A proof of Selberg s orthogonality for automorphic L-functions

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1 A proof of Selberg s orthogonality for automorphic L-functions Jianya Liu, Yonghui Wang 2, and Yangbo Ye 3 Abstract Let π and π be automorphic irreducible cuspidal representations of GL m and GL m, respectively. Assume that π and π are unitary and self contragredient. In this article we will give an unconditional proof of orthogonality of Fourier coefficients of π and π, weighted by the von Mangoldt function Λn and n/. We then remove the weighting factor n/ and prove the Selberg orthogonality conjecture for automorphic L-functions Ls, π and Ls, π, unconditionally for m 4 and m 4, and under the Hypothesis H of Rudnick and Sarnak [8] in other cases. This proof of Selberg s orthogonality removes such an assumption in the computation of superposition distribution of nontrivial zeros of distinct automorphic L-functions by Liu and Ye [0].. Introduction Let π be an irreducible unitary cuspidal representation of GL m Q, and s σ + it C. Then the global L-function attached to π is given by products of local factors for σ > Godement and Jacquet [3]: Ls, π p L p s, π p, Φs, π L s, π Ls, π, where and L p s, π p m j L s, π α πp, j p s, m Γ s + µ π j. j Jianya Liu: Department of Mathematics, Shandong University, Jinan, Shandong 25000, China jyliu@sdu.edu.cn 2 Yonghui Wang: Department of Mathematics, Capital Normal University, Beijing 00037, China yhwang@mail.cnu.edu.cn 3 Yangbo Ye: Department of Mathematics, The University of Iowa, Iowa City, Iowa , USA yey@math.uiowa.edu 2000 Mathematics Subject Classification: F70, M26, M4.

2 Here Γ s π s/2 Γs/2, and α π p, j and µ π j, j,..., m, are comple numbers associated with π p and π, respectively, according to the Langlands correspondence. Denote by a π p k j m α π p, j k the Fourier coefficients of π. Then for σ >, we have d ds log Ls, π Λna π n n s, n where Λn is the von Mangoldt function. If π is an automorphic irreducible cuspidal representation of GL m Q, we define Ls, π, α π p, i, µ π i, and a π p k likewise, for i,..., m. If π and π are equivalent, then m m and {α π p, j} {α π p, i} for any p. Hence a π n a π n for any n p k, when π π. The Selberg orthogonality conjecture for automorphic L-functions Ls, π was proposed by Selberg [20] in 989. See also Ram Murty [6] [7]. Selberg s orthogonality conjecture. i For any automorphic irreducible cuspidal representation π of GL m Q, a π p 2 log log + O.. p p ii For any automorphic irreducible cuspidal representations π of GL m Q and π of GL m Q, if π is not equivalent to π. p a π pā π p p, The asymptotic formula in. was proved by Rudnick and Sarnak [8] under a conjecture on convergence of a series on prime powers: Hypothesis H. For k 2, p a π p k 2 log 2 p p k <. This Hypothesis H is trivial for m, and follows from bounds toward the Ramanujan conjecture for m 2. For m 3 it was proved by Rudnick and Sarnak [8] using Rankin-Selberg theory, while the case of m 4 was proved by 2

3 Kim and Sarnak [8]. Thus. is known unconditionally for m 4. For m > 4, Hypothesis H is an easy consequence of the Ramanujan conjecture. Denote αg detg. In 3, we will prove the following orthogonality. Theorem.. Let π and π be irreducible unitary cuspidal representations of GL m Q and GL m Q, respectively. Assume that at least one of π and π is self-contragredient: π π or π π. Then n Λna π nā π n +iτ 0 + iτ iτ 0 + O{ ep c log } if π π α iτ 0 for some τ 0 R; O{ ep c log } if π π α it for any t R..2 Here and throughout, c is a positive constant, not necessarily the same at each occurrence. that If τ 0 0, i.e. if π π, then a π n a π n, and hence Theorem. states n Λn a π n O{ ep c log }. Now Λn a π n 2 is non-negative. By a classical argument of de la Vallée Poussin, we can remove the weight n/ from Theorem. when π π, to get the following prime number theorem for automorphic representations. Corollary.2. For any automorphic irreducible cuspidal unitary representation π of GL m Q, Λn a π n 2 + O{ ep c log }. In general, we cannot remove the weight n/ from Theorem.. But similar to Theorem., we can establish in Theorem.3i an asymptotic formula for the epression n Λnaπ nā π n, n in which we are able to remove the weight n/ to get Theorem.3ii. Theorem.3. Let π and π be given as in Theorem.. 3

4 i We have n Λnaπ nā π n n log + c + O{ep c log } if π π; iτ 0 iτ 0 + iτ 0 + c 2 + O{ep c log } if π π α iτ 0 for some τ 0 R ; c 2 + O{ep c log } if π π α it for any t R..3 Here c and c 2 are constants depending on π and π : L c lim s 0 L s +, π π, c 2 L s L, π π. ii We can remove the weight n/ in i, getting Λna π nā π n n { log + O if π π; O if π π..4 Note that Theorems. and.3, and Corollary.2 are unconditional results. The reason that we can remove n/ from.3 is that now the main term is of order log when π π, which is substantially bigger than the O of the case of π π. See the proof of Theorem.3ii in 5 for details. Corollary.4. Let π and π be given as in Theorem.. Assume either i m 4 and m 4, or ii Hypothesis H for both π and π. Then p a π pā π p log p p { log + O if π π; O if π π..5 Corollary.5 Selberg s orthogonality. Let π and π be given as in Theorem.. Assume either i m 4 and m 4, or ii Hypothesis H for both π and π. Then p a π pā π p p { log log + O if π π; O if π π..6 Proofs of Corollaries.4 and.5 proceed along standard arguments, based on variations of Abel summation. We will thus not give the proofs here, but only point out that Hypothesis H is used to control sums over prime powers in the 4

5 epression on the left side of.4. This way we can obtain a sum taken over primes as in.5 and.6. An important application of.5 is in the superposition distribution of nontrivial zeros of distinct automorphic L-functions. In Liu and Ye [0] this superposition distribution was computed [0] Theorem 3., pp under the assumption of.5. Now with Corollary.4, we can restate Theorem 3. of [0] without assuming Selberg s conjecture: Theorem 3. now holds unconditionally for m 4, and under Hypothesis H for m > 4. This application is indeed our motivation of the present paper. Under the generalized Ramanujan conjecture, stronger orthogonality relations can be obtained Liu and Ye [2]: log nλna π nā π n +iτ 0 + iτ log +iτ iτ O{ ep c log } if π π α iτ 0 O{ ep c log } for some τ 0 R; if π π α it for any t R,.7 and p a π pā π p p log log + c 3 + O{ep c log } if π π; c 4 + Eiiτ 0 log + O{ep c log } if π π α iτ 0 for some τ 0 R ; c 5 + O{ep c log } if π π α it for any t R,.8 where c 3, c 4, and c 5 are constants possibly depending on π and π. Here Ei is the eponential integral, and Eiiτ 0 log iτ 0 iτ 0 log n k0 k! iτ 0 log k + O log n+2 It is interesting to compare.2 with.7, and compare.6 with.8. Under the Ramanujan conjecture, a remarkable feature of.7 is that it is indeed a prime number theorem weighted by Fourier coefficients of automorphic cuspidal representations, while.8 describes orthogonality of a π n and a π n in three cases with different main terms. 5.

6 Without assuming the Ramanujan conjecture, an unconditional proof of a weighted orthogonality similar to.3 was given in Liu and Ye []: n log nλnaπ nā π n log, n when π and π are not equivalent. From this weighted orthogonality, the uniqueness of factorization of automorphic L-functions for GL m Q was proved in []. 2. Rankin-Selberg L-functions We will use the Rankin-Selberg L-functions Ls, π π as developed by Jacquet, Piatetski-Shapiro, and Shalika [4], Shahidi [2], and Moeglin and Waldspurger [3], where π and π are automorphic irreducible cuspidal representations of GL m and GL m, respectively, over Q with unitary central characters. This L-function is given by local factors: Ls, π π p L p s, π p π p 2. where L p s, π p π p m m j k α πp, jᾱ π p, k p s. The Archimedean local factor L s, π π is defined by L s, π π m m j k Γ s + µ π π j, k where the comple numbers µ π π j, k satisfy the trivial bound Re µ π π j, k >. 2.2 Denote Φs, π π L s, π π Ls, π π. We will need the following properties of the L-functions Ls, π π and Φs, π π. RS. The Euler product for Ls, π π in 2. converges absolutely for σ > Jacquet and Shalika [5]. RS2. The complete L-function Φs, π π has an analytic continuation to the entire comple plane and satisfies a functional equation Φs, π π εs, π π Φ s, π π 6

7 with εs, π π τπ π Q s π π, where Q π π > 0 and τπ π ±Q /2 π π Shahidi [2], [22], [23], and [24]. RS3. Denote αg detg. When π π α it for any t R, Φs, π π is holomorphic. When m m and π π α iτ 0 for some τ 0 R, the only poles of Φs, π π are simple poles at s iτ 0 and + iτ 0 coming from Ls, π π Jacquet and Shalika [5] and [6], Moeglin and Waldspurger [3]. RS4. Φs, π π is meromorphic of order one away from its poles, and bounded in vertical strips Gelbart and Shahidi [2]. RS5. Φs, π π and Ls, π π are non-zero in σ Shahidi [2]. In addition to the above RS-RS5, we will also need the structure of Cm, m, defined as the comple plane with the discs s 2n + µ π π j, k <, n 0, j m, k m 8mm ecluded. For j,, m and j,, m, denote by βj, k the fractional part of Reµ π π j, k. In addition we let β0, 0 0 and βm +, m +. Then all βj, k [0, ], and hence there eist βj, k, βj 2, k 2 such that βj 2, k 2 βj, k /3mm and there is no βj, k lying between βj, k and βj 2, k 2. It follows that the strip S 0 {s : βj, k +/8mm σ βj 2, k 2 /8mm } is contained in Cm, m. Consequently, for all n 0,, 2,, the strips S n {s : n + βj, k + /8mm σ n + βj 2, k 2 /8mm } 2.3 are subsets of Cm, m. This structure of Cm, m will be used later. In [0] and [], Liu and Ye proved the following lemmas. Lemma 2.. Let s σ + it with 2 σ 2, t > 2. i Assume m m and π π α iτ 0 for some nonzero τ 0 R. If s Cm, m is not a zero of Ls, π π, then d ds log Ls, π π t γ ii If π π α it for any t R, then d ds log Ls, π π s ρ + O{logQ π π t }. s iτ 0 s iτ 0 t γ 7 s ρ + O{logQ π π t }.

8 Lemma 2.2. i For T > 2, there eists τ with T τ T + such that when 2 σ 2, d ds log Lσ + iτ, π π log 2 Q π π τ. ii If s is in some strip S n as in 2.3 with n 2, then d ds log Ls, π π. Furthermore, we need a zero-free region for the Rankin-Selberg L-function Ls, π π which was proved by Moreno [4] and [5]. See also Gelbart, Lapid, and Sarnak [], and Sarnak [9]. Lemma 2.3. Let π and π be as in Theorem.. Then there are effectively computable positive constants Q, c, and c such that there is no zero of Ls, π π in the region σ c log, Q t c t, and at most one eceptional zero in the region σ c log, Q2 + c t. The eceptional zero, if eists, must be real. 3. Proof of Theorem. We prove Theorem. when π π α iτ 0 for some τ 0 R. The proof for case of π and π being not twisted equivalent is the same with all terms related to τ 0 removed. By RS, we have for σ > that Js d ds log Ls, π π n Λna π nā π n n s. By RS3 and RS5, Js is holomorphic in σ >. On σ, Ls, π π is nonzero RS5 and has only a simple pole at s + iτ 0. Thus Js s iτ 0 + Gs 3. has only a simple pole in σ, where Gs is analytic for σ. On C, Js also has a simple pole at each of the pole at iτ 0, trivial zeros, and nontrivial zeros of Ls, π π. 8

9 Note that y s { /y if y, 2πi b ss + ds 0 if 0 < y <, where b means the line σ b > 0. Then we have n Λna π nā π n Js 2πi 2 ss + ds 2+iT 2 it + + 2πi 2 it 2 i 2+i 2+iT. The last two integrals are clearly bounded by T 2 /t 2 dt 2 /T. Thus, n Λna π nā π n 2+iT 2πi 2 it 2 Js ss + ds + O T. Choose a with 2 < a < such that the line σ a is contained in the strip S 2 Cm, m ; this is guaranteed by the structure of Cm, m. Without loss of generality, let T > 0 be a large number such that T and T can be taken as the τ in Lemma 2.2i. Now we consider the contour C : 2 σ a, t T ; C 2 : σ a, T t T ; C 3 : a σ 2, t T. Note that the two poles, some trivial zeros, and certain nontrivial zeros ρ σ + iγ of Ls, π π, as well as s 0, are passed by the shifting of the contour. The trivial zeros can be determined by the functional equation in RS2: s µ π π j, k with a < Reµ π π j, k < and s 2 µ π π j, k with a + 2 < Reµ π π j, k <. Here we have used 2.2 and 2 < a <. Then 9

10 we have 2πi 2+iT 2 it Js ss + ds 2πi By Lemma 2.2i, we get 2 C a + + C C 2 C 3 Js s+iτ 0,iτ 0,0, + Res + + ss + Res Js s µ a< Reµ π π j,k< π π j,k ss + Res Js s 2 µ a+2< Reµ π π j,k< π π j,k + γ T ss + Res Js sρ ss log 2 Q π π T σ T 2 dσ 2 log 2 Q π π T T 2, and the same upper bound also holds for the integral on C 3. By Lemma 2.2ii, then T C 2 T a t + 2 dt. On taking T, the three integrals on C, C 2, C 3 are If τ 0 R, then log 2 Q π π. 3.3 Res Js s+iτ 0,iτ 0,0, ss + +iτ 0 + iτ iτ 0 + iτ0 iτ 0 + iτ 0 +J0 J +iτ 0 + Olog iτ iτ 0 If τ 0 0, then s 0 is a double pole. Computing the residues similarly, we see that the final estimate 3.4 is still true. Near a trivial zero s µ π π j, k of order l, we can epress Js as l/s + µ π π j, k plus an analytic function, like in 3.. The residues at these trivial zeros can therefore be computed similarly to what we have done in 3.4. By 0

11 2.2, we know that Re µ π π j, k δ for some δ > 0. Consequently, Res Js s µ a< Reµ π π j,k< π π j,k Res Js s 2 µ a+2< Reµ π π j,k< π π j,k ss + δ, 3.5 ss + δ. 3.6 To compute the residues corresponding to nontrivial zeros, we note that Φs, π π is of order RS4, and Φ, π π 0 RS5, and hence ρρ + <. Consequently, γ T ρ Res Js sρ ss + γ T γ T γ T ρ E Res sρ ρ ρρ + + γ T ρ / E where E is the set of eceptional zeros in Lemma 2.3ii. s ρ ss + σ ρρ +, 3.7 By Lemma 2.3ii, E, and therefore the sum over ρ E is clearly δ for some δ > 0. By Lemma 2.3i, the sum over ρ E is ep c log log T ρρ + ep c log, 3.8 by taking T ep log + d for some d with 0 < d <. bounded by ep c log. ρ Theorem. then follows from applying and 3.8 to Proof of Theorem.3 i Hence 3.7 is The proof follows the same line as that of Theorem., so we only indicate the differences. With the same Js, we have n Λnaπ nā π n n 2πi 2πi Js + ss + ds +it it it + + i +i +it.

12 The last two integrals are clearly bounded by T /t2 dt /T. Let a and T be as before. Then Define n Λnaπ nā π n n +it 2πi it D : σ a, t T ; D 2 : σ a, T t T ; D 3 : a σ, t T. Js + ss + ds + O T Note that the two poles, some trivial zeros, and certain nontrivial zeros ρ σ+iγ of Ls +, π π, as well as s 0, are passed by the shifting of the contour. The trivial zeros can be determined by the functional equation in RS2: s + µ π π j, k with a < Reµ π π j, k < and s + 2 µ π π j, k with a + 2 < Reµ π π j, k <. Then we have 2πi 2πi +it it Js + ss + ds D + D 2 + D 3 + Res Js + s0,iτ 0,, +iτ 0 ss + + Res Js + ss + + s µ a< Reµ π π j,k< π π j,k Res s 3 µ a+2< Reµ π π j,k< π π j,k + γ T Res sρ s s Js + ss +. Js + ss Similar to the previous argument, we can prove that, ecept the residues at 0, iτ 0, everything on the right of 4. is ep c log 4.2 by taking T ep log + d for some d with 0 < d <. In particular, log 2 Q π π T σ T 2 dσ log2 Q π π T T 2. D a To compute the residues at 0, iτ 0, we have to distinguish three cases. If π π, i.e. τ 0 0, then s 0 is a double pole of Js + /{ss + }, and the residue 2

13 is lim s 0 { d ds If π π α iτ 0 with residues s 2 } Js + ss + log + lim Js +. s 0 s for some τ 0 R, then s 0, iτ 0 are two different simple poles J + iτ 0 iτ 0 + iτ 0. If π π α it for any t R, then the terms involving τ 0 disappear, and only s 0 is a simple pole, with residue J. Inserting these and 4.2 into 4. gives part i of Theorem Weight removal Now we prove Theorem.3ii using a technique of Landau [9]. For simplicity, we denote cn Λna π nā π n/n, and C cn, D n cn. Then 0 Ctdt ncn D. We begin with an observation that +v Ctdt + vd + v D vd + v + D + v Dv, 5. where v > 0 is any real number, but for later use we specify that v. By Theorem.3i, we have vd + v v The last term in 5. is D + v Dv + { log + O if π π; O if π π. 5.2 n <+v : E + E 2, 3 n cn + v n cn + v

14 say. We have E v + v n cn v + v Λn a π n 2 + a π n 2 v by Corollary.2. On the other hand, v E 2 cn + v v + v <+v <+v v + v <+v n cn Λn a π n 2 + a π n 2 v. 5.3 Putting 5.2 and 5.3 into 5., we get +v { log + O if π π; Ctdt v O if π 5.4 π. Now we consider the difference +v Ctdt vc +v v v v, Ct Cdt v <+v <+v n cn <+v Λn a π n 2 + a π n 2 cn by the same argument as in 5.3. The desired result.4 now follows from this and 5.4. Acknowledgments This project is sponsored in part by China NSF Grant #0250 J. Liu, by China NSF Tianyuan Youth Grant #02600 Y. Wang, and by the USA National Security Agency under Grant Number MDA Y. Ye. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein. The last author wishes to acknowledge support services provided by the Obermann Center for Advanced Studies at the University of Iowa. References [] S.S. Gelbart, E.M. Lapid, and P. Sarnak, A new method for lower bounds of L-functions, C.R. Acad. Sci. Paris, Ser. I , [2] S.S. Gelbart and F. Shahidi, Boundedness of automorphic L-functions in vertical strips, J. Amer. Math. Soc., 4 200,

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