k 0, the estimate
g(k)(u) k
√
x
qQ
k
(4.10)
holds uniformly for 0 u
√
x . Let f(u) = 4πiuz. Obviously, (4.10) holds for
k = 0. Since g (u) = g(u)f(u), we see that (4.10) holds for k = 1. For k > 1, by
Leibniz’s formula, we get
g(k+1)(u) = (f(u)g(u))(k) = f(u)g(k)(u) + kf
(u)g(k−1)(u),
and hence,
|g(k+1)(u)| |f(u)g(k)(u)| + k|f
(u)g(k−1)(u)|
4π
√
x |z| |g(k)(u)| + 4kπ|z| |g(k−1)(u)|. (4.11)
Now, (4.10) follows from (4.11) by
1
qQ
√
x
qQ
2
.
Inserting (4.6) and (4.10) into (4.5), we get
q
r=1
e
r2a
q

2

√
q log(q + 1) +
√
q
l
j=1
√
x
Q
j
+
√
x
√
x
Q
l+1
.
Taking l = [2/ε] and using the value of Q, we get
√
q
l
j=1
√
x
Q
j

√
q,
√
x
√
x
Q
l+1
1.
From the above two estimates, we have
q
r=1
e
r2a
q

2

√
q log(q + 1). (4.12)
From (4.1)–(4.3) and (4.12), we get the following lemma.
Quadratic forms connected with Fourier coefficients of Maass cusp forms 1109
Lemma 4.1 For α = a
q + z ∈M with |z| 1
qQ and (a, q) = 1, we have
S1(α;
√
x) = G(a, 0, q)
q
√
x
1
0
e(xzv2)dv + O(
√
q log(q + 1)). (4.13)
4.2 Estimate of S1(α;
√
x ) on minor arc
For α = a
q + z ∈ C(M) with |z| 1
qQ and (a, q) = 1, we have
|S1(α;
√
x )|2 =

1 m,n
√
x
e((m2 −n2)α) = [
√
x−1]+T(
√
x)+T(
√
x ) , (4.14)
where
T(
√
x) =

1 n<m
√
x
e((m2 − n2)α).
By Lemmas 3.6 and 3.7, we have
T(
√
x) =

1 n<m
√
x
e((m + n)(m − n)α)
=

1 v
√
x−1
e(v2α)

1 n
√
x−v
e(2nvα)


1 v
√
x
min
√
x,
1
2vα



1 v 2
√
x
min
√
x,
1
vα

(
√
x + q log q)

1 +
√
x
q

xq
−1 + q log q +
√
x log q. (4.15)
From (4.14) and (4.15), we get the following lemma.
Lemma 4.2 For α = a
q + z ∈ C(M) with |z| 1
qQ and (a, q) = 1, we have
S1(α;
√
x ) x1/2q
−1/2 + q1/2 log1/2 q + x1/4 log1/2 q x1/2P
−1/2.
5 Estimation of T (−α; x) on major arc
Suppose α = a
q +z ∈ M with |z| 1
qQ and (a, q) = 1. Applying the [9, Theorem
1], we have

1 n x
τ (n)Λ(n)e(nα) x11/2(xq
−1/2 + x1/2q1/2 + x5/6) logc1 x,
1110 Liqun HU
where τ (n) is Ramanujan’s τ -function and c1 is a suitable constant.
Because λ(n) is the normalized Fourier coefficients of Maass cusp forms, we
can get the following lemma by normalizing τ (n).
Lemma 5.1 Suppose α = a
q + z ∈ M with |z| 1
qQ and (a, q) = 1. Then
T(−α; x) (xq
−1/2 + x1/2q1/2 + x5/6) logc1 x, (5.1)
where c1 is a suitable constant.
6 Proofs of Theorems 1 and 2
Proof of Theorem 1 We first treat the integral on the major arc. We have

M
S3
1(α;
√
x )T(−α; x)dα =

1 q P

0 a<q
(a,q)=1
a
q+ 1
qQ
a
q
− 1
qQ
S3
1 (α;
√
x )T(−α; x)dα.
(6.1)
By Lemmas 4.1 and 5.1, after some simplification we get
S3
1(α;
√
x )T(−α; x)


x3/2G3(a, 0, q)
q3
1
0
e(xzv2)dv
3
+ O

x
G2(a, 0, q)
q3/2
log(q + 1)
1
0
e(xzv2)dv
2
+ O

x1/2G(a, 0, q) log2(q + 1)
1
0
e(xzv2)dv

+ O(q3/2 log3(q + 1))

(xq
−1/2 + x1/2q1/2 + x5/6) logc1 x
x5/2q
−2 logc1 x + x2q
−1 logc1+1 x + x3/2 logc1+1 x + xq logc1+1 x.
Thus,
a
q+ 1
qQ
a
q
− 1
qQ
S3
1(α;
√
x )T(−α; x)dα
x5/2q
−3Q
−1 logc1 x + x2q
−2Q
−1 logc1+1 x
+x3/2q
−1Q
−1 logc1+1 x + xQ
−1 logc1+1 x. (6.2)
Combining (6.1) and (6.2), we get

M
S3
1(α;
√
x )T(−α; x)dα
Quadratic forms connected with Fourier coefficients of Maass cusp forms 1111


1 q P

0 a<q
(a,q)=1
(x5/2q
−3Q
−1 logc1 x + x2q
−2Q
−1 logc1+1 x
+x3/2q
−1Q
−1 logc1+1 x + xQ
−1 logc1+1 x)
x5/2Q
−1 logc1 x + x2Q
−1 logc1+1 x
+x3/2PQ
−1 logc1+1 x + xP2Q
−1 logc1+1 x. (6.3)
Inserting (2.4) into (6.3), we can get

M
S3
1(α;
√
x )T(−α; x)dα x3/2 logc x, (6.4)
where c = c1 + 6 is a fixed constant.
Now, we study the integral on the minor arc. By Cauchy’s inequality, we
have

C (M)
S3
1(α;
√
x )T(−α; x)dα
max
α∈C (M)
|S1(α;
√
x )|
1
0
|S1(α;
√
x )|2|T(−α; x)|)dα
max
α∈C (M)
|S1(α;
√
x )|
1
0
|S1(α;
√
x )|4dα
1/2 1
0
|T(−α; x)|2dα
1/2
max
α∈C (M)
|S1(α;
√
x )|


m21
+m22
=m23
+m24
1 m1,m2,m3,m4

√
x
1
1/2
1 n x
λ2(n)Λ2(n)
1/2
max
α∈C (M)
|S1(α;
√
x )|


m21
−m23
=m24
−m22
1 m1,m2,m3,m4

√
x
1
1/2
1 n x
λ2(n)Λ2(n)
1/2
max
α∈C (M)
|S1(α;
√
x )|

n x
d2(n)
1/2
1 n x
λ2(n)Λ2(n)
1/2
max
α∈C (M)
|S1(α;
√
x )|x log5/2 x
x3/2P
−1/2 log5/2 x, (6.5)
where we use Lemma 4.2 and the well-known estimates

n x
d2(n) x log3 x,

n x
λ2(n)Λ2(n) x log2 x.
From (2.4), (6.4), and (6.5), the proof is complete.
The proof of Theorem 2 is similar to Theorem 1, because we can also deal
with the normalized Fourier coefficients of holomorphic cusp forms a(n) by
normalizing τ (n). So we omit it here.
1112 Liqun HU
Acknowledgements This work was supported in part by the Natural Science Foundation
of Jiangxi Province (Nos. 2012ZBAB211001, 20132BAB2010031).
References
1. Chamizo F, Iwaniec H. On the sphere problem. Rev Mat Iberoam, 1995, 11: 417–429
2. Chen J. Improvement of asymptotic formulas for the number of lattice points in a
region of three dimensions (II). Sci Sin, 1963, 12: 751–764
3. Friedlander J B, Iwaniec H. Hyperbolic prime number theorem. Acta Math, 2009, 202:
1–19
4. Guo Ruting, Zhai Wenguang. Some problem about the ternary quadratic form
m21
+ m22
+m23
. Acta Arith, 2012, 156(2): 101–121
5. Heath-Brown D R. The Pjateckii-Shapiro prime number theorem. J Number Theory,
1983, 16: 242–266
6. Heath-Brown D R. Lattice points in the sphere. In: Number Theory in Progress.
Berlin: Walter de Gruyter, 1999, 883–892
7. Hua L K. Introduction to Number Theory. Beijing: Science Press, 1957 (in Chinese)
8. Pan Chengdong, Pan Chengbiao. Goldbach Conjecture. Beijing: Science Press, 1981
(in Chinese)
9. Perelli A. On some exponential sums connected with Ramanujan’s τ -function.
Mathematika, 1984, 31(1): 150–158
10. Vinogradov I M. On the number of integer points in a sphere. Izv Akad Nauk SSSR
Ser Mat, 1963, 27: 957–968