Ramanujan J DOI 10.1007/s11139-014-9611-6
The average order of Hecke eigenvalues of Siegel cusp forms of genus 2 Yujiao Jiang · Guangshi Lü
Received: 18 February 2014 / Accepted: 3 July 2014 © Springer Science+Business Media New York 2014
Abstract Let λ(n) be the n-th normalized Hecke eigenvalue of a Siegel cusp form F of integral weight k on the group Sp4 (Z). We establish asymptotic formulae for the summatory functions
λ(n)2 and
n≤x
λ(n 2 )
n≤x
as x → ∞, in which k grows with x in a definite way. Keywords
Hecke eigenvalues · Siegel cusp forms · Mean value of L-functions
Mathematics Subject Classification
11F46
1 Introduction Let Sk denote the space of Siegel cusp forms of integral weight k on the group Sp4 (Z). Let F ∈ Sk be not a Saito–Kurokawa lift, but an eigenfunction of all Hecke operators Tn with eigenvalues λ F (n). It is well known that λ F (n) are real and multiplicative. We define the normalized Hecke eigenvalue of F by λ(n) :=
λ F (n) . n k−3/2
Y. Jiang (B) · G. Lü Department of Mathematics, Shandong University, Jinan 250100, Shandong, China e-mail:
[email protected] G. Lü e-mail:
[email protected]
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Y. Jiang, G. Lü
Recently, the asymptotic behavior of λ(n) has interested to some authors. Royer et al. [10] showed that
3
a F (n) F,ε x 5 +ε ,
(1.1)
n≤x
where a F (n) are coefficients of the spinor zeta function of F. By using the relation (2.10), one can easily derive
3
λ(n) F,ε x 5 +ε .
(1.2)
n≤x
Moreover, Das et al. [2] treated the second power sum of λ(n) and proved that
5 31 λ(n)2 = c F x + Oε k 16 x 32 +ε ,
(1.3)
n≤x
where c F is a constant depending on F. Our aim here is to improve the result (1.3) and estimate the average order of λ(n) over the sparse sequence {n 2 }. The main results are the following. Theorem 1.1 For any ε > 0, we have
41 30 11 λ(n)2 = c F x + Oε x 47 +ε + k 17 x 17 +ε ,
(1.4)
n≤x
where c F is a constant depending on F. 15 We note that Landau’s Lemma (see [7]) implies that the error is Oε,F x 17 +ε . It can be easily seen that the estimate of Theorem 1.1 is stronger and gives the dependence on the Siegel cusp form explicitly. Theorem 1.2 For any ε > 0, we have
41
27
14
λ(n 2 ) ε x 50 +ε + k 23 x 23 +ε .
(1.5)
n≤x
2 Preliminaries In this section we will briefly recall some fundamental facts about Siegel cusp forms and their L-functions. Let F ∈ Sk be not a Saito–Kurokawa lift and an eigenfunction of all Hecke operators. The incomplete degree-4 spinor zeta function of F is Z (s, F) =
p<∞
123
Z p ( p −s , F),
(2.1)
Hecke eigenvalues of Siegel cusp forms of genus 2
where Z p (t, F)−1 = (1 − α0, p t)(1 − α0, p α1, p t)(1 − α0, p α2, p t)(1 − α0, p α1, p α2, p t). The incomplete degree-5 standard zeta function of F is defined as Z St (s, F) =
−s Z St p ( p , F),
(2.2)
p<∞
where −1 −1 −1 = (1 − α1, p t)(1 − α2, p t)(1 − t)(1 − α1, Z St p (t, F) p t)(1 − α2, p t).
Here α0, p , α1, p , α2, p are the Satake p-parameters attached to F and satisfy 2 α0, p α1, p α2, p = 1.
(2.3)
The Ramanujan–Petersson conjecture, which has been proved by Weissauer, [11], which asserts that for all primes p α0, p = α1, p = α2, p = 1.
(2.4)
This shows that the spinor zeta function Z (s, F) and the standard zeta function Z St (s, F) converge absolutely for s > 1. Then we can define symmetric square L-function by the degree 10 Euler product L(s, sym2 F) =
−1 1 − βi, p β j, p p −s ,
(2.5)
p 1≤i≤ j≤4
where β1, p = α0, p , β2, p = α0, p α1, p , β3, p = α0, p α2, p , β4, p = α0, p α1, p α2, p . It follows from [9, Theorem 5.1.2 and Theorem 5.1.5] that there exist cuspidal automorphic representations 4 of GL4 (A) and 5 of GL5 (A) such that Z (s, F) = L(s, 4 ) and Z St (s, F) = L(s, 5 ). Moreover, we know from [9, Theorem 5.2.1] that the completed L-functions of Z (s, F), Z St (s, F), and L(s, sym2 F) are entire, bounded in vertical strips and satisfy the expected functional equations. The corresponding Archimedean factors are all given explicitly in [9]. The detailed results are summarized in the following lemma.
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Lemma 2.1 The completed L-functions of Z (s, F), Z St (s, F), and L(s, sym2 F) are defined as
1 3 C s + k − Z (s, F) (s, F, spin) := C s + 2 2 (s, F, stan) := R (s) C (s + k − 2) C (s + k − 1) Z St (s, F) (s, sym2 F) := R (s + 1)2 C (s + 1) C (s + k − 2) C (s + k − 1) (2.6) C (s + 2k − 3) L(s, sym2 F), where R (s) = π −s/2 (s/2), C (s) = (2π )−s (s). Then they are entire, bounded in vertical strips and satisfy the functional equations (s, F, spin) = (−1)k (1 − s, F, spin) (s, F, stan) = (1 − s, F, stan) (s, sym2 F) =
(1 − s, sym2 F).
(2.7)
2.1 Decomposition of D j (s) and S j (s) Let F ∈ Sk be not a Saito–Kurokawa lift, but be a Hecke eigenform. Define the related Dirichlet series as D j (s) =
∞
j −s
λ(n) n
, and S j (s) =
n=1
∞
λ(n j )n −s
(2.8)
n=1
for j ∈ N. The aim is to decompose them into some functions whose properties are well known. In the case j = 1, we know from [1] that D1 (s) = S1 (s) = Z (s, F)ζ (2s + 1)−1 .
(2.9)
Let a F (n) be the n-th coefficient of Z (s, F). It follows from Weissauer’ bound (2.4) that |a F (n)| ≤ d4 (n) for all n ≥ 1, where dk (n) is the k-dimensional divisor function. Then by using the relation (2.9), we see that λ(n) =
μ(d) a F (m). d 2
(2.10)
n=md
Furthermore, we have for any ε > 0 |λ(n)| ≤ d5 (n) n ε .
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(2.11)
Hecke eigenvalues of Siegel cusp forms of genus 2
Hence, D j (s) and S j (s) converge absolutely for s > 1. We shall decompose them similar to (2.9) in the following manner. Lemma 2.2 With notations as above and s > 1, for any ε > 0 we have D2 (s) = ζ (s)Z St (s, F)L(s, sym2 F)H (s)
(2.12)
where the function H (s) := p H p ( p −s ). H p (X ) is a polynomial of degree ≤ 15, whose coefficients are polynomials in the number βi and are absolutely bounded. Furthermore, H (s) converges absolutely in s ≥ 1/2 + ε and is free of zeros on the line s = 1. Proof We shall use the notations as those in [9]. It follows from [2, Lemma 3.1] that D2 (s) = L(s, F ⊗ F)H (s) = L(s, ρ4 ⊗ ρ4 )H (s), where L(s, F ⊗ F) is the Rankin–Selberg L-function, and ρ4 is the 4-dimensional irreducible representation of Sp4 (Z). On the other hand, we have the canonical decomposition (see [3]) ρ4 ⊗ ρ4 = sym2 ρ4 ⊗ ∧2 ρ4 = sym2 ρ4 ⊗ ρ5 ⊗ C, where ∧2 ρ4 is the exterior square representation of ρ4 , and ρ5 is the 5-dimensional irreducible representation of Sp4 (Z). This completes the proof. Lemma 2.3 For s > 1, ε > 0 we have S2 (s) = L(s, sym2 F)U (s)
(2.13)
where the function U (s) := p U p ( p −s ). U p (X ) is a polynomial of degree ≤ 9, whose coefficients are absolutely bounded. Furthermore, U (s) converges absolutely in s ≥ 1/2 + ε. Proof From [1, Theorem 1.3.2] and (2.1), we have Z p (X, F)−1 =
1 − βi, p X 1≤i≤4
= 1 − λ( p)X + λ( p)2 − λ( p 2 ) − p −1 X 2 − λ( p)X 3 + X 4 . By expanding the product, it can be easily seen that λ( p) =
βi, p , λ( p)2 − λ( p 2 ) − p −1 =
1≤i≤4
βi, p β j, p .
1≤i< j≤4
Then it follows from a straightforward calculation λ( p 2 ) =
βi, p β j, p − p −1 .
(2.14)
1≤i≤ j≤4
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Let S2 (s) be expressed as the Euler product S2 (s) =
S2, p ( p
−s
p
λ( p 2 ) λ( p 4 ) λ( p 2k ) 1+ )= + 2s + · · · + + ··· . ps p p ks p
Thus, we obtain from (2.14)
U p (X ) = S2, p (X )
1 − βi, p β j, p X
1≤i≤ j≤4
= 1 − p −1 X + b2 ( p)X 2 + · · · + bk ( p)X k + · · · .
(2.15)
where bi ( p) are rational functions in the β j, p and λ( pl ) for any i ≥ 2. Here 1 ≤ j ≤ 4 and l ≤ i. Clearly, bi ( p) is absolutely bounded. Also, U (s) converges absolutely in s ≥ 1/2 + ε. In fact, we can claim that U p (X ) is a polynomial of degree ≤ 9. We know from [2, Lemma 3.1] that ri βi,δ p λ( p δ ) = 1≤i≤4
where ri = ci 1 − βi,−2p / p , ci are rational functions in the βi, p . S2, p (X ) =
∞
λ( p 2δ )X δ =
δ=0
∞
ri βi,2δp X δ =
δ=0 1≤i≤4
ri 1 − βi,2 p X 1≤i≤4
Hence from the expression (2.15), the degree of U p (X ) in X is at most 9.
2.2 Convexity bounds and mean values Following the book of Iwaniec and Kowalski [5, Chapter 5], define an L-function as given by a Dirichlet series with Euler product of degree d ≥ 1, L( f, s) =
∞ λ f (n) n=1
ns
αi ( p) 1− . = ps p 1≤i≤d
The series and Euler product are absolutely convergent for σ = (s) > 1. The complete L-function s
( f, s) = q( f ) 2 γ ( f, s)L( f, s) satisfies the functional equation ( f, s) = ε( f )( f , 1 − s),
123
(2.16)
Hecke eigenvalues of Siegel cusp forms of genus 2
where f is the dual of f, ε( f ) is a complex number of absolute value of 1, and γ ( f, s) = π
−ds/2
d
j=1
s + κj 2
.
The other concrete conditions and parameters are omitted here. Then we define the analytic conductor as q( f, t) = q( f )q∞ (t) = q( f )
d it + κ j + 3 .
(2.17)
j=1
We may write (2.16) as L( f, s) = χ ( f, s)L( f , 1 − s).
(2.18)
In this paper we suppose that κ j are real for 1 ≤ j ≤ d. Then for −1/2 ≤ σ ≤ 2 we obtain from the estimate [5, eqn. (5.115)] 1
χ ( f, s) = q( f ) 2 −s
1 γ ( f, 1 − s) q( f, t) 2 −σ . γ ( f, s)
(2.19)
The next lemma gives the convexity bound for L( f, s) in the critical strip. Lemma 2.4 For any ε > 0, we have L( f, σ + it) q( f, t)max{(1−σ )/2,0}+ε
(2.20)
uniformly in −ε ≤ σ ≤ 1 + ε and | (s)| ≥ 1. Proof It follows from [5, eqn. (5.20)].
The mean value for the L-function has been extensively studied by many authors (see [6], [8]). The upper bounds in their results only contain the level q( f ) and the height | s| of the complex variable s. Our bounds are uniform in terms of the spectral parameter, so we need to study the mean value for the L-function uniformly in all aspects of the analytic conductor. First we show the following approximate functional equation for L( f, s) by the reflection principle (see [4] ). Lemma 2.5 Let h be a fixed positive constant such that 1 − 1/ h > 0, and let Y and M be positive parameters. For s = σ + it with 0 < σ < 1, we have L( f, s) = S − I1 − I2 2 +O Y 1+ε + Y −σ M ε q( f, t)1/2+ε + M σ +ε q( f, t)1/2−σ +ε e−c log |t| , (2.21)
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where S=
∞ λ f (n) n=1
ns
e−(n/Y ) , h
⎛ ⎞ dw w w Y χ ( f, s + w) ⎝ , 1+ λ f (n)n −1+s+w ⎠ w=−ε h w n≤M | w|≤log2 |t| 1 w w −1+s+w dw I2 = Y χ ( f, s + w) , 1+ λ f (n)n 2πi w=−σ 2−ε h w
1 I1 = 2πi
| w|≤log |t|
n>M
and c is some positive constant. Proof We follow the argument analogous to Section 4.4 of Ivi´c [4]. For any ε > 1, we have ∞ λ f (n) n=1
ns
e−(n/Y ) = h
1 2πi
w w dw Y L( f, s + w) . 1+ h w (1+ε)
(2.22)
We truncate the line of integration at | w| = log2 |t|. By using the Stirling formula and the absolute convergence of L( f, s + w)at the line, the integrals along the half lines 2 with | w| > log2 |t| are estimated as O Y 1+ε e−c log |t| for some positive constant c. We move the remaining line of integration to w = −σ − ε. Since the height of this rectangle is log2 |t| and (1 + w/ h) = 1 − (σ + ε)/ h > 0 by choosing a small number ε > 0, it is seen that the only residue coming from the pole w = 0 is L( f, s). The integral on the horizontal segment −σ − ε ≤ w ≤ 1 + ε, | w| = log2 |t| 2 is estimated as O Y 1+ε + Y −σ q( f, t)1/2+ε e−c log |t| by Stirling’s formula and convexity bound (2.20). Thus we have S = L( f, s) + I + O
1 2 Y 1+ε + Y −σ q( f, t) 2 +ε e−c log |t| ,
where I =
1 2πi
w=−σ −ε | w|≤log2 |t|
w w dw Y L( f, s + w) . 1+ h w
To transform I we substitute the functional equation (2.18) and divide the sum L( f , 1 − s − w) =
∞
λ f (n)n −1+s+w
n=1
into two parts I1 with n ≤ M and I2 with n > M. The second part I2 is equal to I2 in our assertion. For the part I1 , we move the line of integration back to w = −ε.
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Hecke eigenvalues of Siegel cusp forms of genus 2
The integrand is regular in the region −σ − ε ≤ w ≤ −ε, | w| ≤ log2 |t|. By Stirling’s segment is estimated formula and (2.19), the integral along the horizontal 2 |t| −σ ε 1/2 σ +ε 1/2−σ −c log . Thus it yields q( f, t) e as O Y M q( f, t) + M I1 = I1 + O
2 Y −σ M ε q( f, t)1/2+ε + M σ +ε q( f, t)1/2−σ +ε e−c log |t| .
This proves the lemma. Lemma 2.6 For any ε > 0, assume that have
T
1
|L( f, s)|2 dt ε
n≤x
λ f (n)2 x 1+ε , d ≥ 2. Then we
T q( f, T )1−2σ +ε + q( f, T )1−σ +ε , if 0 < σ ≤ 21 , T + q( f, T )1−σ +ε , if 21 < σ < 1.
(2.23)
Proof It is clear that
T
T
|L( f, s)| dt 2
1
|S|2 + |I1 |2 + |I2 |2 + |E|2 dt,
(2.24)
1
where 2 E = Y 1+ε + Y −σ M ε q( f, t)1/2+ε + M σ +ε q( f, t)1/2−σ +ε e−c log |t| . If we split the right integral dyadically, then it suffices to compute the mean squares of S, I1 , I2 , and E over the segment T ≤ t ≤ 2T. First, applying the Montgomery– Vaughan’s mean value theorem to the expression of S, we obtain
2T T
|S| dt = 2
2 ∞ λ f (n) n=1
n 2σ
e−2(n/Y ) (T + O(n)) . h
Dividing the sum on the right-hand side into two parts S1 with n ≤ Y and S2 with h n > Y and substituting the approximation e−2(n/Y ) = O(1) in the former and h e−2(n/Y ) = O (Y/n)lh in the latter, where l is any fixed positive integer, we have λ f (n)2 S1 (T + n) n 2σ n≤Y T Y 1−2σ +ε + Y 2−2σ +ε , T + Y 2−2σ +ε ,
if 0 < σ ≤ 21 ; if 21 < σ < 1.
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Y. Jiang, G. Lü
and λ f (n)2 Y lh S2 (T + n) n 2σ n n>Y 1−2σ +ε
TY
+ Y 2−2σ +ε .
Combining the estimates above, we conclude that
2T
|S| dt 2
T
T Y 1−2σ +ε + Y 2−2σ +ε , T + Y 2−2σ +ε ,
if 0 < σ ≤ 21 , if 21 < σ < 1.
(2.25)
Second, we consider the mean square of I1 . By the Cauchy–Schwarz inequality and (2.19), we get
2T
|I1 |2 dt Y −2ε q( f, T )1−2(σ −ε)
w=−ε | w|≤log2 (2T )
T
2T T
2 −1+s+w λ f (n)n dtdw. n≤M
If we apply the Montgomery–Vaughan’s mean value theorem to the inner integral, then it can be controlled by λ f (n)2 n −2+2σ −2ε (T + n)
n≤M
T + M 2σ −ε , T M 2σ −1−ε + M 2σ −ε ,
if 0 < σ ≤ 21 , if 21 < σ < 1.
Hence we have
2T
|I1 | dt 2
T
q( f, T )1−2σ +ε T + M 2σ −ε , q( f, T )1−2σ +ε T M 2σ −1−ε + M 2σ −ε ,
if 0 < σ ≤ 21 , if 21 < σ < 1.
(2.26)
Similarly, we can know the mean square of I2 ,
2T
|I2 | dt Y 2
−2σ
q( f, T )
1+ε
T
Y
−2σ
q( f, T )
1+ε
w=−σ −ε | w|≤log2 (2T )
T M −1 + 1 .
2T T
2 −1+s+w λ f (n)n dtdw n>M
(2.27)
In the case 0 < σ ≤ 21 , choosing M = T 1/(2σ ) , Y = q( f, T )1/2 , we have
2T T
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|S|2 + |I1 |2 + |I2 |2 dt T q( f, T )1−2σ +ε + q( f, T )1−σ +ε .
(2.28)
Hecke eigenvalues of Siegel cusp forms of genus 2
In the case
1 2
< σ ≤ 1, choosing M = Y = q( f, T )1/2 , we have
2T
|S|2 + |I1 |2 + |I2 |2 dt T + q( f, T )1−σ +ε .
(2.29)
T
Next the mean square of E is O T −A if q( f, 0) ≤ T A , where A is an arbitrarily large constant. Then the assertion follows immediately from (2.28) and (2.29) in this moment. If q( f, 0) > T A , it follows directly from (2.20). Now we can establish the convexity bounds and mean values of L-functions attached to Siegel cusp forms as the application of Lemma 2.4 and Lemma 2.6. Lemma 2.7 For any ε > 0, we have 1−2σ +ε 5 1−σ +ε T 5 4 St 2 + T + k4T , if 0 < σ ≤ 21 , Z (s, F) dt ε T T + k T 1−σ +ε 5 4 T + T +k T , if 21 < σ ≤ 1. 1 (2.30) uniformly for 0 < σ < 1 and T ≥ 1, and max {(1−σ )/2,0}+ε Z St (s, F) ε |t|5 + k 4 |t|
(2.31)
uniformly for −ε ≤ σ ≤ 1 + ε and |t| ≥ 1. Proof From Lemma 2.1, we know q(F, stan, t) = (|t| + 3) (|it + k − 2| + 3)2 (|it + k − 1| + 3)2 |t|5 + k 4 |t| + 1. Then we obtain the results by Lemma 2.4 and Lemma 2.6.
Lemma 2.8 For any ε > 0, we have
2 L(s, sym2 F) dt 1 1−2σ +ε 10 1−σ +ε T T 10 + k 6 T 4 + T + k6T 4 , if 0 < σ ≤ 21 , ε 10 1−σ +ε , if 21 < σ ≤ 1. T + T + k6T 4 T
(2.32)
uniformly for 0 < σ < 1 and T ≥ 1, and max {1−σ,0}+ε L(s, sym2 F) ε |t|5 + k 3 |t|2
(2.33)
uniformly for −ε ≤ σ ≤ 1 + ε and |t| ≥ 1. Proof It follows similarly from the fact q(sym2 F, t) |t|10 + k 6 |t|4 + 1.
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We also need the following lemma (see Ivi´c [4]). Lemma 2.9 For any ε > 0, we have 1
ζ (σ + it) ε (|t| + 1)max { 3 (1−σ ),0}+ε
(2.34)
uniformly for 1/2 ≤ σ ≤ 1 + ε, |t| ≥ 1. 3 Proof of Theorem 1.1 In this section we give the proof of Theorem 1.1. Applying the Perron formula [5, Proposition 5.54] for the function (2.12), we get due to Weissauer’ bound (2.11),
1 λ(n) = 2πi n≤x 2
1+ε+i T
1+ε−i T
xs D2 (s) ds + Oε s
x 1+ε T
,
uniformly for 2 ≤ T ≤ x. From Lemma 2.2, the point s = 1 is the only pole of the integrand in the rectangle κ ≤ σ ≤ 1 + ε and |t| ≤ T for any κ ∈ [1/2 + ε, 1), and the order is one. Thus, we obtain by the residue theorem 1+ε
x 1 xs , λ(n) = c F x − D2 (s) ds + Oε 2πi s T C n≤x
2
where c F = ress=1 D2 (s), and C is the contour joining 1 + ε + i T, κ + i T, κ − i T, 1 + ε − i T with straight lines. In addition we know that H (s) converges absolutely and uniformly in the weight k and s ≥ 1/2 + ε, it follows that n≤x
λ(n)2 = c F x + O(I1 ) + O(I2 ) + Oε
x 1+ε T
,
(3.1)
where
ζ (s)Z St (s, F)L(s, sym2 F) x σ dσ, κ T dt 5(1−κ) κ+ε κ+ε x +x I2 := k ζ (s)Z St (s, F)L(s, sym2 F) t 1 2T1 1 k 5(1−κ) x κ+ε + x κ+ε sup ζ (s)Z St (s, F)L(s, sym2 F) dt 1≤T1 ≤T T1 T1 2T1 1 St 5(1−κ) κ+ε κ+ε 2 k x +x sup (s, F)L(s, sym F) dt. Z (2+κ)/3 T1 1≤T1 ≤T T1 I1 :=
123
1 T
1+ε
Hecke eigenvalues of Siegel cusp forms of genus 2
First we will estimate I1 . By inserting the upper bounds (2.31), (2.33) and (2.34), we can show (1−σ )+ε 1 1 1+ε 1 5 T 3 T 2 + k2T 2 T 5 + k3T 2 x σ dσ T κ σ 1+ε 41 11 x 5 T 6 +k T 6 dσ 47 17 κ T 6 + k5T 6
I1
x 1+ε , T
(3.2)
47
17
provided that T 6 + k 5 T 6 ≤ x. Next we will estimate I2 . Taking κ = 1/2 + ε and applying the Cauchy–Schwarz inequality, we obtain 5
1
− 56
1
I2 k 2 x 2 +ε + x 2 +ε sup T1 1≤T1 ≤T
1
1
I2,1 (T1 ) 2 I2,2 (T1 ) 2 ,
(3.3)
where
2 St 1 Z dt, + ε + it, F 2 T1
2 2T1 1 2 I2,2 (T1 ) := L 2 + ε + it, sym F dt. T1
I2,1 (T1 ) :=
2T1
It follows from (2.30) and (2.32) that 5
I2,1 (T1 ) T12
+ε
1
+ k 2+ε T12
+ε
,
I2,2 (T1 ) T15+ε + k 3+ε T12+ε .
It yields by inserting these into (3.3) 1 35 5 5 I2 T 12 + k 2 T 12 x 2 +ε .
(3.4) 6
Combining (3.2), (3.4), and (3.1), we get Theorem 1.1 by choosing T = x 47 if k ≤ T, 30 6 otherwise choosing T = k − 17 x 17 . 4 Proof of Theorem 1.2 Let L(s, sym2 F), U (s) be expressed as Dirichlet series L(s, sym2 F) =
∞ λsym2 F (n) n=1
ns
, U (s) =
∞ a(n) n=1
ns
.
(4.1)
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Similar to the proof of Theorem 1.1, applying the Perron formula for the function L(s, sym2 F), we have
λsym2 F (n) =
n≤x
1 2πi
1+ε+i T
L(s, sym2 F)
1+ε−i T
xs ds + Oε s
x 1+ε T
.
There is no pole of the integrand in the rectangle κ ≤ σ ≤ 1 + ε and |τ | ≤ T for any κ ∈ (0, 1), Thus, we obtain by the residue theorem 1+ε
x 1 xs 2 , λsym2 F (n) = − L(s, sym F) ds + Oε 2πi D s T n≤x
where D is the contour joining 1 + ε + i T, κ + i T, κ − i T, 1 + ε − i T with straight lines. It follows
λsym2 F (n) = O(J1 ) + O(J2 ) + Oε
n≤x
x 1+ε T
,
(4.2)
where
L(s, sym2 F) x σ dσ, κ T dt J2 := k 3(1−κ) x κ+ε + x κ+ε L(s, sym2 F) t 1 2T1 1 k 3(1−κ) x κ+ε + x κ+ε sup L(s, sym2 F) dt. T 1≤T1 ≤T 1 T1 J1 :=
1 T
1+ε
For the estimate of J1 , we deduce by (2.33) (1−σ )+ε 1 1+ε 5 T + k3T 2 x σ dσ J1 T κ
σ 1+ε x 4 3 dσ T +k T T 5 + k3T 2 κ x 1+ε , T provided that T 5 + k 3 T 2 ≤ x. For the estimate of J2 , we take the optimal value κ = Schwarz inequality, we obtain from (2.32)
123
(4.3)
1 10 . By applying the Cauchy–
Hecke eigenvalues of Siegel cusp forms of genus 2
27
1
− 21
1
J2 k 10 x 10 +ε + x 10 +ε sup T1 1≤T1 ≤T
27 10
k x
1 10 +ε
+x
27 10
T4 + k T
1 10 +ε
13 10
sup
1≤T1 ≤T
x
1 10 +ε
−1 T1 2
2T1 T1
T19
2 21 2 sym F) dt L(s, 27 5
18 5
1
+ k T1
2
.
(4.4) 9
Inserting (4.3), (4.4) into (4.2) and choosing T = x 50 if k ≤ T, otherwise choosing 27 9 T = k − 23 x 23 , we obtain
41
27
14
λsym2 F (n) x 50 +ε + k 23 x 23 +ε .
(4.5)
n≤x
Moreover, we know from (2.13) and (4.1) the convolution λ(n 2 ) =
λsym2 F (u)a(v).
(4.6)
n=uv
By Lemma 2.3, it follows from the absolute convergence of U (s) in the regions s ≥ 1/2 + ε that n≤x
λ(n 2 ) =
v≤x 41
a(v)
x 50 +ε
|a(v)| v≤x
41
λsym2 F (u)
u≤x/v
v
41 50 +ε
27
14
27
14
+ k 23 x 23 +ε
x 50 +ε + k 23 x 23 +ε .
|a(v)| v≤x
14
v 23 +ε (4.7)
This completes the proof of Theorem 1.2. References 1. Andrianov, A.: Euler products associated with Siegel modular forms of degree 2. Russ. Math. Surv. 29(3), 45–116 (1974) 2. Das, S., Kohnen, W., Sengupta, J.: On a convolution series attached to a Siegel Hecke cusp form of degree 2. Ramanujan J. 33, 367–378 (2014) 3. Fulton, W., Harris, J.: Representation Theory. A First Course, volume 129 of Graduate Texts in Mathematics. Springer, New York (1991) 4. Ivi´c, A.: The Riemann Zeta-Function Theory and Applications. Dover Publications Inc., Mineola, New York (2003) 5. Iwaniec, H., Kowalski, E.: Analytic Number Theory, vol. 53, American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI (2004) 6. Kanemitsu, S., Sankaranarayanan, A., Tanigawa, Y.: A mean value theorem for Dirichlet series and a general divisor problem. Monatsh. Math. 136, 17–34 (2002) 7. Michel, P.: Analytic Number Theory and Families of Automorphic L-functions. IAS/Park City Mathematics Series, vol. 12, pp. 181–295. American Mathematical Society, Providence (2007) 8. Perelli, A.: General L-functions. Ann. Mat. Pura Appl. 130, 287–306 (1982)
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Y. Jiang, G. Lü 9. Pitale, A., Saha, A., Schmidt, R.: Transfer of Siegel Cusp Forms of Degree 2. arXiv:1106.5611v3 (2013) 10. Royer, E., Sengupta, J., Wu, J.: Sign changes in short intervals of coefficients of spinor zeta function of a Siegel cusp form of genus 2. Int. J. Number Theory 10, 327–339 (2014) 11. Weissauer, R.: Endoscopy for GSp(4) and the Cohomology of Siegel Modular Threefolds. Lecture Notes in Mathematics, vol. 1968. Springer, Berlin (2009)
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