Averages of shifted convolution sums for $GL(3) \times GL(2)$

0.17 MB·

by  Qingfeng Sun

GL GL
Averages of shifted convolution sums for (3) (2)
×
Qingfeng Sun
Abstract Let A (1,n) be the normalized Fourier coefficients of a GL(3) Maass cusp form f and let a (n)
f g
be the normalized Fourier coefficients of a GL(2) cusp form g. Let λ(n) be either A (1,n) or the triple
f
divisor function d3(n). It is proved that for any ǫ>0, any integer r 1 and r5/2X1/4+7δ/2 H X with
7 ≥ ≤ ≤
1 δ >0,
0 1 h n
2 W λ(n)ag(rn+h)V X1−δ+ǫ,
H H X ≪
n hX≥1 (cid:18) (cid:19)nX≥1 (cid:16) (cid:17)
a
where V and W are smooth compactly supported functions, and the implied constants depend only on the
J
associated forms and ǫ.
8

Keywords Averages, shifted convolution sums, GL(3) GL(2)
]
×
T
N Mathematics Subject Classification (2010) 11F30, 11F37, 11F66.
.
h
t
a
m 1 Introduction
[

1 Theshiftedconvolutionsumproblemshavealonghistoryinanalyticnumbertheory. Nontrivial
v bounds of various shifted convolution sums have been playing important roles in many central
8
problems, such as quantum unique ergodicity, subconvexity and power moments of L-functions
1
0 (see for example [1], [4], [6], [8], [12], [16], [22]). The first shifted convolution sum involving GL(3)
2
Fourier coefficients was studied in [21] by Pitt who considered the shifted convolution sum of d (n)
0 3
1. with the Fourier coefficients ag(n) of a holomorphic cusp form g, where d3(n) = 1 is the
0 lj∈l1Nl2P,jl=3=1,n2,3
7
triple divisor function which is the n-th coefficient of the cube of the Riemann zeta function ζ3(s).
1
: Recently, Munshi [18] studied the general GL(3) GL(2) shifted convolution sum
v ×
i
X n
(X) = A (1,n)a (n+h)V ,
r Dh f g X
a
nX≥1 (cid:16) (cid:17)
where A (1,n) are the Fourier coefficients of a GL(3) Maass cusp form f, a (n) are those of a
f g
GL(2) Maass or holomorphic cusp form g, 1 h X1+ǫ an integer, and V is a smooth compactly
≤ ≤
supported function, and succeeded in showing that
h(X) f,g,ǫ X1−210+ǫ
D ≪
by using the idea of factorizable moduli with the circle method of Jutila’s version. As Munshi
remarked in his paper, “it is expected that extra cancellation can be obtained by averaging over
h”, which will be the main concern of this paper. In fact, we shall consider the following averages
of GL(3) GL(2) shifted convolution sums
×
1 h n
(H,X) = W λ(n)a (rn+h)V , (1.1)
g
S H H X
Xh≥1 (cid:18) (cid:19)nX≥1 (cid:16) (cid:17)
1
where W is another smooth compactly supported function, r 1 is an integer, λ(n) is either
≥
A (1,n) or d (n). Here f is a Maass cusp form for SL(3,Z) and g is a Maass or holomorphic cusp
f 3
form for SL(2,Z). Our main result is the following theorem.
Theorem 1 For any ǫ > 0, any integer r 1 and (rX)1/2+ǫ H X, we have
≥ ≤ ≤
(H,X) X−A
S ≪
for any A > 0. For any ǫ > 0, any integer r 1 and r5/2X1/4+7δ/2 H (rX)1/2+ǫ with δ > 0,
≥ ≤ ≤
we have
(H,X) X1−δ+ǫ.
S ≪
Here the implied constants depend only on the associated forms and ǫ.
Recently, averages of shifted convolution sums for GL(2) cusp forms have been studied in
[2],[14] and [23]. We note that for the shifted convolution sum in (1.1) without averaging and
λ(n) = d (n), Munshi’sapproachfor (X)canalsobeapplied(see[19]). Moreover, sinced (n)
3 h 3
D ≪
nǫ for any ǫ > 0, we can remove the smooth weight V in (1.1).
Theorem 2 Assume that a (n) nθ+ǫ for any ǫ > 0. For any ǫ > 0, any integer r 1 and
g
≪ ≥
(rX)1/2+ǫ H X, we have
≤ ≤
1 h
W d (n)a (rn+h) X−A
3 g
H H ≪
h≥1 (cid:18) (cid:19)n≤X
X X
for any A > 0. For any ǫ > 0, any integer r 1 and r5/2X1/4+6δ(rX)5θ/2 H (rX)1/2+ǫ with
≥ ≤ ≤
δ > 0, we have
1 h
W d (n)a (rn+h) X1−δ+ǫ.
3 g g,ǫ
H H ≪
h≥1 (cid:18) (cid:19)n≤X
X X
Note that we can take θ = 0 for g a holomorphic cusp form and θ = 7/64 for g a Maass cusp
form (see [11]).
2 The circle method and Voronoi formulas
2.1 The circle method
1, if n = 0,
As usual, denote δ(n) =
( 0, otherwise.
Lemma 1 ([7]) For any P > 1 there is a positive constant c , and a smooth function h(x,y)
P
defined on (0, ) R, such that
∞ ×
∞
cP ∗ cn q n
δ(n) = e h ,
P2 q P P2
Xq=1cmXodq (cid:18) (cid:19) (cid:16) (cid:17)
2
for n Z. Here the over the sum indicates that c and q are coprime. The constant c =
P
∈ ∗
1+O (P−A) for any A> 0. Moreover, h(x,y) x−1 for all y, and h(x,y) is nonzero only when
A
≪
x max 1,2y . The smooth function h(x,y) satisfies
≤ { | |}
∂ih ∂h
xi (x,y) x−1 and (x,y) = 0 (2.1)
∂xi ≪i ∂y
for x 1 and y x/2. And also for y x/2, we have
≤ | | ≤ | | ≥
∂i+jh
xiyj (x,y) x−1. (2.2)
∂xi∂yj ≪i,j
We will apply Lemma 1 for larger H using the fact that we can choose P = √Y to detect the
equation n = 0 for integers in the range n Y. For small H, Lemma 1 is not efficient to obtain
| | ≤
savings (for small q) in our problem and we will apply Jutila’s variation of the circle method ([10])
1, x [0,1],
which gives an approximation for I (x) = ∈ where I (x) is the characteristic
[0,1] S
( 0, otherwise,
function of the set S. We have the following result (for a proof see [18], Lemma 4).
Lemma 2 Let Q [1,Q], Q >0 and Q−2 η Q−1. Define
⊂ ≤ ≤
1 ∗
IQ,η(x) = 2ηL I c−η,c+η (x), (2.3)
q∈Qcmodq hq q i
X X
e
where L = φ(q). Then for any ǫ > 0,
q∈Q
P
1
2 Q2+ǫ
1−IQ,η(β) dβ ≪ ηL2 . (2.4)
Z0 (cid:12) (cid:12)
(cid:12) (cid:12)
e
(cid:12) (cid:12)
2.2 GL(2) Voronoi formulas
For notational simplicity, we assume that g is a Hecke-Maass cusp form for SL(2,Z) with
Laplace eigenvalue 1/4+µ2 and normalized Fourier coefficients a (m).
g
Lemma 3 ([17]) Let ψ(y) C∞(0, ). For (c,q) = 1, we have
∈ c ∞
∞ ∞
cm 1 cm m
a (m)e ψ(m) = a ( m)e Ψ± ,
g q q g ∓ ± q q2
m=1 (cid:18) (cid:19) ± m=1 (cid:18) (cid:19) (cid:18) (cid:19)
X X X
where c denote the multiplicative inverse of c modq, and
π ∞
Ψ−(y) = ψ(v)(Y +Y )(4π√yv)dv, (2.5)
2iµ −2iµ
−cosh(πµ)
Z0
∞
Ψ+(y) = 4cosh(πµ) ψ(v)K (4π√yv)dv. (2.6)
2iµ
Z0
3
If ψ(y) is a smooth function of compact support in [AY,BY], where Y > 0 and B > A > 0,
satisfying ψ(j)(y) Y−j for any integer j 0, then for any fixed ǫ > 0 and yY Yǫ, Ψ±(y)
A,B,j
≪ ≥ ≫
are negligibly small. For yY Yǫ, we have the trivial bound Ψ±(y) Y1+ǫ.
g,ǫ
≪ ≪
2.3 GL(3) Voronoi formulas
Let f be a Hecke-Maass cusp form of type (ν ,ν ) for SL(3,Z) with normalized Fourier coef-
1 2
ficients A (n ,n ). Denote µ = ν 2ν +1,µ = ν +ν ,µ = ν +ν 1. The generalized
f 1 2 1 1 2 2 1 2 3 1 2
− − − −
Ramanujan conjecture asserts that Re(µ ) = 0, 1 j 3, while the current record bound due to
j
≤ ≤
Luo, Rudnick and Sarnak [15] is Re(µ ) 1 1 ,1 j 3.
| j | ≤ 2 − 10 ≤ ≤
Let ϕ(y) be a smooth function compactly supported on (0, ) and denote by ϕ(s) the Mellin
∞
transform of ϕ(y). For k = 0,1, we define
e
3 Γ 1+s+µj+2k
2
Φ (y):= (π3y)−s ϕ( s k)ds (2.7)
k (cid:16)Γ −s−µj (cid:17) − −
Re(Zs)=σ jY=1 2
(cid:16) (cid:17) e
with σ > max 1 Re(µ ) 2k . Set
j
1≤j≤3{− − − }
1
Φ±(y)= Φ (y) Φ (y). (2.8)
0 ± iπ3y 1
Then we have the following Voronoi formula.
Lemma 4 ([5], [20]) Let ϕ(y) C∞(0, ). For (c,q) = 1 we have
∈ c ∞
A (1,n)e cn ϕ(n) = qπ−25 ∞ Af(n2,n1)S c, n ; q Φ± n21n2 ,
f q 4i n n ± 2 n q3
nX≥1 (cid:18) (cid:19) X± Xn1|qnX2=1 1 2 (cid:18) 1(cid:19) (cid:18) (cid:19)
where c denote the multiplicative inverse of c mod q and S(m,n;c) is the classical Kloosterman
sum.
Next we state the Voronoi formula for d (n) in Li’s version (see [13]). Set σ (k,l) =
3 0,0
1.Letγ := lim ζ(s) 1 betheEulerconstantandγ := d ζ(s) 1
ddP11>|l0 d2>0d,2P(|dd2l1,k)=1 s→1(cid:16) − s−1(cid:17) 1 −ds (cid:16) − s−1(cid:17)(cid:12)(cid:12)s=1
(cid:12)
be the Stieltjes constant. For ω(y) C (0, ), k = 0,1 and σ > 1 2k, set
c
∈ ∞ − −
1 Γ 1+s+2k 3
Ω (y) = π3y −s 2 ω( s k)ds (2.9)
k 2πi Z (cid:0)Γ −2s 3(cid:1) − −
Re(s)=σ (cid:0) (cid:1)
(cid:0) (cid:1) e
with ω(s)= ∞ω(u)us−1du the Mellin transform of ω, and
0
R
1
e Ω±(y)= Ω (y) Ω (y). (2.10)
0 ± iπ3y 1
4
Lemma 5 Let ω(y) C∞(0, ). For (c,q) = 1 and cc 1(modq) we have
∈ c ∞ ≡
cn
d (n)e ω(n)
3
q
n≥1 (cid:18) (cid:19)
X
q 1 n q mn2
= σ ,m S m,c; Ω±
2π32 X± Xn|q mX≥1 nmnX1|nnX2|nn1 0,0(cid:18)n1n2 (cid:19) (cid:16)± n(cid:17) (cid:18) q3 (cid:19)
1 q
+ ω(1) nτ(n)P (n,q)S 0,c;
2q2 2 n
Xn|q (cid:16) (cid:17)
1 e q
+ ω′(1) nτ(n)P (n,q)S 0,c;
2q2 1 n
Xn|q (cid:16) (cid:17)
1 e q
+ ω′′(1) nτ(n)S 0,c; ,
4q2 n
Xn|q (cid:16) (cid:17)
e
where P (n,q)= 5 logn 3logq+3γ 1 logd, and
1 3 − − 3τ(n) d|n
P
9
P (n,q) = (logn)2 5logqlogn+ (logq)2+3γ2 3γ +7γlogn 9γlogq
2 1
− 2 − −
1 3
+ (logn+logq 5γ) logd (logd)2 .
τ(n) − − 2 
d|n d|n
X X
 
The functions Φ±(y) (also Ω±(y)) have the following properties (see Sun [24] for proof).
Lemma 6 Suppose that ϕ(y) is a smooth function of compact support in [AY,BY], where Y > 0
and B > A> 0, satisfying ϕ(j)(y) Pj for any integer j 0. Then for y > 0 and any integer
A,B,j
≪ ≥
ℓ 0, we have
≥
y −ℓ
Φ±(y) (yY)−ǫ(PY)3 .
≪A,B,ℓ,ǫ P3Y2
(cid:16) (cid:17)
By Lemma 6, for any fixed ǫ > 0 and yY Yǫ(PY)3, Φ±(y) are negligibly small. For
≫
yY Yǫ(PY)3, we can shift the contour of integration in (2.7) to σ = 3/5+ǫ with ǫ > 0 to get
≪ −
Φ±(y) (yY)35−ǫPY. (2.11)
≪
3 Proof of Theorem 1
We write
1 h n m
(H,X) = W λ(n)V a (m)φ δ(rn+h m), (3.1)
g
S H H X rX +h −
Xh≥1 (cid:18) (cid:19)nX≥1 (cid:16) (cid:17)mX≥1 (cid:18) (cid:19)
5
where φ(y) is a smooth function compactly supported in [1/2,5/2], which equals 1 on [1,2] and
satisfies φ(j)(y) 1. Taking P = √6rX and applying the circle method in Lemma 1, we have
j
≪
cP ∗ crn n cm
(H,X) = λ(n)e V a (m)e
S HP2 q X g − q
qX≤P cmXodqnX≥1 (cid:18) (cid:19) (cid:16) (cid:17)mX≥1 (cid:18) (cid:19)
ch h m q rn+h m
e W φ h , − .
q H rX +h P P2
h≥1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
X
Applying Poisson summation to the h-sum (see Theorem 4.4 in [9]), we have
cγ h m q rn+h m
h-sum = e W φ h , −
q H rX +h P P2
γmodq (cid:18) (cid:19)h≡γmodq (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
X X
1 cγ hγ x m q rn+x m hx
= e e W φ h , − e dx
q q q H rX +x P P2 − q
γXmodq (cid:18) (cid:19)Xh∈Z (cid:18) (cid:19)ZR (cid:16) (cid:17) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
= H (h,n,m,q),
I
h∈Z
X
h≡−cmodq
where
m q rn+Hx m hHx
(h,n,m,q) = W (x)φ h , − e dx.
I rX +Hx P P2 − q
ZR (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Note that the condition h cmod q implies that (h,q) = 1. Then for h = 0, we have q = 1. For
≡ −
h = 0, by partial integration j times and (2.1)-(2.2), we have
6
−j j j
P hH H P H P P
(h,n,m,q) | | 1+ + .
I ≪j q q rX +H q P2 ≪j q H h
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) | |(cid:19)
Thus the contribution from h P1+ǫ/H is negligible. In particular, if H > (rX)12+ǫ, we have
| | ≥
c n
P
(H,X) = λ(n)V a (m) (0,n,m,1)
S P2 X g I
Xn≥1 (cid:16) (cid:17)mX≥1
c n m 1 rn+Hx m
P
= W (x) λ(n)V a (m)φ h , − dx
P2 X g rX +Hx P P2
ZR nX≥1 (cid:16) (cid:17)mX≥1 (cid:18) (cid:19) (cid:18) (cid:19)
X−A
≪
for any A > 0. Here we have used the fact of Booker [3] that for π an automorphic representation
of GL (A ) whose L-function L(s,π) = λ (n)n−s is entire, and F a Schwartz function on
r Q n≥1 π
(0, ),
P
∞
n
λ (n)F X−A (3.2)
π π,A,F
X ≪
nX≥1 (cid:16) (cid:17)
for any A> 0.
6
For H (rX)12+ǫ, we write (3.1) as
≤
1 h n m 1
(H,X) = W λ(n)V a (m)φ e((rn+h m)α)dα
g
S H H X rX +h −
Xh≥1 (cid:18) (cid:19)nX≥1 (cid:16) (cid:17)mX≥1 (cid:18) (cid:19)Z0
1 h 1 n m
= W e(αh) λ(n)e(αrn)V a (m)e( αm)φ dα.
g
H H X − rX +h
Xh≥1 (cid:18) (cid:19)Z0 nX≥1 (cid:16) (cid:17)mX≥1 (cid:18) (cid:19)
By Lemma 2 we shall approximate (H,X) by
S
1 h 1 n
(H,X) = W IQ,η(α)e(αh) λ(n)e(αrn)V
S H H X
Xh≥1 (cid:18) (cid:19)Z0 nX≥1 (cid:16) (cid:17)
e e
m
a (m)e( αm)φ dα,
g
− rX +h
m≥1 (cid:18) (cid:19)
X
where IQ,η(α)(x) is defined in (2.3). Then by Cauchy’s inequality and (2.4),
e 1 h 1
(H,X) (H,X) W 1 IQ,η(α)
S −S ≪ H H −
Xh≥1 (cid:18) (cid:19)Z0 (cid:12) (cid:12)
(cid:12) (cid:12)
e e
(cid:12) (cid:12)
n m
λ(n)e(αrn)V a (m)e( αm)φ dα
g
(cid:12) X (cid:12)(cid:12) − rX +h (cid:12)
(cid:12)(cid:12)Xn≥1 (cid:16) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)mX≥1 (cid:18) (cid:19)(cid:12)(cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) 1 (cid:12)(cid:12) 2 1/2 (cid:12)
g,ǫ (cid:12)(rX)12+ǫ 1 IQ,η((cid:12)α(cid:12)) dα (cid:12)
≪ −
(cid:18)Z0 (cid:12) (cid:12) (cid:19)
(cid:12) e (cid:12) 1/2
1 (cid:12) n(cid:12)
λ(n)e(αrn)V dα
 (cid:12) X (cid:12) 
Z0 (cid:12)(cid:12)Xn≥1 (cid:16) (cid:17)(cid:12)(cid:12)
 (cid:12) (cid:12) 
(cid:12) Q2+ǫ 1/2 (cid:12)
(rX)ǫ(cid:12)r1/2X (cid:12)
≪f,g,ǫ ηL2
(cid:18) (cid:19)
r1/2X
(rX)ǫ
f,g,ǫ
≪ √ηQ
since L Q2−ǫ, where we have used the Rankin-Selberg estimate A (1,n)2 Y1+ǫ and
≫ n≤Y | f | ≪f,ǫ
the uniform bound in α R
∈ P
m
a (m)e( αm)φ Y1/2+ǫ.
g g,ǫ
− Y ≪
mX≥1 (cid:16) (cid:17)
Taking η = (rX +H)−1 we obtain
3
(H,X) = (H,X)+O (rX)ǫrX2 . (3.3)
S S Q
!
e
7
Then we only need to estimate (H,X). Changing variable α c +β, we have
S → q
e 1 η
(H,X) := (H,X)dβ,
β
S 2η S
Z−η
e e
where
1 ∗ crn n cm
(H,X) = λ(n)e V e(βrn) a (m)e e( βm)
β g
S HL q X − q −
qX∈QcmXodqnX≥1 (cid:18) (cid:19) (cid:16) (cid:17) mX≥1 (cid:18) (cid:19)
e
ch h m
e W φ e(βh). (3.4)
q H rX +h
h≥1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
X
Applying Poisson summation to the h-sum, we have
cγ h m
h-sum = e W φ e(βh)
q H rX +h
γmodq (cid:18) (cid:19)h≡γmodq (cid:18) (cid:19) (cid:18) (cid:19)
X X
1 cγ hγ x m hx
= e e W φ e(βx)e dx
q q q H rX +x − q
γXmodq (cid:18) (cid:19)hX∈Z (cid:18) (cid:19)ZR (cid:16) (cid:17) (cid:18) (cid:19) (cid:18) (cid:19)
= H I (h,m,q),
β
h∈Z
X
h≡−cmodq
where
m hHx
I (h,m,q) = W (x)φ e(βHx)e dx.
β
rX +Hx − q
ZR (cid:18) (cid:19) (cid:18) (cid:19)
Now we choose the set of moduli Q as the prime set
Q = q : q [Q/2,Q] is prime and(q,r) = 1 .
{ ∈ }
Then the requirement L Q2−ǫ is satisfied and h = 0 since the condition h cmod q implies
ǫ
≫ 6 ≡ −
that (h,q) = 1. By partial integration j times we have I (h,m,q) (hH/q)−j since β η =
β j
≪ | | | | ≤
(rX +H)−1. Thus contribution from h Q1+ǫ/H is negligible and
| | ≥
h-sum = H I (h,m,q)+O((rX)−A) (3.5)
β
1≤|hX|≤Q1H+ǫ
h≡−cmodq
for any A> 0. Plugging (3.5) into (3.4), we need to estimate
1 hrn n hm
λ(n)e V e(βrn) a (m)e e( βm)I (h,m,q)
g β
L − q X q −
1≤|hX|≤Q1+ǫ Xq∈Q nX≥1 (cid:18) (cid:19) (cid:16) (cid:17) mX≥1 (cid:18) (cid:19)
H (q,h)=1
:= W (x)e(βHx) (H,X)dx,
β,x
S
R
Z
e
8
where
1 hHx hrn n
(H,X) = e λ(n)e V e(βrn)
β,x
S L − q − q X
1≤|hX|≤Q1+ǫ Xq∈Q (cid:18) (cid:19)Xn≥1 (cid:18) (cid:19) (cid:16) (cid:17)
e H (q,h)=1
hm m
a (m)e φ e( βm). (3.6)
g
q rX +Hx −
m≥1 (cid:18) (cid:19) (cid:18) (cid:19)
X
We first apply GL(2) Voronoi formula in Lemma 3 to the m-sum in (3.6) to get
∞
1 hm m
m-sum = a ( m)e Ψ± , (3.7)
q g ∓ ± q β,x q2
± m=1 (cid:18) (cid:19) (cid:18) (cid:19)
X X
where Ψ± (y) are defined in (2.5)-(2.6) with ψ(y) = φ y e( βy). Note that
β,x rX+Hx −
(cid:16) (cid:17)
dj y 1 j 1 j
φ e( βy) + β .
dyj rX +Hx − ≪ rX +H | | ≪ rX
(cid:26) (cid:18) (cid:19) (cid:27) (cid:18) (cid:19) (cid:18) (cid:19)
Thus the contribution from m Q2(rX)ǫ/(rX) in (3.7) is negligible. For m Q2(rX)ǫ/(rX),
| | ≫ | |≪
we have the trivial bound Ψ± m (rX)1+ǫ.
β,x q2 ≪
Next we want to apply the(cid:16)Vor(cid:17)onoi formulas to the n-sum in (3.6).
Case (i) λ(n)= A (1,n). We apply the GL(3) Voronoi formula in Lemma 4 to the n-sum in
f
(3.6) to get
hrn n
A (1,n)e V e(βrn)
f
− q X
nX≥1 (cid:18) (cid:19) (cid:16) (cid:17)
= qπ−25 ∞ Af(n2,n1)S hr, n ; q Φ± n21n2 , (3.8)
4i n n − ± 2 n β q3
X± Xn1|qnX2=1 1 2 (cid:18) 1(cid:19) (cid:18) (cid:19)
whereΦ±(y)aredefinedin(2.7)-(2.8)withϕ(y) = V (y/X)e(βry). Notethat dj V y e(βry)
β dyj X ≪
X−j for any j ≥ 0. By Lemma 6, one sees that the contribution from n21n2 ≫ Q3(cid:8)Xǫ/(cid:0)X(cid:1)in (3.8)(cid:9)is
negligible. For n2n Q3Xǫ/X, by (2.11) we get
1 2 ≪
Ψ± n21n2 Xn21n2 53−ǫ. (3.9)
β q3 ≪f,ǫ q3
(cid:18) (cid:19) (cid:18) (cid:19)
9
By (3.6)-(3.9) and Weil’s bound for Kloosterman sums, we conclude that
1
(H,X) a ( m)(rX)1+ǫ
β,x f,g,ǫ g
S ≪ L | ∓ |
X± 1≤|hX|≤Q1+ǫ qX∈Q|m|≪Q2X(rX)ǫ/(rX)
e H
|Af(n2,n1)| q 1/2 Xn21n2 35−ǫ
n n n q3
Xn1|q n2≪Q3XXǫ/(n21X) 1 2 (cid:18) 1(cid:19) (cid:18) (cid:19)
≪f,g,ǫ X35H+ǫQ q−1103 n−1 130 |Af(n2,n1)|n2−52
qX∈Q Xn1|q n2≪Q3XXǫ/(n21X)
(rX)ǫQ5/2
, (3.10)
f,g,ǫ
≪ H
wherewehaveusedtheRankin-Selbergestimates a ( m) N and A (n ,n )
|m|≪N | g ∓ |≪g n2≤N | f 1 2 | ≪f
N n . Taking
1 P P
| |
Q = (rH)2/7X3/7.
Then for λ(n)= A (1,n) Theorem 1 follows from (3.3) and (3.10).
f
Case (ii) λ(n) = d (n). Applying Lemma 5 to the n-sum in (3.6) we get
3
hrn n
d (n)e V e(βrn)
3
− q X
Xn≥1 (cid:18) (cid:19) (cid:16) (cid:17)
q 1 n q n2l
= σ ,l S hr, l; Ω±
2π32 X± Xn|q Xl≥1 nl nX1|nnX2|nn1 0,0(cid:18)n1n2 (cid:19) (cid:16)− ± n(cid:17) β (cid:18) q3 (cid:19)
1 q 1 q
+ ω(1) nτ(n)P (n,q)µ + ω′(1) nτ(n)P (n,q)µ
2q2 2 n 2q2 1 n
Xn|q (cid:16) (cid:17) Xn|q (cid:16) (cid:17)
1 e q e
+ ω′′(1) nτ(n)µ , (3.11)
4q2 n
Xn|q (cid:16) (cid:17)
e
where Ω±(y) are defined in (2.9)-(2.10) with ω(y) = V (y/X)e(βry). As in the Case (i) the first
β
term in (3.11) is essentially supported on n2l Q3Xǫ/X and the contribution from the first
≪
(rX)ǫQ5/2
term of (3.11) can be bounded similarly as that in the Case (i), which is at most with
H
Q = (rH)2/7X3/7. For the remaining terms in (3.11), we have trivially
∞
ω(j)(1) = ω(u)(logu)jdu X(logX)j,
j
≪
Z0
and they contribute (3.6)eby
1 1 X1+ǫ X
a ( m)(rX)1+ǫ nτ(n)log2(nq) (rX)ǫ
L q | g ∓ | q2 ≪ H
X± 1≤|hX|≤Q1+ǫ qX∈Q |m|≪Q2X(rX)ǫ/(rX) Xn|q
H
which is (rX)ǫQ5/2 for Q = (rH)2/7X3/7. This finishes the proof of Theorem 1.
≪ H
10