aa r X i v : . [ m a t h . N T ] J un O N TH E H ECKE E IGENVALU E S OF M AASS F ORMS
Wenzhi Luo ∗ and Fan ZhouJune 4, 2018 Abstract
Let φ denote a primitive Hecke-Maass cusp form for Γ o ( N ) with the Laplacianeigenvalue λ φ = + t φ . In this work we show that there exists a prime p suchthat p ∤ N , | α p | = | β p | =
1, and p ≪ ( N ( + | t φ | )) c , where α p , β p are the Satakeparameters of φ at p , and c is an absolute constant with 0 < c <
1. In fact, c can betaken as 0.27332. In addition, we prove that the natural density of such primes p ( p ∤ N and | α p | = | β p | =
1) is at least 34/35.MSC: 11F30 (Primary) 11F41, 11F12 (Secondary)
The celebrated Ramanujan-Petersson conjecture for an elliptic cuspidal Hecke eigen-form f of weight k ≥ N asserts that for any prime p ∤ N , | λ f ( p ) | ≤ p k − ,where λ f ( p ) denotes the p -th Hecke eigenvalue of f . This conjecture has been solvedaffirmatively by Deligne in [De1] and [De2] as a consequence of his proof of the Weilconjectures.Now let φ denote a primitive Hecke-Maass cusp form for Γ o ( N ) and Dirichlet char-acter χ φ with the Laplacian eigenvalue λ φ = + t φ . Denote the n -th Hecke eigen-value of φ by λ φ ( n ) for n ∈ N . The generalized Ramanujan-Petersson conjecturepredicts that for p ∤ N , | λ φ ( p ) | ≤ | α p | = | β p | =
1, where { α p , β p } are the Satake parameters of φ at p , i.e. the local component of φ at p is tempered.This is an outstanding unsolved problem in number theory, which would follow fromthe Langlands functoriality conjectures. Currently the record of individual boundstowards this conjecture is due to Kim-Sarnak [KS] | λ φ ( p ) | ≤ p + p − , ∗ Research of W. Luo is partially supported by NSF grant DMS-1160647. φ as above, this conjecture (i.e., | α p | = | β p | =
1) is true for (unramified) primeswith the lower
Dirichlet density at least 9/10. This lower Dirichlet density is later im-proved to 34/35 in [KSh]. For simplicity the primes at which the Ramanujan conjec-ture holds are referred as the
Ramanujan primes of φ , so the Ramanujan conjecture isequivalent to the statement that all (unramified) primes are Ramanujan primes of φ .Note that the method in [Ram] (and [KSh]) is ineffective, and does not provide anyquantitative bound, for example, for the occurrence of the least Ramanujan prime fora given Maass form φ .The main purpose of this paper is to show that the least Ramanujan prime of φ isbounded by (cid:0) N ( + | t φ | ) (cid:1) c for some constant c >
0, and in fact we can prove a ’sub-convexity’ bound with c < L -functions on GL ( ) in the eigenvalue aspect. Furthermore, the Lindel ¨of hypothe-sis (a consequence of the Riemann Hypothesis) for the adjoint L -function of φ (see (1)below) would imply that the exponent c > p is not a Ramanujan prime of φ , then (see Lemma 1.1 below) λ φ ( p i ) χ φ ( p i ) > i + i ≥
1, where χ φ is the central character of φ . Thus thefollowing adjoint (square) L -function associated to φ comes into play (see [GJ]), L ( s , Ad φ ) = L ( s , φ × φ ) ζ ( s ) = ζ ( N ) ( s ) ∞ ∑ n = χ φ ( n ) λ φ ( n ) n − s , (1)where ζ ( N ) ( s ) , as usual, stands for the partial zeta function with local factors at p | N removed from ζ ( s ) . Then naturally we can relate our goal of bounding the least un-ramified Ramanujan prime for Maass form φ to the sieving idea in the work [IKS] (aswell as its further refinements in [KLSW] and [Mat]), which study the first negativeHecke eigenvalue for a holomorphic Hecke eigenform based on the Deligne’s resolu-tion of Ramanujan-Petersson conjecture in the case of elliptic modular forms.It turns out that the sieving idea in [IKS] (also in [KLSW] and [Mat]) works well inthe current quite different setting, even though the Deligne-type bound is not availableyet for Maass form φ .We present two proofs with different exponents c . The first proof (Section 2) illus-trates our basic ideas via the simple case of level 1. The second proof obtains signifi-cantly better (smaller) exponent c . In Section 4, we refine the density results in [Ram]and [KSh] from the Dirichlet density to the natural density.We end the Introduction by stating the following Lemma 1.1, which will be used inthe proofs of the following sections, and a part of it is also an ingredient in [Ram]. Lemma 1.1.
Let { α p , β p } denote the Satake parameters at p ∤ N of a primitive Hecke-Maasscusp form φ for Γ ( N ) with Dirichlet character χ φ . Then the Satake parameters at p forL ( s , Ad φ ) are given by { α p / β p , 1, β p / α p } . For any unramified p ∤ N, we have | λ φ ( p ) | = λ φ ( p ) χ φ ( p ) = λ φ ( p ) χ φ ( p ) + n particular λ φ ( p ) χ φ ( p ) is real and λ φ ( p ) χ φ ( p ) ≥ − . If p is not a Ramanujan prime of φ , i.e., | α p / β p | 6 = , then we have | λ φ ( p ) | > and α p / β p is real and > and for n ≥ λ φ ( p n ) χ φ ( p n ) = (cid:16)q α p β p (cid:17) n + − (cid:18)r β p α p (cid:19) n + q α p β p − r β p α p > d ( p n ) = n + where d is the divisor function.Proof. The first assertion follows from the definition of L ( s , Ad φ ) and the fact that theSatake parameters at p for the contragredient form φ are { α − p , β − p } . For p ∤ N , wehave λ φ ( p ) = χ φ ( p ) λ φ ( p ) .By Hecke relation, we have λ φ ( p ) = λ φ ( p ) − χ φ ( p ) . Then we have λ φ ( p ) χ φ ( p ) = λ φ ( p ) χ φ ( p ) − = λ φ ( p ) λ φ ( p ) − λ φ ( p ) χ φ ( p ) is real and ≥ − p ∤ N , we have α p + β p = λ φ ( p ) and α p β p = χ φ ( p ) .Then we get α p β p + β p α p = | λ φ ( p ) | − ≥ − α p β p · β p α p = { α p / β p , β p / α p } are the roots of the quadratic equation X − ( | λ φ ( p ) | − ) X + = p ∤ N is not a Ramanujan prime of φ , i.e., | α p / β p | 6 =
1, this implies that { α p / β p , β p / α p } are two real positive distinct roots. Because their product is 1, one ofthem is > <
1. Also, we have | λ φ ( p ) | >
2. From λ φ ( p n ) = α n + p − β n + p α p − β p and α p β p = χ φ ( p ) ,we get the last assertion. In this section, to illustrate quickly and clearly the main ideas of this paper, we con-sider the simplest case of level 1. Thus φ is a Hecke-Maass cusp form for SL ( Z ) ,with the Laplacian eigenvalue λ φ = + t φ and the n -th Hecke eigenvalue λ φ ( n ) .The goal of this section is to prove the following theorem. Theorem 2.1.
Let φ be a Hecke-Maass cusp form for SL ( Z ) as above. Then for any ǫ > ,there exists a prime p such that | λ φ ( p ) | ≤ and p ≪ t + ǫφ , where the implied constantdepends on ǫ > alone. emark 2.2. It is clear from the proof that the same argument is in fact still valid forany primitive Hecke-Maass cusp form φ on Γ o ( N ) with the central character χ φ , bysimply replacing λ φ ( p ) by λ φ ( p ) χ φ ( p ) . Proof.
Assume p is not a Ramanujan prime of φ for all primes p ≤ y . Then by theLemma 1.1 we have λ φ ( d ) > < d ≤ y . Take x = yz and z = y δ with0 < δ < S ( x ) = ∑ d < x λ φ ( d ) log xd = S + ( x ) + S − ( x ) ,where S + ( x ) and S − ( x ) denote the partial sums over the positive and negative eigen-values λ φ ( d ) respectively.If λ φ ( d ) < S − ( x ) , then d = mp with λ φ ( m ) > λ φ ( p ) <
0, where all theprime divisors of m do not exceed y , and p > y . From λ φ ( p ) = λ φ ( p ) − ≥ −
1, wededuce that S − ( x ) = ∑ pm < x , p > y , λ φ ( p ) < λ φ (( pm ) ) log (cid:18) xpm (cid:19) ≥ − ∑ m < z λ φ ( m ) ∑ p ≤ x / m log (cid:18) xpm (cid:19) ≥ − ∑ m < z λ φ ( m ) m ! x log y (cid:18) + O (cid:18) y (cid:19)(cid:19) , (2)in view of the asymptotics π ( x ) log x − ∑ p ≤ x log p = x log x + O x log x ! ,by the Prime Number Theorem (see [Pra]).Next we bound S + ( x ) . By positivity, S + ( x ) ≥ ∑ m < z λ φ ( m ) ∑ l < x / mp | l ⇒ z < p ≤ y λ φ ( l ) log (cid:16) xlm (cid:17) ≥ ∑ m < z λ φ ( m ) Φ ′ ( x / m , y , z ) , (3)where Φ ′ ( X , Y , Z ) = ∑ < l < Xp | l ⇒ Z < p ≤ Y log (cid:18) Xl (cid:19) . Lemma 2.3.
If Z is large, Z < Y and Y < X ≤ YZ, then Φ ′ ( X , Y , Z ) > X Z − X log Y + O Z log Y log Z + X log Z ! .4 roof. Define Φ ( X , Y , Z ) = ∑ < l < Xp | l ⇒ Z < p ≤ Y Φ ( X , Z ) = ∑ < l < Xp | l ⇒ Z < p Φ ′ ( X , Y , Z ) = Z XY Φ ( t , Y , Z ) d tt + Z YZ Φ ( t , Z ) d tt .For Y < t ≤ YZ , it is easy to see that Φ ( t , Y , Z ) = Φ ( t , Z ) − Φ ( t , Y ) .Recall the asymptotic formula of Φ ( X , Z ) , X ≥ Z ≥ Φ ( X , Z ) = ω (cid:18) log X log Z (cid:19) X log Z − Z log Z + O X log Z ! , (4)where ω ( u ) is the Buchstab function, that is the continuous solution to the difference-differential equation u ω ( u ) = ( ≤ u ≤ ) , ( u ω ( u )) ′ = ω ( u − ) ( u > ) .Moreover the range of the Buchstab function is 1/2 ≤ ω ( u ) ≤
1. We infer that Φ ′ ( X , Y , Z ) = Z XZ Φ ( t , Z ) d tt − Z XY Φ ( t , Y ) d tt ≥ Z XZ (cid:18) t log Z − Z log Z (cid:19) d tt − Z XY (cid:18) t log Y − Y log Y (cid:19) d tt + O X log Z ! ≥ X Z − X log Y + O Z log Y log Z + X log Z ! .This completes the proof of Lemma 2.3.By Lemma 2.3, we have Φ ′ ( x / m , y , z ) > (cid:18) δ − + O (cid:18) y (cid:19)(cid:19) xm log y ,and S + ( x ) > (cid:18) δ − + O (cid:18) y (cid:19)(cid:19) ∑ m < z λ φ ( m ) m ! x log y from (3). Consequently, after combining with the lower bound of S − ( x ) in (2), wededuce that S ( x ) > (cid:18) δ − + O (cid:18) y (cid:19)(cid:19) ∑ m < z λ φ ( m ) m ! x log y .5herefore S ( x ) ≫ x log x , (5)on choosing δ = − ǫ , provided y ≫ σ > ǫ >
0, we have S ( x ) = ∑ d < x λ φ ( d ) log (cid:16) xd (cid:17) = π i Z ( σ ) L ( s , Ad φ ) ζ ( s ) x s s d s = π i Z ( ) L ( s , Ad φ ) ζ ( s ) x s s d s ≪ t + ηφ x , (6)by shifting the line of integration to ℜ ( s ) = L ( s , Ad φ ) on the critical line.Comparing (5) and (6), we obtain x = y + δ ≪ t + ηφ ,i.e. y ≪ t + ǫφ ,for any ǫ >
0. This completes the proof of Theorem 2.1.
Remark 2.4.
A hypothetical subconvexity bound of L ( s , Ad φ ) in the eigenvalue aspecton the critical line ℜ ( s ) = L ( + it , Ad φ ) ≪ t − δφ t + ǫ ,for some δ >
0. It is clear that this in turn would immediately lead to y ≪ t − δφ . In this section we refine the approach in Section 2 to obtain a better exponent. Themethod employs the theory of multiplicative functions.Let φ be a primitive Hecke-Maass cusp form for Γ o ( N ) ⊂ SL ( Z ) with Dirichletcharacter χ φ : ( Z / N Z ) ∗ → C . It has Laplacian eigenvalue 1/4 + t φ with the parame-ter t φ lying in R ∪ [ − i /64, 7 i /64 ] . We assume that φ is not of dihedral type, otherwisethe full Ramanujan conjecture is known. The standard L -function of φ is given by L ( s , φ ) = ∞ ∑ n = λ φ ( n ) n s ,where λ φ ( n ) ’s are normalized Hecke eigenvalues with λ φ ( ) = T n φ = λ φ ( n ) φ for n ∈ Z . 6ur main tool is the adjoint L -function of φ mentioned in the Introduction andLemma 1.1 L ( s , Ad φ ) = ζ ( N ) ( s ) ∞ ∑ n = λ φ ( n ) χ φ ( n ) n s = ∞ ∑ n = A φ ( n ) n s ,where A φ ( n ) = ∑ k | n λ φ ( n / k ) χ φ ( n / k ) for ( n , N ) =
1. As in [IS], we denote theanalytic conductor by Q = Q ( Ad φ ) .We have Q ( Ad φ ) ≤ N ( + | t φ | ) .Lemma 1.1 implies that for a prime p ∤ N then A φ ( p ) is real and ≥ −
1. It also implies A φ ( p ) > p is not a Ramanujan prime of φ , i.e., | λ φ ( p ) | > p is not a Ramanujan prime of φ for all p ≤ y and p ∤ N . Thuswe have A φ ( p ) > p ≤ y . Define S ♭ ( x ) = ∑ ♭ n ≤ x ( n , N )= A φ ( n ) log (cid:16) xn (cid:17) where the summation ∑ ♭ is taken over squarefree numbers. Lemma 3.1.
We have S ♭ ( x ) ≪ x Q + ǫ . Proof.
Define G ( s ) = ∏ p ∤ N (cid:18) − A φ ( p ) p s + A φ ( p ) p s − p s (cid:19) (cid:18) + A φ ( p ) p s (cid:19) .The analytic function G ( s ) is absolutely convergent in {ℜ ( s ) > + ǫ } , and uni-formly bounded by Q ǫ with any ǫ >
0, in view of the Rankin-Selberg convolution ofAd φ × Ad φ . Now L ( s , Ad φ ) G ( s ) = ∞ ∑ ♭ n = ( n , N )= A φ ( n ) n s is absolutely convergent in {ℜ ( s ) > } . For c > S ♭ ( x ) = π i Z ( c ) L ( N ) ( s , Ad φ ) G ( s ) x s s d s = π i Z ( ) L ( N ) ( s , Ad φ ) G ( s ) x s s d s . (7)By using the convexity bound L ( s , Ad φ ) ≪ ǫ ( Qt ) ( −ℜ ( s )) /2 + ǫ ,we obtain S ♭ ( x ) ≪ x Q + ǫ . 7efine a multiplicative function supported on squarefree numbers with h ( p ) = ( p ≤ y , − p > y .It extends to all squarefree numbers. For convenience, we define h ( n ) = n is notsquarefree. Define S ♭ ( x ) = ∑ ♭ n ≤ x ( n , N )= A φ ( n ) . Lemma 3.2. If ∑ n ≤ t ( n , N )= h ( n ) ≥ for all t ≤ x, we have S ♭ ( x ) ≥ ∑ n ≤ x ( n , N )= h ( n ) . (8) Proof.
The proof follows [KLSW]. Let us define a multiplicative function g defined bythe Dirichlet convolution A φ = h ∗ g , or A φ ( n ) = ∑ d | n h ( d ) g (cid:16) nd (cid:17) .We have g ( p ) = A φ ( p ) − h ( p ) ≥ p ∤ N . Then we have S ♭ ( x ) = ∑ ♭ n ≤ x ( n , N )= A φ ( n )= ∑ ♭ n ≤ x ( n , N )= ∑ d | n h ( d ) g (cid:16) nd (cid:17) = ∑ ♭ d ≤ x ( d , N )= g ( d ) ∑ b ≤ x / d ( b , N )= h ( b ) ≥ ∑ n ≤ x ( n , N )= h ( n ) Both g ( d ) and ∑ h ( b ) are non-negative. We have g ( ) = Lemma 3.3. If ∑ n ≤ t ( n , N )= h ( n ) ≥ for all t ≤ x, we haveS ♭ ( x ) ≥ ∑ n ≤ x ( n , N )= h ( n ) log (cid:16) xn (cid:17) .8 roof. It follows from the formula S ♭ ( x ) = Z x S ♭ ( t ) d tt and Lemma 3.2.The following lemma evaluates the mean of the multiplicative function h ( n ) overa long range 1 ≤ n ≤ x where x equals y u for some u >
1. The special case of thislemma appears in [KLSW] and a more elaborate version is available in [Mat].
Lemma 3.4.
Let U ≥ and let h ( n ) be as above. We have ∑ n ≤ y u ( n , N )= h ( n ) = c ( N )( σ ( u ) + o U ( ))( log y ) y u uniformly for u ∈ [ U , U ] , where lim y → ∞ o U ( ) = and c ( N ) = (cid:16) φ ( N ) N (cid:17) ∏ p ∤ N ( − p ) ( + p ) ≫ ( log log N ) − . The constant σ ( u ) is the continuous function of u ∈ ( ∞ ) uniquelydetermined by the differential-difference equation σ ( u ) = u , 0 < u ≤ ( u − σ ( u )) ′ = − σ ( u − ) u , u > Proof.
In Lemma 6 of [Mat], take K = x = x = χ = χ = − q =
1. Thefunction σ ( u ) can be computed from Lemma 8 of [Mat]. Lemma 3.5.
Let u > be such that σ ( u ) > for < u ≤ u . We have for y ≫ u , ∑ n ≤ y u ( n , N )= h ( n ) log (cid:18) y u n (cid:19) ≫ u c ( N ) y u . Proof.
Define H ( x ) = ∑ n ≤ x ( n , N )= h ( n ) .We have ∑ n ≤ y u ( n , N )= h ( n ) log (cid:18) y u n (cid:19) = Z y u H ( t ) d tt = Z u H ( y u ) log y d u ≥ Z u u H ( y u ) log y d u By Lemma 3.4, we have for 1/ u ≤ u ≤ u uniformly H ( y u ) = c ( N )( σ ( u ) + o u ( ))( log y ) y u .9or y ≫ u
1, we hence have Z u u H ( y u ) log y d u ≫ u c ( N ) y u and this completes the proof.Let u be the same as defined in Lemma 3.5. We have c ( N ) ≫ Q − ǫ for ǫ > y u Q − ǫ ≪ u ∑ n ≤ y u ( n , N )= h ( n ) log (cid:18) y u n (cid:19) ≪ S ♭ ( y u ) ≪ ( y u ) Q + ǫ and this in turn gives y ≪ u Q u + ǫ . (9)By numerical computation of Mathematica , we find the smallest zero of σ ( u ) is ap-proximately 3.65887. Then taking u to be microscopically less than 3.65887 we get: Theorem 3.6.
For any primitive Hecke-Maass cusp form φ for Γ o ( N ) with character χ φ andLaplace eigenvalue + t φ , there exists a prime number p ∤ N with p ≪ ( N ( + | t φ | )) such that the Ramanujan conjecture holds for φ at p. Remark 3.7.
In Lemma 3.1, the line of integration in (7) may be taken on {ℜ ( s ) = σ } instead of {ℜ ( s ) = } for 1/2 < σ <
1. This will result in a different version ofLemma 3.1, i.e., S ♭ ( x ) ≪ x σ Q ( − σ ) /2 + ǫ .However, this change has no impact on the final exponent in Theorem 3.6. Remark 3.8.
To estimate the smallest zero of σ ( u ) without numerical computation, wehave from Lemma 3.4 σ ( u ) = u − u + − u log u for 1 ≤ u ≤
2. It is not hard to prove that σ ( u ) is monotone for 1 ≤ u ≤ σ ( u ) is positive for 1 ≤ u ≤
2. Without numerical computation,we can have 1/4 as the exponent in Theorem 3.6.For 2 ≤ u ≤
3, we have σ ( u ) = u Li ( − u ) + ( π u ) /3 + u − u log ( u − ) + u log ( u − ) log ( u ) − u log ( u ) − u + u log ( u − ) − ( u − ) + Li is the famous dilogarithm function (see [Zag]). We leave to the reader toverify that σ ( u ) is positive for 2 ≤ u ≤
3. 10
Natural Density of Ramanujan Primes
Let φ be a primitve Maass form for Γ o ( N ) with character χ φ and with Hecke eigenval-ues λ φ ( n ) , following the same notations of the previous sections. We assume that φ isnot of Artin type, since otherwise the full Ramanujan conjecture is known ([KSh]).In this section, we refine the density results of the Ramanujan primes in [Ram] and[KSh] from Dirichlet density to natural density . We achieve the same constant by em-ploying a similar but different method. We will first quickly indicate how our methodleads directly to the fact that the lower natural density of the Ramanujan primes of φ isat least 9/10, and then improve it further to 34/35 by a more elaborate argument.The adjoint (Gelbart-Jacquet) lift (see [GJ]) of φ , with its L -function defined by L ( s , Ad φ ) = ∑ ∞ n = A φ ( n ) / n s , ℜ ( s ) > A φ ( p ) = λ φ ( p ) χ φ ( p ) , is a cuspidalautomorphic representation of GL ( ) . The symmetric cube lift Sym φ and the twistedsymmetric fourth power lift Sym φ × χ φ are cuspidal automorphic representations of GL ( ) and GL ( ) respectively (see [KSh2] and [Ki]). Let L ( s , Sym φ ) = ∞ ∑ n = A [ ] φ ( n ) n s and L ( s , Sym φ × χ φ ) = ∞ ∑ n = A [ ] φ ( n ) n s be their L -functions and { α p , β p } be the Satake parameters associated with φ at an un-ramified prime p . The Satake parameters of Sym φ are given by { α p , α p β p , α p β p , β p } ,while those of Sym φ × χ φ are given by { α p / β p , α p / β p , 1, β p / α p , β p / α p } .In light of the standard zero-free region of L ( s , Ad φ ) and L ( s , Sym φ × χ φ ) , thefollowing Prime Number Theorem for L -functions holds. (see Theorem 5.13 of [IK]). Lemma 4.1.
We have ∑ p ≤ X A φ ( p ) = o (cid:18) X log X (cid:19) and ∑ p ≤ X A [ ] φ ( p ) = o (cid:18) X log X (cid:19) , as X → ∞ . For the result on the natural density, let us first consider the sum S ( X ) = ∑ p ≤ X ( + A φ ( p )) .On one hand, we have S ( X ) > { p ≤ X , | λ φ ( p ) | > } .On the other hand, we have S ( X ) = ∑ p ≤ X ( + A φ ( p ) + A φ ( p ) )= ∑ p ≤ X ( + A φ ( p ) + ( A [ ] φ ( p ) + A φ ( p ) + ))= ∑ p ≤ X ( + A φ ( p ) + A [ ] φ ( p ))= π ( X ) + o ( π ( X )) ,11y Lemma 4.1. Hence we get { p ≤ X , | λ φ ( p ) | > } ≤ π ( X ) + o ( π ( X )) ,or equivalently { p ≤ X , | λ φ ( p ) | ≤ } ≥ π ( X ) + o ( π ( X )) ,i.e., the lower natural density of the Ramanujan primes of φ is at least 9/10.Next we turn to the improvement of the above density result. A zero-free region ofRankin-Selberg L -functions has been established by Moreno in [Mor]. By the Taube-rian theorem of Wiener and Ikehara (Theorem 1, page 311, [Lan]) for L ′ / L ( s ) , where L ( s ) = L ( s , Π × Π ) , and Π = Sym φ or Sym φ , we obtain the Prime Number Theoremfor L ( s ) . Lemma 4.2.
Let Λ be the von Mangoldt function. We have ∑ n ≤ X Λ ( n ) | A [ ] φ ( n ) | ∼ X and ∑ n ≤ X Λ ( n ) | A [ ] φ ( n ) | ∼ X , as X → ∞ . Remark 4.3.
The previous lemma implieslim sup X → ∞ ∑ p ≤ X | A [ ] φ ( p ) | π ( X ) ≤ X → ∞ ∑ p ≤ X | A [ ] φ ( p ) | π ( X ) ≤ Theorem 4.4.
We have lim inf X → ∞ { p ≤ X , | λ φ ( p ) | ≤ } π ( X ) ≥ Proof.
Define U ( p ) = (cid:16) + A φ ( p ) + A [ ] φ ( p ) (cid:17) . Obviously we have U ( p ) ≥ p is not a Ramanujan prime, we have U ( p ) > .By the Hecke relations A φ ( p ) A [ ] φ ( p ) = | A [ ] φ ( p ) | − A φ ( p ) = A [ ] φ ( p ) + A φ ( p ) + U ( p ) = + A φ ( p ) + A [ ] φ ( p ) + A φ ( p ) + A [ ] φ ( p ) + A φ ( p ) A [ ] φ ( p )= − + A φ ( p ) + A [ ] φ ( p ) + | A [ ] φ ( p ) | + A [ ] φ ( p ) .By the previous two lemmas, we havelim sup X → ∞ ∑ p ≤ X U ( p ) π ( X ) ≤
35. (10)12e have ∑ p ≤ X U ( p ) π ( X ) ≥ ( π ( X ) − { p ≤ X , | λ φ ( p ) | ≤ } ) π ( X ) and then { p ≤ X , | λ φ ( p ) | ≤ } π ( X ) ≥ − ∑ p ≤ X U ( p ) π ( X ) .Hence by (10) we get lim inf X → ∞ { p ≤ X , | λ φ ( p ) | ≤ } / π ( X ) ≥ References [De1] Deligne, Pierre. ”Formes modulaires et repr´esentations l -adiques.” In S´eminaireBourbaki vol. 1968/69 Expos´es 347-363 , pp. 139-172. Springer Berlin Heidelberg, 1971.[De2] Deligne, Pierre. ”La conjecture de Weil. I.”
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