Analytic ranks of automorphic L-functions and Landau-Siegel zeros
aa r X i v : . [ m a t h . N T ] F e b ANALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS ANDLANDAU-SIEGEL ZEROS HUNG M. BUI, KYLE PRATT AND ALEXANDRU ZAHARESCU
Abstract.
We relate the study of Landau-Siegel zeros to the ranks of Jacobians J ( q ) ofmodular curves for large primes q . By a conjecture of Brumer-Murty, the rank should beequal to half of the dimension. Equivalently, almost all newforms of weight two and level q have analytic rank ≤
1. We show that either Landau-Siegel zeros do not exist, or thatalmost all such newforms have analytic rank ≤
2. In particular, almost all odd newformshave analytic rank equal to one. Additionally, for a sparse set of primes q we show the rankof J ( q ) is asymptotically equal to the rank predicted by the Brumer-Murty conjecture. Contents
1. Introduction 22. Set up and main propositions 32.1. The mollifier 42.2. Main propositions 53. Auxiliary lemmas 64. The first moments - Initial manipulations 115. The first moments - The singular series 126. The second moments - Initial manipulations 167. The second moments - The singular series 198. The second moments - The error term E k c M ODk T − M,N,h ( c ) 3310.1. Shifted convolution sum 3310.2. Evaluating T − M,N,h ( c ) 3611. The off-off-diagonal M OODk - Initial manipulations 3712. Evaluating M OOD k h -sum 4012.4. Moving the contours 4113. Evaluating M OOD k Mathematics Subject Classification.
Keywords and phrases : Landau-Siegel zeros, exceptional characters, automorphic L -functions, analyticranks, ranks of Jacobians, Birch and Swinnerton-Dyer conjecture, nonvanishing, mollfier.
14. Proofs of the main theorems 43References 501.
Introduction
Given a newform f of weight two and prime level q (i.e. f ∈ S ∗ ( q )), we denote itscorresponding L -function by L ( f, s ). We are interested in these L -functions because, amongother reasons, the L -function of the Jacobian of the curve X ( q ) is given by a product over L -functions L ( f, s ) (see [ ]), L ( J ( q ) , s ) = Y f ∈ S ∗ ( q ) L ( f, s ) . (1.1)A generalization of the Birch and Swinnerton-Dyer conjecture then relates the order ofvanishing of L ( f, s ) at s = 1 / X ( q ). Wedefine the analytic rank r f to be the order of vanishing of L ( f, s ) at s = 1 / L -function L ( f, s ) is self-dual, and therefore its root number ε f satisfies ε f ∈ {± } .If the root number is +1 then we say that f or L ( f, s ) is even , and if the root number is − f or L ( f, s ) is odd . Accordingly, we may speak of the parity of f or of L ( f, s ).The parity of f has an effect on r f . Specifically, if f is even then r f is even, and if f isodd then r f is odd. As r f is a nonnegative integer, we observe that if r f is odd then r f = 0.As q → ∞ asymptotically half of the elements of S ∗ ( q ) are even and the other half are odd.Brumer [ ] and Murty [ ] conjectured that almost all even forms f have r f = 0, and thatalmost all odd forms f have r f = 1. Thus, we expect that 50% of the forms have analyticrank zero, 50% of the forms have analytic rank one, and 0% of the forms have analytic ranktwo or more.This conjecture, if true, has important consequences. For example, Iwaniec and Sarnakrelated the proportion of forms with analytic rank zero to the Landau-Siegel zero problem[ ]. In particular, they showed that if one could show strictly more than 50% of the even forms were nonzero and the central values are not too small (this is an important condition),then one could rule out the existence of Landau-Siegel zeros. We expect almost all evenforms to have analytic rank zero and to have central values that are not too small [ ], butthere seems to be a barrier to going past 50%.In some sense, we approach things from another, or converse, direction. Assuming the existence of Landau-Siegel zeros, what can one deduce about vanishing or nonvanishing ofcentral values of automorphic L -functions? Roughly speaking, under such a hypothesis, wecan confirm the “odd part” of the Brumer-Murty conjecture, and we can establish the “evenpart” of the conjecture up to a factor of two. Theorem 1.1.
Let D be large and let ψ be a real, odd, primitive Dirichlet character modulo D . Then one of the following two possibilities must hold: (A) L (1 , ψ ) ≥ (log D ) − , or (B) for every fixed C ≥ and every prime q satisfying D ≤ q ≤ D C we have rank( J ( q )) = (cid:18)
12 + O (cid:18)s log log log q log log q (cid:19)(cid:19) dim( J ( q )) NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 3 if ψ ( q ) = 1 , and (cid:18)
12 + O (cid:16) log log log q log log q (cid:17)(cid:19) dim( J ( q )) ≤ rank( J ( q )) ≤ (cid:18) O (cid:18)s log log log q log log q (cid:19)(cid:19) dim( J ( q )) if ψ ( q ) = − . This theorem surpasses what is known even under the Generalized Riemann Hypothesis[ ]. Kowalski, Michel and VanderKam [ ] showed unconditionally thatrank( J ( q )) ≤ ( c + o (1))dim( J ( q )) , where c = 1 . c < Theorem 1.2.
Let C ≥ be a fixed real number. Let D be large and let ψ be a real, odd,primitive Dirichlet character modulo D . Then for any ε > and any prime q satisfying D ≤ q ≤ D C we have | S ∗ ( q ) | X f ∈ S ∗ ( q ) r f ≤ O ε ( L (1 , ψ )(log q ) ε ) + O ε ( L (1 , ψ ) (log q ) ε ) + O A ((log q ) − A ) if ψ ( q ) = 1 , and | S ∗ ( q ) | X f ∈ S ∗ ( q ) r f ≤ O (cid:18) log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) (cid:19) + O ε ( L (1 , ψ )(log q ) ε )+ O ε ( L (1 , ψ ) (log q ) ε ) if ψ ( q ) = − and L (1 , ψ )(log D ) = o (1) . Remark.
For convenience we work only with weight two forms, but one could prove ananalogue of Theorem 1.2 in which the weight is any fixed, even k ≥ Set up and main propositions
Let S ∗ ( q ) be the set of primitive Hecke eigenforms of weight 2 and level q ( q prime).Throughout we let D be large and let ψ be a real, odd, primitive Dirichlet character mod-ulo D . We think of D as being quite small compared to q . We assume throughout that L (1 , ψ )(log D ) = o (1), where o (1) denotes a quantity that tends to zero as D → ∞ . We let ε denote a sufficiently small positive constant, the size of which might change from one lineto the next.For f ∈ S ∗ ( q ), the Fourier expansion of f at infinity takes the form f ( z ) = X n ≥ √ nλ f ( n ) e ( nz )with λ f (1) = 1. The L -function associated to f , L ( f, s ) = X n ≥ λ f ( n ) n s , HUNG M. BUI, KYLE PRATT AND ALEXANDRU ZAHARESCU satisfies a functional equationΛ (cid:16) f,
12 + s (cid:17) := (cid:16) √ q π (cid:17) s Γ(1 + s ) L (cid:16) f,
12 + s (cid:17) = ǫ f Λ (cid:16) f, − s (cid:17) , (2.1)where ǫ f = q / λ f ( q ) = ± . An eigenform f is said to be even (resp. odd ) if ǫ f = 1 (resp. ǫ f = − L -function.For ψ a primitive character modulo D , the twist of f by ψ ,( f ⊗ ψ )( z ) = X n ≥ √ nψ ( n ) λ f ( n ) e ( nz ) , is a cuspidal modular form of level qD . Given that ( D, q ) = 1, this is a primitive form. Thetwisted L -function is defined by L ( f ⊗ ψ, s ) = X n ≥ ψ ( n ) λ f ( n ) n s . This has a functional equation [ ]Λ (cid:16) f ⊗ ψ,
12 + s (cid:17) := (cid:16) √ qD π (cid:17) s Γ(1 + s ) L (cid:16) f ⊗ ψ,
12 + s (cid:17) = − ψ ( q ) ǫ f Λ (cid:16) f ⊗ ψ, − s (cid:17) . Hence Λ f,ψ (cid:16)
12 + s (cid:17) := Λ (cid:16) f,
12 + s (cid:17) Λ (cid:16) f ⊗ ψ,
12 + s (cid:17) = − ψ ( q )Λ f,ψ (cid:16) − s (cid:17) . (2.2)We are interested in the nonvanishing of Λ ′ f,ψ (1 /
2) and Λ ′′ f,ψ (1 / L -functions bestows certain technical advantages compared tostudying products of derivatives of the L -functions themselves.2.1. The mollifier.
The proof of Theorem 1.1 goes through studying nonvanishing of centralvalues of L -functions. The relevant connection is given by (1.1), which is due to Eichler andShimura [ ]. We study the first and second mollified moments of Λ ′ f,ψ (1 /
2) and Λ ′′ f,ψ (1 / X mn ≤ X α ( m ) µ ( m ) α ( n ) µ ( n ) ψ ( n ) λ f ( m ) λ f ( n ) √ mn with X ≍ q and α ( n ) = Y p | n (cid:18) p (cid:19) − . NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 5 Using the Hecke relation in Lemma 3.1 below this is X mn ≤ X α ( m ) µ ( m ) α ( n ) µ ( n ) ψ ( n ) λ f ( mn ) √ mn X d ≤ √ X/mn ( d,mn )=1 α ( d ) µ ( d ) ψ ( d ) d . By lacunarity, we can extend the sum on d to all ( d, mn ) = 1, so we choose the mollifier tobe M f,ψ = X a ≤ X ρ ( a ) λ f ( a ) √ a , (2.3)where ρ ( a ) = (cid:0) ( αµ ) ⋆ ( αµψ ) (cid:1) ( a ) h ( a )and h ( n ) = h ψ ( n ) = Y p | n (cid:18) α ( p ) ψ ( p ) p (cid:19) − . In particular we have ρ ( a ) ≪ ε (log q ) ε τ ( a ).Applying Lemma 3.1 one more time we get M f,ψ = X d ≤ X d X a ,a ≤ X/d ρ ( da ) ρ ( da ) λ f ( a a ) √ a a = X a ≤ X ρ ( a ) λ f ( a ) √ a , (2.4)where ρ ( a ) = X d ≤ X d X a a = aa ,a ≤ X/d ρ ( da ) ρ ( da ) . (2.5)In particular we have ρ ( a ) ≪ ε (log q ) ε τ ( a ).2.2. Main propositions.
We state here the key propositions we need. For technical con-venience we work with the so-called “harmonic average.” For complex numbers α f we write X hf ∈ S ∗ ( q ) α f = X f ∈ S ∗ ( q ) ω f α f , where ω f = 1 / π h f, f i and h· , ·i is the Petersson inner product on Γ ( q ) \ H . We shall discussin Section 14 how to remove the weights ω f and thereby state results for the “natural average”1 | S ∗ ( q ) | X f ∈ S ∗ ( q ) α f . We note that X hf ∈ S ∗ ( q ) O ( q − / ) . HUNG M. BUI, KYLE PRATT AND ALEXANDRU ZAHARESCU
Proposition 2.1.
Provided that q, X ≥ D we have X hf ∈ S ∗ ( q ) Λ ′ f,ψ (cid:16) (cid:17) M f,ψ = 2 (cid:0) ψ ( q ) (cid:1) S L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O A (cid:0) (log q ) − A (cid:1) + O ε (cid:0) q − / ε DX (cid:1) and X hf ∈ S ∗ ( q ) Λ ′′ f,ψ (cid:16) (cid:17) M f,ψ = 4 (cid:0) − ψ ( q ) (cid:1) S (cid:18) O (cid:16) log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) (cid:17)(cid:19)(cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O ε (cid:0) q − / ε DX (cid:1) , where S = Y p | D (cid:18) p (cid:19) − Y p ≤ Xp ∤ D (cid:18)(cid:16) p (cid:17) + ψ ( p ) p (cid:19) − (cid:18) − ψ ( p ) p (cid:19) . (2.6) Proposition 2.2.
Provided that X ≥ D and D X ≪ q − ε we have X hf ∈ S ∗ ( q ) Λ ′ f,ψ (cid:16) (cid:17) M f,ψ = 4 (cid:0) ψ ( q ) (cid:1) S L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O A ((log q ) − A )+ O ε (cid:0) q − / ε D / X / (cid:1) + O ε (cid:0) q − / ε D / X / (cid:1) , where S = S = Y p | D (cid:18) p (cid:19) − Y p ≤ Xp ∤ D (cid:18)(cid:16) p (cid:17) + ψ ( p ) p (cid:19) − (cid:18) − ψ ( p ) p (cid:19) . (2.7) Proposition 2.3.
Provided that X ≥ D and D X ≪ q − ε we have X hf ∈ S ∗ ( q ) Λ ′′ f,ψ (cid:16) (cid:17) M f,ψ = 16 (cid:0) − ψ ( q ) (cid:1) S (cid:18) O (cid:16) log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) (cid:17)(cid:19)(cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + O ε ( L (1 , ψ )(log q ) ε ) + O ε (cid:0) q − / ε D / X / (cid:1) + O ε (cid:0) q − / ε D / X / (cid:1) . Auxiliary lemmas
The first lemma is the Hecke relation.
Lemma 3.1.
For m, n ≥ , we have λ f ( m ) λ f ( n ) = X d | ( m,n )( d,q )=1 λ f (cid:16) mnd (cid:17) . NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 7 Let S ( m, n ; c ) be the Kloosterman sum, S ( m, n ; c ) = X ∗ u (mod c ) e (cid:16) mu + nuc (cid:17) , and given a character χ we denote by S χ ( m, n ; c ) the hybrid Gauss-Kloosterman sum, S χ ( m, n ; c ) = X ∗ u (mod c ) χ ( u ) e (cid:16) mu + nuc (cid:17) . Lemma 3.2.
For m, n ≥ , we have X hf ∈ S ∗ ( q ) λ f ( m ) λ f ( n ) = m = n − πq X c ≥ S ( m, n ; cq ) c J (cid:16) π √ mncq (cid:17) . As a result, X hf ∈ S ∗ ( q ) λ f ( m ) λ f ( n ) = m = n + O ε (cid:0) q − / ( m, n, q ) / ( mn ) / ε (cid:1) . Proof.
The first statement is a particular case of Petersson’s trace formula. The last estimatefollows easily from the bound J ( x ) ≪ x and the Weil bound on Kloosterman sums. (cid:3) Lemma 3.3.
Let χ be a character modulo d . Let c = c c and m = m m , where c m | d ∞ and ( c m , d ) = 1 . Then S χ ( m, c ) vanishes unless dm = c , and in that case we have S χ ( m, c ) = m τ ( χ ) χ ( c m ) X r | ( c ,m ) µ (cid:16) c r (cid:17) r. Proof.
See [ ; Lemma 2.7]. (cid:3) The next lemma is the approximate functional equation for Λ ( k ) f,ψ (1 / Lemma 3.4.
Let k ∈ N . Let G ( u ) be an even entire function of rapid decay in any fixedstrip | Re( u ) | ≤ C satisfying G (0) = 1 , G (1) = 0 and G ( j ) (0) = 0 for ≤ j ≤ k . Let V k ( x ) = 12 πi Z (1) G ( u )Γ(1 + u ) x − u duu k +1 . (3.1) Then we have Λ ( k ) f,ψ (cid:16) (cid:17) = k ! (cid:0) − k +1 ψ ( q ) (cid:1) X ( d,q )=1 ψ ( d ) d X n (1 ⋆ ψ )( n ) λ f ( n ) √ n V k (cid:16) d nQ (cid:17) , where Q = qD π . Proof. As G (0) = 1 and G ( j ) (0) = 0 for 1 ≤ j ≤ k , by Cauchy’s theorem we have12 πi Z (1) G ( u )Λ f,ψ (cid:16)
12 + u (cid:17) duu k +1 = Λ ( k ) f,ψ (1 / k ! + 12 πi Z ( − G ( u )Λ f,ψ (cid:16)
12 + u (cid:17) duu k +1 . By a change of variables u → − u and using (2.2), we then obtainΛ ( k ) f,ψ (1 / k ! = 1 + ( − k +1 ψ ( q )2 πi Z (1) G ( u )Λ f,ψ (cid:16)
12 + u (cid:17) duu k +1 . HUNG M. BUI, KYLE PRATT AND ALEXANDRU ZAHARESCU
Writing Λ f,ψ in terms of Dirichlet series and then integrating term-by-term givesΛ ( k ) f,ψ (cid:16) (cid:17) = k ! (cid:0) − k +1 ψ ( q ) (cid:1) X m,n ψ ( n ) λ f ( m ) λ f ( n ) √ mn V k (cid:16) mnQ (cid:17) . The lemma now follows by applying Lemma 3.1. (cid:3)
Remarks. • We can move the line of integration in (3.1) to Re( u ) = u for any fixed u > V k ( x ). • For k ≤ G in Lemma 3.4 is G ( u ) = e u ( u − , say, but there is no need to specify G . We have not used the condition G (1) = 0,but this is to cancel out certain poles of some Gamma functions, which appear in theevaluation of the off-off-diagonal terms in Section 12. This substantially simplifiesour later calculations.The next result is fundamental to our work. It is a quantitative statement of “lacunarity”. Lemma 3.5.
For any x ≥ D we have X D Apply [ ; (22.109)] twice and subtract the expressions. This gives X D Lemma 3.6. Let k ∈ N , B > and N ≥ be fixed. Let D ≤ Z ≤ D N and define T = exp((log Z )(log log Z ) ) . Then we have X D A,k (log Z ) k +1 X D ̟ t ! (cid:0) k +3 log log log Z + 2 k +1 log B + o k (1) (cid:1) t ≪ B,k (log Z ) O k (1) X t>̟ (2 k +4 log log log Z ) t t ! ≪ A,B,k (log Z ) − A . We therefore have T ≪ ε,A,B,k L (1 , ψ )(log Z ) k +1 +1+ ε + (log Z ) − A , which completes the proof. (cid:3) In order to bound some logarithmic derivatives we will need estimates for certain sumsover primes. The key point is that lacunarity enables a small amount of savings over thetrivial bound. Lemma 3.7. For any x ≥ D we have X p ≤ xψ ( p )=1 log pp ≪ log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) log D + L (1 , ψ ) / (log x ) / . Proof. We use lacunarity to reduce the range of p to p ≤ D . Since our basic lacunarityestimate Lemma 3.5 does not quite apply in this shorter range, we use a crude “tensorpower” trick. We have (cid:16) X D √ D p + X D 20, say. We split the sum over p ≤ D as X p ≤ Dψ ( p )=1 log pp = X p ≤ D κ ψ ( p )=1 log pp + X D κ
We have L ′ (1 , ψ ) = X n ≤ D (1 ⋆ ψ )( n ) n + O (cid:0) L (1 , ψ )(log D ) (cid:1) and L ′′ (1 , ψ ) = − γL ′ (1 , ψ ) − X n ≤ D (1 ⋆ ψ )( n )(log n ) n + O (cid:0) L (1 , ψ )(log D ) (cid:1) . Proof. The expression for L ′ (1 , ψ ) comes from using [ ; (22.109)] and the lower bound L (1 , ψ ) ≫ D − / . The expression for L ′′ (1 , ψ ) comes from [ ; Exercise 2 on p.527] and asimilar argument. (cid:3) The first moments - Initial manipulations In this section we make some initial manipulations in the first moment calculation. Theerror term analysis is easy, and the bulk of the effort goes into analysing the main term.This is in direct contrast to the second moment, which we discuss later.From Lemma 3.4 we have X hf ∈ S ∗ ( q ) Λ ( k ) f,ψ (cid:16) (cid:17) M f,ψ = k ! (cid:0) − k +1 ψ ( q ) (cid:1) X ( d,q )=1 ψ ( d ) d X a ≤ X X n ρ ( a )(1 ⋆ ψ )( n ) √ an V k (cid:16) d nQ (cid:17) X hf ∈ S ∗ ( q ) λ f ( a ) λ f ( n ) . The condition ( d, q ) = 1 may be removed at a negligible cost due to the decay of the function V k . In view of Lemma 3.2 and (3.1), this is equal to k ! (cid:0) − k +1 ψ ( q ) (cid:1) X d ψ ( d ) d X n ≤ X ρ ( n )(1 ⋆ ψ )( n ) n V k (cid:16) d nQ (cid:17) + O ε (cid:0) q − / ε DX (cid:1) = k ! (cid:0) − k +1 ψ ( q ) (cid:1) πi Z (1) G ( u )Γ(1 + u ) Q u L (1 + 2 u, ψ ) S ( u ) duu k +1 + O ε (cid:0) q − / ε DX (cid:1) , where S ( u ) = X n ≤ X ρ ( n )(1 ⋆ ψ )( n ) n u . (4.1)Note that S ( u ) ≪ ε X − Re( u )+ ε for Re( u ) ≪ / log q and trivially we have S ( j )1 (0) ≪ ε,j (log q ) j +4+ ε for any j ≥ 0. We move the line of integration to Re( u ) = − / ε , crossing apole of order ( k + 1) at u = 0, and the new integral is ≪ ε q ε (cid:16) QX (cid:17) − / D / ≪ ε q − / ε X / . The contribution of the residue is2 (cid:0) ψ ( q ) (cid:1) S (0) L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) if k = 1, and is 4 (cid:0) − ψ ( q ) (cid:1) S (0) (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + 4 (cid:0) − ψ ( q ) (cid:1) S ′ (0) L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) if k = 2, since Γ ′ (1) = − γ . Thus we are left to study the singular series S (0) and its firstderivative. 5. The first moments - The singular series In this section we finish the main term analysis of the first moment. It is helpful to recallthat ρ ( a ) = (cid:0) ( αµ ) ⋆ ( αµψ ) (cid:1) ( a ) h ( a ) , where h ( n ) = h ψ ( n ) = Y p | n (cid:18) α ( p ) ψ ( p ) p (cid:19) − and ρ ( a ) ≪ ε (log q ) ε τ ( a ). The following two lemmas and our results in Section 4 will proveProposition 2.1. Lemma 5.1. Provided that X ≥ D we have S (0) = S + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O A (cid:0) (log q ) − A (cid:1) , where S is given by (2.6) .Proof. Our basic idea is to use lacunarity to show that the sum S (0) = X n ≤ X ρ ( n )(1 ⋆ ψ )( n ) n is, up to an acceptable error, equal to the Euler product P := Y p ≤ X (cid:18) X n ≥ ρ ( p n )(1 ⋆ ψ )( p n ) p n (cid:19) . We actually start with P and show how to trim it down to get S (0).Observe that P = X p | n ⇒ p ≤ X ρ ( n )(1 ⋆ ψ )( n ) n . We first use a crude estimate to truncate the sum. Define T = exp((log X )(log log X ) ) and δ = 1 / log X . The contribution to P from n > T is ≤ X p | n ⇒ p ≤ Xn>T | ρ ( n ) | τ ( n ) n ≤ T − δ X p | n ⇒ p ≤ X | ρ ( n ) | τ ( n ) n − δ = T − δ Y p ≤ X (cid:18) | ρ ( p ) | p δ p + 3 | ρ ( p ) | p δ p (cid:19) ≪ T − δ Y p ≤ X (cid:18) α ( p ) h ( p ) p δ p (cid:19) ≪ T − δ Y p ≤ X (cid:18) p δ p (cid:19) ≤ T − δ Y p ≤ X (cid:18) p (cid:19) ≪ (log X ) T − δ , where the penultimate inequality follows since p δ ≤ e < 3. Since T δ = exp((log log X ) ) ≫ A (log X ) A , NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 13 we see that the contribution from n > T contributes an error of size O A ((log q ) − A ). We havetherefore deduced that P = X n ≤ Tp | n ⇒ p ≤ X ρ ( n )(1 ⋆ ψ )( n ) n + O A ((log q ) − A ) . We now use the lacunarity of (1 ⋆ ψ ) to replace the condition n ≤ T by n ≤ X . For n ≤ T we have h ( n ) ≪ (log q ) ε , and therefore the contribution of n > X is ≪ ε (log q ) ε X X 1, then X n ≥ ρ ( p n )(1 ⋆ ψ )( p n ) p n = 1 + ρ ( p ) p = 1 − α ( p ) h ( p ) p = p + pp + p + 1= (cid:18)(cid:16) p (cid:17) + ψ ( p ) p (cid:19) − (cid:18) − ψ ( p ) p (cid:19) . Lastly, if ψ ( p ) = 1, then X n ≥ ρ ( p n )(1 ⋆ ψ )( p n ) p n = 1 − α ( p ) h ( p ) p + 3 α ( p ) h ( p ) p = p − pp + 3 p + 1= (cid:18)(cid:16) p (cid:17) + ψ ( p ) p (cid:19) − (cid:18) − ψ ( p ) p (cid:19) . (cid:3) Lemma 5.2. With S given by (4.1) and X ≥ D we have S ′ (0) ≪ ε,A S log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) log D + L (1 , ψ )(log q ) ε . Proof. By Cauchy’s integral formula, we have S ′ (0) = 12 πi I S ( u ) duu , where u runs over the circle centered at the origin of radius (log q ) − − ε and ε > S ( u ) = Y p ≤ X (cid:18) X n ≥ ρ ( p n )(1 ⋆ ψ )( p n ) p n (1+ u ) (cid:19) + O ε ( L (1 , ψ )(log q ) ε ) + O A ((log q ) − A )= P ( u ) + O ε ( L (1 , ψ )(log q ) ε ) + O A ((log q ) − A ) , say. The choice of | u | ensures that T | u | = 1 + o (1), where T is as in Lemma 5.1. Hence S ′ (0) = 12 πi I P ( u ) duu + O ε ( L (1 , ψ )(log q ) ε + ε ) + O A ((log q ) − A ) . Noting that ε can be arbitrarily small, this gives some of the error terms in the statementof the lemma.We have 12 πi I P ( u ) duu = P ′ (0) = Q (0) P (0) , say, where Q ( u ) = P ′ ( u ) P ( u ) = ddu log P ( u )is a sum over primes p ≤ X . By simple computations analogous to those in the proof ofLemma 5.1 we deduce that P ( u ) = Y p | D (cid:18) − α ( p ) p u (cid:19) Y p ≤ Xψ ( p )= − (cid:18) − α ( p ) h ( p ) p u ) (cid:19) Y p ≤ Xψ ( p )=1 (cid:18) − α ( p ) h ( p ) p u + 3 α ( p ) h ( p ) p u ) (cid:19) , and by logarithmic differentiation we derive Q (0) = X p | D α ( p )(log p ) p − α ( p ) + X p ≤ Xψ ( p )= − α ( p ) h ( p )(log p ) p − α ( p ) h ( p )+ X p ≤ Xψ ( p )=1 α ( p ) h ( p ) p (log p ) − α ( p ) h ( p )(log p ) p − α ( p ) h ( p ) p + 3 α ( p ) h ( p ) . We observe that the denominators in all the fractions here are uniformly bounded away fromzero. Since X p | D log pp ≪ log log D and X p log pp ≪ NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 15 we see that Q (0) ≪ log log D + X p ≤ Xψ ( p )=1 log pp . An appeal to Lemma 3.7 finishes the proof, once we notice thatlog log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) log D ≫ log log D by the lower bound L (1 , ψ ) ≫ D − / , and S ≤ (cid:3) Proof of Proposition 2.1. By the work of Section 4, we have X hf ∈ S ∗ ( q ) Λ ′ f,ψ (cid:16) (cid:17) M f,ψ = 2 (cid:0) ψ ( q ) (cid:1) S (0) L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O ε (cid:0) q − / ε DX (cid:1) . Lemma 5.1 and the trivial bound L ′ (1 , ψ ) ≪ (log D ) together yield X hf ∈ S ∗ ( q ) Λ ′ f,ψ (cid:16) (cid:17) M f,ψ = 2 (cid:0) ψ ( q ) (cid:1) S L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O A (cid:0) (log q ) − A (cid:1) + O ε (cid:0) q − / ε DX (cid:1) , which is the first part of the proposition.We also have X hf ∈ S ∗ ( q ) Λ ′′ f,ψ (cid:16) (cid:17) M f,ψ = 4 (cid:0) − ψ ( q ) (cid:1) S (0) (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + 4 (cid:0) − ψ ( q ) (cid:1) S ′ (0) L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O ε (cid:0) q − / ε DX (cid:1) . We utilize the trivial bounds L ′ (1 , ψ ) ≪ (log D ) , L ′′ (1 , ψ ) ≪ (log D ) and Lemmas 5.1 and5.2 to derive X hf ∈ S ∗ ( q ) Λ ′′ f,ψ (cid:16) (cid:17) M f,ψ = 4 (cid:0) − ψ ( q ) (cid:1) S (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + O (cid:18) S | L ′ (1 , ψ ) | log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) log D (cid:19) + O ε ( L (1 , ψ )(log q ) ε ) (5.1)+ O ε (cid:0) q − / ε DX (cid:1) . We can simplify things nicely if we know something about the size of L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) . We might expect that L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) ≫ (log Q ) L ′ (1 , ψ ) , at least if L ′ (1 , ψ ) is not too small. We can show that this does indeed occur using lacunarity.Applying Lemma 3.8 we obtain L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) = (log Q − γ ) X n ≤ D (1 ⋆ ψ )( n ) n − X n ≤ D (1 ⋆ ψ )( n )(log n ) n + O ( L (1 , ψ )(log q ) ) . We also note that(log Q − γ ) X n ≤ D (1 ⋆ ψ )( n ) n − X n ≤ D (1 ⋆ ψ )( n )(log n ) n ≥ (log Q − D − γ ) X n ≤ D (1 ⋆ ψ )( n ) n ≥ log Q X n ≤ D (1 ⋆ ψ )( n ) n , the last inequality holding provided q ≥ D , say, since Q = qD/ π . The sum over n here is ≥ ⋆ ψ )( n ) is nonnegative and equal to 1 at n = 1.Since S ≤ 1, we derive X hf ∈ S ∗ ( q ) Λ ′′ f,ψ (cid:16) (cid:17) M f,ψ = 4 (cid:0) − ψ ( q ) (cid:1) S (cid:18) O (cid:16) log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) (cid:17)(cid:19)(cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O ε (cid:0) q − / ε DX (cid:1) . Note that the second term in the second line of (5.1) has been dropped using Cauchy-Schwarz’s inequality. (cid:3) The second moments - Initial manipulations The second moment analysis is much more complicated than the first moment analysis.A distinction in the main term analysis arises between k = 1 and k = 2, necessitating someadditional arguments. Furthermore, the error term analysis is now highly non-trivial, andspills over several sections.As before, we can remove the condition ( d, q ) = 1 in the expression for Λ ( k ) f,ψ (1 / 2) in Lemma3.4 at a negligible cost due to the decay of the function V k . We use (2.4) and apply Lemma3.1 to getΛ ( k ) f,ψ (cid:16) (cid:17) M f,ψ = k ! (cid:0) − k +1 ψ ( q ) (cid:1) X d,b ψ ( d ) db X a ≤ X /b X n ρ ( ab )(1 ⋆ ψ )( bn ) λ f ( an ) √ an V k (cid:16) d bnQ (cid:17) . Making use of the recursion formula(1 ⋆ ψ )( bn ) = X g | ( b,n ) µ ( g )(1 ⋆ ψ ) (cid:16) bg (cid:17) (1 ⋆ ψ ) (cid:16) ng (cid:17) (6.1)we obtain thatΛ ( k ) f,ψ (cid:16) (cid:17) M f,ψ = k ! (cid:0) − k +1 ψ ( q ) (cid:1)X d,b,g ψ ( d ) µ ( g )(1 ⋆ ψ )( b ) dbg / X a ≤ X /bg X n ρ ( abg )(1 ⋆ ψ )( n ) λ f ( agn ) √ an V k (cid:16) d bg nQ (cid:17) . NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 17 Lemma 3.2 then yields X hf ∈ S ∗ ( q ) Λ ( k ) f,ψ (cid:16) (cid:17) M f,ψ = M Dk + O (cid:0) | E k | (cid:1) , where M Dk = ( k !) (cid:0) − k +1 ψ ( q ) (cid:1) X d ,dabg ≤ X ψ ( d ) ψ ( d ) µ ( g )(1 ⋆ ψ )( b ) d dbg / X m = agn ρ ( abg )(1 ⋆ ψ )( m )(1 ⋆ ψ )( n ) √ amn V k (cid:16) d mQ (cid:17) V k (cid:16) d bg nQ (cid:17) and E k = q − X d ,dabg ≤ X ψ ( d ) ψ ( d ) µ ( g )(1 ⋆ ψ )( b ) ρ ( abg ) d d √ abg / X c ≥ T ( c ) c , (6.2)with T ( c ) = cq X m,n (1 ⋆ ψ )( m )(1 ⋆ ψ )( n ) √ mn S ( m, agn ; cq ) J (cid:16) π √ agmncq (cid:17) V k (cid:16) d mQ (cid:17) V k (cid:16) d bg nQ (cid:17) . (6.3)Proposition 2.2 and Proposition 2.3 are immediate consequences of the following two results. Proposition 6.1. Provided that q, X ≥ D we have M D = 4 (cid:0) ψ ( q ) (cid:1) S L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O A ((log q ) − A )+ O ε ( q − / ε D − / X / ) and M D = 16 (cid:0) − ψ ( q ) (cid:1) S (cid:18) O (cid:16) log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) (cid:17)(cid:19) × (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + O ε ( L (1 , ψ )(log q ) ε ) + O ε ( q − / ε D − / X / ) . Proposition 6.2. Provided that D X ≪ q − ε we have E k ≪ ε L (1 , ψ )(log q ) k +24+ ε + q − / ε D / X / + q − / ε D / X / . We shall estimate the error term E k in Sections 8–13. We finish this section with theinitial evaluation of the diagonal contribution M Dk . Inverting (6.1) and using (3.1) we get M Dk = ( k !) (cid:0) − k +1 ψ ( q ) (cid:1) X d ,dab ≤ X ψ ( d ) ψ ( d ) ρ ( ab ) d dab X n (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n V k (cid:16) d anQ (cid:17) V k (cid:16) d bnQ (cid:17) = ( k !) (cid:0) − k +1 ψ ( q ) (cid:1) πi ) Z (1) Z (1) G ( u ) G ( v )Γ(1 + u ) Γ(1 + v ) Q u + v L (1 + 2 u, ψ ) L (1 + 2 v, ψ ) W ( u, v ) dudvu k +1 v k +1 , where W ( u, v ) = X ab ≤ X ρ ( ab ) a u b v X n ≥ (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n u + v . We move the lines of integration to Re( u ) = Re( v ) = δ = 1 / log X without encounteringany poles. To proceed further we truncate the sum over n in W ( u, v ) to be finite. Setting T = exp((log X )(log log X ) ), we find by trivial estimation that W ( u, v ) = X ab ≤ X ρ ( ab ) a u b v X n ≤ T (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n u + v + O A ((log q ) − A ) . Showing that the contribution to W ( u, v ) from X < n ≤ T is negligible is more challengingand requires an in-depth analysis. We do not provide the details here, but all the necessarytools can be found in the proof of Lemma 7.1 below (see also the remark after Lemma 7.1).Naturally, using the lacunarity of (1 ⋆ ψ ) is the main idea. We content ourselves here withstating that W ( u, v ) = S ( u, v ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O A ((log q ) − A ) , (6.4)where S ( u, v ) = X ab ≤ X ρ ( ab ) a u b v X n ≤ X (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n u + v . (6.5)The contribution from the error terms in (6.4) to M Dk is ≪ ε,A L (1 , ψ )(log q ) k + ε + (log q ) − A . Observe that we have the bound S ( u, v ) ≪ ε X − u + v )+ ε for Re( u ) , Re( v ) ≪ / log q . Wealso have the bounds ∂ i + j ∂u i ∂v j S ( u, v ) ≪ ε,i,j (log q ) i + j + ε for Re( u ) , Re( v ) ≥ i, j ≥ 0. It hence suffices to study( k !) (cid:0) − k +1 ψ ( q ) (cid:1) πi ) Z ( δ ) Z ( δ ) G ( u ) G ( v )Γ(1 + u ) Γ(1 + v ) Q u + v × L (1 + 2 u, ψ ) L (1 + 2 v, ψ ) S ( u, v ) dudvu k +1 v k +1 . We move the v -contour to Re( v ) = − / ε , crossing a pole of order ( k + 1) at v = 0with residue R = 2 (cid:0) ψ ( q ) (cid:1) L ′ (1 , ψ )2 πi Z ( δ ) G ( u )Γ(1 + u ) Q u L (1 + 2 u, ψ ) S ( u, duu + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) if k = 1, and R = 8 (cid:0) − ψ ( q ) (cid:1) πi Z ( δ ) G ( u )Γ(1 + u ) Q u L (1 + 2 u, ψ ) S ( u, NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 19 (cid:18) L ′′ (1 , ψ ) + (cid:16) log Q − γ + ∂ v S S ( u, (cid:17) L ′ (1 , ψ ) (cid:19) duu + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) if k = 2. The contribution of the new integral is bounded trivially by ≪ ε Q − / ε X / ≪ ε q − / ε D − / X / . Regarding R and R , we move the line of integration to Re( u ) = − / ε , crossing apole at u = 0 with residue4 (cid:0) ψ ( q ) (cid:1) S (0 , L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) if k = 1, and16(1 − ψ ( q )) S (0 , (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + 32 (cid:0) − ψ ( q ) (cid:1) ∂ u S (0 , L ′ (1 , ψ ) (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + 16 (cid:0) − ψ ( q ) (cid:1) ∂ uv S (0 , L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) . if k = 2, as S is symmetric in u and v . The final integrals, like before, are bounded triviallyby O ε ( q − / ε D − / X / ). We therefore obtain M D = 4 (cid:0) ψ ( q ) (cid:1) S (0 , L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O A ((log q ) − A )+ O ε ( q − / ε D − / X / ) (6.6)and M D = 16 (cid:0) − ψ ( q ) (cid:1) S (0 , (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + 32 (cid:0) − ψ ( q ) (cid:1) ∂ u S (0 , L ′ (1 , ψ ) (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + 16 (cid:0) − ψ ( q ) (cid:1) ∂ uv S (0 , L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O A ((log q ) − A ) + O ε ( q − / ε D − / X / ) . (6.7)7. The second moments - The singular series In this section we prove Proposition 6.1. In view of (6.6) and (6.7), we need to study S and some of its derivatives.Recall from (2.5) that ρ ( a ) = X d ≤ X d X a a = aa ,a ≤ X/d ρ ( da ) ρ ( da )and ρ ( a ) ≪ ε (log q ) ε τ ( a ). In this section we shall prove the following two lemmas. Lemma 7.1. Provided that X ≥ D we have S (0 , 0) = S + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O A (cid:0) (log q ) − A (cid:1) , where S is given by (2.7) . Proof. We wish to deduce an asymptotic formula for the sum S (0 , 0) = X ab ≤ X ab X d ≤ X d X m m = abdm i ≤ X ρ ( dm ) ρ ( dm ) X n ≤ X (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n . Here we have written m i instead of n i to reduce visual similarity between the variables. Wewish to compare S (0 , 0) with the Euler product P := Y p ≤ X X a,b,d,m ,m ,n ≥ m + m = a + b ρ ( p d + m ) ρ ( p d + m )(1 ⋆ ψ )( p a + n )(1 ⋆ ψ )( p b + n ) p a + b + d + n = X p | ab ⇒ p ≤ X ab X p | d ⇒ p ≤ X d X m m = ab ρ ( dm ) ρ ( dm ) X p | n ⇒ p ≤ X (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n . Our strategy is the same as that in Lemma 5.1. We first use smooth number estimates totruncate the variables a, b, d, n . We then use lacunarity to reduce the size of the variablesmuch further and show that P is equal to S (0 , P .We begin with the smooth number estimates. Let T = exp((log X )(log log X ) ). We wishto show that the contribution to P from any variable larger than T is negligible. Let usconsider the contribution from n > T , say. Set δ = 1 / log X . Then by the triangle inequalitythe contribution to P from n > T is ≪ ε (log q ) ε X p | ab ⇒ p ≤ X τ ( a ) τ ( b ) ab X m m = ab τ ( m ) τ ( m ) X p | d ⇒ p ≤ X τ ( d ) d X n>Tp | n ⇒ p ≤ X τ ( n ) n ≪ ε (log q ) ε X p | ab ⇒ p ≤ X τ ( a ) τ ( b ) ab X m m = ab τ ( m ) τ ( m ) X n>Tp | n ⇒ p ≤ X τ ( n ) n ≪ ε (log q ) ε X n>Tp | n ⇒ p ≤ X τ ( n ) n ≪ ε T − δ (log q ) ε X p | n ⇒ p ≤ X τ ( n ) n − δ ≪ ε T − δ (log q ) ε Y p ≤ X (cid:18) p δ p (cid:19) ≪ ε T − δ (log q ) ε Y p ≤ X (cid:18) p (cid:19) ≪ ε (log q ) ε T − δ , where the penultimate inequality follows since p δ ≤ e and 4 e < 11. By the definition of T and δ we have T δ = exp((log log X ) ) ≫ A (log q ) A , so the contribution from n > T to P is O A ((log q ) − A ). By similar arguments we show thatthe cost of truncating to a, b or d ≤ T is O A ((log q ) − A ). Hence P = X a,b ≤ Tp | ab ⇒ p ≤ X ab X d ≤ Tp | d ⇒ p ≤ X d X m m = ab ρ ( dm ) ρ ( dm ) X n ≤ Tp | n ⇒ p ≤ X (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n + O A ((log q ) − A ) . NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 21 Now we use lacunarity to cut down the size of the variables much further. These argumentsare rather more complicated than those we have heretofore seen, since the variables are quiteentangled with each other.We begin with the n variable. We wish to show that, by lacunarity, the contribution to P from X < n ≤ T is negligible; call this contribution C n . By the triangle inequality C n ≪ ε (log q ) ε X p | ab ⇒ p ≤ X ab X m m = ab τ ( m ) τ ( m ) X p | d ⇒ p ≤ X τ ( d ) d X X 4. Observe that n , n | a ∞ and n | b ∞ . Therefore, by the symmetry between a and b we may factor n = n a m , say, where n a > X / , whence C n ≪ ε (log q ) ε X p | ab ⇒ p ≤ X τ ( a ) τ ( b ) ab X m m = ab τ ( m ) τ ( m ) X n a >X / n a | a ∞ τ ( n a ) n a X p | m ⇒ p ≤ X τ ( m ) m ≪ ε q ε X p | ab ⇒ p ≤ X ab X n a >X / n a | a ∞ τ ( n a ) n a ≪ ε X − / ε X p | ab ⇒ p ≤ X ab X n a | a ∞ τ ( n a ) √ n a ≪ ε X − / ε X p | ab ⇒ p ≤ X τ ( a ) ab ≪ ε X − / ε . The above arguments thus yield P = X a,b ≤ Tp | ab ⇒ p ≤ X ab X d ≤ Tp | d ⇒ p ≤ X d X m m = ab ρ ( dm ) ρ ( dm ) X n ≤ X (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n + O ε ( L (1 , ψ )(log q ) ε ) + O A ((log q ) − A ) . We next wish to impose the conditions dm i ≤ X . Note that if dm ≤ X and dm ≤ X then ab ≤ X . By symmetry it suffices to consider the contribution C d from dm > X . Bythe triangle inequality, C d ≪ ε (log q ) ε X a,b ≤ Tp | ab ⇒ p ≤ X τ ( a ) τ ( b ) τ ( ab ) ab X d ≤ Tp | d ⇒ p ≤ X τ ( d ) d X m | abdm >X | ρ ( dm ) | X n ≤ X τ ( n ) n ≪ ε (log q ) ε X a,b ≤ Tp | ab ⇒ p ≤ X τ ( a ) τ ( b ) τ ( ab ) ab X d ≤ Tp | d ⇒ p ≤ X τ ( d ) d X m | abdm >X | ρ ( dm ) | . NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 23 We change the order of summation to put the summations over a and b as the innermostsums, and then change variables to obtain C d ≪ ε (log q ) ε X d ≤ T,m ≤ T dm >Xp | dm ⇒ p ≤ X τ ( d ) | ρ ( dm ) | d X p | t ⇒ p ≤ Xm | t τ ( t ) t X ab = t τ ( a ) τ ( b ) ≪ ε (log q ) ε X d ≤ T,m ≤ T dm >Xp | dm ⇒ p ≤ X τ ( d ) | ρ ( dm ) | d X p | t ⇒ p ≤ Xm | t τ ( t ) τ ( t ) t ≪ ε (log q ) ε X d ≤ T,m ≤ T dm >Xp | dm ⇒ p ≤ X τ ( d ) τ ( m ) τ ( m ) | ρ ( dm ) | dm . We change variables again, writing ℓ = dm , to derive C d ≪ ε (log q ) ε X X<ℓ ≤ T p | ℓ ⇒ p ≤ X | ρ ( ℓ ) | ℓ X dm = ℓ τ ( d ) τ ( m ) τ ( m ) = (log q ) ε X X<ℓ ≤ T p | ℓ ⇒ p ≤ X | ρ ( ℓ ) | f ( ℓ ) ℓ , where f ( ℓ ) is a multiplicative function supported on cube-free integers with f ( p ) = 10 and f ( p ) = 49 (recall that ρ is supported on cube-free integers). To handle the factor of ρ ( ℓ )we factor ℓ = ℓ ℓ , where ℓ and ℓ are squarefree and coprime to each other. We must have ℓ > X / or ℓ > X / . The contribution from ℓ > X / is easily seen to be O ( q − ε ), so wefocus on the case when ℓ > X / . In this case we find that C d ≪ ε (log q ) ε X X / <ℓ ≤ T p | ℓ ⇒ p ≤ X µ ( ℓ )(1 ⋆ ψ )( ℓ ) τ ( ℓ ) log 10 / log 2 ℓ + q − ε ≪ ε,A L (1 , ψ )(log q ) ε + (log q ) − A , by Lemma 3.6. Therefore P = X ab ≤ X ab X d ≤ X d X m m = abdm i ≤ X ρ ( dm ) ρ ( dm ) X n ≤ X (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n + O ε ( L (1 , ψ )(log q ) ε ) + O A ((log q ) − A ) , or in other words, we have S (0 , 0) equals P up to an acceptable error.The local factor for each p ≤ X in P is X a,b,d,m ,m ,n ≥ m + m = a + b ρ ( p d + m ) ρ ( p d + m )(1 ⋆ ψ )( p a + n )(1 ⋆ ψ )( p b + n ) p a + b + d + n . There are three different cases to consider, depending on the value of ψ ( p ). When ψ ( p ) = 0 the local factor is X a,b,d,m ,m ,n ≥ m + m = a + bd + m i ≤ α ( p d + m ) µ ( p d + m ) α ( p d + m ) µ ( p d + m ) p a + b + d + n = (cid:18) − p (cid:19) − X d,m ,m ≥ d + m i ≤ ( − m + m α ( p d + m ) α ( p d + m ) p d + m + m X a,b ≥ a + b = m + m (cid:18) − p (cid:19) − X d,m ,m ≥ d + m i ≤ ( − m + m α ( p d + m ) α ( p d + m )( m + m + 1) p d + m + m = (cid:18) − p (cid:19) − (cid:16) − α ( p ) p + 3 α ( p ) p + α ( p ) p (cid:17) = (cid:18) p (cid:19) − . When ψ ( p ) = − d , and then the values of m and m , to see that the local factor is X a,b,d,m ,m ,n ≥ m + m = a + bd + m i ∈{ , } a + n,b + n even h ( p d + m ) α ( p d + m ) ψ ( p ( d + m ) / ) h ( p d + m ) α ( p d + m ) ψ ( p ( d + m ) / ) p a + b + d + n = (cid:18) − p (cid:19) − − h ( p ) α ( p ) p X n ≥ − n p n + h ( p ) α ( p ) p X n ≥ − n p n + h ( p ) α ( p ) p X n ≥ − n p n + h ( p ) α ( p ) p (cid:18) − p (cid:19) − = (cid:18) − p (cid:19) − − h ( p ) α ( p ) p p (2 p + 1)( p − p + 1) + h ( p ) α ( p ) p p (3 p + 2)( p − p + 1)+ h ( p ) α ( p ) p p (2 p + 1)( p − p + 1) + h ( p ) α ( p ) p (cid:18) − p (cid:19) − = p ( p + 1) ( p + p + 1) = (cid:18) p (cid:19) + ψ ( p ) p ! − (cid:18) − ψ ( p ) p (cid:19) . Lastly, we consider the local factor when ψ ( p ) = 1. The analysis here is just slightly morecomplicated because ρ ( p ) and ρ ( p ) are both nonzero. Here it will be helpful for us to notethat for a positive integer M we have X a,b ≥ a + b = M ( a + n + 1)( b + n + 1) = M ( M − M + 1)6 + ( n + 1) M ( M + 1) + ( n + 1) ( M + 1) . NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 25 The terms corresponding to d = 0 are (breaking up m and m according to whether theyare equal or distinct and using symmetries) X a,b,m ,m ,n ≥ m + m = a + bm i ≤ ρ ( p m ) ρ ( p m )( a + n + 1)( b + n + 1) p m + m + n = X n ≥ ( n + 1) p n + 4 h ( p ) α ( p ) p X n ≥ n + 1) + 6( n + 1) + 1 p n + h ( p ) α ( p ) p X n ≥ n + 1) + 20( n + 1) + 10 p n − h ( p ) α ( p ) p X n ≥ n + 1) + 2( n + 1) p n + 2 h ( p ) α ( p ) p X n ≥ n + 1) + 6( n + 1) + 1 p n − h ( p ) α ( p ) p X n ≥ n + 1) + 12( n + 1) + 4 p n . The terms corresponding to d = 1 are1 p X a,b,m ,m ,n ≥ m + m = a + bm i ≤ ρ ( p m +1 ) ρ ( p m +1 )( a + n + 1)( b + n + 1) p m + m + n = 4 h ( p ) α ( p ) p X n ≥ ( n + 1) p n + h ( p ) α ( p ) p X n ≥ n + 1) + 6( n + 1) + 1 p n − h ( p ) α ( p ) p X n ≥ n + 1) + 2( n + 1) p n and the term coming from d = 2 is h ( p ) α ( p ) p X n ≥ ( n + 1) p n . When we combine the terms from d = 0 , d = 1 and d = 2, we find that their sum is equal to p ( p − ( p + 3 p + 1) = (cid:18) p (cid:19) + ψ ( p ) p ! − (cid:18) − ψ ( p ) p (cid:19) , which completes the proof of the lemma. (cid:3) Remark. In order to obtain (6.4) one uses essentially the same argument that we used tobound C n in Lemma 7.1. Lemma 7.2. With S being given by (6.5) and X ≥ D we have ∂ u S (0 , ≪ ε,A S log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) log D + L (1 , ψ )(log q ) ε and ∂ uv S (0 , ≪ ε,A S log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) (log D ) + L (1 , ψ )(log q ) ε . Proof. The proof is similar to the proof of Lemma 5.2, but more tedious. We give somedetails, but leave the patient reader to work out the full argument.We prove the second statement, since the first is similar and easier. Let ε > ∂ uv S (0 , 0) = 1(2 πi ) I I S ( u, v ) dudvu v , where u and v run over the circle centered at the origin of radius (log q ) − − ε . A slightadjustment of the proof of Lemma 7.1 shows that S ( u, v ) = P ( u, v ) + O ε ( L (1 , ψ )(log q ) ε ) + O A ((log q ) − A ) , where P ( u, v ) = Y p ≤ X X a,b,d,m ,m ,n ≥ m + m = a + b ρ ( p d + m ) ρ ( p d + m )(1 ⋆ ψ )( p a + n )(1 ⋆ ψ )( p b + n ) p a (1+ u )+ b (1+ v )+ d + n (1+ u + v ) . By integration we have12 πi I P ( u, v ) dvv = ∂∂v P ( u, v ) (cid:12)(cid:12)(cid:12) v =0 = Q ( u ) P ( u, , say, where Q ( u ) = 1 P ( u, ∂∂v P ( u, v ) (cid:12)(cid:12)(cid:12) v =0 is a sum over primes p ≤ X . As P ( u, v ) = P ( v, u ) we then deduce that1(2 πi ) I I P ( u, v ) dudvu v = (cid:0) Q ′ (0) + Q (0) (cid:1) P (0 , 0) = (cid:0) Q ′ (0) + Q (0) (cid:1) S . It then suffices to bound Q (0) and Q ′ (0).To bound these sums over primes we again discriminate according to the value of ψ ( p ).Since X p | D (log p ) O (1) p ≪ (log log D ) O (1) and X p (log p ) O (1) p ≪ , it suffices to bound X p ≤ Xψ ( p )=1 (log p ) p and X p ≤ Xψ ( p )=1 (log p ) p ≪ (log q ) X p ≤ Xψ ( p )=1 (log p ) p . Now apply Lemma 3.7 and the lemma follows. (cid:3) NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 27 Proof of Proposition 6.1. We derive the expression for M D immediately from (6.6), the triv-ial bound L ′ (1 , ψ ) ≪ (log D ) and Lemma 7.1.Recall from (6.7) that M D = 16 (cid:0) − ψ ( q ) (cid:1) S (0 , (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + 32 (cid:0) − ψ ( q ) (cid:1) ∂ u S (0 , L ′ (1 , ψ ) (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + 16 (cid:0) − ψ ( q ) (cid:1) ∂ uv S (0 , L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O A ((log q ) − A ) + O ε ( q − / ε D − / X / ) . By Lemma 7.1 and the trivial bounds L ′ (1 , ψ ) ≪ (log D ) , L ′′ (1 , ψ ) ≪ (log D ) , the firstterm is equal to16 (cid:0) − ψ ( q ) (cid:1) S (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + O ε ( L (1 , ψ )(log q ) ε ) + O A ((log q ) − A ) . We use Lemma 7.2 to obtain that the second term is ≪ ε,A S | L ′ (1 , ψ ) | (cid:12)(cid:12)(cid:12) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:12)(cid:12)(cid:12) log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) log D + L (1 , ψ )(log q ) ε . We similarly see that the third term is ≪ ε,A S L ′ (1 , ψ ) log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) (log D ) + L (1 , ψ )(log q ) ε . Using Lemma 3.8 and the trivial bound S ≤ M D = 16 (cid:0) − ψ ( q ) (cid:1) S (cid:18) O (cid:16) log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) (cid:17)(cid:19) × (cid:16) L ′′ (1 , ψ ) + (log Q − γ ) L ′ (1 , ψ ) (cid:17) + O ε ( L (1 , ψ )(log q ) ε ) + O ε ( q − / ε D − / X / ) , as claimed. (cid:3) The second moments - The error term E k In Sections 8–13 we shall prove Proposition 6.2.8.1. Smooth partition of unity. We apply a dyadic partition of unity to the sum over m, n in T ( c ) in (6.3). Let ω be a smooth non-negative function supported in [1 , 2] such that X M ω (cid:16) xM (cid:17) = 1 , where M runs over a sequence of real numbers with { M : M ≤ Z } ≪ log Z . Then T ( c ) = X M,N T M,N ( c ) , where T M,N ( c ) = cq √ M N X m,n (1 ⋆ ψ )( m )(1 ⋆ ψ )( n ) S ( m, agn ; cq ) (8.1) J (cid:16) π √ agmncq (cid:17) V k (cid:16) d mQ (cid:17) V k (cid:16) d bg nQ (cid:17) ω (cid:16) mM (cid:17) ω (cid:16) nN (cid:17) and ω ( x ) = x − / ω ( x ) . Due to the decay of the function V k , we can restrict M, N to M, N ≪ Q ε with a negligible error term.8.2. Removing large values of c . It is convenient to remove large values of c . We shalldo this by appealing to Proposition 1 in [ ]. Lemma 8.1. For any C ≥ q − √ M N X we have X c ≥ C T M,N ( c ) c ≪ ε ( Cq ) ε q / D ( ag ) / (cid:16) √ M NC (cid:17) / . Proof. Consider the summation over a dyadic interval, X C ≤ c ≤ C T M,N ( c ) c = q √ M N X C ≤ c ≤ C X m,n (1 ⋆ ψ )( m )(1 ⋆ ψ )( n ) S ( m, agn ; cq ) cJ (cid:16) π √ agmncq (cid:17) V k (cid:16) d mQ (cid:17) V k (cid:16) d bg nQ (cid:17) ω (cid:16) mM (cid:17) ω (cid:16) nN (cid:17) . By [ ; Proposition 1] this is bounded by ≪ ε ( Cq ) ε q √ M N (cid:16) √ agM NCq (cid:17) / ( q + M ) / ( q + agN ) / √ M N ≪ ε ( Cq ) ε q / D ( ag ) / (cid:16) √ M NC (cid:17) / . Summing over all the dyadic intervals we obtain the result. (cid:3) The above lemma implies that the contribution of the terms c ≥ C to E k is ≪ ε ( Cq ) ε DX / (cid:16) √ M NCq (cid:17) / . (8.2)The value of C is chosen by making this estimate equal to the first summand in (11.2),namely DX / (cid:16) √ M NCq (cid:17) / = CD / M − / N / X / , which gives C = q − / D − / √ M (8.3)(note that the entire sum on c is dropped when M < q / D / ). To make sure the condition C ≥ q − √ M N X is satisfied we shall need D X ≪ q − ε . NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 29 Performing the dyadic summation over M and N we therefore find that the sum over c in E k can be truncated to 1 ≤ c < C at the cost of an error term of size ≪ ε q ε DX / (cid:0) q − / D / p Q (cid:1) / ≪ ε q − / ε D / X / . (8.4)8.3. Voronoi summation formulas. We need the following Voronoi summation formulas.They are both special cases of Proposition 3.3 in [ ] (see also [ ; (15.1)] for the first one). Lemma 8.2. Let ( a, c ) = 1 and let ψ = ψ ψ , where ψ and ψ are real characters modulo D = ( c, D ) and D = D/D , respectively. Then for any smooth function g compactlysupported in R > we have X n (1 ⋆ ψ )( n ) e (cid:16) anc (cid:17) g ( n ) = ρ ( a, c ) L (1 , ψ ) Z ∞ g ( x ) dx + T ( a, c ) , where ρ ( a, c ) = 1 c (cid:16) ψ ( c ) + D | c τ ( ψ ) ψ ( a ) (cid:17) and T ( a, c ) = 2 πiψ ( a ) ψ ( c ) c √ D X m ( ψ ⋆ ψ )( m ) e (cid:16) − aD mc (cid:17) Z ∞ J (cid:16) π √ mxc √ D (cid:17) g ( x ) dx. Lemma 8.3. Let ( a, ℓ ) = 1 and let ψ = ψ ψ , where ψ , ψ are real characters modulo D , D . Let D ♭ = ( D , ℓ ) , D ′ = D /D ♭ , D ♭ = ( D , ℓ ) , D ′ = D /D ♭ and let ψ = ψ D ♭ D ♭ ψ D ′ D ′ = ψ D ♭ D ′ ψ D ′ D ♭ be the corresponding real characters. Then for anysmooth function g compactly supported in R > we have X n ( ψ ⋆ ψ )( n ) e (cid:16) anℓ (cid:17) g ( n ) = ρ ( a, ℓ ) L (1 , ψ ) Z ∞ g ( x ) dx + T ( a, ℓ ) , where ρ ( a, ℓ ) = 1 ℓ (cid:18) τ ( ψ ) ψ ( a ) ψ (cid:16) ℓD (cid:17) + τ ( ψ ) ψ ( a ) ψ (cid:16) ℓD (cid:17)(cid:19) and T ( a, ℓ ) = 2 πǫ ℓ ψ D ♭ D ♭ ( a ) ℓ p D ′ D ′ X m ( ψ D ♭ D ′ ⋆ ψ D ′ D ♭ )( m ) e (cid:16) − aD ′ D ′ mℓ (cid:17) Z ∞ J (cid:16) π √ mxℓ p D ′ D ′ (cid:17) g ( x ) dx, for some ǫ ℓ independent on a and | ǫ ℓ | = 1 . Initial manipulations. Let g ( m, n ) := V k (cid:16) d mQ (cid:17) V k (cid:16) d bg nQ (cid:17) ω (cid:16) mM (cid:17) ω (cid:16) nN (cid:17) . Opening the Kloosterman sum in (8.1) and applying Lemma 8.2 to the sum on m we obtain T M,N ( c ) = T ∗ M,N ( c ) + T − M,N ( c ) , where T ∗ M,N ( c ) = L (1 , ψ ) √ M N X n (1 ⋆ ψ )( n ) (cid:16) ψ ( cq ) S ( agn, cq ) + D | c τ ( ψ ) S ψ ( agn, cq ) (cid:17) G ∗ ( n ) and T − M,N ( c ) = 2 πiψ ( − D ) ψ ( cq ) √ D M N X m ′ ,n ( ψ ⋆ ψ )( m ′ )(1 ⋆ ψ )( n ) S ψ ( m ′ − agD n, cq ) G − ( m ′ , n ) , with D = ( c, D ), D = D/D and ψ = ψ ψ being the corresponding real characters. Here G ∗ ( n ) = Z ∞ J (cid:16) π √ agxncq (cid:17) g ( x, n ) dx,G − ( m ′ , n ) = Z ∞ J (cid:16) π √ m ′ xcq √ D (cid:17) J (cid:16) π √ agxncq (cid:17) g ( x, n ) dx. The contribution of the term T ∗ M,N ( c ) to E k is easy to handle. Using Lemma 3.3 and thebound G ∗ ( n ) ≪ M , that is bounded by ≪ ε L (1 , ψ ) q − ε X √ M N ≪ ε q − ε DX. For T − M,N ( c ) we write T − M,N ( c ) = X h ∈ Z S ψ ( h, cq ) T − M,N,h ( c )= X h =0 + X h =0 = T ODM,N ( c ) + T OODM,N ( c ) , (8.5)say, where T − M,N,h ( c ) = 2 πiψ ( − D ) ψ ( cq ) √ D M N X m ′ − agD n = h ( ψ ⋆ ψ )( m ′ )(1 ⋆ ψ )( n ) G − ( m ′ , n ) . (8.6)We denote the contributions of T ODM,N ( c ) and T OODM,N ( c ) by M ODk and M OODk , respectively, andhence E k = M ODk + M OODk + O ε (cid:0) q − / ε D / X / (cid:1) , (8.7)by (8.4). Note that the sums over c in these terms are restricted to 1 ≤ c < C .9. The off-diagonal M ODk We have T ODM,N ( c ) = 2 πiψ ( − D ) ψ ( cq ) S ψ (0 , cq ) √ D M N X n ( ψ ⋆ ψ )( agD n )(1 ⋆ ψ )( n ) G − ( agD n, n ) . Note that S ψ (0 , cq ) = ( ϕ ( cq ) if D = 1 , D > , (9.1)and so T ODM,N ( c ) = 2 πiψ ( cq ) ϕ ( cq ) √ DM N X n (1 ⋆ ψ )( agn )(1 ⋆ ψ )( n ) G − ( agDn, n )= 2 πiψ ( cq ) ϕ ( cq ) √ DM N X n (1 ⋆ ψ )( agn )(1 ⋆ ψ )( n ) V k (cid:16) d bg nQ (cid:17) ω (cid:16) nN (cid:17) NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 31 Z ∞ J (cid:16) π √ agnxcq (cid:17) J (cid:16) π √ agnxcq (cid:17) V k (cid:16) d xQ (cid:17) ω (cid:16) xM (cid:17) dx. We now put this into (6.2). At this step we would like to remove the restriction c < C .Using the bound J ( x ) J ( x ) ≪ x this can be done at the cost of an error term of size ≪ ε q − ε D − / M NC X ≪ ε q − / ε D / X , by (8.3). Once the restriction on c is removed we execute the dyadic summation over M and N . After using (6.1) and a change of variables we obtain M ODk = iq √ D X dab ≤ X ψ ( d ) ρ ( ab ) dab X n (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n V k (cid:16) d bnQ (cid:17) S ( n )+ O ε (cid:0) q − / ε D / X (cid:1) , where S ( n ) = X d ψ ( d ) d X c ≥ ψ ( cq ) ϕ ( cq ) c Z ∞ J ( x ) J ( x ) V k (cid:16) d c q x π anQ (cid:17) dx. We use (3.1) to obtain S ( n ) = 12 πi Z (1 − ε ) G ( u )Γ(1 + u ) (cid:16) π anQq (cid:17) u L (1 + 2 u, ψ ) (cid:18) X c ≥ ψ ( cq ) ϕ ( cq ) c u (cid:19)(cid:18) Z ∞ J ( x ) J ( x ) x − u dx (cid:19) duu k +1 . The sum on c is ψ ( q ) q L (2 u, ψ ) L (1 + 2 u, ψ ) + O (1) . (9.2)On the other hand, we also have [ ; 6.574(2)] Z ∞ J ( x ) J ( x ) x − u dx = Γ(1 / u )Γ(1 − u )2 √ π Γ(1 + u ) , (9.3)and so S ( n ) = ψ ( q ) q π √ πi Z (1 − ε ) G ( u )Γ (cid:16) 12 + u (cid:17) Γ(1 − u ) (cid:16) π anQq (cid:17) u L (2 u, ψ ) duu k +1 + O ε (cid:0) q ε D X (cid:1) . It follows that M ODk = ψ ( q )4 π √ πD X dab ≤ X ψ ( d ) ρ ( ab ) dab X n (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n V k (cid:16) d bnQ (cid:17) (9.4) Z (1 − ε ) G ( u )Γ (cid:16) 12 + u (cid:17) Γ(1 − u ) (cid:16) π anQq (cid:17) u L (2 u, ψ ) duu k +1 + O ε (cid:0) q − / ε D / X (cid:1) . We now show that the first term in (9.4) is small for every fixed k ∈ N , provided that L (1 , ψ ) is small. By multiplicativity we have X n ≥ (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n s = f a,b ( s ) Y p (cid:18) X j ≥ (1 ⋆ ψ )( p j ) p js (cid:19) for Re( s ) > 1, where f a,b ( s ) = Y p | ab (cid:18) X j ≥ (1 ⋆ ψ )( p j ) p js (cid:19) − Y p ap || ap ∤ b (cid:18) (1 ⋆ ψ )( p a p ) + X j ≥ (1 ⋆ ψ )( p a p + j )(1 ⋆ ψ )( p j ) p js (cid:19)Y p bp || bp ∤ a (cid:18) (1 ⋆ ψ )( p b p ) + X j ≥ (1 ⋆ ψ )( p j )(1 ⋆ ψ )( p b p + j ) p js (cid:19)Y p ap || ap bp || b (cid:18) (1 ⋆ ψ )( p a p )(1 ⋆ ψ )( p b p ) + X j ≥ (1 ⋆ ψ )( p a p + j )(1 ⋆ ψ )( p b p + j ) p js (cid:19) . We also have [ ; p.12] Y p (cid:18) X j ≥ (1 ⋆ ψ )( p j ) p js (cid:19) = Y p | D (cid:18) − p s (cid:19) − Y ψ ( p )= − (cid:18) − p s (cid:19) − Y ψ ( p )=1 (cid:18) p s (cid:19)(cid:18) − p s (cid:19) − , and hence Y p (cid:18) X j ≥ (1 ⋆ ψ )( p j ) p js (cid:19) = α D ( s ) L ( s, ψ ) ζ ( s ) ζ (2 s ) , where α D ( s ) = Y p | D (cid:18) p s (cid:19) − . Thus from (3.1) the first term in (9.4) is ψ ( q )8 π i √ πD X ab ≤ X ρ ( ab ) ab Z (1) Z (1 − ε ) G ( u ) G ( v )Γ (cid:16) 12 + u (cid:17) Γ(1 − u )Γ(1 + v ) (cid:16) π aQq (cid:17) u (cid:16) Qb (cid:17) v L (2 u, ψ ) L (1 + 2 v, ψ ) X n ≥ (1 ⋆ ψ )( an )(1 ⋆ ψ )( bn ) n − u + v dudvu k +1 v k +1 = ψ ( q )8 π i √ πD X ab ≤ X ρ ( ab ) ab Z (1) Z (1 − ε ) G ( u ) G ( v )Γ (cid:16) 12 + u (cid:17) Γ(1 − u )Γ(1 + v ) (cid:16) π aQq (cid:17) u (cid:16) Qb (cid:17) v L (2 u, ψ ) L (1 + 2 v, ψ ) L (1 − u + v, ψ ) ζ (1 − u + v ) ( α D f a,b )(1 − u + v ) dudvu k +1 v k +1 ζ (cid:0) − u + v ) (cid:1) . NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 33 We move the v -contour to Re( v ) = 1 / ε , crossing a double pole at v = u , and the newintegral is trivially bounded by ≪ ε q − / ε D / X , by using the bounds α D ≪ ε D ε and f a,b ≪ ε ( ab ) ε along these contours. For the residue at v = u , we move the line of integration to Re( u ) = 1 / log q and see that the contribution ofthe residue is bounded by ≪ ε L (1 , ψ ) (log q ) k +23+ ε + L (1 , ψ ) L ′ (1 , ψ )(log q ) k +22+ ε ≪ ε L (1 , ψ )(log q ) k +24+ ε , as f a,b (1) ≪ τ ( a ) τ ( b ). So M ODk ≪ ε L (1 , ψ )(log q ) k +24+ ε + q − / ε D / X . (9.5)10. Evaluating T − M,N,h ( c )10.1. Shifted convolution sum. We first study the shifted convolution sum X am − bn = h ( ψ ⋆ ψ )( m )(1 ⋆ ψ )( n )over dyadic intervals. Proposition 10.1. Let h ∈ Z \{ } , M, N, P ≥ . Suppose ψ , ψ are real characters modulo D , D and ψ = ψ ψ . Let f be a smooth function supported on R + × R + such that m i n j f ( ij ) ( m, n ) ≪ (cid:16) mM (cid:17) − A (cid:16) nN (cid:17) − A P i + j for any fixed A, i, j ≥ . Let S a,b,h = X am − bn = h ( ψ ⋆ ψ )( m )(1 ⋆ ψ )( n ) f ( m, n ) . (10.1) Then S a,b,h = L (1 , ψ ) S a,b ( h ) 1 ab Z f (cid:16) y + ha , yb (cid:17) dy + O ε (cid:0) ( abhDM N ) ε DP / ( abM N ) / ( aM + bN ) / (cid:1) , where S a,b ( h ) = X ℓ ≥ ℓ ′ a ℓ ′ b (cid:18) τ ( ψ ) ψ ( ℓ ′ b ) ψ ( − a ′ ) ψ (cid:16) ℓ ′ a D (cid:17) S ψ ( h, ℓ )+ τ ( ψ ) ψ ( ℓ ′ b ) ψ ( − a ′ ) ψ (cid:16) ℓ ′ a D (cid:17) S ψ ( h, ℓ )+ D | ℓ ′ b τ ( ψ ) τ ( ψ ) ψ ( b ′ ) ψ ( − a ′ ) ψ (cid:16) ℓ ′ a D (cid:17) S ψψ ( h, ℓ )+ D | ℓ ′ b τ ( ψ ) τ ( ψ ) ψ ( b ′ ) ψ ( − a ′ ) ψ (cid:16) ℓ ′ a D (cid:17) S ψψ ( h, ℓ ) (cid:19) . (10.2) Here a ′ = a/ ( a, ℓ ) , b ′ = b/ ( b, ℓ ) and ℓ ′ a = ℓ/ ( a, ℓ ) , ℓ ′ b = ℓ/ ( b, ℓ ) . Proof. We use the delta method, as developed by Duke, Friedlander and Iwaniec in [ ]. Asusual, let δ (0) = 1 and δ ( n ) = 0 for n = 0. Let L ≤ P − / min {√ aM , √ bN } . Then δ ( n ) = X ℓ ≥ X ∗ k (mod ℓ ) e (cid:16) knℓ (cid:17) ∆ ℓ ( n ) , (10.3)where ∆ ℓ ( u ) is some smooth function that vanishes if | u | ≤ U = L and ℓ ≥ L (see [ ;Section 4]) and satisfies (see [ ; Lemma 2])∆ ℓ ( u ) ≪ (cid:0) ℓL + L (cid:1) − + (cid:0) ℓL + | u | (cid:1) − . (10.4)We also attach to both sides of (10.3) a redundant factor φ ( n ), where φ ( u ) is a smoothfunction supported on | u | < U satisfying φ (0) = 1 and φ ( j ) ( u ) ≪ U − j for any fixed j ≥ S a,b,h = X ℓ< L X ∗ k (mod ℓ ) e (cid:16) − hkℓ (cid:17) X m,n ( ψ ⋆ ψ )( m )(1 ⋆ ψ )( n ) e (cid:16) k ( am − bn ) ℓ (cid:17) F ( m, n ) , (10.5)where F ( m, n ) = ∆ ℓ ( am − bn − h ) φ ( am − bn − h ) f ( m, n ) . Let a ′ = a/ ( a, ℓ ), b ′ = b/ ( b, ℓ ) and ℓ ′ a = ℓ/ ( a, ℓ ), ℓ ′ b = ℓ/ ( b, ℓ ). Let D ♭ = ( D , ℓ ′ a ) , D ′ = D /D ♭ , D ♭ = ( D , ℓ ′ a ) , D ′ = D /D ♭ ,D ∗ = ( ℓ ′ b , D ) , D ∗ = D/D ∗ and let ψ = ψ D ♭ D ♭ ψ D ′ D ′ = ψ D ♭ D ′ ψ D ′ D ♭ = ψ ∗ ψ ∗ be the corresponding real characters. Weapply the Voronoi summation formulas in Lemma 8.2 and Lemma 8.3 to the sums over n and m in (10.5). In doing so, we can write S a,b,h as a principal term plus three error terms, S a,b,h = M a,b,h + E a,b,h + E a,b,h + E a,b,h . (10.6)We first deal with the error terms. All the three error terms can be treated similarly, sowe only focus here on one of them, E a,b,h = X ℓ< L X ∗ k (mod ℓ ) ψ D ♭ D ♭ ( k ) ψ ∗ ( k ) e (cid:16) − hkℓ (cid:17)X m ′ ,n ′ (cid:0) ψ D ♭ D ′ ⋆ ψ D ′ D ♭ (cid:1) ( m ′ ) (cid:0) ψ ∗ ⋆ ψ ∗ (cid:1) ( n ′ ) e (cid:16) − ka ′ D ′ D ′ m ′ ℓ ′ a + kb ′ D ∗ n ′ ℓ ′ b (cid:17) H ( m ′ , n ′ ) , where H ( m ′ , n ′ ) = 4 π ǫ a,b,ℓ ℓ ′ a ℓ ′ b p D ′ D ′ D ∗ Z ∞ Z ∞ J (cid:16) π √ m ′ xℓ ′ a p D ′ D ′ (cid:17) J (cid:16) π √ n ′ yℓ ′ b p D ∗ (cid:17) F ( x, y ) dxdy for some | ǫ a,b,ℓ | = 1. The sum over k is S ψ D♭ D♭ ψ ∗ ( − h, ∗ ; ℓ ), which is bounded by ≪ ε ( h, ℓ ) / ℓ / ε , by the Weil bound [ ; Lemma 3]. Hence E a,b,h ≪ ε X ℓ< L ( h, ℓ ) / ℓ / ε X m ′ ,n ′ τ ( m ′ ) τ ( n ′ ) (cid:12)(cid:12) H ( m ′ , n ′ ) (cid:12)(cid:12) . NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 35 We have F ( ij ) ≪ i,j ℓL (cid:16) abℓL (cid:17) i + j for any fixed i, j ≥ 0. Using the recurrence formula ( x ν J ν ( x )) ′ = x ν J ν − ( x ), integration byparts then implies that the sums are negligible unless m ′ ≪ ε ( abDM N ) ε a ′ M D ′ D ′ L and n ′ ≪ ε ( abDM N ) ε b ′ N D ∗ L . (10.7)For m ′ , n ′ in these ranges we bound H ( m ′ , n ′ ) trivially using J ( x ) ≪ x − / and get H ( m ′ , n ′ ) ≪ ℓ ′ a ℓ ′ b ) / ( D ′ D ′ D ∗ m ′ n ′ M N ) / Z Z (cid:12)(cid:12)(cid:12)(cid:12) ∆ ℓ ( ax − by − h ) φ ( ax − by − h ) f ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) dxdy ≪ ε ( abM N ) ε ab ( ℓ ′ a ℓ ′ b ) / ( D ′ D ′ D ∗ m ′ n ′ M N ) / min { aM, bN } Z (cid:12)(cid:12) ∆ ℓ ( u ) (cid:12)(cid:12) du ≪ ε ( abM N ) ε ( M N ) / ( ℓ ′ a ℓ ′ b ) / ( D ′ D ′ D ∗ m ′ n ′ ) / ( aM + bN ) , by (10.4). So summing over m ′ , n ′ in the range (10.7) we obtain X m ′ ,n ′ τ ( m ′ ) τ ( n ′ ) (cid:12)(cid:12) H ( m ′ , n ′ ) (cid:12)(cid:12) ≪ ε ( abDM N ) ε L − ( D ′ D ′ D ∗ ) / ( a ′ b ′ M N ) / ( ℓ ′ a ℓ ′ b ) / ( aM + bN ) , and hence E a,b,h ≪ ε ( abhDM N ) ε L − / D ( abM N ) / aM + bN . (10.8)We now return to the principal term M a,h in (10.6). This corresponds to the product ofthe two constant terms after the applications of Lemma 8.2 and Lemma 8.3, and hence M a,h = L (1 , ψ ) X ℓ< L X ∗ k (mod ℓ ) ρ ( − kb ′ , ℓ ′ b ) ρ ( ka ′ , ℓ ′ a ) e (cid:16) − hkℓ (cid:17)Z ∞ Z ∞ ∆ ℓ ( ax − by − h ) φ ( ax − by − h ) f ( x, y ) dxdy. (10.9)We have ρ ( − kb ′ , ℓ ′ b ) ρ ( ka ′ , ℓ ′ a )= 1 ℓ ′ a ℓ ′ b (cid:0) ψ ( ℓ ′ b ) + D | ℓ ′ b τ ( ψ ) ψ ( − kb ′ ) (cid:1)(cid:18) τ ( ψ ) ψ ( ka ′ ) ψ (cid:16) ℓ ′ a D (cid:17) + τ ( ψ ) ψ ( ka ′ ) ψ (cid:16) ℓ ′ a D (cid:17)(cid:19) . So X ∗ k (mod ℓ ) ρ ( − kb ′ , ℓ ′ b ) ρ ( ka ′ , ℓ ′ a ) e (cid:16) − hkℓ (cid:17) = 1 ℓ ′ a ℓ ′ b (cid:18) τ ( ψ ) ψ ( ℓ ′ b ) ψ ( − a ′ ) ψ (cid:16) ℓ ′ a D (cid:17) S ψ ( h, ℓ ) + τ ( ψ ) ψ ( ℓ ′ b ) ψ ( − a ′ ) ψ (cid:16) ℓ ′ a D (cid:17) S ψ ( h, ℓ )+ D | ℓ ′ b τ ( ψ ) τ ( ψ ) ψ ( b ′ ) ψ ( − a ′ ) ψ (cid:16) ℓ ′ a D (cid:17) S ψψ ( h, ℓ ) + D | ℓ ′ b τ ( ψ ) τ ( ψ ) ψ ( b ′ ) ψ ( − a ′ ) ψ (cid:16) ℓ ′ a D (cid:17) S ψψ ( h, ℓ ) (cid:19) . (10.10)For the double integrals in (10.9), a change of variables gives Z ∞ Z ∞ ∆ ℓ ( ax − by − h ) φ ( ax − by − h ) f ( x, y ) dxdy = 1 ab Z Z ∆ ℓ ( u ) φ ( u ) f (cid:16) y + u + ha , yb (cid:17) dydu. By (10.4) this is bounded by ≪ ε ( abM N ) ε min { aM, bN } ab ≪ ε ( abM N ) ε M NaM + bN . Moreover, if ℓ ≪ L − ε , then by [ ; (18)], this is equal to1 ab Z f (cid:16) y + ha , yb (cid:17) dy + O A (cid:0) L − A (cid:1) for any fixed A > 0. So in view of (10.10) and the Weil bound, we can first restricted thesum over ℓ in (10.9) to ℓ ≪ L − ε , and then extend it to all ℓ ≥ O ε (cid:0) ( abhM N ) ε L (1 , ψ ) ( h, D ) / L − / M N ( aM + bN ) − (cid:1) . Hence M a,b,h = L (1 , ψ ) S a,b ( h ) 1 ab Z f (cid:16) y + ha , yb (cid:17) dy + O ε (cid:0) ( abhM N ) ε L (1 , ψ ) ( h, D ) / L − / M N ( aM + bN ) − (cid:1) , (10.11)where S a,b ( h ) is defined in (10.2).Finally, combining (10.8) and (10.11), and choosing L = P − / min {√ aM , √ bN } we ob-tain the proposition. (cid:3) Evaluating T − M,N,h ( c ) . Recall that G − ( m ′ , n ) = Z ∞ J (cid:16) π √ m ′ xcq √ D (cid:17) J (cid:16) π √ agnxcq (cid:17) g ( x, n ) dx. Let P = 1 + √ agM Ncq and M ′ = c q D P M . Note that M ′ > agD N . With respect to x , we do nothing if m ′ ≤ M ′ , but if m ′ > M ′ weintegrate by parts in x several times. Then we differentiate i times in m ′ and j times in n .Using the bounds J ( x ) ≪ (1 + x ) − / and J ( x ) ≪ x (1 + x ) − / we get m ′ i n j G − ( ij ) ( m ′ , n ) ≪ A,i,j (cid:16) m ′ M ′ (cid:17) − A (cid:16) nN (cid:17) − A M √ agM Ncq P i + j − (10.12)for any fixed A, i, j ≥ 0. Applying Proposition 10.1 we obtain T − M,N,h ( c ) = 2 πiL (1 , ψ ) ψ ( − D ) ψ ( cq ) √ D M N S ,agD ( h ) W ( h ) + O ε (cid:0) q ε D / ( ag ) / M / ( N P ) / (cid:1) , (10.13) NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 37 where S a,b ( h ) is given in (10.2) and W ( h ) = 1 agD Z Z J (cid:16) π p x ( y + h ) cq √ D (cid:17) J (cid:16) π √ xycq √ D (cid:17) g (cid:16) x, yagD (cid:17) dxdy. Note that as in the proof of Proposition 10.1 we can truncate the sum over ℓ in S ,agD ( h )to ℓ < cq and the resulting error is absorbed in the above error term.11. The off-off-diagonal M OODk - Initial manipulations In view of (10.13) and (10.2) we write T − M,N,h ( c ) = X j =1 T − j ; M,N,h ( c ) + O ε (cid:0) q ε D / ( ag ) / M / ( N P ) / (cid:1) . (11.1)Observe from (10.12) that the contribution of the terms with m ′ ≫ q ε M ′ is negligible. Wealso have M ′ > agD N so we can restrict | h | ≤ H = q ε M ′ in (8.5) at the cost of a negligibleerror term. Hence, using Lemma 3.3 the contribution of the O -term in (11.1) to M OODk is ≪ ε q ε D / (cid:0) CM − / N / X / + q − / M / N / X / (cid:1) ≪ ε q − / ε D / X / + q − / ε D / X / , (11.2)by (8.3).All the four terms T − j ; M,N,h ( c ) are in similar forms, so we only illustrate how to estimatethe contribution of T − M,N,h ( c ) to M OODk . We have X h =0 S ψ ( h, cq ) T − M,N,h ( c )= 2 πiL (1 , ψ ) τ ( ψ ) ψ ( D ) ψ ( cq ) √ D M N X ℓ 2, but note that the above conditions force u + v + w < 2. We therefore have to move the contours of integration several timesto make sure the h -sum converges absolutely. The extra factor G ( u ) in the integralexpression of V k in (3.1) is crucial in this step.12. Evaluating M OOD k Bringing in the Mellin transforms. Removing the partition of unity we obtain X M,N W ( h ) = 1 √ agD Z Z J (cid:16) π p x ( y + h ) cq √ D (cid:17) J (cid:16) π √ xycq √ D (cid:17) V k (cid:16) d xQ (cid:17) V k (cid:16) d bgyaD Q (cid:17) dxdy √ xy . We bring in the Mellin transforms of J ( x ) and J ( x ): J ( x ) = 12 πi Z ( v ) Γ( v )Γ(1 − v ) (cid:16) x (cid:17) − v dv and J ( x ) = 12 πi Z ( w ) Γ( w )Γ(2 − w ) (cid:16) x (cid:17) − w dw for v , w > ; 6.561(14), 8.412(4)]). These together with the integralformula (3.1) for the second factor V k imply that X M,N W ( h ) = 1 √ agD πi ) Z ( w ) Z ( v ) Z ( u ) G ( u ) Γ(1 + u ) Γ( v )Γ( w )Γ(1 − v )Γ(2 − w ) (cid:16) d bgaD Q (cid:17) − u (cid:16) πcq √ D (cid:17) − v − w (cid:18) Z x − ( v + w ) V k (cid:16) d xQ (cid:17) dx (cid:19)(cid:18) Z ( y + h ) − v y − ( u + w ) dy (cid:19) dudvdwu k +1 . NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 39 By (3.1) and the inversion Mellin transform, the x -integral is (cid:16) Qd (cid:17) − v − w G (1 − v − w ) Γ(2 − v − w ) (1 − v − w ) k +1 if v + w < . (12.1)For the y -integral, note that if h < y > − h , and if h > y > 0. For absolute convergence we need to impose the conditions ( u + v + w > ,u + w < . (12.2)Under these assumptions, the y -integral is equal to (see, for instance, [ ; 17.43(21), 17.43(22)]) | h | − u − v − w × ( Γ( u + v + w − − v )Γ( u + w ) if h < , Γ( u + v + w − − u − w )Γ( v ) if h > . Hence provided that u , v , w satisfy conditions (12.1) and (12.2) we have M OOD k = L (1 , ψ ) X h> πi ) Z ( w ) Z ( v ) Z ( u ) γ ( u, v, w ) I h ( u, v, w ) dudvdwu k +1 (1 − v/ − w ) k +1 , (12.3)where γ ( u, v, w ) = G ( u ) G (1 − v − w ) Γ(1 + u ) Γ( w )Γ(2 − v − w ) Γ( u + v + w − − w ) (cid:16) Γ( v )Γ( u + w ) + Γ(1 − u − w )Γ(1 − v ) (cid:17) (12.4)and I h ( u, v, w ) = L (1 + 2 u, ψ ) L (3 − v − w, ψ ) (2 π ) − v − w q − v − w Q u − v − w X abg ≤ X µ ( g )(1 ⋆ ψ )( b ) ρ ( abg ) a − u b u g u X c ≥ τ ( ψ ) ψ ( D ) ψ ( cq ) c − v − w D / − u − v − w X ℓ ≥ ψ ( ℓ ′ ) ψ ( ℓ/D ) ℓℓ ′ S ψ ( h, cq ) S ψ ( h, ℓ ) h − u − v − w . To make sure the sums over a and b are controllable we want u ≍ / log q . Later we alsowant (1 − w ) ≍ / log q . So we choose u = 2log q , v = 2log q and w = 1 − q . Converting the Gauss-Ramanujan sums. We write c = c c , ℓ = ℓ ℓ and h = h h , where c ℓ h | D ∞ and ( c ℓ h , D ) = 1. Using Lemma 3.3 the product of the Gauss-Ramanujan sums vanishes unless D h = c = ℓ , and in that case we have S ψ ( h, cq ) = h τ ( ψ ) ψ ( c qh ) X rc ′ = c qr | h µ ( c ′ ) r (12.5) and S ψ ( h, ℓ ) = h τ ( ψ ) ψ ( ℓ h ) X sℓ ′ = ℓ s | h µ ( ℓ ′ ) s. (12.6)There are two cases: ( q, c ′ ) = 1, which will force q | r | h , or q | c ′ . The former leads to twodifferent cases to consider: ( q, s ) = 1 or q | s | h .12.3. The h -sum. We consider the terms with ( q, c ′ s ) = 1. The other cases can be dealtwith in the same way and can be adsorbed in the expression (12.7) below. By (12.5) and(12.6) we get X c ≥ τ ( ψ ) ψ ( D ) ψ ( cq ) c − v − w D / − u − v − w X ℓ 0, Re( v ) = v and Re( w ) = w .For convergence we need ( Re( u ) > Re( v + w ) , Re( u + v + w ) > . (12.9)Hence in order to be able to use (12.7) we need to move the contours of integration severaltimes. NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 41 Moving the contours. We first move the u -contour in (12.3) to the right to Re( u ) =1 + 2 / log q , encountering a simple pole at u = 1 − w from the gamma factor Γ(1 − u − w )in γ ( u, v, w ). In doing so we obtain M OOD k = M OOD ′ k − R OOD , where M OOD ′ k is the new integral and R OOD = − L (1 , ψ ) X h> πi ) Z ( w ) Z ( v ) G (1 − w ) G (1 − v − w ) (12.10)Γ(2 − w )Γ( w )Γ(2 − v − w ) Γ( v )Γ(1 − v ) I h (1 − w, v, w ) dvdw (1 − w ) k +1 (1 − v − w ) k +1 . With M OOD ′ k we can move the sum over h inside the integrals and replace the sum over h using (12.7) (see condition (12.9)). We then shift the u -contour to Re( u ) = 4 / log q , crossingno poles. Note that the pole of ζ ( u + v/ w − 1) is cancelled as γ (2 − v/ − w, v, w ) = 0.Hence M OOD ′ k = L (1 , ψ ) πi ) Z ( w ) Z ( v ) Z (4 / log q ) γ ( u, v, w ) D − / u L (1 + 2 u, ψ ) L (1 + u − v − w, ψ ) L q ( u + v + w, ψ ) ζ qD (1 + 2 u ) ζ qD (2 + u − v − w ) ζ ( u + v + w − J ( u, v, w ) dudvdwu k +1 (1 − v − w ) k +1 . Bounding this trivially using (12.8) we get M OOD ′ k ≪ ε L (1 , ψ ) (log q ) k +20+ ε . For R OOD we move the w -contour in (12.10) to Re( w ) = − / log q , followed by movingthe v -contour to Re( v ) = 1 + 1 / log q . Both times we cross no poles as the pole at w = 0 ofΓ( w ) is cancelled by the factor G (1 − w ). Along these new contours, conditions (12.9) arenow satisfied and hence the sum over h can be moved inside the integrals and replaced by(12.7). So R OOD = − L (1 , ψ ) πi ) Z ( − / log q ) Z (1+1 / log q ) G (1 − w ) G (1 − v − w )Γ(2 − w )Γ( w )Γ(2 − v − w ) Γ( v )Γ(1 − v ) D / − w L (3 − w, ψ ) L (2 − v − w, ψ ) L q (1 + v, ψ ) ζ qD (3 − w ) ζ qD (3 − v − w ) ζ ( v ) J (1 − w, v, w ) dvdw (1 − w ) k +1 (1 − v/ − w ) k +1 . We next move the v -contour back to Re( v ) = v , and then the w -contour back to Re( w ) = w . Again there are no poles as G (1 − w )Γ( w ) has no pole at w = 0 and ζ ( v ) / Γ(1 − v ) hasno pole at v = 1. We bound the resulting integral trivially using (12.8) and get R OOD ≪ ε L (1 , ψ ) (log q ) k +19+ ε . It follows that M OOD k ≪ ε L (1 , ψ ) (log q ) k +20+ ε . (12.11) Evaluating M OOD k To estimate M OOD k we shall use the following lemma. Lemma 13.1. Let c , c ∈ N with c = c and let f be a smooth compactly supported functionon R . Then we have X h S χ ( h, c ) S χ ( h, c ) f ( h ) = X u =0 b f (cid:16) u [ c , c ] (cid:17) ν c ,c ( u ) , with | ν c ,c ( u ) | ≤ ϕ (( c , c )) .Proof. We split the sum into progressions h ≡ a (mod [ c , c ]) and then apply the Poissonsummation formula. In doing so we obtain1[ c , c ] X a (mod [ c ,c ]) S χ ( a, c ) S χ ( a, c ) X u e (cid:16) − au [ c , c ] (cid:17) b f (cid:16) u [ c , c ] (cid:17) = X u b f (cid:16) u [ c , c ] (cid:17) ν c ,c ( u ) , where ν c ,c ( u ) = 1[ c , c ] X ∗ v (mod c ) X ∗ w (mod c ) χ ( vw ) X a (mod [ c ,c ]) e (cid:18) a (cid:16) vc + wc − u [ c , c ] (cid:17)(cid:19) = X ∗ v (mod c ) X ∗ w (mod c ) v [ c ,c ] /c + w [ c ,c ] /c ≡ u (mod [ c ,c ]) χ ( vw ) . Observe that the congruence equation has no solutions unless ( u, [ c c ] / ( c , c )) = 1 (so, inparticular, u = 0 as c = c ), and in that case the number of solutions is less than or equalto ϕ (( c , c )). This completes the proof of the lemma. (cid:3) In order to apply this we first write W ( h ) = 1 agD Z ∞ M Z agD MN/xagD MN/x J (cid:16) π p x ( y + 1) cq √ D (cid:17) J (cid:16) π √ xycq √ D (cid:17) f ( h ) dydx, where f ( h ) = g (cid:16) xh , yhagD (cid:17) , if h > 0, and a a similar representation holds for h < 0. Integrating by parts twice we seethat the Fourier transform of f is bounded by b f ( u ) ≪ (cid:0) | u | xM − (cid:1) − xM − . Hence by Lemma 13.1 we obtain X h> S ψ ( h, cq ) S ψ ( h, ℓ ) f ( h ) ≪ cqℓ. Moreover we have Z ∞ M Z agD MN/xagD MN/x (cid:12)(cid:12)(cid:12)(cid:12) J (cid:16) π p x ( y + 1) cq √ D (cid:17) J (cid:16) π √ xycq √ D (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) dydx ≪ ( agD M N ) / cq √ D (log q ) , NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 43 by using the bounds J ( x ) ≪ (1 + x ) − / and J ( x ) ≪ x . Thus X h> S ψ ( h, cq ) S ψ ( h, ℓ ) W ( h ) ≪ ( ag ) / ( M N ) / ℓ (log q ) . The same bound holds for the sum over h < 0. Putting this into M OOD k we obtain M OOD k ≪ ε L (1 , ψ ) q − ε M NC √ D X ≪ ε q − / ε D / X , by (8.3). This together with (8.7), (9.5), (11.3) and (12.11) complete the proof of Proposition6.2. 14. Proofs of the main theorems As mentioned in the introduction, Theorem 1.1 is a straightforward consequence of The-orem 1.2. Here we give the easy proof. Proof of Theorem 1.1 assuming Theorem 1.2. We may assume L (1 , ψ ) ≤ (log D ) − . By thework of Gross and Zagier [ ], the product Y f ∈ S ∗ ( q ) r f =1 L ( f, s )is the L -function of a quotient of J ( q ) with rank exactly equal to its dimension. Hence thelower bound rank( J ( q )) ≥ (cid:18) 12 + O (cid:16) log log log q log log q (cid:17)(cid:19) dim( J ( q ))follows from Theorem 1.2, since the proportion of odd forms is 1 / O ( q − ε ).For the upper bound, it follows from Theorem 1.2 that it suffices to prove1 | S ∗ ( q ) | X f ∈ S ∗ ( q ) r f ≥ r f ≪ s log log log q log log q . By Cauchy-Schwarz’s inequality the left-hand side is ≤ | S ∗ ( q ) | (cid:18) X f ∈ S ∗ ( q ) r f ≥ (cid:19) / (cid:18) X f ∈ S ∗ ( q ) r f (cid:19) / ≪ q − / (cid:18) X f ∈ S ∗ ( q ) r f ≥ (cid:19) / , by Theorem 8.1 in [ ]. By Theorem 1.2 again, this is ≪ s log log log q log log q , as required. (cid:3) As stated above, Theorem 1.1 is a result about the natural average of analytic ranks of L -functions. We first prove a result for the harmonic average, and then describe the necessarymodifications to obtain Theorem 1.1. Theorem 14.1. Let C ≥ be a fixed real number. Let D be large, and let ψ be a real,odd, primitive Dirichlet character modulo D .Then for any ε > and any prime q satisfying D ≤ q ≤ D C we have X hf ∈ S ∗ ( q ) r f ≤ O ε ( L (1 , ψ )(log q ) ε ) + O ε ( L (1 , ψ ) (log q ) ε ) + O A ((log q ) − A ) if ψ ( q ) = 1 , and X hf ∈ S ∗ ( q ) r f ≤ O (cid:18) log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) (cid:19) + O ε ( L (1 , ψ )(log q ) ε )+ O ε ( L (1 , ψ ) (log q ) ε ) if ψ ( q ) = − and L (1 , ψ )(log D ) = o (1) .Proof. We prove the second part of the theorem, since the first is similar and easier. Define S = X hf ∈ S ∗ ( q ) Λ ′′ f,ψ (cid:16) (cid:17) M f,ψ and S = X hf ∈ S ∗ ( q ) Λ ′′ f,ψ (cid:16) (cid:17) M f,ψ . By Cauchy-Schwarz’s inequality, we have X hf ∈ S ∗ ( q )Λ ′′ f,ψ (1 / =0 ≥ S S . Letting X = D , together Propositions 2.1 and 2.3 imply that X hf ∈ S ∗ ( q )Λ ′′ f,ψ (1 / =0 O (cid:18) log log(1 /L (1 , ψ ) log D )log(1 /L (1 , ψ ) log D ) (cid:19) + O ε ( L (1 , ψ )(log q ) ε )+ O ε ( L (1 , ψ ) (log q ) ε ) . By the product rule we see thatΛ ′′ f,ψ (cid:16) (cid:17) = Λ ′′ (cid:16) f, (cid:17) Λ (cid:16) f ⊗ ψ, (cid:17) + 2Λ ′ (cid:16) f, (cid:17) Λ ′ (cid:16) f ⊗ ψ, (cid:17) + Λ (cid:16) f, (cid:17) Λ ′′ (cid:16) f ⊗ ψ, (cid:17) . If f is odd then by root number considerations we see thatΛ ′′ f,ψ (cid:16) (cid:17) = 2Λ ′ (cid:16) f, (cid:17) Λ ′ (cid:16) f ⊗ ψ, (cid:17) . If the left-hand side is nonzero then we must have Λ ′ ( f, / = 0. Again by root numberconsiderations we see that Λ ′ ( f, / ′ = L ′ ( f, / r f = 1. NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 45 If f is even, thenΛ ′′ f,ψ (cid:16) (cid:17) = Λ ′′ (cid:16) f, (cid:17) Λ (cid:16) f ⊗ ψ, (cid:17) + Λ (cid:16) f, (cid:17) Λ ′′ (cid:16) f ⊗ ψ, (cid:17) , and therefore at least one of Λ( f, / 2) or Λ ′′ ( f, / 2) is nonzero. If Λ( f, / = 0 then r f = 0, so suppose that Λ( f, / 2) = 0 and Λ ′′ ( f, / = 0. Taking derivatives and notingthat L ( f, / 2) = L ′ ( f, / 2) = 0, we deduce that Λ ′′ ( f, / 2) = L ′′ ( f, / r f = 2. (cid:3) To deduce Theorem 1.2 we just need to remove the harmonic weights w f in Propositions2.1–2.3 so that the main propositions also hold for the natural average, X nf ∈ S ∗ ( q ) A f := 1 | S ∗ ( q ) | X f ∈ S ∗ ( q ) A f . This technique has been done several times (see, for instance, [ , , ]), so here we shallonly illustrate the method for the mollified first moment of Λ ′ f,ψ (1 / X nf ∈ S ∗ ( q ) Λ ′ f,ψ (cid:16) (cid:17) M f,ψ . In doing this we need the following general lemma from [ ]. Lemma 14.2. Let ( A f ) f ∈ S ∗ ( q ) be a family of complex numbers satisfying X hf ∈ S ∗ ( q ) | A f | ≪ (log q ) A (14.1) and max f ∈ S ∗ ( q ) w f | A f | ≪ q − δ (14.2) for some absolute A, δ > . Then for all κ > , there exists ε = ε ( A, δ ) > such that X nf ∈ S ∗ ( q ) A f = 1 ζ (2) X hf ∈ S ∗ ( q ) w f ( q κ ) A f + O (cid:0) q − ε (cid:1) , where w f ( q κ ) = X ℓm ≤ q κ λ f ( ℓ ) ℓm . We shall apply this lemma for A f = Λ ′ f,ψ (1 / M f,ψ . Condition (14.1) follows immediatelyfrom Proposition 2.2 and Cauchy-Schwarz’s inequality. For condition (14.2), it is known that w f ≪ log q/q [ ]. Hence (14.2) is satisfied provided that DX ≪ q − δ for some δ > 0, usingthe convexity bound Λ ′ f,ψ (1 / ≪ ε ( qD ) / ε and the trivial bound M f,ψ ≪ ε X / ε . Thuswe are left with estimating the sum I = 1 ζ (2) X hf ∈ S ∗ ( q ) w f ( q κ )Λ ′ f,ψ (cid:16) (cid:17) M f,ψ . As before, we can remove the condition ( d, q ) = 1 in the expression for Λ ′ f,ψ (1 / 2) in Lemma3.4 due to the decay of the function V . Using (2.3) and applying Lemma 3.1 we getΛ ′ f,ψ (cid:16) (cid:17) M f,ψ = (cid:0) ψ ( q ) (cid:1) X dab ≤ X ψ ( d ) ρ ( ab ) d √ ab X n (1 ⋆ ψ )( bn ) λ f ( an ) √ n V (cid:16) d bnQ (cid:17) . Hence I = 1 + ψ ( q ) ζ (2) X ℓm ≤ q κ X dab ≤ X ψ ( d ) ρ ( ab ) d √ ab X n (1 ⋆ ψ )( bn ) ℓm √ n V (cid:16) d bnQ (cid:17) X hf ∈ S ∗ ( q ) λ f ( ℓ ) λ f ( an ) . From Lemma 3.2 we obtain I = 1 + ψ ( q ) ζ (2) X ℓm ≤ q κ X dab ≤ Xan = ℓ ψ ( d ) ρ ( ab )(1 ⋆ ψ )( bn ) dabm n V (cid:16) d bnQ (cid:17) + O ε (cid:0) q − / κ + ε DX (cid:1) . By (3.1) we can write the main term as1 + ψ ( q ) ζ (2) 12 πi Z (1) Γ(1 + u ) Q u L (1 + 2 u, ψ ) S ( u ) duu , where S ( u ) = X ℓm ≤ q κ X ab ≤ Xan = ℓ ρ ( ab )(1 ⋆ ψ )( bn ) ab u m n u . It is clear that S ( u ) ≪ ε X − Re( u )+ ε and trivially we have S ′ (0) ≪ ε (log q ) ε . We move thecontour to Re( u ) = − / ε , crossing a double pole at u = 0, and the new integral is ≪ ε q ε (cid:16) QX (cid:17) − / D / ≪ ε q − / ε X / . The contribution of the residue is2 (cid:0) ψ ( q ) (cid:1) ζ (2) S (0) L ′ (1 , ψ ) + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) . Thus it remains to prove the following lemma. Lemma 14.3. We have S (0) = ζ (2) S + O ε (cid:0) L (1 , ψ )(log q ) ε (cid:1) + O A (cid:0) (log q ) − A (cid:1) , where S is given by (2.6) .Proof. The sum on m in S (0) should obviously yield the factor of ζ (2) out in front. To executethis sum rigorously, we first separate ℓ and m from each other. By trivial estimation, thecontribution from m > q ε is ≪ ε (log q ) ε X m>q ε m X a,b,n ≤ q τ ( ab ) τ ( bn ) abn ≪ ε q − ε (cid:18) X a ≤ q τ ( a ) a (cid:19) (cid:18) X b ≤ q τ ( b ) b (cid:19) ≪ ε q − ε . NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 47 With the size of m reduced, we now wish to extend the sum on ℓ to infinity. Of course,the sum on ℓ is not really infinite because we have the condition an = ℓ , but it is useful tothink in these terms. The contribution from ℓ > q κ /m ≥ q κ − ε is ≪ ε (log q ) ε X b ≤ X τ ( b ) b X ℓ>q κ − ε ℓ X an = ℓ τ ( a ) τ ( n ) ≪ ε (log q ) ε X ℓ>q κ − ε τ ( ℓ ) ℓ ≪ ε q − ε . We therefore have S (0) = X m ≤ q ε m X ab ≤ X ρ ( ab )(1 ⋆ ψ )( bn ) abn X ℓ ≥ ℓ = an O ε ( q − ε )= X m ≤ q ε m X ab ≤ Xan = (cid:3) ρ ( ab )(1 ⋆ ψ )( bn ) abn + O ε ( q − ε )= ζ (2) X ab ≤ Xan = (cid:3) ρ ( ab )(1 ⋆ ψ )( bn ) abn + O ε ( q − ε ) . We wish to compare the triple sum over a, b , and n with the Euler product P := Y p ≤ X X a,b,n ≥ a + n even ρ ( p a + b )(1 ⋆ ψ )( p b + n ) p a + b + n = X p | abn ⇒ p ≤ Xan = (cid:3) ρ ( ab )(1 ⋆ ψ )( bn ) abn . It shall follow from lacunarity that P is a good approximation to S (0).We use a crude smooth number estimate, as in the proof of Lemma 7.1, to first restrict thesize of ab . If we set T = exp((log X )(log log X ) ), then one finds, as in the proof of Lemma7.1, that the contribution to P from ab > T is O A ((log q ) − A ). It follows that P = X p | abn ⇒ p ≤ Xan = (cid:3) ab ≤ T ρ ( ab )(1 ⋆ ψ )( bn ) abn + O A ((log q ) − A ) . The next step is to use lacunarity to show that the error from restricting to ab ≤ X isnegligible. We recall here that ρ is supported on cube-free integers, and that ρ ( p ) = h ( p ) α ( p ) µ ( p )(1 ⋆ ψ )( p ) ,ρ ( p ) = h ( p ) α ( p ) ψ ( p ) . The contribution to P from ab > X is ≤ X p | abn ⇒ p ≤ XX 1. In this case we have X a,b,n ≥ a + n even ρ ( p a + b )(1 ⋆ ψ )( p b + n ) p a + b + n = X a,b,n ≥ a + n even b + n even a + b ∈{ , } h ( p a + b ) α ( p a + b ) ψ ( p ( a + b ) / ) p a + b + n NALYTIC RANKS OF AUTOMORPHIC L -FUNCTIONS AND LANDAU-SIEGEL ZEROS 49 = X n even p n − h ( p ) α ( p ) p X n ≥ p n − h ( p ) α ( p ) p X n even p n = p p − − h ( p ) α ( p ) p ( p − − h ( p ) α ( p ) p − p + pp + p + 1= (cid:18) p (cid:19) + ψ ( p ) p ! − (cid:18) − ψ ( p ) p (cid:19) . Lastly, we examine the local factor when ψ ( p ) = 1: X a,b,n ≥ a + n even ρ ( p a + b )(1 ⋆ ψ )( p b + n ) p a + b + n = X a,b,n ≥ a + n even ρ ( p a + b )( b + n + 1) p a + b + n = X n even n + 1 p n − h ( p ) α ( p ) p X n ≥ n + 1 p n − h ( p ) α ( p ) p X n even n + 2 p n + h ( p ) α ( p ) p X n ≥ n + 2 p n + h ( p ) α ( p ) p X n even n + 1 p n + h ( p ) α ( p ) p X n even n + 3 p n = p ( p + 1)( p − − h ( p ) α ( p ) p ( p − − h ( p ) α ( p ) p ( p − + h ( p ) α ( p ) (3 p − p ( p − + h ( p ) α ( p ) ( p + 1)( p − + h ( p ) α ( p ) (3 p − p − = p − pp + 3 p + 1 = (cid:18) p (cid:19) + ψ ( p ) p ! − (cid:18) − ψ ( p ) p (cid:19) . We see that the local factors match in each case with those in S , and this completes theproof. (cid:3) The argument for the second moment is similar, but the convexity bound is no longersufficient to obtain (14.2). Instead, we use the subconvexity bound L (cid:16) f, (cid:17) ≪ ε q / ϑ/ ε due to Blomer and Khan [ ; Theorem 4]. Here ϑ is an admissible bound towards the Ra-manujan conjecture, and it is known that ϑ ≤ / 64 is admissible (see [ ] for more discussion).This yields the subconvexity bound L (cid:16) f, (cid:17) ≪ ε q / ε . For the central value L ( f ⊗ ψ, / 2) we use the convexity bound L (cid:16) f ⊗ ψ, (cid:17) ≪ ε ( qD ) / ε . Recalling that we choose X in the definition of the mollifier so that X = D , we see that(14.2) is satisfied if q > D . References [1] S. Bettin, V. Chandee, Trilinear forms with Kloosterman fractions , Adv. Math. (2018), 1234–1262.[2] V. Blomer, F. Brumley, The role of the Ramanujan conjecture in analytic number theory , Bull. Amer.Math. Soc. (N.S.) (2013), 267–320.[3] V. Blomer, R. Khan, Twisted moments of L -functions and spectral reciprocity , Duke Math. J. (2019), 1109–1177.[4] A. Brumer, The rank of J ( N ), Ast´erisque (1995), 41–68.[5] H. M. Bui, K. Pratt, A. Zaharescu, Exceptional characters and nonvanishing of Dirichlet L -functions ,to appear in Math. 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Department of Mathematics, University of Manchester, Manchester M13 9PL, UK Email address : [email protected] All Souls College, Oxford OX1 4AL, UK Email address : [email protected], [email protected] Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 WestGreen Street, Urbana, IL 61801, USA and Simion Stoilow Institute of Mathematics ofthe Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania Email address :: S ψ ( h, cq ) S ψ ( h, ℓ ) h − u − v − w = ψ ( q ) q − u − v − w L (3 − v − w, ψ ) D / − u − v − w X D D = DD | ag τ ( ψ ) ψ ( D ) ψ ( D ) D / u − v − w X h | D ∞ h u − v − w X ( h ,D )=1 h − u − v − w X r | h s | h ( s,q )=1 ψ ( rs ) r − v − w ) s X ℓ ≥ µ ( l ) ψ ( h sℓ/ ( ag/D , h sℓ )) ψ ( ℓ )( ag/D , h sℓ ) ℓ . Writing X ( h ,D )=1 X r | h s | h ( s,q )=1 = X r,s ≥ s,q )=1 X [ r,s ] | h ( h ,D )=1 and using the Mobius inversion formula we see that X h> I h ( u, v, w ) = D − / u L (1 + 2 u, ψ ) L (1 + u − v − w, ψ ) L q ( u + v + w, ψ ) ζ qD (1 + 2 u ) ζ qD (2 + u − v − w ) ζ ( u + v + w − J ( u, v, w ) , (12.7)where J ( u, v, w ) ≪ ε (log q ) ε X u ) (12.8)uniformly for Re( u ) >