Monatsh Math DOI 10.1007/s00605-016-0887-z
Estimates for coefficients of certain L-functions Guangshi Lü1
Received: 21 June 2015 / Accepted: 18 February 2016 © Springer-Verlag Wien 2016
Abstract Let πϕ (or πψ ) be an automorphic cuspidal representation of GL2 (AQ ) associated to a primitive Maass cusp form ϕ (or ψ), and sym j πϕ be the jth symmetric power lift of πϕ . Let asym j πϕ (n) denote the nth Dirichlet series coefficient of the principal L-function associated to sym j πϕ . In this paper, we study first moments of Dirichlet series coefficients of automorphic representations sym3 πϕ of GL4 (AQ ), and πψ ⊗ sym2 πϕ of GL6 (AQ ). For 3 ≤ j ≤ 8, estimates for |asym j πϕ (n)| on average over a short interval have also been established. Keywords Dirichlet series coefficients · Maass cusp form · Symmetric power L-function Mathematics Subject Classification
11F30 · 11F12 · 11F66
1 Introduction and main results Let π be an irreducible cuspidal automorphic representation of GL(m, A), where m ≥ 2 and A is the Adele ring of Q. Let aπ (n) denote the nth Dirichlet series coefficient of the principal L-function associated to π . Recently Goldfeld and Sengupta [8] studied
Communicated by A. Constantin. This work is supported in part by the key project of the National Natural Science Foundation of China (11531008) and IRT1264.
B 1
Guangshi Lü
[email protected] School of Mathematics, Shandong University, Jinan, Shandong 250100, China
123
G. Lü
the first moment have
n≤x
aπ (n) as x → ∞. They proved that for any fixed ε > 0, we
m 3 −1
aπ (n) ε,π x 1+m+m 2 +m 3
+ε
.
(1.1)
n≤x
In 2009, for L-functions associated to irreducible cuspidal automorphic representations of GL(m, A) that are unramified at every finite place, the author [16] has proved a stronger result, which states that for any ε > 0
aπ (n) ε,π
n≤x
71
x 192 +ε x
m 2 −m +ε m 2 +1
if m = 2, if m ≥ 3.
(1.2)
In fact, from the proof of (1.1) and (1.2), one can easily find that the result (1.2) also holds for the general case studied by Goldfeld and Sengupta. Let ϕ be a primitive Maass cusp form, which admits the expansion √ aϕ (n)K iκϕ (2π |n|y)eϕ (nx), ϕ(z) = ρϕ (1) y n≥1
where K ν is the K -Bessel function and eϕ (x) is defined as 2 cos(x) if ϕ is even, or 2i sin(x) if ϕ is odd (i.e. according as the eigenvalues +1 or −1 for the reflection operator). The numbers ρϕ (1) and κϕ depend on the spectral parameter (i.e. the eigenvalue of the Laplace operator) for ϕ, and aϕ (n) is the nth eigenvalue of the Hecke operator. The best known bound on aϕ (n) is due to Kim and Sarnak [12] 7
|aϕ (n)| ≤ n 64 d(n).
(1.3)
It is well-known that for every primitive Maass cusp form ϕ, there is associated an automorphic cuspidal representation πϕ of GL2 (AQ ). Let sym j πϕ be the jth symmetric power lift of πϕ . As a part of the far-reaching Langlands program, it is conjectured that sym j πϕ is an automorphic cuspidal self-dual representation of GL j+1 (AQ ). Thanking to the work of Gelbart and Jacquet [7], Kim and Shahidi [13], [14], and Kim [12], the automorphy of jth symmetric power lift (up to 4), has been established. Let asym j πϕ (n) denote the nth Dirichlet series coefficient of the principal L-function associated to sym j πϕ (from now on, we use aπϕ (n) instead of aϕ (n) to coincide with the notation asym j πϕ (n)). Therefore our result (1.2) (see also Lau and Lü [15]) implies that unconditionally for j = 2, 3, 4, we have n≤x
123
j ( j+1)
asym j πϕ (n) x ( j+1)2 +1
+ε
.
(1.4)
Estimates for coefficients of certain L-functions
For j = 1, one earlier result of Hafner and Ivi´c [9] states that
aπϕ (n) =
n≤x
2
asym1 πϕ (n) ϕ x 5 .
n≤x
Our result (1.2) with m = 2 gives
aπϕ (n) =
n≤x
71
asym1 πϕ (n) ϕ x 192 +ε .
(1.5)
n≤x
Very recently, the author [17] proved that for any ε > 0, we have
1027
aπϕ (n) ϕ,ε x 2827 +ε ,
n≤x
489
asym2 πϕ (n) ϕ,ε x 861 +ε ,
n≤x
which improves (1.4) (with j = 2) and (1.5). In this paper, our first aim is to improve the result on the first moment of Dirichlet series coefficients of automorphic representation sym3 πϕ of GL4 (AQ ). Theorem 1.1 Let ϕ be a primitive Maass cusp form. Then we have
asym3 πϕ (n) ϕ x 0.700947 .
n≤x
Suppose that ψ is another primitive Maass cusp form such that πψ ⊗ sym2 πϕ , the tensor product transfer from automorphic representations on GL2 × GL3 to GL6 , is cuspidal. (This assumption is reasonable due to the cuspidality criterion for the functorial product GL2 × GL3 proved by Ramakrishnan and Wang [18]). Similar to the proof of Theorem 1.1, we are able to prove the following result. Theorem 1.2 Let ϕ be a primitive Maass cusp form. Suppose that ψ is another primitive Maass cusp form such that πψ ⊗ sym2 πϕ is an automorphic cuspidal representation GL6 (AQ ). Then we have
aπψ ⊗sym2 πϕ (n) ϕ,ψ x 0.794345 .
n≤x
Since the automorphy of the jth symmetric power lift ( j > 4) has not been established, it seems rather difficult to obtain nontrivial estimates for
asym j πϕ (n),
if
j > 4.
n≤x
We shall establish estimates for |asym j πϕ (n)| on average over a short interval, when 3 ≤ j ≤ 8.
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G. Lü
Theorem 1.3 For = 2, 3, 4, if x
1−
2 +ε ( +1)2 +2
< y < x, we have
asym2 πϕ (n) yx ε .
x
For = 2, 3, 4, if x
1−
2 +ε ( +1)2 +2
< y < x, we have asym2 −1 πϕ (n) yx ε .
x
9
Theorem 1.3 with j = 2 − 1 = 3 (i.e. x 11 < y < x) improves one earlier result 5 proved by Duke and Iwaniec [3], which states that if x 6 < y < x, asym3 πϕ (n) yx ε .
x
2 Preliminaries The automorphic cuspidal representation πϕ of GL2 (AQ ) factors as a restricted tensor product of local GL2 representations πϕ = ⊗v πϕ,v , where v runs over all places of Q. If v = p is finite, πϕ, p is an unramified principal series representation, and one associates to it a semi-simple SL2 (C)-conjugacy class gϕ ( p)
=
αp 0 0 βp
.
Here α p + β p = aπϕ ( p), α p β p = 1.
(2.1)
For s > 1, the automorphic L-function associated to πϕ is defined by L(πϕ , s) =
det(I − p −s gϕ ( p))−1 ,
(2.2)
p
which coincides with the classical definition ∞
aπϕ (n)n −s =
(1 − aπϕ ( p) p −s + p −2s )−1 p
n=1
=
p
123
a πϕ ( p k ) aπϕ ( p) + · · · + ··· 1+ ps p ks
Estimates for coefficients of certain L-functions
α p −1 β p −1 = . 1− s 1− s p p p Here
a πϕ ( p k ) =
α ip1 β ip2 =
k
i 1 +i 2 =k
i α k−i p βp.
i=0
The jth symmetric power L-function attached to πϕ is defined by L(sym j πϕ , s) =
−1 det I − p −s sym j (gϕ ( p))
p
=
j
j−m m −s −1 βp p )
(1 − α p
p m=0
=
∞ a sym j πϕ (n) n=1
ns
(2.3)
for s 1. Here asym j πϕ (n) is multiplicative, and asym j πϕ ( p k ) =
i0 i1 i j j j−1 j−1 i j−1 j αp αp βp · · · αpβp βp .
i 0 +i 1 +···+i j =k
In particular, for any positive integer j, we have asym j πϕ ( p) = aπϕ ( p j ). The Rankin-Selberg L-function L(symi πψ ⊗ sym j πϕ , s) attached to symi πψ and sym j πϕ is defined as L(symi πψ ⊗ sym j πϕ , s) =
−1 det I − p −s symi (gψ ( p)) ⊗ sym j (gϕ ( p))
p
=
j i
⎛ ⎝1 −
ps
p m=0 m =0
=
∞ a symi πψ ⊗sym j πϕ (n)
ns
n=1
⎞−1 j−m m βp ⎠
˜m α˜ i−m p βp αp
(2.4)
.
Here gψ ( p) belongs to the semi-simple SL2 (C)-conjugacy class
gψ ( p) =
α˜ p 0 0 β˜ p
,
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and asymi πψ ⊗sym j πϕ (n) is multiplicative. In particular, for any positive integer j, we have asym j πϕ ⊗sym j πϕ ( p) = asym j πϕ ( p)2 = aπϕ ( p j )2 . Lemma 2.1 (i) Let ψ and ϕ be two primitive Maass cusp forms. Then we have 2 aπψ ⊗sym2 πϕ (n) ≤ aπψ ⊗πψ (n)asym2 πϕ ⊗sym2 πϕ (n).
(2.5)
(ii) For any ε > 0, we have ⎛ ⎞1 ⎛ ⎞1 8 4 ε⎝ 4⎠ ⎝ 2⎠ n (n) a (d) a (d) . (2.6) asym3 πϕ πϕ ⊗πϕ sym2 πϕ ⊗sym2 πϕ d|n
d|n
Proof Firstly, by a standard identity concerning the Schur functions and then applying the Cauchy’s inequality, one can obtain (i). See Page 31, Line 13 in Brumley [4] for a detailed proof. In order to prove (ii), it suffices to show the inequality over prime powers. Its proof is based on the following well known identity L(πϕ ⊗ sym2 πϕ , s) = L(πϕ , s)L(sym3 πϕ , s)
(2.7)
for s 1. If prime p is such that the Ramanujan conjecture holds. Then |α p | = |β p | = 1. For
s 1, we have L(πϕ , s)−1 :=
∞ μπϕ (n) n=1
ns
.
It is easy to see that for this kind of p’s |μπϕ ( p k )| ≤ d( p k ). where d(n) is the divisor function. In fact μπϕ ( p k ) ≡ 0 for k > 2. This gives k p ( p kε ) aπϕ ⊗sym2 πϕ (d)μπϕ 2 π (d) . asym3 πϕ ( p k ) = a π ⊗sym ϕ ϕ d d| pk d| p k (2.8)
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Estimates for coefficients of certain L-functions
If the Ramanujan conjecture does not hold for p. Then from (2.1) obviously α p and β p are real numbers with the same signs. From (2.7), we have
1+
p
=
aπϕ ⊗sym2 πϕ ( p)
p
ps
1+
+ ··· +
aπϕ ⊗sym2 πϕ ( p k ) p ks
aπϕ ( p k ) aπϕ ( p) + ··· + + ··· s p p ks
+ ···
1+
asym3 πϕ ( p) ps
+ ··· +
asym3 πϕ ( p k ) p ks
+ ···
.
Hence aπϕ ⊗sym2 πϕ ( p k ) = =
i 1 +i 2 =k
i 1 +i 2 =k
aπϕ ( pi1 )asym3 πϕ ( pi2 ) ⎛ ⎝
⎞⎛
αlp1 β lp2 ⎠ ⎝
l1 +l2 =i 1
(2.9)
⎞ 1 s2 s3 3s4 ⎠ . α 3s p αp βp βp
s1 +s2 +s3 +s4 =i 2
If α p and β p are positive real numbers, aπϕ ( pi1 ) and asym3 πϕ ( pi2 ) are both positive. Then by choosing only one term (with i 1 = 0 and i 2 = k) on the right-hand side of (2.9), we have asym3 πϕ ( p k ) ≤ |aπϕ ⊗sym2 πϕ ( p k )|.
(2.10)
If α p and β p are negative real numbers, the sign of aπϕ ( pi1 ) = l1 +l2 =i1 αlp1 β lp2 1 s2 s3 3s4 i2 is (−1)i1 , and the sign of asym3 πϕ ( pi2 ) = s1 +s2 +s3 +s4 =i2 α 3s p α p β p β p is (−1) . So that every term aπϕ ( pi1 )asym3 πϕ ( pi2 ) with i 1 + i 2 = k in (2.9) has the sign (−1)k . Hence aπϕ ( pi1 )asym3 πϕ ( pi2 ) aπϕ ⊗sym2 πϕ ( p k ) = i1 +i2 =k = aπϕ ( pi1 )asym3 πϕ ( pi2 ) i 1 +i 2 =k
=
aπϕ ( pi1 ) asym3 πϕ ( pi2 ) .
i 1 +i 2 =k
By choosing only one term (with i 1 = 0 and i 2 = k) on the right-hand side of the above sum, we also have asym3 πϕ ( p k ) ≤ |aπϕ ⊗sym2 πϕ ( p k )|.
(2.11)
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From (2.8), (2.10) and (2.11), we have asym3 πϕ (n) n ε aπϕ ⊗sym2 πϕ (d) .
(2.12)
d|n
Hence by (i), we have 1 1 aπ ⊗π (d) 2 asym2 π ⊗sym2 π (d) 2 asym3 πϕ (n) n ε ϕ ϕ ϕ ϕ d|n
⎞5 ⎛ ⎞1 ⎛ ⎞1 ⎛ 8 4 8 ε⎝ 4⎠ ⎝ 2⎠ ⎝ ⎠ n aπϕ ⊗πϕ (d) asym2 πϕ ⊗sym2 πϕ (d) 1 d|n
d|n
d|n
⎛ ⎞1 ⎛ ⎞1 8 4 ε⎝ 4⎠ ⎝ 2⎠ n aπϕ ⊗πϕ (d) asym2 πϕ ⊗sym2 πϕ (d) . d|n
d|n
This completes the proof of Lemma 2.1. Define ∞ aπϕ ⊗πϕ (n)4
L 1 (s) =
ns
n=1
,
L 2 (s) =
∞ a 2 sym2 πϕ ⊗sym2 πϕ (n)
ns
n=1
,
and L 3 (s) =
∞
4 d|n aπϕ ⊗πϕ (d)
ns
n=1
,
L 4 (s) =
∞
n=1
2 d|n asym2 πϕ ⊗sym2 πϕ (d)
ns
Lemma 2.2 We have L 1 (s) = ζ (s)7 L(sym2 πϕ , s)21 L(sym4 πϕ , s)13 × L(sym3 πϕ ⊗ sym3 πϕ , s)6 L(sym4 πϕ ⊗ sym4 πϕ , s)H1 (s).
L 2 (s) = L(sym2 πϕ , s)3 L(sym4 πϕ , s)3 L(sym4 πϕ ⊗ sym4 πϕ , s) L(sym3 πϕ ⊗ sym3 πϕ , s)2 H2 (s)
L 3 (s) = ζ (s)8 L(sym2 πϕ , s)21 L(sym4 πϕ , s)13 × L(sym3 πϕ ⊗ sym3 πϕ , s)6 L(sym4 πϕ ⊗ sym4 πϕ , s)H3 (s).
123
.
Estimates for coefficients of certain L-functions
L 4 (s) = ζ (s)L(sym2 πϕ , s)3 L(sym4 πϕ , s)3 L(sym4 πϕ ⊗ sym4 πϕ , s) × L(sym3 πϕ ⊗ sym3 πϕ , s)2 H4 (s). Here H1 (s) and H3 (s) are Dirichlet series absolutely convergent in s > 95/96, and H2 (s) and H4 (s) are Dirichlet series absolutely convergent in s > 23/32. Proof It is well known that in order to probe analytic properties of L j (s) (1 ≤ j ≤ 4), one can use symmetric power L-functions and their Rankin-Selberg L-functions to represent L j (s). This can be done by only comparing the coefficients of p −s in their Euler products. Hence the proof is based on Kim and Sarnak’s bound (1.3), and the following identities: 4 aπϕ ⊗πϕ ( p)4 = aπϕ ( p)2 = aπϕ ( p)8 = 7 + 21asym2 πϕ ( p) + 13asym4 πϕ ( p) + 6asym3 πϕ ⊗sym3 πϕ ( p) + asym4 πϕ ⊗sym4 πϕ ( p),
asym2 πϕ ⊗sym2 πϕ ( p)2 = asym2 πϕ ( p)4 = aπϕ ( p 2 )4 = 3asym2 πϕ ( p) + 3asym4 πϕ ( p) + 2asym3 πϕ ⊗sym3 πϕ ( p) +asym4 πϕ ⊗sym4 πϕ ( p),
and
aπϕ ⊗πϕ (d)4 = 1 + aπϕ ⊗πϕ ( p)4 ,
d| p
asym2 πϕ ⊗sym2 πϕ (d)2 = 1 + asym2 πϕ ⊗sym2 πϕ ( p)2 .
d| p
3 Proof of Theorems 1.1 and 1.2. Let us begin with a general setting. Let L( f, s) be a Dirichlet series (associated to an object f ) that admits an Euler product of degree m ≥ 1, given as L( f, s) =
∞
λ f (n)n
−s
=
m p<∞ j=1
n=1
α f ( p, j) 1− ps
−1
,
where α f ( p, j), j = 1, . . . , m, are the local parameters of L( f, s) at (finite) prime p. Suppose this series and Euler product are absolutely convergent for s > 1. We denote the gamma factor by L ∞ ( f, s) =
m j=1
π−
s+μ f ( j) 2
s + μ f ( j) , 2
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G. Lü
with the local parameters μ f ( j), j = 1, . . . , m, of L( f, s) at ∞. The complete Lfunction ( f, s) is defined as s
( f, s) = q( f ) 2 L ∞ ( f, s)L( f, s), where q( f ) is the conductor of L( f, s). We assume that ( f, s) admits an analytic continuation to the whole complex plane C and is holomorphic everywhere on C except possibly poles of finite order at s = 0, 1. Moreover it satisfies (hypothetically) a functional equation of Riemann type ( f, s) = f ( f˜, 1 − s) where f is the root number with | f | = 1, and f˜ is the dual of f such that λ f˜ (n) = λ f (n), L ∞ ( f˜, s) = L ∞ ( f, s), and q( f˜) = q( f ). We say that L( f, s) ∈ F if it is endowed with the above conditions. For L( f, s) ∈ F, we have the following two lemmas. Lemma 3.1 Assume L( f, s) in F is entire. Then for every η ≥ 0 we have
1
1
m
1
λ f (n) f x 2 − 2m +( 2 − 2 )η +
n≤x
|λ f (n)|.
1 x
Proof This is a special case of Theorem 4.1 in Chandrasekharan and Narasimhan [2] with δ = 1,
A=
1 1 m , β = 1, u = − and q = −∞. 2 2 2m
Lemma 3.2 Let L( f, s) ∈ F. Assume the coefficients λ f (n) ≥ 0. Then for any ε > 0, we have m−1 λ f (n) = x P(log x) + O f,ε x m+1 +ε , n≤x
where P is some polynomial of degree ords=1 L( f, s) − 1 and depends only on f . Proof This is a refined version of Landau’s lemma, see Barthel and Ramakrishnan [1].
From Lemma 3.2, we have the following results. Lemma 3.3 When ϕ is a primitive Maass cusp form, we have n≤x
123
255 aπϕ ⊗πϕ (n)4 = x P1 (log x) + Oϕ,ε x 257 +ε ,
(3.1)
Estimates for coefficients of certain L-functions
40 asym2 πϕ ⊗sym2 πϕ (n)2 = x P2 (log x) + Oϕ,ε x 41 +ε ,
(3.2)
n≤x
128 aπϕ ⊗πϕ (d)4 = x P3 (log x) + Oϕ,ε x 129 +ε ,
(3.3)
81 asym2 πϕ ⊗sym2 πϕ (d)2 = x P4 (log x) + Oϕ,ε x 83 +ε ,
(3.4)
n≤x d|n
n≤x d|n
where deg P1 (t) = 13, deg P2 (t) = 2, deg P3 (t) = 14, and deg P4 (t) = 3. Proof Since from the work of Gelbart and Jacquet [7] for j = 2, and that of Kim and Shahidi [13,14] and Kim [12] for j = 3, 4, we learn that for 1 ≤ j ≤ 4 the jth symmetric power L-function L(sym j πϕ , s) agrees with the principal L-function associated to an automorphic cuspidal self-dual representation of GL j+1 (AQ ),and hence its analytic properties follows. Furthermore from the works of Jacquet and Shalika [10,11], Shahidi [21–24], and the reformulation of Rudnick and Sarnak [19], we know the analytic properties for the Rankin-Selberg L-functions L(symi πϕ × sym j πϕ , s) with i, j = 1, 2, 3, 4. Therefore ζ (s)7 L(sym2 πϕ , s)21 L(sym4 πϕ , s)13 L(sym3 πϕ × sym3 πϕ , s)6 L(sym4 πϕ × sym4 πϕ , s) belongs to F with m = 256, and s = 1 is its only pole of order 14. Suppose that for s > 1 ζ (s)7 L(sym2 πϕ , s)21 L(sym4 πϕ , s)13 L(sym3 πϕ × sym3 πϕ , s)6 L(sym4 πϕ × sym4 πϕ , s) :=
∞
bϕ (n)n −s .
n=1
Then by Lemma 3.2, we have
255 bϕ (n) = x Q(log x) + Oϕ,ε x 257 +ε ,
n≤x
where deg Q(t) = 13. By the first formula in Lemma 2.2, we have the convolution aπϕ ⊗πϕ (n)4 =
bϕ (u)c(v).
(3.5)
n=uv
and v≥1
|c(v)|v −σ σ 1
(∀ σ >
95 96 ).
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G. Lü
Hence by the easy convolution arguments, we have
255 aπϕ ⊗πϕ (n)4 = x P1 (log x) + Oϕ,ε x 257 +ε ,
n≤x
where deg P1 (t) = 13. Another three asymptotic formulae can be established similarly by noting that L(sym2 πϕ , s)3 L(sym4 πϕ , s)3 L(sym4 πϕ ⊗ sym4 πϕ , s)L(sym3 πϕ ⊗ sym3 πϕ , s)2 , ζ (s)8 L(sym2 πϕ , s)21 L(sym4 πϕ , s)13 L(sym3 πϕ ⊗ sym3 πϕ , s)6 L(sym4 πϕ ⊗ sym4 πϕ , s),
and ζ (s)L(sym2 πϕ , s)3 L(sym4 πϕ , s)3 L(sym4 πϕ ⊗ sym4 πϕ , s)L(sym3 πϕ ⊗ sym3 πϕ , s)2
belong to F with m = 81, 257, and 82 respectively.
Now we complete the proof of Theorems 1.1 and 1.2. Since sym3 πϕ is an automorphic cuspidal self-dual representation of GL4 (AQ ), L(sym3 πϕ , s) is an L-function of degree 4, which satisfies Lemma 3.1. Therefore we have
3
3
asym3 πϕ (n) x 8 + 2 η +
n≤x
x
|asym3 πϕ (n)|,
(3.6)
3 4 −η
where η ≥ 0 is a parameter to be chosen later. By (2.6) in Lemma 2.1, we have
asym3 πϕ (n)
n≤x
x
3 3 8+2η
+ xε
⎛ ⎞1 ⎛ ⎞1 8 4 4⎠ ⎝ 2⎠ ⎝ aπϕ ⊗πϕ (d) asym2 πϕ ⊗sym2 πϕ (d) .
3
x
d|n
d|n
(3.7)
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Estimates for coefficients of certain L-functions
By the Hölder inequality, we have
asym3 πϕ (n)
n≤x
⎛
x
3 3 8+2η
⎞5 ⎛
⎜ + xε ⎝
3
⎜ ×⎝
3
x
3
5
⎟ ⎜ 1⎠ ⎝
3
x
⎛
⎞1
8
x
8
⎟ aπϕ ⊗πϕ (d)4 ⎠
d|n
⎞1 4
⎟ asym2 πϕ ⊗sym2 πϕ (d)2 ⎠
d|n
3
1
128
1
81
x 8 + 2 η + x 8 ×( 4 −η)+ 8 × 129 + 4 × 83 +ε . On taking η = 0.217298, we have
asym3 πϕ (n) x 0.700947 .
n≤x
Similarly, by Lemma 3.1 we have 5
5
aπψ ⊗sym2 πϕ (n) x 12 + 2 ϑ +
x
|aπψ ⊗sym2 πϕ (n)|,
(3.8)
5 −ϑ 6
where ϑ ≥ 0 is a parameter to be chosen later. Then by Lemmas 2.1 and 3.3, we have aπψ ⊗sym2 πϕ (n) 5
5
x 12 + 2 ϑ ⎛
⎞5 ⎛
⎜ +⎝
⎟ aπψ ⊗πψ (n)4 ⎠ 5
x
x
⎞1 4
⎜ ×⎝
8
⎟ ⎜ 1⎠ ⎝ 5
⎛
⎞1
8
⎟ asym2 πϕ ⊗sym2 πϕ (n)2 ⎠ 5
x
5
5
5
1
255
1
40
x 12 + 2 ϑ + x 8 ×( 6 −ϑ)+ 8 × 257 + 4 × 41 +ε . On taking ϑ = 0.151071, we have
aπψ ⊗sym2 πϕ (n) x 0.794345 .
n≤x
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G. Lü
4 Proof of Theorem 1.3. Firstly, we establish some general inequalities about coefficients of L-functions, which generalize the inequality (2.12). Lemma 4.1 For = 1, 2, 3, 4, we have asym πϕ ⊗sym πϕ (d), asym2 πϕ (n) n ε d|n
and for = 2, 3, 4 asym2 −1 πϕ (n) n ε asym −1 πϕ ⊗sym πϕ (d) . d|n
Proof It is well known that L(sym1 πϕ ⊗ sym1 πϕ , s) = L(πϕ × πϕ , s) = ζ (s)L(sym2 πϕ , s) for s > 1. This gives n ≤ = a (n) μ (d) aπϕ ×πϕ (d). asym2 πϕ πϕ ⊗πϕ d d|n d|n
(4.1)
For other cases, it suffices to show the inequalities over prime powers. Consider the standard representation C2 of the group S L 2 (C), i.e. ρ : S L 2 (C) → C2 , g → ρ(g)v = gv where gv is the usual matrix multiplication. We identify, whenever no confusion arises, ρ = ρ(g)
where g = diag(α p , β p ) =
αp
βp
for any prime p, and denote tr (ρ) to be the trace of ρ = ρ(g). Then aπϕ ( p) = tr (ρ), asym j πϕ ( p) = tr (sym j ρ), asymi πϕ ⊗sym j πϕ ( p) = tr (symi ρ ⊗ sym j ρ),
where sym j is the jth symmetric power and ⊗ is the tensor product. Moreover, the plocal factors of the associated L-functions are given by the reciprocal of characteristic polynomials, L(ρ, X ) = det(I − ρ X )−1 , L(sym j ρ, X ) = det(I − sym j ρ X )−1 , L(symi ρ ⊗ sym j ρ, X ) = det(I − symi ρ ⊗ sym j ρ X )−1 with X = p −s .
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Estimates for coefficients of certain L-functions
The following are well-known identities: for j ≥ 1, ⊗2 sym j ρ := sym j ρ ⊗ sym j ρ = sym2r ρ,
(4.2)
0≤r ≤ j
or more generally, for a ≥ b, syma ρ ⊗ symb ρ =
syma+b−2r ρ,
(4.3)
0≤r ≤b
([6, p.151]) where sym0 ρ denotes the 1-dimensional trivial representation. These identities give asym πϕ ⊗sym πϕ ( p) =
r =0
asym2r πϕ ( p), asym −1 πϕ ⊗sym πϕ ( p) =
−1 r =0
asym2 −1−2r πϕ ( p),
(4.4) L(sym πϕ ⊗ sym πϕ , s) =
L(sym2r πϕ , s),
(4.5)
r =0
and L(sym −1 πϕ ⊗ sym πϕ , s) =
−1
L(sym2 −1−2r πϕ , s).
(4.6)
r =0
If prime p is such that the Ramanujan conjecture holds. Then |α p | = |β p | = 1. Similar to (4.1), our inequalities over powers of p will follow from identities (4.5) and (4.6). If the Ramanujan conjecture does not hold for p. Then obviously α p and β p are real numbers with the same signs. From (4.5) and (4.6), we have asym πϕ ⊗sym πϕ ( p k ) =
i 0 +i 1 +···+i =k
asym0 πϕ ( pi0 )asym2 πϕ ( pi1 )asym4 πϕ ( pi2 ) · · · asym2 πϕ ( pi ),
(4.7) and asym −1 πϕ ⊗sym πϕ ( p k ) =
i 0 +i 1 +···+i −1 =k
asym2 −1 πϕ ( pi0 )asym2 −3 πϕ ( pi1 ) · · · asym1 πϕ ( pi −1 ).
(4.8) On noting that α p and β p are real numbers with the same signs, we can easily find from (2.3) that every asym2r πϕ ( pir ), (0 ≤ r ≤ ) is positive. Then by only choosing
123
G. Lü
one term (with i 0 = i 1 · · · = i −2 = 0 and i −1 = k) on the right-hand side of (4.7), we have (4.9) asym2 πϕ ( p k ) ≤ asym πϕ ⊗sym πϕ ( p k ) . If α p and β p are positive real numbers, similarly we have asym2 −1 πϕ ( p k ) ≤ asym −1 πϕ ⊗sym πϕ ( p k ) . Then from now on, we assume that α p and β p are both negative real numbers. Although asym2 −1−2r πϕ ( pir ) (0 ≤ r ≤ − 1) may not share the same sign, we shall show that for chosen -tuple (i 0 , i 1 , i 2 , . . . , i −1 ) satisfying i 0 + i 1 + . . . + i −1 = k every product asym2 −1 πϕ ( pi0 )asym2 −3 πϕ ( pi1 ) . . . asym1 πϕ ( pi −1 ) has the same sign (−1)k . In fact, from the identity asym2 −1−2r πϕ ( pir ) =
(2 j−1−2r −2)s2
)s0 α (2 −1−2r αp p
(2 −1−2r )s2 −1−2r
· · · βp
,
s0 +s1 +...+s2 −1−2r =ir
one can easily find that asym2 −1−2r πϕ ( pir ) has the sign (−1)ir . Therefore k i i i asym2 −1 πϕ ( p 0 )asym2 −3 πϕ ( p 1 ) · · · asym1 πϕ ( p −1 ) asym −1 πϕ ⊗sym πϕ ( p ) = i0 +i1 +···+i −1 =k = asym2 −1 πϕ ( pi0 )asym2 −3 πϕ ( pi1 ) · · · asym1 πϕ ( pi −1 ) i 0 +i 1 +···+i −1 =k
=
i 0 +i 1 +···+i −1 =k
asym2 −1 πϕ ( pi0 ) asym2 −3 πϕ ( pi1 ) · · · asym1 πϕ ( pi −1 ) .
This obviously gives asym2 −1 ϕ ( p k ) ≤ asym −1 ϕ⊗sym ϕ ( p k ) .
This completes the proof of Lemma 3.1. Similar to Lemma 3.3, we have Lemma 4.2 When ϕ is a primitive Maass cusp form, we have that for = 2, 3, 4
2 1− +ε , asym πϕ ⊗sym πϕ (d) = x P (log x) + Oϕ,ε x ( +1)2 +2
n≤x d|n
where deg(P ) = 1.
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(4.10)
Estimates for coefficients of certain L-functions
Proof In fact, the nice part of the generating function of the arithmetic function 2 a d|n sym πϕ ⊗sym πϕ (d) is ζ (s)L(sym πϕ ⊗sym πϕ , s), whose degree is ( +1) +1. Hence Lemma 4.2 follows from Lemma 3.2.
By Lemma 4.1, we have asym2 πϕ (n) x ε
x
asym ϕ⊗sym ϕ (d),
x
and asym2 −1 πϕ (n)
x
xε
asym −1 πϕ ⊗sym πϕ (d)
x
x
ε
1
1
asym −1 πϕ ×sym −1 πϕ (d) 2 asym πϕ ⊗sym πϕ (d) 2
x
⎛ x ⎝ ε
⎞1 ⎛ 2
asym −1 πϕ ⊗sym −1 πϕ (d)⎠ ⎝
x
⎞1 2
asym πϕ ⊗sym πϕ (d)⎠ .
x
Hence Theorem 1.3 follows from Lemma 4.2. Acknowledgments
The author would like to thank the referee for valuable comments.
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