aa r X i v : . [ m a t h . N T ] N ov EFFECTIVE FORMS OF THE SATO–TATE CONJECTURE
JESSE THORNER
Abstract.
We prove effective forms of the Sato–Tate conjecture for holomorphic cuspidalnewforms which improve on the author’s previous work (solo and joint with Lemke Oliver).We also prove an effective form of the joint Sato–Tate distribution for two twist-inequivalentnewforms. Our results are unconditional because of recent work of Newton and Thorne. Introduction and statement of results
Let f ( z ) = P ∞ n =1 a f ( n ) n k − e πinz ∈ S new k (Γ ( q )) be a cusp form (normalized so that a f (1) = 1 ) with trivial nebentypus. If f is also an eigenform for all of the Hecke opera-tors and all of the Atkin–Lehner involutions | k W ( q ) and | k W ( Q p ) for each prime p | q , then f is a newform (see [27, Section 2.5]). Throughout, we assume that f does not have complexmultiplication (CM), so there is no imaginary quadratic field K such that for p ∤ q , p is inertin K if and only if a f ( p ) = 0 . Deligne’s proof of the Weil conjectures implies that for eachprime p , there exists an angle θ p ∈ [0 , π ] such that a f ( p ) = 2 cos θ p . Serre’s extension ofthe Sato–Tate conjecture [31], originally proposed for f attached to a non-CM elliptic curveby modularity, asserts that if f is non-CM, then the sequence { θ p } is equidistributed in theinterval [0 , π ] with respect to the measure dµ ST := (2 /π ) sin θdθ . Equivalently, one has(1.1) π f,I ( x ) := { p ≤ x : θ p ∈ I, p ∤ q } ∼ µ ST ( I ) π ( x ) as x → ∞ ,where π ( x ) = { p ≤ x } and I = [ α, β ] ⊆ [0 , π ] . Barnet-Lamb, Geraghty, Harris, and Taylor[2] proved Serre’s extension of the Sato–Tate conjecture. See [21] for an excellent overiew.In [33], the author bounded the error term in (1.1) assuming that the m -th symmetricpower lift Sym m f corresponds with a cuspidal automorphic representation of GL m +1 ( A ) foreach m ≥ , where A denotes the ring of adeles over Q . This implies that the symmetricpower L -functions L ( s, Sym m f ) have an analytic continuation and functional equation of theexpected type for all m ≥ . With this hypothesis, the author proved for fixed f and I thatfor all ε > , there exist constants c ε , C ε > such that(1.2) | π f,I ( x ) − µ ST ( I ) π ( x ) | ≤ c ε π ( x )(log x ) − + ε , x > C ε . Until recently, it was known that
Sym m f corresponds with a cuspidal automorphic repre-sentation of GL m +1 ( A ) only for m ≤ [6, 10, 14, 15]. Recently, Newton and Thorne [25, 26]proved that Sym m f corresponds with a cuspidal automorphic representation of GL m +1 ( A ) for all m ≥ . This inspired the author to improve the quality of (1.2) in the x -aspect andspecify the uniformity with respect to f and I . In what follows, c , c , c , . . . denotes asequence of positive, absolute, and effectively computable constants. Theorem 1.1.
Let f ∈ S new k (Γ ( q )) be a non-CM newform as above, and let I = [ α, β ] ⊆ [0 , π ] . There exists a constant c such that (1.3) | π f,I ( x ) − µ ST ( I ) π ( x ) | ≤ c π ( x ) log( kq log x ) √ log x , x ≥ . Furthermore, there exist constants c , c , and c such that (with µ = µ ST ( I ) ) (1.4) c µ ST ( I ) π ( x ) ≤ π f,I ( x ) ≤ c µ ST ( I ) π ( x ) , x ≥ ( kq/µ ) c log( e/µ ) /µ . Remark.
The error term in (1.3) is uniform enough to accommodate some small-scale equidis-tribution for the sequence { θ p } . For instance, if x ≥ e kq and F ( x ) is a monotonically increas-ing function with lim x →∞ F ( x ) = ∞ , then the sequence { θ p } is equidistributed with respectto µ ST in intervals with µ ST ( I ) as small as (log log x ) F ( x ) / √ log x . Remark.
The proof can be modified for when the nebentypus character ψ (mod q ) of f isnontrivial. However, if ω is a root of unity with ω in the image of ψ , then one must restrictconsideration to the primes p such that ψ ( p ) = ω . Then a f ( p ) /ω ∈ R , and we can define θ p ∈ [0 , π ] by a f ( p ) = 2 ω cos θ p . The ensuing changes to our proofs are not purely cosmetic.In particular, the m -dependence in Corollary 4.2 would become worse since the cuspidalautomorphic representation of GL ( A F ) corresponding with f is no longer guaranteed to beself-dual. This directly affects the power of log x that is saved in Theorem 1.1. Remark.
The bounds in (1.4) improve work of Lemke Oliver and the author [18, Thm 1.6].If I is fixed, then as f varies, (1.4) gives an upper bound on π f,I ( x ) commensurate with theBrun–Titchmarsh theorem as well as an upper bound on the least p ∤ q such that θ p ∈ I commensurate with Linnik’s bound on the least prime in an arithmetic progression [19].Even though the error term in asymptotic in Theorem 1.1 saves less than a full power of log x over π ( x ) , some arithmetically significant consequences still follow from Theorem 1.1.For example, Luca, Radziwiłł, and Shparlinski [20, Thm 1.1] proved that the inequality | a f ( n ) | ≤ (log n ) − + o (1) (where o (1) denotes a quantity, possibly depending on f , whichtends to zero as n → ∞ ) holds for a density one subset of integers n . This improves on astandard argument which achieves the same bound with log 2 replacing − . One might askwhether the exponent − might be lowered any further for a density one subset of n . In thesame paper, Luca, Radziwiłł, and Shparlinski [20, Cor 1.5] proved that Theorem 1.1 sufficesto show that if v ∈ R is fixed, then(1.5) lim x →∞ n ≤ x : a f ( n ) = 0 , log | a f ( n ) | + log log n q ( + π ) log log n ≥ v { n ≤ x : a f ( n ) = 0 } = 1 √ π Z ∞ v e − u / du. Thus the exponent − cannot be lowered any further for a density one subset of n . Morerecently, Klurman and Mangerel [16] used Theorem 1.1 to prove a multidimensional versionof (1.5), which enabled them to prove that if N ≥ , A f ( n ) = a f ( n ) n k − , and b , . . . , b N aredistinct nonnegative integers such that A f ( b j ) = 0 for all ≤ j ≤ N , then(1.6) lim x →∞ { n ≥ x : 0 < | A f ( n + b ) | < | A f ( n + b ) | < · · · < | A f ( n + b N ) |} { n ≤ x : A f ( n + b ) · · · A f ( n + b N ) = 0 } = 1 N ! . In both (1.5) and (1.6), if x is sufficiently large, then there exists a constant c f > suchthat the counting function in the denominator is bounded below by c f x . This lower boundis a direct consequence of the fact that there exists a constant δ > such that the densityof primes p ≤ x such that a f ( p ) = 0 is at most (log x ) − δ [30, 34].For i = 1 , , let f i ∈ S new k i ( q i ) be a non-CM newform, let { θ ( i ) p } be the sequence of anglesin the Sato–Tate conjecture for f i , and let I i be the indicator function of the interval I i = [ α i , β i ] ⊆ [0 , π ] . Suppose that there exists no primitive character χ satisfying theproperty that f = f ⊗ χ , in which case f and f are not twist-equivalent. We denote thisby f f . If f f , then it is natural to ask whether the Sato–Tate distributions for f and f are independent, as quantified by the proposed existence of the asymptotic(1.7) π f ,f ,I ,I ( x ) := X p ≤ xp ∤ q q I ( θ (1) p ) I ( θ (2) p ) ∼ µ ST ( I ) µ ST ( I ) π ( x ) as x → ∞ ,where I , I ⊆ [0 , π ] are two intervals. This question was posed independently by Katz andMazur for f i corresponding to elliptic curves by modularity. Harris [11] proved the asymptotic(1.7) for non-CM newforms associated to pairs of twist-inequivalent elliptic curves. See Wong[36] for a generalization to all pairs of non-CM newforms and even pairs of Hilbert modularforms over totally real fields. The work of Newton and Thorne [25, 26] and the ideas leadingto Theorem 1.1 permit us to effectively quantify this independence. Theorem 1.2.
For i = 1 , , let f i ∈ S k i (Γ ( q i )) be a newform as in Theorem 1.1, and let I i = [ α i , β i ] ⊆ [0 , π ] . If f f , then there exists a constant c such that | π f ,f ,I ,I ( x ) − µ ST ( I ) µ ST ( I ) π ( x ) | ≤ c π ( x ) log( k q k q log log x ) √ log log x , x ≥ . Remark.
One could establish a much stronger ineffective error term, where the ineffectivityarises from the possible existence of a Landau–Siegel zero (see Sections 4 and 6). See Molteni[22] (and also his PhD thesis) for a discussion on how to bound such zeros in our setting.As one sees in Section 6, Landau–Siegel zeros do not plague the error term in Theorem 1.1.If one assumes the generalized Riemann hypothesis (GRH) for each symmetric power L -function L ( s, Sym m f ) , then Theorem 1.1 improves as follows. Theorem 1.3.
Let f ∈ S k (Γ ( q )) and I be as in Theorem 1.1, and let µ = µ ST ( I ) . If L ( s, Sym m f ) satisfies GRH for all m ≥ , then there exists a constant c such that | π f,I ( x ) − µ ST ( I ) π ( x ) | ≤ c x log( kqx )log x , x ≥ . Also, there exist a constant c and a prime p ∤ q which satisfies θ p ∈ I and p ≤ c µ (log kqµ ) .Proof. Rouse and the author [29] proved the first result for squarefree levels. Chen, Park,and Swaminathan [5] proved the second result when f corresponds with an elliptic curveby modularity, again for squarefree levels. Their proof extends to other f with little addi-tional effort using Theorem 6.1 below. The work in [5, 29] assumed the automorphy of thesymmetric powers of f , which is now known unconditionally [25, 26]. (cid:3) Remark.
The authors in [5, 29] assumed a squarefree level in order to produce strong explicitbounds on c and c . The orders of magnitude do not change when q is not squarefree, butthe task of obtaining strong bounds on the implied constants seems difficult. Remark.
An error term of size c f,ε x + ε is expected for all fixed ε > . See [1, Thm 1.4] forsome compelling on-average results in this direction.If one assumes GRH for the Rankin–Selberg L -functions associated to the tensor productsof Sym m f and Sym m f for all m , m ≥ , then Theorem 1.2 improves as follows. JESSE THORNER
Theorem 1.4.
For i = 1 , , let f i ∈ S new k i (Γ ( q i )) and I i be as in Theorem 1.2. If f f and the Rankin–Selberg L -functions L ( s, Sym m f × Sym m f ) satisfy GRH for all integers m , m ≥ , then there exists a constant c such that | π f ,f ,I ,I ( x ) − µ ST ( I ) µ ST ( I ) π ( x ) | ≤ c x log( k q k q x ) , x ≥ . Proof.
Bucur and Kedlaya [3] proved this when f and f correspond with elliptic curves bymodularity. One can extend their proof to other f and f with little additional effort. (cid:3) Acknowledgements.
I thank Maksym Radziwiłł, Jeremy Rouse, Jack Thorne, John Voight,and Peng-Jie Wong for helpful discussions and the anonymous referees.2.
Proof of Theorems 1.1 and 1.2
In this section, we reduce the proofs of Theorems 1.1 and 1.2 to the task of proving auniform version of the prime number theorem for certain classes of L -functions. We usethe Vinogradov notation F ≪ G to denote the existence of an absolute and effectivelycomputable constant c > (not necessarily the same in each occurrence) such that | F | ≤ c | G | in the range indicated. We write F = G + O ( H ) to denote that | F − G | ≪ H .2.1. Proof of Theorem 1.1.
Let I = [ α, β ] ⊆ [0 , π ] be an interval. Let I be the indicatorfunction of the interval I , and define π f,I ( x ) as in (1.1). Let U m (cos θ p ) = sin(( m + 1) θ p )sin θ p = m X j =0 e (2 j − m ) iθ p be the m -th Chebyshev polynomial of the second type. These polynomials form an orthonor-mal basis for L ([0 , π ] , µ ST ) with respect to the usual inner product h f, g i = R π f ( θ ) g ( θ ) dµ ST .Montgomery [23, Chapter 1] used the work of Beurling and Selberg to efficiently majorizeand minorize the indicator function of a subinterval of [0 , using carefully constructedtrigonometric polynomials. One performs a suitable change of variables to handle subinter-vals of [0 , π ] and a change of basis to express the trigonometric polynomials in terms of theChebyshev polynomials U m (cos θ ) (see [29, Section 3]). For any integer M ≥ , we find that(2.1) | π f,I ( x ) − µ ST ( I ) π ( x ) | ≪ π ( x ) M + M X m =1 m (cid:12)(cid:12)(cid:12) X p ≤ xp ∤ q U m (cos θ p ) (cid:12)(cid:12)(cid:12) , x ≥ . Note that π ( x ) ∼ x/ log x by the prime number theorem. By partial summation, we have(2.2) X p ≤ xp ∤ q U m (cos θ p ) = θ f,m ( x )log x − Z x θ f,m ( t ) t (log t ) dt, θ f,m ( x ) := X p ≤ xp ∤ q U m (cos θ p ) log p. Proposition 2.1.
Let f be as in Theorem 1.1. There exist constants c (suitably large) and c and c (suitably small) such that if ≤ m ≤ c √ log x/ p log( kq log x ) , then | θ f,m ( x ) | ≪ m x − c m + m x (cid:16) exp h − c log xm log( kqm ) i + exp h − c √ log x √ m i(cid:17) . By (2.1), (2.2), and Proposition 2.1, if ≤ M ≤ c √ log x/ p log( kq log x ) , then(2.3) | π f,I ( x ) − µ ST ( I ) π ( x ) |≪ π ( x ) (cid:16) M + M X m =1 m (cid:16) x − c m + exp h − c xm log( kqm ) i + exp h − c √ log x √ m i(cid:17)(cid:17) . Proof of Theorem 1.1.
Write µ = µ ST ( I ) . There exists a suitably large constant c suchthat if x ≥ exp( c ( µ log kqµ ) ) , then the bound (1.3) follows from (2.3) by choosing c > to be a sufficiently small constant and M = ⌈ c √ log x/ log( kq log x ) ⌉ ≥ . For all remainingvalues of x , (1.3) is trivial. The bounds in (1.4) follow from (2.3) by choosing c > to bea sufficiently large constant, M = ⌈ c /µ ST ( I ) ⌉ , and x ≥ ( kqM ) c M log( M ) /c . (cid:3) Proof of Theorem 1.2.
For i = 1 , , let µ i = µ ST ( I i ) and let f i be as in Theorem 1.2with Sato–Tate angles { θ ( i ) p } . Let π f ,f ,I ,I ( x ) be as in (1.7). Cochrane [7] carried out a ver-sion of Montgomery’s analysis which constructs trigonometric polynomials which efficientlymajorize and minorize the indicator function of R × R , where R i = [ a i , b i ] ⊆ [0 , . Uponperforming a change of variables and a change of basis similar to [29, Section 3] to expressthese polynomials in terms of Chebyshev polynomials, we find for any integer M ≥ that(2.4) | π f ,f ,I ,I ( x ) − µ ST ( I ) µ ST ( I ) π ( x ) |≪ π ( x ) M + X ≤ m ,m ≤ Mm m =0 m + 1)( m + 1) (cid:12)(cid:12)(cid:12) X p ≤ xp ∤ q q U m (cos θ (1) p ) U m (cos θ (2) p ) (cid:12)(cid:12)(cid:12) , x ≥ . By partial summation, we have(2.5) X p ≤ xp ∤ q q U m (cos θ (1) p ) U m (cos θ (2) p ) = θ f ,f ,m ,m ( x )log x − Z x θ f ,f ,m ,m ( t ) t (log t ) dt, where(2.6) θ f ,f ,m ,m ( x ) := X p ≤ xp ∤ q q U m (cos θ (1) p ) U m (cos θ (2) p ) log p. Proposition 2.2.
For i = 1 , , let f i be as in Theorem 1.2. There exist constants c and c (suitably large) and c , c , and c (suitably small) such that if (2.7) ≤ m , m ≤ M ≤ c p log log x/ log( k q k q log log x ) , then | θ f ,f ,m ,m ( x ) | ≪ x − c k q k q M ) c M + ( m m ) x − c M + ( m m ) x (cid:16) exp h − c log xM log( k q k q M ) i + exp h − c √ log xM i(cid:17) . Note that θ f ,f , ,m ( x ) = θ f ,m ( x ) and θ f ,f ,m , ( x ) = θ f ,m ( x ) . Therefore, by Proposi-tion 2.1, (2.4), (2.5), (2.6), and Proposition 2.2, we find that if (2.7) holds, then(2.8) (cid:12)(cid:12)(cid:12) π f ,f ,I ,I ( x ) − µ ST ( I ) µ ST ( I ) π ( x ) (cid:12)(cid:12)(cid:12) JESSE THORNER ≪ π ( x ) n M + M x − c k q k q M ) c M + M x − c M + M (cid:16) exp h − c log x M log( k q k q M ) i + exp h − c √ log x M i(cid:17)o . Proof of Theorem 1.2.
Write µ = µ ST ( I ) µ ST ( I ) . There exists a suitably large constant c such that if x ≥ exp exp( c ( µ log k q k q µ ) ) , then the theorem follows from (2.8) bychoosing c > to be a sufficiently small constant and M = ⌈ c √ log log x log( k q k q log log x ) ⌉ ≥ . Forall remaining values of x , the theorem is trivial. (cid:3) Outline for proofs of Propositions 2.1 and 2.2.
Propositions 2.1 and 2.2 willfollow from a sufficiently uniform unconditional prime number theorem for the L -functionsassociated to symmetric powers of the cuspidal automorphic resepresentations associated tonewforms and the Rankin–Selberg convolutions of said symmetric powers. To prove suchprime number theorems, we first review well-known properties of L -functions in Section 3.We prove zero-free regions and log-free zero density estimates for L -functions which satisfythe generalized Ramanujan conjecture in Section 4. In Section 5, we use the results inSection 4 to prove a highly uniform prime number theorem for L -functions satisfying thegeneralized Ramanujan conjecture. Finally, in Section 6, we use the aforementioned workof Newton and Thorne [25, 26] to show how L -functions of symmetric powers and theirconvolutions fit into the framework of Section 5. We then prove Propositions 2.1 and 2.2.Our work requires careful attention to the degree dependence in several L -functions es-timates. Other problems in analytic number theory typically do not require such care, butin our setting, the degree aspect of our estimates is the aspect that matters most since itdirectly determines the quality of our results. In order to make the degree dependencies inthe necessary estimates as strong as we can, we will refine the work in [12, Appendix] and[32] to prove the necessary zero-free regions and log-free zero density estimates.3. Properties of L -functions Standard L -functions. Let A be the ring of adeles of Q , and F m be the set of cuspidalautomorphic representations of GL m ( A ) with unitary central character, which assume to benormalized so that it is trivial on the diagonally embedded copy of the positive real numbers.Given π ∈ F m , let e π be the the representation which is contragredient to π , and let q π ≥ bethe conductor of π . Write the finite part of π as a tensor product ⊗ p π p of local representationsover primes p . For each p , there exist Satake parameters α ,π ( p ) , . . . , α m,π ( p ) ∈ C such thatthe local L -function L ( s, π p ) is given by L ( s, π p ) = m Y j =1 (cid:16) − α j,π ( p ) p s (cid:17) − = ∞ X j =0 a π ( p j ) p js . We have α j,π ( p ) = 0 for all j when p ∤ q π , and some of the α j,π ( p ) might equal zero when p | q π . The standard L -function L ( s, π ) attached to π is L ( s, π ) = Y p L ( s, π p ) = ∞ X n =1 a π ( n ) n s , which converges absolutely for Re ( s ) > . The gamma factor corresponding to the infinite place of Q is given by L ( s, π ∞ ) = m Y j =1 Γ R ( s + µ π ( j )) , Γ R ( s ) = π − s/ Γ( s/ , where µ π (1) , . . . , µ π ( n ) ∈ C are the Langlands parameters. The bounds(3.1) | α j,π ( p ) | ≤ θ m , Re ( µ π ( j )) ≥ − θ m hold for some ≤ θ m ≤ − m +1 . The generalized Ramanujan conjecture and generalizedSelberg eigenvalue conjectures assert that the above inequalities hold with θ m = 0 .Let r π ∈ { , } be the order of the pole of L ( s, π ) at s = 1 , where r π = 1 if and only if π isthe trivial representation of GL ( A ) whose L -function is the Riemann zeta function ζ ( s ) .The function Λ( s, π ) = ( s ( s − r π q s/ π L ( s, π ) L ( s, π ∞ ) is entire of order one. There exists acomplex number W ( π ) of modulus one such that Λ( s, π ) = W ( π )Λ(1 − s, e π ) , where e π ∈ F m is the contragredient representation. We have the equalities of sets { α j, e π ( p ) } = { α j,π ( p ) } , { µ e π ( j ) } = { µ π ( j ) } , and q e π = q π . We define the analytic conductor C ( π ) by C ( π, t ) = q π n Y j =1 (1 + | µ π ( j ) + it | ) , C ( π ) = C ( π, . Define the numbers Λ π ( n ) by the Dirichlet series identity ∞ X n =1 Λ π ( n ) n s = − L ′ L ( s, π ) = X p ∞ X ℓ =1 P mj =1 α j,π ( p ) ℓ log pp ℓs , Re ( s ) > . It was proved in the discussion following [32, Lem 2.3] that for all η > , we have(3.2) ∞ X n =1 | Λ π ( n ) | n η ≤ η + m log C ( π ) + O ( m ) . Rankin–Selberg L -functions. Let π ∈ F m and π ′ ∈ F m ′ . For each prime p , we let L ( s, π p × π ′ p ) = m Y j =1 m ′ Y j ′ =1 (cid:16) − α j,j ′ ,π × π ′ ( p ) p s (cid:17) − = 1 + ∞ X j =1 a π × π ′ ( p j ) p js for suitable complex numbers α j,j ′ ,π × π ′ ( p ) . If p ∤ q π q π ′ , then we have the equality of sets { α j,j ′ ,π × π ′ ( p ) } = { α j,π ( p ) α j ′ ,π ′ ( p ) } . A complete description of α j,j ′ ,π × π ′ ( p ) is given in [32,Appendix]. From these Satake parameters, one defines the Rankin–Selberg L -function L ( s, π × π ′ ) = Y p L ( s, π p × π ′ p ) = ∞ X n =1 a π × π ′ ( n ) n s associated to the tensor product π ⊗ π ′ , which converges absolutely for Re ( s ) > . We write q π × π ′ for the conductor of π ⊗ π ′ . Bushnell and Henniart [4] proved that q π × π ′ | q n ′ π q nπ ′ .The gamma factor corresponding to the infinite place of Q is given by L ( s, π ∞ × π ′∞ ) = m Y j =1 m ′ Y j ′ =1 Γ R ( s + µ π × π ′ ( j, j ′ )) JESSE THORNER for suitable complex numbers µ π × π ′ ( j, j ′ ) . If both π and π ′ are unramified at the infiniteplace of Q , then we have the equality of sets { µ π × π ′ ( j, j ′ ) } = { µ π ( j ) + µ π ′ ( j ′ ) } . A completedescription of the numbers can be found in [32, Proof of Lemma 2.1]. From the explicitdescriptions of the numbers α j,j ′ ,π × π ′ ( p ) and µ π × π ′ ( j, j ′ ) , we find that(3.3) | α j,j ′ ,π × π ′ ( p ) | ≤ θ m + θ m ′ , Re ( µ π × π ′ ( j, j ′ )) ≥ − θ m − θ m ′ . Let r π × π ′ ∈ { , } be the order of the pole of L ( s, π × π ′ ) at s = 1 . We have that r π × π ′ = 1 if and only if π ′ = e π . The function Λ( s, π × π ′ ) = ( s ( s − r π × π ′ q s/ π × π ′ L ( s, π × π ′ ) L ( s, π ∞ × π ′∞ ) is entire of order one. There exists a complex number W ( π × π ′ ) of modulus one such that Λ( s, π × π ′ ) = W ( π × π ′ )Λ(1 − s, e π × e π ′ ) . We define the analytic conductor C ( π × π ′ ) to be C ( π × π ′ , t ) = q π × π ′ m Y j =1 m ′ Y j ′ =1 (1 + | µ π × π ′ ( j, j ′ ) + it | ) , C ( π × π ′ ) = C ( π × π ′ , . The combined work of Bushnell and Henniart [4] and Brumley [12, Appendix] proves that(3.4) C ( π × π ′ , t ) ≪ C ( π × π ′ )(1 + | t | ) m ′ m , C ( π × π ′ ) ≤ e O ( m ′ m ) C ( π ) m ′ C ( π ′ ) m . Define the numbers Λ π × π ′ ( n ) by the Dirichlet series identity ∞ X n =1 Λ π × π ′ ( n ) n s = − L ′ L ( s, π × π ′ ) = X p ∞ X ℓ =1 P mj =1 P m ′ j ′ =1 α j,j ′ ,π × π ′ ( p ) ℓ log pp ℓs , Re ( s ) > . It was proved in the discussion following [32, Lem 2.3] that for all η > , we have(3.5) ∞ X n =1 | Λ π × π ′ ( n ) | n η ≤ η + m ′ m log C ( π × π ′ ) + O (( m ′ m ) ) . Isobaric automorphic representations.
Let d ≥ be an integer; let m , . . . , m d bepositive integers; let r = P di =1 m i ; let t , . . . , t d ∈ R ; and let π i ∈ F m i with ≤ i ≤ d .Consider the isobaric automorphic representation Π of GL r ( A ) given by the isobaric sum Π = π ⊗ | det | it ⊞ · · · ⊞ π d ⊗ | det | it d . The L -function associated to Π is L ( s, Π) = Q dj =1 L ( s + it j , π j ) , with analytic conductor C (Π , t ) = d Y j =1 C ( π j , t + t j ) , C (Π) = C (Π , . Let d ′ ≥ be an integer; let m ′ , . . . , m ′ d ′ be positive integers; let r ′ = P d ′ i ′ =1 m ′ i ′ ; let t ′ , . . . , t ′ d ∈ R ; and let π ′ i ′ ∈ F m ′ i ′ with ≤ i ′ ≤ d ′ . Consider the isobaric automorphicrepresentation Π ′ of GL r ′ ( A ) given by the isobaric sum Π ′ = π ′ ⊗| det | it ′ ⊞ · · · ⊞ π d ′ ⊗| det | it ′ d ′ .The Rankin–Selberg L -function associated to Π ⊗ Π ′ is L ( s, Π × Π ′ ) = d Y j =1 d ′ Y j ′ =1 L ( s + it j + it ′ j ′ , π j × π ′ j ′ ) , and its analytic conductor is C (Π × Π ′ , t ) = d Y j =1 d ′ Y j ′ =1 C ( π j × π ′ j ′ , t + t j + t ′ j ′ ) , C (Π × Π ′ ) = C (Π × Π ′ , . Zeros of L -functions We require two results on the distribution of zeros of standard L -functions and Rankin–Selberg L -functions. First, we require a standard zero-free region. We present a modificationto the work of Brumley [12, Appendix] which will improve the degree dependence. We willuse this improved degree dependence to prove our second result, namely, the log-free zerodensity estimate of Soundararajan and the author [32] with improved degree dependence.4.1. Zero-free regions.
A proof of the following proposition is sketched in [18, Lem 3.1].We give a complete proof here.
Proposition 4.1.
Let Π be an isobaric automorphic representation of GL r ( A ) . If L ( s, Π × e Π) has a pole of order r Π × e Π ≥ at s = 1 , then L (1 , Π × e Π) = 0 , and there exists a constant c > such that L ( s, Π × e Π) has at most r Π × e Π real zeros in the interval (4.1) s ≥ − c ( r Π × e Π + 1) log C (Π × e Π) . Remark.
Our proof removes the extraneous factor of d in the denominator in [13, Lem 5.9]when f = Π × e Π . While this may seem like a small improvement, the quality of our maintheorems depends heavily on it. Proof.
By proceeding as in the proof of [13, Equations 5.28 and 5.37], we find that X − c / log C (Π × e Π) <β ≤ L ( β, Π × e Π)=0 σ − β < r Π × e Π σ − Re (cid:16) L ′ L ( σ, Π × e Π) (cid:17) + O (log C (Π × e Π)) , where s = σ ≥ . Define Λ Π × e Π ( n ) by the Dirichlet series identity ∞ X n =1 Λ Π × e Π ( n ) n s = − L ′ L ( s, Π × e Π) . The work in [32, Section A.2] leading up to Equation A.9 shows that Λ Π × e Π ( n ) ≥ for all n ≥ (even if n shares a prime factor with a conductor of one of the constituents of Π ).Thus by nonnegativity, we find that(4.2) X − c / log C (Π × e Π) <β ≤ L ( β, Π × e Π)=0 σ − β < r Π × e Π σ − O (log C (Π × e Π)) . Let N be the number of real zeros in the sum on the left hand side of (4.2). We choose σ = 1 + 2 c / log C (Π × e Π) and conclude that N log C (Π × e Π)2 c + c r Π × e Π +1 < (cid:16) r Π × e Π c + O (1) (cid:17) log C (Π × e Π) . This implies that
N < r Π × e Π + r Π × e Π r Π × e Π +1) + O ( c ) , so N ≤ r Π × e Π when c is small enough.The possibility that L (1 , Π × e Π) = 0 is ruled out by [17, Thm A.1]. (cid:3)
Corollary 4.2.
Let π ∈ F m and π ′ ∈ F m ′ . Suppose that both π and π ′ are self-dual (that is, π = e π and π ′ = e π ′ ). There exists a constant c > such that the following results hold. (1) L ( s, π ) = 0 in the region Re ( s ) ≥ − c m log( C ( π )(3 + | Im ( s ) | )) apart from at most one zero. If the exceptional zero exists, then it is real and simple.(2) L ( s, π × π ′ ) = 0 in the region (4.3) Re ( s ) ≥ − c ( m + m ′ ) log( C ( π ) C ( π ′ )(3 + | Im ( s ) | ) min { m,m ′ } ) apart from at most one zero. If the exceptional zero exists, then it is real and simple.Remark. This improves the denominator ( m + m ′ ) log( C ( π ) C ( π ′ )(3 + | Im ( s ) | ) m ) in [12, ThmA.1] when F = Q and both π and π ′ are self-dual. A similar improvement holds when π and π ′ are defined over number fields. Proof.
We prove the second part; the proof of the first part is the same once we choose π ′ = . Without loss, suppose that m ′ ≤ m . First, let γ = 0 , and suppose to the contrarythat ρ = β + iγ is a zero in the region (4.3). We apply Proposition 4.1 to the choice of Π = π ′ ⊗ | det | iγ ⊞ e π ′ ⊗ | det | − iγ ⊞ π , in which case (since π and π ′ are self-dual) L ( s, Π × e Π) = L ( s, π × e π ) L ( s, π ′ × e π ′ ) L ( s + it, π × π ′ ) L ( s − it, π × π ′ ) × L ( s + 2 iγ, π ′ × e π ′ ) L ( s − iγ, π ′ × e π ′ ) . Since π and π ′ are self-dual, it follows that ρ is a zero of L ( s, π × π ′ ) if and only if ρ is. Thusif ρ is a zero of L ( s, π ) , then L ( s, Π × e Π) has a real zero at s = β of order 4 in the region (4.1).This contradicts Proposition 4.1 since r Π × e Π = 3 when γ = 0 . The desired result follows fromthe bound log C (Π × e Π) ≪ ( m + m ′ ) log( C ( π ) C ( π ′ )(3 + | γ | ) m ′ ) , which holds via (3.4).Second, suppose that γ = 0 . The same arguments as when γ = 0 hold, except that now r Π × e Π = 5 . Since the presence of a single zero of L ( s, π × π ′ ) in the claimed region contributesa zero of order 4 to L ( s, Π × e Π) , we must conclude that if a real zero of L ( s, π × π ′ ) existsin the region (4.3), then such a zero must be simple. (cid:3) We also record a bound on the exceptional zero in Part (2) of Corollary 4.2 (if it exists).
Lemma 4.3.
There exists a constant c > such that the exceptional zero in Part (2) ofCorollary 4.2 when π = π ′ is bounded by − c ( C ( π ) C ( π ′ )) − m − m ′ .Proof. This is an immediate consequence of [17, Thm A.1]. (cid:3)
Log-free zero density estimates.
Our log-free zero density estimates are as follows.
Proposition 4.4.
Let π ∈ F m and π ′ ∈ F m ′ . There exists a constant c > such that thefollowing are true.(1) Suppose that π satisfies GRC at all primes p ∤ q π . If ≤ σ ≤ and T ≥ , then N π ( σ, T ) := { ρ = β + iγ : β ≥ σ, | γ | ≤ T, L ( ρ, π ) = 0 } ≪ m ( C ( π ) T ) c m (1 − σ ) . (2) If ≤ σ ≤ , T ≥ , m, m ′ ≤ M , and π and π ′ satisfy GRC at all p ∤ q π q π ′ , then N π × π ′ ( σ, T ) := { ρ = β + iγ : β ≥ σ, | γ | ≤ T, L ( ρ, π × π ′ ) = 0 } ≪ M (( C ( π ) C ( π ′ )) M T ) c M (1 − σ ) . For the sake of brevity, we will only prove Part (1). There are no structural differencesin the proof of Part (2) except that m is replaced by m ′ m ≤ M and C ( π ) is replaced by C ( π ) O ( m ′ ) C ( π ′ ) O ( m ) (which is an upper bound for C ( π × π ′ ) ). Our proof runs parallel to thatof [32, Thm 1.2], so we only point out the key differences. We begin with some adjustmentsto the lemmas in [32] which will improve the degree dependence. Lemma 4.5. If t ∈ R and < η ≤ , then X ρ η − β | η + it − ρ | ≤ m log C ( π ) + m log(2 + | t | ) + 2 η + O ( m ) so that { ρ : | ρ − (1 + it ) | ≤ η } ≤ mη log C ( π ) + 5 mη log(2 + | t | ) + O ( m η + 1) .Proof. The proof proceeds just as in [32, Lem 3.1], but we use (3.2) instead of [32, Equation1.9], and we use the fact that r π ≤ in our case. (cid:3) Lemma 4.6.
Let T ≥ , and let τ ∈ R satisfy η ≤ | τ | ≤ T . If c / ( m log( C ( π ) T )) ≤ η ≤ (200 m ) − and s = 1 + η + iτ , then ( − k k ! (cid:16) L ′ L ( s, π ) (cid:17) ( k ) = X ρ | s − ρ |≤ η s − ρ ) k +1 + O (cid:16) m log( C ( π ) T )(200 η ) k (cid:17) . Proof.
The proof is the same as that of [32, Equation 4.1] with three changes. First, we havethat r π ≤ . Second, we widen the range of η all the way to the edge of the zero-free regionin Corollary 4.2. Finally, we use Lemma 4.5 instead of [32, Lem 3.1]. (cid:3) Lemma 4.7.
Let T ≥ , η ≤ | τ | ≤ T , and c / ( m log( C ( π ) T )) < η ≤ (200 m ) − . Let K > ⌈ mη log( C ( π ) T )+ O ( m η +1) ⌉ and s = 1+ η + iτ . If L ( s, π ) has a zero ρ satisfying | ρ − (1 + iτ ) | ≤ η , then (cid:12)(cid:12)(cid:12) X ρ | s − ρ |≤ η s − ρ ) k +1 (cid:12)(cid:12)(cid:12) ≥ (cid:16) η (cid:17) k +1 . Proof.
The proof is the same as that of [32, Lem 4.2], but we use Lemmas 4.5 and 4.6 insteadof [32, Equations 3.6 and 4.1]. (cid:3)
Lemma 4.8.
Let η and τ be real numbers satisfying c / ( m log( C ( π ) T )) < η ≤ (200 m ) − and η ≤ | τ | ≤ T . Let K ≥ be an integer, and let N = exp( K/ (300 η )) and N =exp((40 K ) /η ) . Let s = 1 + η + iτ , If K ≤ k ≤ K , then (cid:12)(cid:12)(cid:12) η k +1 k ! (cid:16) L ′ L ( s, π ) (cid:17) ( k ) (cid:12)(cid:12)(cid:12) ≤ η Z N N (cid:12)(cid:12)(cid:12) Λ π ( n ) n iτ (cid:12)(cid:12)(cid:12) duu + O (cid:16) mη log( C ( π ) T )(110) k (cid:17) . Proof.
The proof is the same as that of [32, Lem 4.3] except that it incorporates (3.2) insteadof [32, Equation 19] as well as our wider range of η . (cid:3) Proof of Proposition 4.4.
We reiterate that we have only given the details for L ( s, π ) , butthe details for L ( s, π × π ′ ) run parallel apart from bounding C ( π × π ′ ) . Compare with [32].Let η, τ ∈ R satisfy c / ( m log( C ( π ) T )) < η ≤ (200 m ) − and η ≤ | τ | ≤ T . Let K = 2000 ηm log( C ( π ) T ) + O ( m η + 1) , N = exp( K/ (300 η )) , N = exp(40 K/η ) , where the implied constant in our definition of K is sufficiently large. We proceed as in theproof of [32, Thm 1.2], but we use Lemmas 4.7 and 4.8 instead of [32, Lem 4.2 and 4.3]. Weconclude that { ρ = β + iγ : β ≥ − η/ , | γ | ≤ T }≪ K η m log( C ( π ) T ) T Z N N /e (cid:12)(cid:12)(cid:12) X x We will prove a uniform version of the prime number theorem for the L -functions L ( s, π ) and L ( s, π × π ′ ) which satisfy the hypotheses of Corollary 4.2 and Proposition 4.4. The use ofProposition 4.4 helps us to improve the range of x (which important for (1.4)) and simplifycertain aspects of the proof. We define the weighted prime counting functions θ π ( x ) := X p ≤ xp ∤ q π a π ( p ) log p, θ π × π ′ ( x ) := X p ≤ xp ∤ q π q π ′ a π × π ′ ( p ) log p. Proposition 5.1. Let π ∈ F m and π ′ ∈ F m ′ . Suppose that π and π ′ satisfy the hypothesesof Corollary 4.2 and Proposition 4.4.(1) Suppose that either µ π ( j ) = 0 or Re ( µ π ( j )) ≥ for each j . If ≤ C ( π ) m ≤ x / (36 c ) and c mx − / (32 c m ) < , then (cid:12)(cid:12)(cid:12) θ π ( x ) − r π x + x β β (cid:12)(cid:12)(cid:12) ≪ m x − c m + m x (cid:16) exp h − c log x m log C ( π ) i + exp h − √ c log x √ m i(cid:17) . We omit the β term if the exceptional zero in Corollary 4.2 (Part 1) does not exist.(2) Let m, m ′ ≤ M . Suppose that either µ π × π ′ ( j, j ′ ) = 0 or Re ( µ π × π ′ ( j, j ′ )) ≥ . If ≤ ( C ( π ) C ( π ′ )) M ≤ x / (36 c ) and c M x − / (32 c M ) < , then (cid:12)(cid:12)(cid:12) θ π × π ′ ( x ) − r π × π ′ x + x β β (cid:12)(cid:12)(cid:12) ≪ ( m ′ m ) x − c M + ( m ′ m ) x (cid:16) exp h − c log x M log( C ( π ) C ( π ′ )) i + exp h − √ c log x M i(cid:17) . We omit the β term if the exceptional zero in Corollary 4.2 (Part 2) does not exist. As with our proof of Proposition 4.4, we will only prove Part (1) of Proposition 5.1 since theproof of Part (2) runs entirely parallel apart from an application of (3.4) to bound C ( π × π ′ ) .To begin our work toward Proposition 5.1, we introduce a carefully chosen smooth weightfor sums over prime powers. Lemma 5.2. Choose x ≥ , ε ∈ (0 , / , and an integer ℓ ≥ . Define A = ε/ (2 ℓ log x ) .There exists a continuous function φ ( t ) = φ ( t ; x, ℓ, ε ) which satisfies the following properties:(1) ≤ φ ( t ) ≤ for all t ∈ R , and φ ( t ) ≡ for ≤ t ≤ .(2) The support of φ is contained in the interval [ − ε log x , ε log x ] .(3) Its Laplace transform Φ( z ) = R R φ ( t ) e − zt dt is entire and is given by (5.1) Φ( z ) = e − (1+2 ℓA ) z · (cid:16) − e ( +2 ℓA ) z − z (cid:17)(cid:16) − e Az − Az (cid:17) ℓ . (4) Let s = σ + it, σ > , t ∈ R and α be any real number satisfying ≤ α ≤ ℓ . Then | Φ( − s log x ) | ≤ e σε x σ | s | log x · (cid:0) x − σ/ (cid:1) · (cid:16) ℓε | s | (cid:17) α . Moreover, | Φ( − s log x ) | ≤ e σε x σ and < Φ(0) < .(5) If < σ ≤ , x ≥ , and δ π, , r π ∈ { , } , then r π Φ( − log x ) log x − δ π, Φ( − σ log x ) log x = (cid:16) r π x − δ π, x σ σ (cid:17) (1 + O ( ε )) + O ( √ x ) . (6) Let s = − + it with t ∈ R . Then | Φ( − s log x ) | ≪ x − / log x (cid:16) ℓε (cid:17) ℓ ( + t ) − ℓ/ . Proof. This is [35, Lem 2.2], except that item (5) is slightly more general and in item (6),we have Re ( s ) = − instead of − . The proofs proceed in exactly the same way. (cid:3) Define ψ π ( x ) = ψ π ( x, φ ) = ∞ X n =1 φ (cid:16) log n log x (cid:17) Λ π ( n ) . The next lemma shows that ψ π ( x ) closely approximates θ π ( x ) . Lemma 5.3. If ℓ ≥ is an integer, x satisfies the hypotheses of Proposition 2.1, and ε ∈ ( x − , ) , then θ π ( x ) = ψ π ( x ) + O ( mx − m + εmx ) . Proof. By Lemma 5.2, we have ψ π ( x ) = X √ x We begin our proof of Proposition 5.1 by shifting the contour to the line Re ( s ) = − ,accumulating contributions from the residues at the nontrivial zeros of L ( s, π ) and a trivialzero at s = 0 of order O ( m ) with residue O (1) . We bound the shifted contour integral onthe line Re ( s ) = − using Lemmas 5.2 and 5.4 and conclude that ψ π ( x ) = r π Φ( − log x ) log x − δ π, Φ( − β log x ) log x − (log x ) X ρ = β Φ( − ρ log x )+ O (cid:16) (2 ℓ/ε ) ℓ log C ( π ) x + m (cid:17) = r π x − δ π, x β β − (log x ) X ρ = β Φ( − ρ log x ) + O (cid:16) (2 ℓ/ε ) ℓ log C ( π ) x + m + √ x + εx (cid:17) . (5.2) By [13, Prop 5.7], there are ≪ log C ( π ) nontrivial zeros ρ of L ( s, π ) with | ρ | < . Thus itfollows from Lemma 5.2 that(5.3) X | ρ |≤ ρ = β | Φ( − ρ log x ) | ≪ X | ρ |≤ ρ = β x ≪ x log C ( π ) . Lemma 5.5. Let φ be defined as in Lemma 5.2 with ε = 8 ℓx − ℓ and ℓ = 4 c m . If ≤ C ( π ) m ≤ x / (8 c ) and ε < , then log x X | ρ |≥ ρ = β | Φ( − ρ log x ) | ≪ m xe − η π ( x ) / , η π ( x ) := inf t ≥ (cid:16) c log xm log( C ( π ) t ) + log t (cid:17) . Proof. Let T = 0 and T j = 2 j − for all j ≥ . We consider the sums S j := log x X T j − ≤| γ |≤ T j | ρ |≥ , ρ = β | Φ( − ρ log x ) | . We estimate | F ( − ρ log x ) | for ρ in the sum S j using Lemma 5.2 with α = ℓ (1 − β ) . Ourchoices of ε and ℓ and our restriction ≤ C ( π ) m ≤ x / (8 c ) imply that | Φ( − ρ log x ) | log x ≪ x β | ρ | (cid:16) ℓε | ρ | (cid:17) ℓ (1 − β ) ≪ xT − j ( | γ | + 3) − x − − β ( x T ℓj ) − (1 − β ) ≪ xT − j ( | γ | + 3) − x − − β ( C ( π ) T ) − mc (1 − β ) . By the definition of η π ( x ) and Corollary 4.2, we have the bound ( | γ | + 3) − x − − β = e − (log( | γ | +3)+(1 − β ) log x ) ≤ e − η π ( x ) / . Consequently, we have that S j ≪ xe − η π ( x ) / T j X T j − ≤| γ |≤ T j ( C ( π ) T ) − mc (1 − β ) ≤ xe − η π ( x ) / T j Z ( C ( π ) T ) − mc σ dN m (1 − σ, T j ) . The Stieltjes integral equals ( C ( π ) T ) − c m N π (0 , T j ) + m log( C ( π ) T ) Z ( C ( π ) T ) − mc σ N π (1 − σ, T j ) dσ, which we estimate using Proposition 4.4. We conclude that S j ≪ m xe − η π ( x ) / T − / j , hence log x X | ρ |≥ ρ = β | Φ( − ρ log x ) | ≪ ∞ X j =1 S j ≪ m xe − η π ( x ) / ∞ X j =1 T − j ≪ m xe − η π ( x ) / , as desired. (cid:3) Lemma 5.6. If m ≥ and x ≥ , then e − η π ( x ) / ≤ exp[ − c log x m log C ( π ) ] + exp[ − √ c log x √ m ] .Proof. This is a straightforward optimization problem. (cid:3) Proof of Proposition 5.1. Collect the estimates in Lemma 5.3, (5.2), (5.3), Lemma 5.5, andLemma 5.6 and apply the prescribed choices for ℓ and ε in our range of x . (cid:3) Proof of Propositions 2.1 and 2.2 Let f ∈ S new k (Γ ( q )) be a newform as in the statement of Theorem 1.1. For each prime p , let θ p ∈ [0 , π ] be the unique angle such that a f ( p ) = 2 cos θ p . The modular L -function L ( s, f ) associated to f has the Euler product representation L ( s, f ) = ∞ X n =1 a f ( n ) n s = Y p (cid:16) − a f ( p ) p s + χ ( p ) p s (cid:17) − , Re ( s ) > , where χ is the trivial Dirichlet character modulo q . We rewrite the Euler product as(6.1) L ( s, f ) = Y p | q (cid:16) − ( − λ p p − ) p s (cid:17) Y p ∤ q Y j =0 (cid:16) − e i (2 j − θ p p s (cid:17) − , Re ( s ) > , where λ p ∈ {− , } is the eigenvalue of the Atkin–Lehner operator | k W ( Q p ) .6.1. Standard L -functions. For each m ≥ , we define the Euler product(6.2) L ( s, Sym m f ) = ∞ X n =1 a Sym m f ( n ) n s = Y p | q m Y j =0 (cid:16) − α j, Sym m f ( p ) p s (cid:17) − Y p ∤ q m Y j =0 (cid:16) − e i (2 j − m ) θ p p s (cid:17) − for Re ( s ) > . The values α j, Sym m f ( p ) can be determined using [32, Appendix], but anexplicit description with uniformity in f can be unwieldy when q is not squarefree. Theseexplicit descriptions are not germane to our proofs. We do note that when f correspondswith a non-CM elliptic curve via modularity, a completely explicit and wieldy description of α j, Sym m f ( p ) at p | q can be found in [9, Appendix].We also define(6.3) L ( s, (Sym m f ) ∞ ) = ( q Sym m f Q ( m +1) / j =1 Γ C ( s + ( j − )( k − if m is odd, q Sym m f Γ R ( s + r ) Q m/ j =1 Γ C ( s + j ( k − if m is evenfor a suitable integer q Sym m f , where Γ C ( s ) = Γ R ( s )Γ R ( s + 1) , r = 0 if m ≡ , and r = 1 if m ≡ . Note that L (Sym f ) = L ( s, f ) and L (Sym f ) = ζ ( s ) . One easilychecks via (6.2) that(6.4) a Sym m f ( p ) = U m (cos θ p ) , p ∤ q. Theorem 6.1. Let f ∈ S new k (Γ ( q )) be as in Theorem 1.1, and let π f ∈ F correspond with f . If m ≥ , then L ( s, Sym m f ) is the standard L -function associated to the representation Sym m π f ∈ F m +1 , with L ( s, (Sym m π f ) ∞ ) given by (6.3) . We also have log q Sym m f ≪ m log q .Proof. Let π f be the cuspidal automorphic representation of GL ( A ) with unitary centralcharacter which corresponds with f . Newton and Thorne ([25, Thm B] and [26, ThmA]) recently proved that if m ≥ , then the m -th symmetric power lift Sym m π f is a self-dual cuspidal automorphic representation of GL m +1 ( A ) with trivial central character whosestandard L -function is given by (6.1). Moreno and Shahidi [24] and Cogdell and Michel [8,Section 3] computed L ( s, (Sym m π f ) ∞ ) under the assumption of cuspidality, which we nowhave. When q is squarefree, we have log q Sym m f = m log q [8, Section 3]; otherwise, Rouse,following a suggestion of Serre, proved that log q Sym m f ≪ m log q [28, Section 5]. (cid:3) For m ≥ , a straightforward calculation using (6.3) and Stirling’s formula yields(6.5) log C (Sym m f ) ≪ m log( kqm ) , C (Sym m f ) := C (Sym m π f ) . Rankin–Selberg L -functions. Given f as in Theorem 6.1, let π f be the cuspidalautomorphic representation of GL ( A ) corresponding to f . Given an integer m ≥ , let Sym m π f be the m -th symmetric power lift, which is shown in Theorem 6.1 to be a cuspidalautomorphic representation of GL m +1 ( A ) . (If m = 0 , then Sym m π f = .)For i = 1 , , let f i ∈ S new k i (Γ ( q i )) be a newform as in Theorem 6.1, and let { θ ( j ) p } bethe sequence of Sato–Tate angles for f i . Suppose that π f = π f . For integers m i ≥ , weconsider the tensor product Sym m π f ⊗ Sym m π f , whose Rankin–Selberg L -function is L ( s, Sym m f × Sym m f ) . = Y p ∤ q q m Y j =0 m Y j =0 (cid:16) − e i (2 j − m ) θ (1) p e i (2 j − m ) θ (2) p p s (cid:17) − in view of Theorem 6.1. The . = suppresses the (more complicated) Euler factors at primes p | q q which have an unwieldy (and, for our purposes, unenlightening) explicit description via[32, Appendix]. Instead describing the Euler factors at primes p | q q and the gamma factors,we observe that the bound (3.3) applied to the primes p | q q , while probably very inefficient,is strong enough for us to prove Theorem 1.2, and we can estimate C (Sym m f × Sym m f ) using (3.4) and (6.5). A standard though tedious calculation shows that the Langlandsparameters of L ( s, Sym m f × Sym m f ) satisfy the hypotheses of Proposition 5.1; we omitthis calculation. It is straightforward to check that(6.6) a Sym m f × Sym m f ( p ) = U m (cos θ (1) p ) U m (cos θ (2) p ) , p ∤ q q . Lemma 6.2. If m m = 0 and f f are as in Theorem 6.1, then L ( s, Sym m f × Sym m f ) extends to an entire function with no pole at s = 1 .Proof. This is proved by Harris [11, Theorem 5.3] when f and f are associated to non-CMelliptic curves. The proof is identical for other pairs f f . The assumption of Harris’s“Expected Theorems” is replaced by the automorphy of symmetric powers proved by Newtonand Thorne [25, 26]. (cid:3) Lemma 6.3. There exists a constant c > such that if m ≥ , then L ( s, Sym m f ) = 0 for Re ( s ) ≥ − c m log( kqm (3 + | Im ( s ) | )) . Proof. When Im ( s ) = 0 , this follows from Corollary 4.2 and (6.5) (once c is made suitablysmall). It remains to handle the case where Im ( s ) = 0 . Suppose to the contrary thata real zero in this region exists. Consider the isobaric automorphic representation Π m = ⊞ Sym π f ⊞ Sym m π f . Using the identities Sym m π f ⊗ Sym m π f = ⊞ ( ⊞ mj =1 Sym j π f ) , π f ⊗ Sym π f = π f ⊞ Sym π f , and Sym m π f ⊗ Sym π f = ⊞ j =0 Sym m +2 − j π f for m ≥ , wefind for m ≥ that L ( s, Π m × e Π m ) = ζ ( s ) L ( s, Sym m f ) L ( s, Sym f ) L ( s, Sym f ) L ( s, Sym m +2 f ) × L ( s, Sym m − f ) m Y j =1 L ( s, Sym j f ) (with L ( s, Sym m − f ) omitted when m = 1 ). The bound log C (Π m × e Π m ) ≪ m log( kqm ) follows from (3.4) and (6.5). Note that L ( s, Π m × e Π m ) always has a pole of order exactly3, but a proposed zero of L ( s, Sym m f ) ensures that L ( s, Π m × e Π m ) has a real zero of order at least 4 in the region (4.1). This contradicts Proposition 4.1, hence no such zero can exist(once c is made suitably small). (cid:3) Lemma 6.4. Let ≤ m , m ≤ M . There exists a constant c > such that The Rankin–Selberg L -function L ( s, π × π ′ ) = 0 for Re ( s ) ≥ − c M log( k q k q M (3 + | Im ( s ) | )) apart from at most one zero β m ,m . If β m ,m exists, then it is real and simple, and thereexist constants < c < and c > such that β m ,m ≤ − c ( k q k q M ) − c M .Proof. This follows from Corollary 4.2, Lemma 4.3, and (6.5). (cid:3) Proofs of Propositions 2.1 and 2.2. Proof of Proposition 2.1. This follows from Proposition 5.1(1), (6.4), Theorem 6.1, (6.5), andLemma 6.3. The conditions in Proposition 5.1 are satisfied for m in the claimed range. (cid:3) Proof of Proposition 2.2. This follows from Proposition 5.1(2), Theorem 6.1, (6.5), Lemma 6.2,(6.6), and Lemma 6.4. The two conditions in Proposition 5.1 are satisfied for m and m inthe claimed range. (cid:3) References [1] S. Baier and N. Prabhu. Moments of the error term in the Sato-Tate law for elliptic curves. J. NumberTheory , 194:44–82, 2019.[2] T. Barnet-Lamb, D. Geraghty, M. Harris, and R. Taylor. A family of Calabi-Yau varieties and potentialautomorphy II. Publ. Res. Inst. Math. Sci. , 47(1):29–98, 2011.[3] A. Bucur and K. S. Kedlaya. An application of the effective Sato-Tate conjecture. In Frobenius distri-butions: Lang-Trotter and Sato-Tate conjectures , volume 663 of Contemp. Math. , pages 45–56. Amer.Math. Soc., Providence, RI, 2016.[4] C. J. Bushnell and G. Henniart. An upper bound on conductors for pairs. J. Number Theory , 65(2):183–196, 1997.[5] E. Chen, P. S. Park, and A. A. Swaminathan. Elliptic curve variants of the least quadratic nonresidueproblem and Linnik’s theorem. Int. J. Number Theory , 14(1):255–288, 2018.[6] L. Clozel and J. A. Thorne. Level-raising and symmetric power functoriality, III. Duke Math. J. ,166(2):325–402, 2017.[7] T. Cochrane. Trigonometric approximation and uniform distribution modulo one. Proc. Amer. Math.Soc. , 103(3):695–702, 1988.[8] J. Cogdell and P. Michel. On the complex moments of symmetric power L -functions at s = 1 . Int. Math.Res. Not. , (31):1561–1617, 2004.[9] C. David, A. Gafni, A. Malik, N. Prabhu, and C. L. Turnage-Butterbaugh. Extremal primes for ellipticcurves without complex multiplication. Proc. Amer. Math. Soc. , 148(3):929–943, 2020.[10] S. Gelbart and H. Jacquet. A relation between automorphic representations of GL(2) and GL(3) . Ann.Sci. École Norm. Sup. (4) , 11(4):471–542, 1978.[11] M. Harris. Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applica-tions. In Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II , volume 270 of Progr.Math. , pages 1–21. Birkhäuser Boston, Inc., Boston, MA, 2009.[12] P. Humphries and F. Brumley. Standard zero-free regions for Rankin-Selberg L -functions via sievetheory. Math. Z. , 292(3-4):1105–1122, 2019.[13] H. Iwaniec and E. Kowalski. Analytic number theory , volume 53 of American Mathematical SocietyColloquium Publications . American Mathematical Society, Providence, RI, 2004.[14] H. H. Kim. Functoriality for the exterior square of GL and the symmetric fourth of GL . J. Amer.Math. Soc. , 16(1):139–183, 2003. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kimand Peter Sarnak. [15] H. H. Kim and F. Shahidi. Functorial products for GL × GL and the symmetric cube for GL . Ann.of Math. (2) , 155(3):837–893, 2002. With an appendix by Colin J. Bushnell and Guy Henniart.[16] O. Klurman and A. Mangerel. Monotone chains of Fourier coefficients of Hecke cusp forms. arXiv e-prints , page arXiv:2009.03225, Sept. 2020.[17] E. Lapid. On the Harish-Chandra Schwartz space of G ( F ) \ G ( A ) . In Automorphic representations and L -functions , volume 22 of Tata Inst. Fundam. Res. Stud. Math. , pages 335–377. Tata Inst. Fund. Res.,Mumbai, 2013. With an appendix by Farrell Brumley.[18] R. J. Lemke Oliver and J. Thorner. Effective log-free zero density estimates for automorphic L -functionsand the Sato-Tate conjecture. Int. Math. Res. Not. IMRN , (22):6988–7036, 2019.[19] U. V. Linnik. On the least prime in an arithmetic progression. Rec. Math. [Mat. Sbornik] N.S. ,15(57):139–178,347–368, 1944.[20] F. Luca, M. Radziwiłł, and I. E. Shparlinski. On the typical size and cancellations among the coefficientsof some modular forms. Math. Proc. Cambridge Philos. Soc. , 166(1):173–189, 2019.[21] B. Mazur. Finding meaning in error terms. Bull. Amer. Math. Soc. (N.S.) , 45(2):185–228, 2008.[22] G. Molteni. Upper and lower bounds at s = 1 for certain Dirichlet series with Euler product. DukeMath. J. , 111(1):133–158, 2002.[23] H. L. Montgomery. Ten lectures on the interface between analytic number theory and harmonic analysis ,volume 84 of CBMS Regional Conference Series in Mathematics . Published for the Conference Board ofthe Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI,1994.[24] C. J. Moreno and F. Shahidi. The L -functions L ( s, Sym m ( r ) , π ) . Canad. Math. Bull. , 28(4):405–410,1985.[25] J. Newton and J. A. Thorne. Symmetric power functoriality for holomorphic modular forms. arXive-prints , page arXiv:1912.11261, Dec 2019.[26] J. Newton and J. A. Thorne. Symmetric power functoriality for holomorphic modular forms, II. arXive-prints , page arXiv:2009.07180, Sept. 2020.[27] K. Ono. The web of modularity: arithmetic of the coefficients of modular forms and q -series , volume102 of CBMS Regional Conference Series in Mathematics . Published for the Conference Board of theMathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004.[28] J. Rouse. Atkin-Serre type conjectures for automorphic representations on GL(2) . Math. Res. Lett. ,14(2):189–204, 2007.[29] J. Rouse and J. Thorner. The explicit Sato-Tate conjecture and densities pertaining to Lehmer-typequestions. Trans. Amer. Math. Soc. , 369(5):3575–3604, 2017.[30] J.-P. Serre. Quelques applications du théorème de densité de Chebotarev. Publications Mathématiquesde l’IHÉS , 54:123–201, 1981.[31] J.-P. Serre. Abelian l -adic representations and elliptic curves , volume 7 of Research Notes in Mathemat-ics . A K Peters, Ltd., Wellesley, MA, 1998. With the collaboration of Willem Kuyk and John Labute,Revised reprint of the 1968 original.[32] K. Soundararajan and J. Thorner. Weak subconvexity without a Ramanujan hypothesis. Duke Math.J. , 168:1231–1268, 2019. With an appendix by Farrell Brumley.[33] J. Thorner. The error term in the Sato-Tate conjecture. Arch. Math. (Basel) , 103(2):147–156, 2014.[34] J. Thorner and A. Zaman. A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for theLang-Trotter conjectures. Int. Math. Res. Not. IMRN , (16):4991–5027, 2018.[35] J. Thorner and A. Zaman. A unified and improved Chebotarev density theorem. Algebra Number Theory ,13(5):1039–1068, 2019.[36] P.-J. Wong. On the Chebotarev-Sato-Tate phenomenon. J. Number Theory , 196:272–290, 2019. Department of Mathematics, University of Illinois, Urbana, IL 61801 Email address ::