Zero-density estimates for L-functions attached to cusp forms
aa r X i v : . [ m a t h . N T ] F e b Zero-density estimates for L -functions attached to cusp forms Yoshikatsu Yashiro
Graduate School of Mathematics, Nagoya University,464-8602 Chikusa-ku, Nagoya, JapanE-mail: [email protected]
Abstract
Let S k be the space of holomorphic cusp forms of weight k withrespect to SL ( Z ). Let f ∈ S k be a normalized Hecke eigenform, L f ( s ) the L -function attached to the form f . In this paper we con-sider the distribution of zeros of L f ( s ) in the strip σ ≤ Re s ≤ σ > / N f ( σ, T ) = { ρ ∈ C | L f ( ρ ) = 0 , σ ≤ Re ρ ≤ , < Im ρ ≤ T } for 1 / ≤ σ ≤ T >
0. Using the methods of Karatsubaand Voronin we shall give another proof for Ivi´c’s method.
It is conjectured that non-trivial zeros of the Riemann zeta function ζ ( s )lie on the critical line Re s = 1 / / ≤ σ ≤ T >
0, let N ( σ, T ) be the number of zeros of ζ ( s ) in the region σ ≤ Re s ≤ < Im s ≤ T . If RH is true then N ( σ, T ) = 0 for 1 / < σ ≤ ζ ( s ) is to estimate N ( σ, T ) as Mathematics Subject Classification : Primary 11M26; Secondary 11N75.
Key words and phrases : cusp forms, L -functions, zero-density. Y. Yashiro small as possible to support RH. For this problem Bohr and Landau [2]began to study zero-density for ζ ( s ) and proved N ( σ, T ) ≪ T uniformly for1 / ≤ σ ≤
1. Ingham [9] improved their result to N ( σ, T ) ≪ T − σ (1 − σ ) (log T ) (1 / ≤ σ ≤ . (1.1)Moreover Huxley [8] improved the result (1.1) for σ ≥ / N ( σ, T ) ≪ T σ − (1 − σ ) (log T ) (3 / ≤ σ ≤ . (1.2)The results (1.1) and (1.2) give the following estimate: N ( σ, T ) ≪ T . − σ ) (log T ) (1 / ≤ σ ≤ . (1.3)Karatsuba and Voronin [13] gave alternative proof for (1.3) by using theapproximate functional equation for ζ ( s ) (see Hardy and Littlewood [6]).In this paper we study the distribution of zeros of L -functions associatedto holomorphic cusp forms. Let S k be the space of cusp forms of weight k ∈ Z ≥ with respect to the full modular group SL ( Z ). Let f ∈ S k bea normalized Hecke eigenform, and a f ( n ) the n -th Fourier coefficient of f .It is known that all a f ( n )’s are real numbers (see [1, Chapter 6.14]) andestimated as | a f ( n ) | ≤ d ( n ) n k − by Deligne [3], where d ( n ) is the divisorfunction defined by d ( n ) = P m | n
1. The L -function attatched to f is definedby L f ( s ) = ∞ X n =1 λ f ( n ) n s = Y p :prime − λ f ( p ) p − s + p − s (Re s > , (1.4)where λ f ( n ) = a f ( n ) n − k − . Hecke [7] proved that L f ( s ) has an analyticcontinuation to the whole s -plane and the completed L -functionΛ f ( s ) = (2 π ) − s − k − Γ( s + k − ) L f ( s ) = Z ∞ f ( iy ) y s + k − − dy (1.5)satisfies Λ f ( s ) = ( − k/ Λ f (1 − s ), namely, L f ( s ) = χ f ( s ) L f (1 − s )(1.6) ero-density estimates for L -functions attached to cusp forms s ∈ C , where χ f ( s ) is defined by χ f ( s ) = ( − k/ (2 π ) s − Γ(1 − s + k − )Γ( s + k − ) . By (1.4) and (1.6), L f ( s ) has no zeros in Re s > s < s = − n + 1 / n ∈ Z ≥ k/ ). The zeros of L f ( s ) in the critical strip 0 ≤ Re s ≤ non-trivial zeros . Moreover, by (1.5) and (1.6), the non-trivialzeros are located symmetrically with respect to Im s = 0 and Re s = 1 / L f ( s ) lie on Re s = 1 /
2, whichis called the Generalized Riemann Hypothesis (in short GRH). Let N f ( σ, T )be the function defined by N f ( σ, T ) = { ρ ∈ C | L f ( ρ ) = 0 , σ ≤ Re ρ < , < Im ρ ≤ T } . (1.7)As in the case of the Riemann zeta function, it is important to study thebehavior of N f ( σ, T ). It is Ivi´c [10] who first proved the non-trivial estimatesof N f ( σ, T ).The aim of this paper is to give an alternative proof of Ivi´c’s estimate,namely, Theorem 1.1.
Let f ∈ S k be a normalized Hecke eigenform. For any large T we have N f ( σ, T ) ≪ ( T − σ (1 − σ )+ ε , / ≤ σ ≤ / ,T σ (1 − σ )+ ε , / ≤ σ ≤ uniformly / ≤ σ ≤ . Here and later ε denotes arbitrarily positive smallconstant. Ivi´c [10] proof is based on the second and the sixth power momentsof L f ( s ) due to Good [5, Theorem] and Jutila [11, (4.4.2)]. In this paper,instead of using the higher power moments of L f ( s ), we follow Karatsubaand Voronin’s approach and use only the approximate functional equationof L f ( s ) (see Lemma 2.2) and the well-known estimates of exponential sum(see Lemma 2.6, 2.7).It is important that we can construct a set E of zeros of L f ( s ) such thatthe estimate of R = N f ( σ, T ) − N f ( σ, T /
2) is reduced to that of S ( ρ ), where1 / ≤ σ ≤
1, 2 ≤ T ≤ T and S ( ρ ) is a function obtained by multiplying Y. Yashiro the approximate functional equation of L f ( s ) by 1 /L f ( s ). The existence of E works to obtain an estimate R ≪ (log T ) α +3 X ρ ∈E | S ( ρ ) | α (1.9)where α is any fixed positive integer. The upper bound of sum of (1.9) isobtained by the technique of Karatsuba and Voronin’s calculating, in whichpower moment of L f ( s ) is not needed. To prove Theorem 1.1, we need Lemmas 2.1–2.7. First Lemmas 2.1–2.3 arerequired to show Proposition 3.1 (see Section 3), which is required for theproof of Theorem 1.1.
Lemma 2.1.
If we write L f ( s ) = ∞ X n =1 µ f ( n ) n s (Re s > , then we see that µ f ( n ) is multiplicative and given by µ f ( p r ) = , r = 0 , − λ f ( p ) , r = 1 , , r = 2 , , r ∈ Z ≥ for any prime number p . In addition we have | µ f ( n ) | ≤ d ( n ) for n ∈ Z ≥ .Proof. By expanding the right-hand side of (1.4) and using Deligne’s result,we can obtain the assertion of this lemma.
Lemma 2.2 ( The approximate functional equation of L f ( s ) , [4, KOROL-LAR 2]) . There exist α ∈ (0 , / and β ∈ R > such that X x ≤ n ≤ x (1+ x − α ) | a f ( n ) | ≪ x k − β ero-density estimates for L -functions attached to cusp forms and L f ( s − k − ) = X n ≤ y a f ( n ) n s + (2 π ) s − k Γ ( k − s ) Γ ( s ) X n ≤ y a f ( n ) n k − s +(2.1) + O ( | t | k +12 − σ − α + β ) uniformly for ( k − / ≤ σ ≤ ( k + 1) / where s = σ + it and y = | t | / (2 π ) .Moreover we have L f ( s ) = X n ≤ y λ f ( n ) n s + χ f ( s ) X n ≤ y λ f ( n ) n − s + O ( | t | / − σ + ε )(2.2) where χ f ( s ) is given by (1.6).Proof. We shall show (2.2) by using (2.1). Since | a f ( n ) | ≪ n k − + ε for any n ∈ Z ≥ from Deligne’s result, it follows that X x ≤ n ≤ x (1+ x − α ) | a f ( n ) | ≪ x k − α + ε for α ∈ R > . If we put α = 1 / − ε and β = α − ε , then the O -term in (2.1)becomes O ( | t | k − σ + ε ). Replacing s by s + ( k − / Lemma 2.3 (Moreno [14, Theorem 3.5]) . Let N f ( T ) be the number of zerosof L f ( s ) in the region ≤ Re ρ ≤ and ≤ Im ρ ≤ T . For large T > wehave N f ( T ) = Tπ log T πe + O (log T ) . Furthermore, we have N f ( σ, T + 1) − N f ( σ, T ) ≪ log T uniformly for / ≤ σ ≤ . When we estimate P kr =0 | P N Lemma 2.5 ([13, Lemma IV.1.2]) . Let a ( n ) be an arithmetic function. Theparameters X, X , N and N satisfy < X < X ≤ X and ≤ N < N ≤ N . Then we have Z X X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X N C > such that | f (1) ( x ) | < C < .Then we have X a 2) for 2 ≤ T ≤ T and σ > / 2. Let X = T δ where δ is a positive integer which is chosen later.Let M X ( s ) be a function defined by M X ( s ) = X m ≤ X µ f ( m ) m s . We multiply both sides of (2.2) by M X ( s ) and obtain L f ( s ) M X ( s ) =1 + X X There exists a set E of zeros of L f ( s ) such that | S ( ρ ) | ≥ D ( ρ ∈ E ) , (3.6a) E ≫ RD log T , (3.6b) | Im ρ − Im ρ ′ | ≥ ρ = ρ ′ ∈ E ) , (3.6c) where S ( ρ ) = S ν ( ρ ) with some number ν ∈ { , . . . , D } .Proof. From (3.4) it is clear that U = S ≤ ν ≤ D A ν where A ν = { ρ ∈ U || S ν ( ρ ) | ≥ / (2 D ) } . Then we see that there exists ν such that | S ( ρ ) | ≥ / (2 D ) for ρ ∈ A and A ≥ R/D where A = A ν and S ( ρ ) = S ν ( ρ ).Let ρ m,n be ρ ∈ A such that Im ρ is the n -th minimum number in( T / m, T / m + 1]. By using Lemma 2.3 we can write A = [ j =0 , [ ≤ n ≤ C log T E n,j , E n,j = { ρ j,n , ρ j,n , ρ j,n , . . . , ρ T / j,n } where C is a positive constant. Then there exist n ∈ { , , . . . , [ C log T ] } and j ∈ { , } such that A ≤ C (log T ) P j =0 , E n ,j ≤ C (log T ) E where E = E n ,j . Since E ⊂ A and A ≥ R/D , it follows that (3.6a) and(3.6b) are shown. And (3.6c) is shown because | Im ρ l + j ,n − Im ρ l ′ + j ,n | ≥ l = l ′ ∈ { , , . . . , [ T / } .From (3.6a) and (3.6b) we can reduce the estimate of R to that of S ( ρ ),that is, by taking 2 α -th power of both sides of (3.6a) we get R ≪ D (log T ) E = D (log T ) X ρ ∈E α ≪ (log T ) α +3 X ρ ∈E | S α ( ρ ) | (3.7)where α is any fixed positive integer.First we consider the case that S ( ρ ) is of the form (3.5a). We shall givea preliminary upper bound of R as Proposition 3.2. Let S ( ρ ) in (3.7) has the form S ( ρ ) = X L Proof. From (3.8) the α -th power of S ( ρ ) has the form S α ( ρ ) = X L α We have R ≪ ( T − σ )+ ε , L ∈ F ,T − σ (1 − σ )+ ε , L ∈ S ≤ r ≤ δ F r . Proof. First in the case of L ∈ F , taking α = 1 in (3.9) and choosing ε ≥ /δ , we obtain R ≪ L − σ ) T ε ≪ T ( δ ) (1 − σ )+ ε ≪ T − σ )+ ε . (3.16)Next we consider the upper bound of R in the case of L ∈ S ≤ r ≤ δ F r .Let 2 ≤ A ≤ 4. When r ∈ { , . . . , [ A/ ( A − } , that is, A ≤ r/ ( r − F r to ( T r , T A r ] and ( T A r , T r − ]. Hence L ∈ ( T r , T A r ], taking α = r in (3.9) we have R ≪ L r (1 − σ ) T ε ≪ T A (1 − σ )+ ε . (3.17)2 Y. Yashiro