Smooth-supported multiplicative functions in arithmetic progressions beyond the x^{1/2}-barrier
aa r X i v : . [ m a t h . N T ] J un SMOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS INARITHMETIC PROGRESSIONS BEYOND THE x / -BARRIER SARY DRAPPEAU, ANDREW GRANVILLE, AND XUANCHENG SHAO
Abstract.
We show that smooth-supported multiplicative functions f are well-distributedin arithmetic progressions a a − (mod q ) on average over moduli q ≤ x / − ε with( q, a a ) = 1. In memory of Klaus Roth Introduction
In this paper we prove a Bombieri-Vinogradov type theorem for general multiplicativefunctions supported on smooth numbers, with a fixed member of the residue class. Given amultiplicative function f , we define, whenever ( a, q ) = 1,∆( f, x ; q, a ) := X n ≤ xn ≡ a (mod q ) f ( n ) − ϕ ( q ) X n ≤ x ( n,q )=1 f ( n ) . We wish to prove that, for an arbitrary fixed
A > X q ∼ Q ( a,q )=1 | ∆( f, x ; q, a ) | ≪ x (log x ) A where, here and henceforth, “ q ∼ Q ” denotes the set of integers q in the range Q < q ≤ Q ,for as large values of Q as possible. Let F ( s ) = ∞ X n =1 f ( n ) n s and − F ′ ( s ) F ( s ) = ∞ X n =2 Λ f ( n ) n s , for Re( s ) >
1. Following [5], we restrict attention to the class C of multiplicative functions f for which | Λ f ( n ) | ≤ Λ( n ) for all n ≥ . Date : November 10, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Multiplicative functions, Bombieri-Vinogradov theorem, Siegel zeroes.A.G. has received funding from the European Research Council grant agreement n o This includes most 1-bounded multiplicative functions of interest, including all 1-boundedcompletely multiplicative functions. Two key observations are that if f ∈ C then each | f ( n ) | ≤
1, and if f ∈ C and F ( s ) G ( s ) = 1 then g ∈ C .In [6] the last two authors showed that there are two different reasons that the sum in(1.1) might be ≫ x/ log x . First f might be a character of small conductor (for example f ( n ) = ( n/ f might have beenselected so that f ( p ) works against us for most primes p in the range x/ < p ≤ x . Wehandled these potential pretentious problems as follows.To avoid issues with the values f ( p ) at the large primes p we only allow f to be supportedon y -smooth integers for y = x θ , for some small θ > f correlating with a given character χ , note that thishappens when S f ( X, χ ) := X n ≤ X f ( n ) χ ( n )is “large” (that is, ≫ X , or ≫ X/ (log X ) A ) for some X in the range x / < X ≤ x , in whichcase (1.1) might well be false. We can either assume that this is false for all χ (which isequivalent to what is known as a “Siegel-Walfisz criterion” in the literature), or we can takeaccount of such χ in the “Expected Main Term”. We will begin by doing the latter, andthen deduce the former as a corollary.We start by stating the Siegel-Walfisz criterion: The Siegel-Walfisz criterion : For any fixed
A >
0, we say that f satisfies the A -Siegel-Walfiszcriterion if for any ( a, q ) = 1 and any x ≥ | ∆( f, x ; q, a ) | ≪ A x ) A X n ≤ x | f ( n ) | . We say that f satisfies the Siegel-Walfisz criterion if it satisfies the A -Siegel-Walfisz criterionfor all A > q be the set of those characters (mod q ) whichare induced by the characters in Ξ. Then denote∆ Ξ ( f, x ; q, a ) := X n ≤ xn ≡ a (mod q ) f ( n ) − ϕ ( q ) X χ ∈ Ξ q χ ( a ) S f ( x, χ )In [6] we proved the following result: Theorem 1.1.
Fix δ, B > . Let y = x ε for some ε > sufficiently small in terms of δ . Let f ∈ C be a multiplicative function which is only supported on y -smooth integers. Then thereexists a set, Ξ , of primitive characters, containing ≪ (log x ) B +7+ o (1) elements, such that for That is, integers all of whose prime factors are ≤ y . MOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN APS 3 any ≤ | a | ≪ Q ≤ x − δ , we have X q ∼ Q ( a,q )=1 | ∆ Ξ ( f, x ; q, a ) | ≪ x (log x ) B . Moreover, if f satisfies the Siegel-Walfisz criterion then X q ∼ Q ( a,q )=1 | ∆( f, x ; q, a ) | ≪ x (log x ) B . In this article we develop Theorem 1.1 further, allowing Q as well as y to vary over a muchwider range, and obtaining upper bounds in terms of (the more appropriate) Ψ( x, y ), thenumber of y -smooth integers up to x . Theorem 1.2.
Fix ε, A > . Suppose that f ∈ C , and is only supported on y -smoothnumbers, where (1.2) x δ > y ≥ exp (cid:18) · √ log x log log x √ log log log x (cid:19) for some sufficiently small δ > . Then there exists a set, Ξ , of primitive characters, con-taining ≪ (log x ) A +38 elements, such that if ≤ | a | , | a | ≤ x δ then X q ≤ x / − ε ( q,a a )=1 | ∆ Ξ ( f, x ; q, a a ) | ≪ Ψ( x, y )(log x ) A . Moreover, if f satisfies the Siegel-Walfisz criterion then X q ≤ x / − ε ( q,a a )=1 | ∆( f, x ; q, a a ) | ≪ Ψ( x, y )(log x ) A . It would be interesting to extend the range (1.2) in Theorem 1.2 down to any y ≥ (log x ) C for some large constant C . We discuss the main issue that forces us to restrict the range inTheorem 1.2 to y > exp((log x ) / o (1) ) in Remark 4.3. In our proofs we have used the range y ≥ (log x ) C when we can, as an aid to future research on this topic, and to make clear whatare the sticking points.Fouvry and Tenenbaum (Th´eor`eme 2 in [4]) established such a result when f is the char-acteristic function of the y -smooth integers (with y < x δ ) and a = 1, in the same range q ≤ x / − ε , but with the bound ≪ x/ (log x ) A . This was improved by Drappeau [2] to ≪ Ψ( x, y ) / (log x ) A for (log x ) C < y ≤ x δ .The proof of Theorem 1.2 combines the ideas from our earlier articles [2] and [6]. Perhapsthe most innovative feature of this article, given [2] and [6], comes in Theorem 5.1 in which weprove a version of the classical large sieve inequality (towards which Roth’s work [10] playeda pivotal role) for (the notably sparse) sequences supported on the y -smooth numbers, whichmay be of independent interest. S. DRAPPEAU, A. GRANVILLE, AND X. SHAO Reduction to a larger set of exceptional moduli
We begin by modifying estimates from [2] to prove Theorem 2.1, which is a version ofTheorem 1.2 with a far larger exceptional set of characters. This is key to the proof ofTheorem 1.2 since we now only need to cope with relatively small moduli. We thereforedefine A ( D ) to be the set of all primitive characters of conductor ≤ D . Theorem 2.1.
For fixed ε, A > , there exist C, δ > such that for any y in the range (log x ) C < y ≤ x δ , and any f ∈ C which is only supported on y -smooth numbers, we have X q ≤ x / − ε ( q,a a )=1 | ∆ A ( f, x ; q, a a ) | ≪ A Ψ( x, y )(log x ) A , for any integers a , a for which ≤ | a | , | a | ≤ x δ , with A = A ( D ) where D =( x/ Ψ( x, y )) (log x ) A +20 . We prove this by modifying some of the estimates in [2]. For any D ≥ q ≥ u D ( n ; q ) = n ≡ q − ϕ ( q ) X χ mod q cond( χ ) ≤ D χ ( n ) , so that ∆ A ( f, x ; q, a a ) = X n ≤ x f ( n ) u D ( na a ; q ) . Note that u D ( n ; q ) = 0 unless ( n, q ) = 1 and q > D , in which case(2.2) | u D ( n ; q ) | ≤ n ≡ q + 1 ϕ ( q ) X r ≤ Dr | q ϕ ( r ) ≤ n ≡ q + Dτ ( q ) ϕ ( q ) . For ( n, q ) = 1, since X χ (mod q )cond( χ ) ≤ D χ ( n ) = X s ≤ Ds | q X ψ (mod s ) ψ primitive ψ ( n ) = X s ≤ Ds | q X d | s µ ( s/d ) ϕ ( d ) d | n − , by letting b = s/d we obtain the alternate expression(2.3) u D ( n ; q ) = n ≡ q − ϕ ( q ) X d ≤ Dd | ( q,n − ϕ ( d ) X b ≤ D/db | q/d µ ( b ) . Theorem 2.1 is an immediate consequence of Theorem 2.2.
MOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN APS 5
Theorem 2.2.
For any fixed ε > , there exists C, δ > such that whenever ≤ D ≤ x δ , (log x ) C ≤ y ≤ x δ , we have, uniformly for < | a | , | a | ≤ x δ and f ∈ C , (2.4) X q ≤ x / − ε ( q,a a )=1 (cid:12)(cid:12)(cid:12) X n ∈ S ( x,y ) f ( n ) u D ( na a ; q ) (cid:12)(cid:12)(cid:12) ≪ ε D − x (log x ) . To prove Theorem 2.2, we first prove the following generalisation of Theorem 3 of [2],where the bound D on the conductor is allowed to vary. Lemma 2.3.
Let
M, N, L, R ≥ and ( α m ) , ( β n ) , ( λ ℓ ) be three sequences, bounded in modulusby , supported on integers inside ( M, M ] , ( N, N ] , and ( L, L ] respectively. Let x = M N L .For any fixed ε > , there exists δ > such that whenever either the conditions (3.1), or theconditions (3.2) of [2] are met, we have (2.5) X R We follow closely the arguments of [2]. Roughly speaking, the main point is thatreducing the size of D only reduces the error terms, except in a certain diagonal contributionwhich yields the dominant error term, and which we analyse more carefully. Proceeding asin section 3 of [2], we reduce to the estimation of S − S ) + S , where S is defined inthe first display of [2, page 838], S = X R To deduce Theorem 2.2 from Lemma 2.3, we start with the following special case ofTheorem 2.2. Note that there is a factor ( τ ( d ) log K ) missing in the third display, p.852 of [2]. MOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN APS 7 Proposition 2.4. Theorem 2.2 holds true for functions f supported on squarefree integers.Proof. We extend the arguments of pages 852-853 of [2], renaming the variable q into r .Suppose first R ≥ x / . We restrict n and r to dyadic intervals x < n ≤ x and R < r ≤ R .Choosing the parameters ( M , N , L ) as in [2, p.852, last display], we obtain X R We let K be the set of powerfulnumbers, that is for k ∈ K if prime p divides k then p also divides k . Note that |K ∩ [1 , x ] | ≪ x . Out of every n counted in the left-hand side of (2.4), we extract thelargest powerful divisor k . Then from the triangle inequality and the bound | f ( k ) | ≤ 1, theleft-hand side of (2.4) is at most(2.7) X q ≤ x / − ε ( q,a a )=1 X k ∈K∩ S ( x,y )( k,q )=1 (cid:12)(cid:12)(cid:12) X n ∈ S ( x/k,y )( n,k )=1 µ ( n ) f ( n ) u D ( kna a ; q ) (cid:12)(cid:12)(cid:12) . Let K ≥ k > K ,getting X q ≤ x / − ε X k ∈K k>K X n ≤ x/k (cid:16) ka n ≡ a mod q + Dτ ( q ) ϕ ( q ) (cid:17) = T + T , S. DRAPPEAU, A. GRANVILLE, AND X. SHAO say, where we have separated the contribution of the two summands. Executing the sumover q first, and separating the case kn | a , we find T ≤ X q ≤ x / − ε τ ( | a | ) + X k ∈K k>K X n ≤ x/k τ ( | kna − a | ) ≪ x / + x ε DK − . It is easy to see that T ≪ x ε DK − as well. Next, to each 1 ≤ k ≤ K in (2.7), by hypoth-esis, we may use Proposition 2.4 with x ← x/k , a ← ka and f ( n ) ← ( n,k )=1 µ ( n ) f ( n ),and obtain the existence of C, δ > | a | , | a k | ≤ x δ and (log x ) C ≤ y ≤ x δ , X q ≤ x / − ε ( q,a a )=1 (cid:12)(cid:12)(cid:12) X n ∈ S ( x/k,y )( n,k )=1 µ ( n ) f ( n ) u D ( na ka ; q ) (cid:12)(cid:12)(cid:12) ≪ D − k − x (log x ) . We take δ = δ / K = x δ / , and sum over k ≤ K , using P k ∈K k − < ∞ . By hypothesis D ≤ x δ , so that x / ≪ DK − x ≪ D − x − δ/ , and we find that (2.7) is at most ≪ D − x (log x ) as claimed. (cid:3) Altering the set of exceptional characters To prove Theorem 1.2 we need to reduce the set of exceptional characters from A ( D ) toΞ. We shall set this up in Proposition 3.2.It is convenient to write b = a /a (which is ≡ a a (mod q )) and to define ( q, b ) to mean( q, a a ). Thus in Theorem 2.1 we are working with X q ≤ Q ( q,b )=1 | ∆ A ( f, x ; q, b ) | for Q = x / − ε . Lemma 3.1. Let A = A ( D ) for some D ≥ . Suppose that Ξ ⊂ A . If ( b, q ) = 1 then ∆ Ξ ( f, x ; q, b ) − ∆ A ( f, x ; q, b )= 1 ϕ ( q ) X ℓ ≥ p | ℓ = ⇒ p | q g ( ℓ ) X d ≤ D ( d,ℓ )=1 d | q ϕ ( d )∆ Ξ ( f, x/ℓ ; d, bℓ ) X n ≤ D/dn | q/d µ ( n ) , where g is the multiplicative function with F ( s ) G ( s ) = 1 . MOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN APS 9 Proof. If ( b, q ) = 1 then∆ Ξ ( f, x ; q, b ) − ∆ A ( f, x ; q, b ) = 1 ϕ ( q ) X χ ∈A q χ Ξ q χ ( b ) S f ( x, χ )= 1 ϕ ( q ) X m ≤ Dm | q X χ ∈P ( m ) q χ Ξ q χ ( b ) S f ( x, χ ) , where P ( m ) denotes the set of primitive characters (mod m ), as A is the set of all primitivecharacters of conductor ≤ D . Let C ( m ) denote the set of all characters (mod m ). For m | q we define ∆ Ξ ,q ( f, x ; m, b ) := X n ≤ xn ≡ a (mod m )( n,q )=1 f ( n ) − ϕ ( m ) X χ ∈C ( m ) q ∩ Ξ q χ ( b ) S f ( x, χ )= 1 ϕ ( m ) X χ ∈C ( m ) q χ Ξ q χ ( b ) S f ( x, χ )= 1 ϕ ( m ) X d | m X χ ∈P ( d ) q χ Ξ q χ ( b ) S f ( x, χ ) . By M¨obius inversion we deduce that, for m | q , X χ ∈P ( m ) q χ Ξ q χ ( b ) S f ( x, χ ) = X d | m µ ( m/d ) ϕ ( d )∆ Ξ ,q ( f, x ; d, b ) . Next we wish to better understand ∆ Ξ ,q ( f, x ; m, a ). Let f q ( p k ) = f ( p k ) if p | q, p ∤ m , and f q ( p k ) = 0 otherwise. Define g q from g in a similarly way. Note f q and g q are simply f and g supported on the integers composed from the prime factors of q . If ( a, m ) = 1 then∆ Ξ ( f, x ; m, a ) = X ℓ ≥ ℓ,m )=1 f q ( ℓ )∆ Ξ ,q ( f, x/ℓ ; m, aℓ ) , and since F q G q = 1 we have∆ Ξ ,q ( f, x ; m, a ) = X ℓ ≥ ℓ,m )=1 g q ( ℓ )∆ Ξ ( f, x/ℓ ; m, aℓ )= X ℓ ≥ p | ℓ = ⇒ p | q ( ℓ,m )=1 g ( ℓ )∆ Ξ ( f, x/ℓ ; m, aℓ ) . Substituting this in above then yields X χ ∈P ( m ) q χ Ξ q χ ( b ) S f ( x, χ ) = X d | m µ ( m/d ) ϕ ( d ) X ℓ ≥ p | ℓ = ⇒ p | q, ( ℓ,d )=1 g ( ℓ )∆ Ξ ( f, x/ℓ ; d, bℓ ) , and the result follows writing m = dn . (cid:3) Proposition 3.2. Let the notations and assumptions be as in the statement of Theorem 2.1.Suppose that Ξ ⊂ A . Then X q ≤ x / − ε ( b,q )=1 | ∆ Ξ ( f, x ; q, b ) | ≤ O (cid:18) Ψ( x, y )(log x ) A (cid:19) + (log x ) X ℓ ≤ XL := Q p | ℓ p τ ( L ) ϕ ( L ) X d ≤ D ( d,ℓ )=1 | ∆ Ξ ( f, x/ℓ ; d, bℓ ) | , where X = (1 + β ) (log x ) A +6 with β = β (Ξ) := X ψ (mod r ψ ) ∈ Ξ r ψ . Proof. Set Q = x / − ε . We deduce from Lemma 3.1, as each | g ( ℓ ) | ≤ f, g ∈ C , that X q ≤ Q ( b,q )=1 | ∆ Ξ ( f, x ; q, b ) | ≤ X q ≤ Q ( b,q )=1 | ∆ A ( f, x ; q, b ) | + X ℓ ≥ L := Q p | ℓ p X d ≤ D ( d,ℓ )=1 ϕ ( d ) | ∆ Ξ ( f, x/ℓ ; d, bℓ ) | X q ≤ Q ( b,q )=1 dL | q ϕ ( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ D/dn | q/d µ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The first term on the right-hand side is ≪ A Ψ( x, y ) / (log x ) A by Theorem 2.1. For the sumat the end we have an upper bound ≤ X q ≤ QdL | q τ ∗ ( q/d ) ϕ ( q ) ≤ τ ∗ ( L ) ϕ ( dL ) X r ≤ Q/dL τ ∗ ( r ) ϕ ( r ) ≪ τ ( L ) ϕ ( d ) ϕ ( L ) (log x ) , MOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN APS 11 where τ ∗ ( m ) denotes the number of squarefree divisors of m , writing q = dLr , as L issquarefree. Therefore X q ≤ Q ( b,q )=1 | ∆ Ξ ( f, x ; q, b ) | ≤ O (cid:18) Ψ( x, y )(log x ) A (cid:19) + (log x ) X ℓ ≥ L := Q p | ℓ p τ ( L ) ϕ ( L ) X d ≤ D ( d,ℓ )=1 | ∆ Ξ ( f, x/ℓ ; d, bℓ ) | . We will attack this last sum first by employing relatively trivial bounds for the terms with ℓ that are not too small, so that we only have to consider ℓd that are smallish in furtherdetail. Now Theorem 1 of [3] gives the upper bound(3.1) Ψ q ( x, y ) ≪ ϕ ( q ) q Ψ( x, y )provided x ≥ y ≥ exp((log log x ) ) and q ≤ x . Therefore | ∆ Ξ ( f, x ; q, a ) | ≤ Ψ( x, y ; a, q ) + | Ξ q | ϕ ( q ) Ψ q ( x, y ) ≪ Ψ( x, y ; a, q ) + | Ξ q | q Ψ( x, y )in this range. Substituting in, the upper bound on the ℓ th term above becomes ≪ (log x ) · τ ( L ) ϕ ( L ) X d ≤ D ( d,ℓ )=1 Ψ( x/ℓ, y ; d, bℓ ) + Ψ( x/ℓ, y ) X d ≤ D ( d,ℓ )=1 | Ξ d | d . The second sum over d is therefore X ψ (mod r ψ ) ∈ Ξ X d ≤ D ( d,ℓ )=1 r ψ | d d ≪ β ϕ ( ℓ ) ℓ log D. For the first term we use Theorem 1 of [7] which yields that X d ≤ D ( d,ℓ )=1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ( x/ℓ, y ; d, bℓ ) − Ψ d ( x/ℓ, y ) ϕ ( d ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ Ψ( x/ℓ, y )(log x ) A , as D ≤ p Ψ( x, y ) / (log x ) B since y ≥ (log x ) C . Therefore, expanding the sum and using(3.1), we obtain X d ≤ D ( d,ℓ )=1 Ψ( x/ℓ, y ; d, bℓ ) ≪ X d ≤ D ( d,ℓ )=1 Ψ( x/ℓ, y ) d + Ψ( x/ℓ, y )(log x ) A ≪ ϕ ( ℓ ) ℓ log D · Ψ( x/ℓ, y ) . By Th´eor`eme 2.1 of [1] we have(3.2) Ψ( x/ℓ, y ) ≪ Ψ( x, y ) /ℓ α , where α > / y . Therefore in total the ℓ th term in ≪ τ ( L ) Lℓ α Ψ( x, y )(log x ) (log D ) · (1 + β ) . Summing over ℓ > X , we obtain, taking σ = α − / X ℓ ≥ XL := Q p | ℓ p τ ( L ) Lℓ α ≤ X ℓL := Q p | ℓ p τ ( L ) Lℓ α ( ℓ/X ) σ = X − σ Y p (cid:18) p ( p / − (cid:19) ≪ X − / . Taking X = (1 + β ) (log x ) A +6 , the contribution of the ℓ > X is therefore ≪ Ψ( x, y ) / (log x ) A . Combining all of the above then yields the result. (cid:3) Putting the pieces together In order to prove Theorem 1.2 we need Theorem 2.1, Proposition 3.2, Corollary 6.1 (whichwill be proved in the final two sections), and the following result which is Proposition 5.1 of[6]. Proposition 4.1. Fix B ≥ and < η < . Given (log x ) B +5 ≤ y = x /u ≤ x / − η let R = R ( x, y ) := min { y log log log x u , x η x } ( ≤ y / ) . Suppose that f ∈ C , and is only supported on y -smooth numbers. There exists a set, Ξ , ofprimitive characters ψ (mod r ) with r ≤ R , such that if q ≤ R and ( a, q ) = 1 then | ∆ Ξ ( f, x ; q, a ) | ≪ ϕ ( q ) Ψ( x, y )(log x ) B . Moreover, one may take Ξ to be Ξ = Ξ(2 B +2 ) , where Ξ( C ) is the set of primitive characters ψ (mod r ) with r ≤ R such that there exists x η < X ≤ x for which (4.1) | S f ( X, ψ ) | ≥ Ψ( X, y )( u log u ) (log x ) C . Proof of Theorem 1.2. Let c > D =( x/ Ψ( x, y )) (log x ) A +20 , and one easily verifies that the hypothesis for y implies that(4.2) D ≤ min( R, y c , exp( c log x/ log log x ))from the usual estimate Ψ( x, y ) = xu − u + o ( u ) for smooth numbers. We will prove Theorem 1.2with Ξ = Ξ(2 A + 8 ), where Ξ( C ) is the set of primitive characters ψ (mod r ) with r ≤ D ,such that there exists x / < X ≤ x for which (4.1) holds. By Proposition 4.1 with B = A +3and η = 1 / 4, we have the bound | ∆ Ξ ( f, x ; q, a ) | ≪ ϕ ( q ) Ψ( x, y )(log x ) A +3 MOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN APS 13 whenever q ≤ D and ( a, q ) = 1. Moreover, we have the same bound with x replaced by x/ℓ for any ℓ = x o (1) .The goal of the next two sections will be to prove Corollary 6.1, which implies that | Ξ | ≪ (log x ) A +38 . This implies that β (Ξ) ≪ (log x ) A +19 , and so X ≪ (log x ) A +44 in Proposition 3.2. Thusfor each ℓ ≤ X and d ≤ D , we have | ∆ Ξ ( f, x/ℓ ; d, bℓ ) | ≪ ϕ ( d ) Ψ( x/ℓ, y )(log x ) A +3 ≪ ϕ ( d ) ℓ α Ψ( x, y )(log x ) A +3 , by (3.2). Therefore(log x ) X ℓ ≤ XL := Q p | ℓ p τ ( L ) ϕ ( L ) X d ≤ D ( d,ℓ )=1 | ∆ Ξ ( f, x/ℓ ; d, bℓ ) |≪ Ψ( x, y )(log x ) A +1 X ℓ ≤ XL := Q p | ℓ p τ ( L ) ϕ ( L ) ℓ α X d ≤ D ( d,L )=1 ϕ ( d ) ≪ Ψ( x, y )(log x ) A X ℓ ≤ XL := Q p | ℓ p τ ( L ) Lℓ α ≤ Ψ( x, y )(log x ) A Y p ≤ X (cid:18) p ( p α − (cid:19) ≪ Ψ( x, y )(log x ) A . We therefore deduce from Proposition 3.2 that X q ≤ x / − ε ( q,a a )=1 | ∆ Ξ ( f, x ; q, a a ) | ≪ Ψ( x, y )(log x ) A as desired.To deduce the second part of Theorem 1.2, about functions f satisfying the Siegel-Walfiszcriterion, we use the following variant of Proposition 3.4 in [6]: Proposition 4.2. Fix ε > . Let (log x ) ε ≤ y ≤ x be large. Let f ∈ C be a multiplicativefunction supported on y -smooth integers. Suppose that Ξ is a set of primitive characters,containing ≪ (log x ) C elements, such that X q ∼ Q | ∆ Ξ ( f, x ; q, a q ) | ≪ Ψ( x, y )(log x ) B , for ( a q , q ) = 1 for all q ∼ Q , where Q ≤ x . If the D -Siegel-Walfisz criterion holds for f ,where D ≥ B + C , then X q ∼ Q | ∆( f, x ; q, a q ) | ≪ Ψ( x, y )(log x ) B . Proof. By the definition of ∆ Ξ we have | ∆( f, x ; q, a q ) | ≤ | ∆ Ξ ( f, x ; q, a q ) | + 1 ϕ ( q ) X χ (mod q ) χ ∈ Ξ q ,χ = χ | S f ( x, χ ) | . Summing this over q ∼ Q and using the hypothesis, we deduce that X q ∼ Q | ∆( f, x ; q, a q ) | ≤ X ψ ∈ Ξ ψ =1 X r ψ | q ∼ Qχ (mod q ) induced by ψ | S f ( x, χ ) | ϕ ( q ) + O (cid:18) Ψ( x, y )(log x ) B (cid:19) . It suffices to show that, for each fixed ψ (mod r ) ∈ Ξ with r > 1, we have(4.3) X r | q ∼ Qχ (mod q ) induced by ψ | S f ( x, χ ) | ϕ ( q ) ≪ Ψ( x, y )(log x ) D . The conclusion then follows since | Ξ | ≪ (log x ) C and D ≥ B + C . If χ (mod q ) is inducedby ψ (mod r ), then there is a multiplicative function h supported only on powers of primeswhich divide q but not r , such that h ∗ f ψ = f χ . Note that h ∈ C since f ∈ C , and inparticular h is 1-bounded. It follows that | S f ( x, χ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X m ≤ x h ( m ) S f ( x/m, ψ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X m ≤ xp | m ⇒ p | q,p ∤ r | S f ( x/m, ψ ) | . Since f satisfies the D -Siegel-Walfisz criterion, we have S f ( x/m, ψ ) ϕ ( r ) = 1 ϕ ( r ) X a (mod r ) ψ ( a )∆( f, x/m ; r, a ) ≪ Ψ( x/m, y )(log( x/m )) D . Using the bound Ψ( x/m, y ) ≪ m − α Ψ( x, y ) where α = α ( x, y ) ≥ ε + o (1), we may bound theleft hand side of (4.3) by(4.4) Ψ( x, y ) X r | q ∼ Q ϕ ( r ) ϕ ( q ) X m ≤ xp | m ⇒ p | q,p ∤ r m α (log( x/m )) D . To analyze the inner sum over m , we break it into two pieces depending on whether m ≤ x / or m > x / : X m ≤ xp | m ⇒ p | q,p ∤ r m α (log( x/m )) D ≪ x ) D X m ≤ x / p | m ⇒ p | q,p ∤ r m α + X x / In this section we prove a large sieve inequality for sequences supported on smooth num-bers, a result which may be of independent interest. Theorem 5.1 (Large sieve for smooth numbers) . There exists C, c > such that the follow-ing statement holds. Let (log x ) C ≤ y ≤ x be large, and Q = min( y c , exp( c log x/ log log x )) . For any sequence { a n } we have X q ≤ Q ∗ X χ (mod q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ xP ( n ) ≤ y a n χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ Ψ( x, y ) · X n ≤ xP ( n ) ≤ y | a n | . The upper bound is sharp up to a constant, as may be seen by taking each a n = 1, so thatthe χ = 1 term on the left-hand side equals Ψ( x, y ) , the size of the right-hand side. Thisresult has the advantage over the traditional large sieve inequality that the sequence { a n } issupported on a sparse set (when y = x o (1) ), but the disadvantage that this inequality holdsin a much smaller range for q than the usual q ≪ x / . It may well be that Theorem 5.1holds with Q = Ψ( x, y ) / .5.1. Zero-density estimates. To prove Theorem 5.1, we will use the following two conse-quences of deep zero-density results in the literature. The first is a bound for character sumsover smooth numbers assuming a suitable zero-free region for the associated L -function (seeSection 3 of [7]). Proposition 5.2. There is a small positive constant δ > and a large positive constant κ > such that the following statement holds. Let (log x ) . ≤ y ≤ x be large. Let χ (mod q ) be a non-principal character with q ≤ x and conductor r := cond( χ ) ≤ x δ . If L ( s, χ ) has no zeros in the region (5.1) Re( s ) > − ε, | Im( s ) | ≤ T, where the parameters ε, T satisfy (5.2) κ log y < ε ≤ α ( x, y )2 , y . ε (log x ) ≤ T ≤ x δ , and moreover (5.3) either y ≥ ( T r ) κ or ε ≥ 40 log log( qyT ) / log y. MOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN APS 17 Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ xP ( n ) ≤ y χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ Ψ( x, y ) p (log x )(log y )( x − . ε log T + T − . ) . We also need the following log-free zero-density estimate by Huxley and Jutila, which canbe found in Section 2 of [7]: Proposition 5.3. Let ε ∈ [0 , / , T ≥ and Q ≥ . Then the function G Q ( s ) = Q q ≤ Q Q ∗ χ (mod q ) L ( s, χ ) has ≪ ( Q T ) ε zeros s , counted with multiplicity, inside the region (5.1) . Proof of Theorem 5.1. It suffices to prove its dual form: Proposition 5.4. There exist C, c > such that the following statement holds. Let (log x ) C ≤ y ≤ x be large, and Q = min( y c , exp( c log x/ log log x )) . For any sequence { b χ } we have X n ≤ xP ( n ) ≤ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X q ≤ Q ∗ X χ (mod q ) b χ χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ Ψ( x, y ) · X q ≤ Q ∗ X χ (mod q ) | b χ | . Proof. The left hand side can be bounded by(5.4) X χ X χ b χ b χ X n ≤ xP ( n ) ≤ y ( χ χ )( n ) . Thus we need to understand character sums over smooth numbers. The contribution fromthe diagonal terms with χ = χ is clearly acceptable, and thus we focus on non-diagonalterms. For η ∈ (0 , / η ) to be the set of all non-principal characters χ (mod q )with q ≤ Q , such that η Ψ( x, y ) < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ xP ( n ) ≤ y χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ η Ψ( x, y ) . Furthermore, define Ξ ∗ ( η ) to be the set of primitive characters which induce a character inΞ( η ). The contribution to (5.4) from those χ , χ with χ χ ∈ Ξ( η ) is ≤ η Ψ( x, y ) X χ ,χ χ χ ∈ Ξ( η ) | b χ b χ | ≤ η Ψ( x, y ) X χ ,χ χ χ ∈ Ξ( η ) | b χ | ≤ η Ψ( x, y ) X ψ ∈ Ξ ∗ ( η ) X χ | b χ | X χ χ χ induced by ψ . We claim that for a given χ (mod q ) and a given primitive ψ ∈ Ξ ∗ ( η ), there is at mostone primitive character χ such that χ χ is induced by ψ . To see this, suppose that thereare two primitive characters χ (mod q ) and χ ′ (mod q ′ ) such that both χ χ := χ and χ χ ′ := χ ′ are induced by ψ . It suffices to show that χ ( n ) = χ ′ ( n ) whenever ( n, q q ′ ) = 1,since this would imply that χ χ ′ is the principal character, and thus χ = χ ′ as they areboth primitive.If ( n, q q ′ ) = 1, then we may find an integer k such that ( n + kq q ′ , q q q ′ ) = 1. Thus χ ( n + kq q ′ ) = 0 and χ ( n + kq q ′ ) = χ ′ ( n + kq q ′ ). It follows that χ ( n ) = χ ( n + kq q ′ ) = ( χχ )( n + kq q ′ ) = ( χ ′ χ )( n + kq q ′ ) = χ ′ ( n + kq q ′ ) = χ ′ ( n ) . This completes the proof of the claim.It follows that the contribution to (5.4) from those χ , χ with χ χ ∈ Ξ( η ) is ≪ η Ψ( x, y ) | Ξ ∗ ( η ) | X χ | b χ | . We will show that | Ξ ∗ ( η ) | ≪ η − / so that the result follows by summing over η dyadically.We may assume that η ≥ Q − , as the bound follows for smaller η from the trivial bound | Ξ ∗ ( η ) | ≤ Q .We now use Proposition 5.2 to show that if χ ∈ Ξ( η ) then L ( s, χ ) has a zero in the region(5.1) for suitable values of ε and T . This would imply that L ( s, ψ ) has zero in the region (5.1)for any ψ ∈ Ξ ∗ ( η ).For the purpose of contradiction, let’s assume that χ ∈ Ξ( η ) and L ( s, χ ) has no zero in(5.1) with T = Q . We wish to verify the hypotheses in (5.2) and (5.3). The upper boundon T in (5.2) follows from the definition of Q . Now r ≤ q ≤ Q and so the first alternativeof (5.3) follows by selecting c so that 502 cκ ≤ 1. We define ε = max (cid:18) κ log y , η − + log log x )log x (cid:19) . Since log η − ≤ Q and log log x ≤ log Q , we have the upper bound ε ≤ max (cid:18) κ log y , 108 log Q log x (cid:19) ≪ x , MOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN APS 19 so that ε = o (1) and y ε ≪ Q . The first hypothesis in (5.2) follows immediately. Finally,by selecting C so that cC ≥ Q ≥ (log x ) , so that the lower bound on T in (5.2) follows easily.Now ε ≥ η − + log log x ) / log x so that x − . ε ≤ ( η/ log x ) . , and therefore p (log x )(log y )( x − . ε log T + T − . ) ≤ η . (log x ) − . + Q − log x. Now Q − log x = o ( Q − ) = o ( η ), as Q ≥ (log x ) . Therefore Proposition 5.2 implies that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ xP ( n ) ≤ y χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ o ( η Ψ( x, y )) , contradicting the definition of Ξ( η ).By Proposition 5.3 we now deduce (remembering that characters in Ξ ∗ ( η ) have conductorsat most Q ) that | Ξ ∗ ( η ) | ≪ ( Q T ) ε = Q ε ≪ η − / , which completes the proof. (cid:3) Remark . Assuming the Riemann hypothesis for Dirichlet L -functions, by Proposition 1of [7], we have the bound P n ≤ x,P ( n ) ≤ y χ ( n ) = O ( x − c ) uniformly for χ non-principalmod q , q ≤ x c , (log x ) C ≤ y ≤ x c , for some absolute constants C, c > 0. This impliesan upper bound ≪ (cid:16) Ψ( x, y ) + Q x − c (cid:17) X χ | b χ | for (5.4), for all Q ≤ x c , and Theorem 5.1 would hold with Q = x c/ and C large enough.5.3. A variant of Theorem 5.1. We may extend the range to Q = x c in Theorem 5.1unconditionally if we insert some weights that reduce the effects of characters with largeconductor. Theorem 5.6. There exists C, c > such that the following statement holds. Let (log x ) C ≤ y ≤ x be large, and let Q = x c . For any sequence { a n } we have X q ≤ Q q / ∗ X χ (mod q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ xP ( n ) ≤ y a n χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ Ψ( x, y ) · X n ≤ xP ( n ) ≤ y | a n | . Proof. The proof is similar as the proof of Theorem 5.1. We begin by passing to its dualform, so that we need to prove that X n ≤ xP ( n ) ≤ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X q ≤ Q q / ∗ X χ (mod q ) b χ χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ Ψ( x, y ) · X q ≤ Q q / ∗ X χ (mod q ) | b χ | for any sequence { b χ } , where the summation is over all primitive characters χ (mod q ) with q ≤ Q . Expanding the square, we can bound the left hand side above by(5.5) X χ (mod q ) X χ (mod q ) | b χ b χ | ( q q ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ xP ( n ) ≤ y ( χ χ )( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For η ∈ (0 , / η ) and Ξ ∗ ( η ) as in the proof of Theorem 5.1. If χ χ ∈ Ξ( η ) sothat it is induced by some ψ (mod r ) ∈ Ξ ∗ ( η ), then | b χ b χ | ( q q ) / ≤ r / | b χ | q / + | b χ | q / ! . Thus the contribution to (5.5) from those χ , χ with χ χ ∈ Ξ( η ) is ≪ η Ψ( x, y ) X ψ (mod r ) ∈ Ξ ∗ ( η ) r / X χ (mod q ) | b χ | q / X χ χ χ induced by ψ ≪ η Ψ( x, y ) X ψ (mod r ) ∈ Ξ ∗ ( η ) r / X χ (mod q ) | b χ | q / . Hence it suffices to show that X ψ (mod r ) ∈ Ξ ∗ ( η ) r / ≪ η − / , and then the conclusion follows after dyadically summing over η . For 1 ≤ R ≤ Q , let Ξ( η, R )and Ξ ∗ ( η, R ) be the set of characters in Ξ( η ) and Ξ ∗ ( η ) with conductors ∼ R , respectively.Thus it suffices to show that | Ξ ∗ ( η, R ) | ≪ η − / R / , for each η ∈ (0 , / 2] and 1 ≤ R ≤ Q . We may assume that η ≥ R − , since otherwise thetrivial bound | Ξ ∗ ( η, R ) | ≪ R suffices. From now on fix such η and R .We now use Proposition 5.2 to show that if χ ∈ Ξ( η, R ) then L ( s, χ ) has a zero in theregion (5.1) for suitable values of ε and T . This would imply that L ( s, ψ ) has zero in theregion (5.1) for any ψ ∈ Ξ ∗ ( η, R ).Set T = ( η − log x ) . If R ≤ (log x ) (say), then the first alternative of (5.3) holdsbecause (2 T R ) κ ≤ (log x ) κ ≤ y, provided that C ≥ κ . In this case we will set ε to be exactly the same as before: ε := max (cid:18) κ log y , η − + log log x )log x (cid:19) , if R ≤ (log x ) . MOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN APS 21 Then (5.2) can be easily verified, and the contrapositive of Proposition 5.2 implies that L ( s, χ ) has a zero in the region (5.1) whenever χ ∈ Ξ( η, R ). Hence by Proposition 5.3 wehave | Ξ ∗ ( η, R ) | ≪ ( R T ) ε ≪ η − ε R ε (log x ) ε ≪ η − / R / , since ε ≪ / log log x in this case.It remains to consider the case when (log x ) ≤ R ≤ Q . We set T as above, and we willnow set ε := max (cid:18) κ log y , η − + log log x )log x , 50 log log x log y (cid:19) , so that the second alternative in (5.3) is satisfied. One can still easily verify (5.2), and thusPropositions 5.2 and 5.3 combine to give | Ξ ∗ ( η, R ) | ≪ ( R T ) ε ≪ η − ε R ε (log x ) ε ≪ η − / R / , since ε ≤ / 300 (by choosing C large enough) and log x ≤ R / . This completes theproof. (cid:3) Examining the proof, one easily sees that the weight 1 /q / can be replaced by 1 /q σ forany constant σ > 0, and the statement remains true provided that C is large enough interms of σ . For our purposes, any exponent strictly smaller than 1 suffices.6. Bounding the number of exceptional characters Corollary 6.1. There exist C, c > such that the following statement holds. Let (log x ) C ≤ y ≤ x / be large. Let { a n } be an arbitrary -bounded sequence. For B ≥ , let Ξ( B ) be theset of primitive characters χ (mod r ) with r ≤ Q where Q := min( y c , exp( c log x/ log log x ) , such that there exists x / < X ≤ x for which (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ XP ( n ) ≤ y a n χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ Ψ( X, y )( u log u ) (log x ) B . Then | Ξ( B ) | ≪ (log x ) B +13 .Proof. Let T = ( u log u ) (log x ) B . We begin by partitioning the interval [ x / , x ] using asequence x / = X < X < · · · < X J − < X J = x with J ≍ T log x , such that X j +1 − X j ≍ εX j /T , for some fixed small enough ε > 0, for each 0 ≤ j < J .For each χ ∈ Ξ( B ), there exists some 0 ≤ j < J for which (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X j a n χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ Ψ( X j , y ) T − X X j By choosing ε sufficiently small we deduce that(6.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X j a n χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ Ψ( X j , y )2 T . We deduce that there exists some 0 ≤ j < J such that (6.2) holds for at least | Ξ( B ) | /J characters χ ∈ Ξ( B ). Therefore X χ ∈ Ξ( B ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X j P ( n ) ≤ y a n χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | Ξ( B ) | J · Ψ( X j , y ) T ≫ | Ξ( B ) | Ψ( X j , y ) T log x On the other hand, Theorem 5.1 implies that X χ ∈ Ξ( B ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X j P ( n ) ≤ y a n χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X r ≤ Q ∗ X χ (mod r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X j P ( n ) ≤ y a n χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ Ψ( X j , y ) , and therefore | Ξ( B ) | ≪ T log x = ( u log u ) (log x ) B +1 ≪ (log x ) B +13 , as claimed. (cid:3) We also record the following variant which gives a weighted count of exceptional characters,but now with the wider range Q = x c . Corollary 6.2. There exist C, c > such that the following statement holds. Let (log x ) C ≤ y ≤ x / be large. Let { a n } be an arbitrary -bounded sequence. For B ≥ , let Ξ( B ) be theset of primitive characters χ (mod r ) with r ≤ Q := x c , such that there exists x / < X ≤ x for which (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ XP ( n ) ≤ y a n χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ Ψ( X, y )( u log u ) (log x ) B . MOOTH-SUPPORTED MULTIPLICATIVE FUNCTIONS IN APS 23 Then X ψ (mod q ) ∈ Ξ( B ) q / ≪ (log x ) B +13 . Proof. The proof is the same as above, except that one considers the weighted sum X χ (mod q ) ∈ Ξ( B ) q / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X j P ( n ) ≤ y a n χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and use Theorem 5.6 instead of Theorem 5.1 in the last step. (cid:3) References [1] R. de la Bret`eche and G. Tenenbaum, Propri´et´es statistiques des entiers friables. Ramanujan J. (2005),13–202.[2] S. Drappeau, Th´eor`emes de type Fouvry-Iwaniec pour les entiers friables. Compos. Math. , (2015),828–862.[3] ´E. Fouvry and G. Tenenbaum, Entiers sans grand facteur premier en progressions arithmetiques, Proc.London Math. Soc. , (1991), 449–494.[4] ´E. Fouvry and G. Tenenbaum, R´epartition statistique des entiers sans grand facteur premier dans lesprogressions arithm´etiques, Proc. London Math. Soc. , (1996), 481–514.[5] A. Granville, A. Harper and K. Soundararajan, A new proof of Hal´asz’s Theorem, and some conse-quences. (preprint)[6] A. Granville and X. Shao, Bombieri-Vinogradov for multiplicative functions, and beyond the x / -barrier. (preprint)[7] A. Harper, Bombieri-Vinogradov and Barban-Davenport-Halberstam type theorems for smooth numbers. (preprint)[8] A. Hildebrand, Integers free of large prime divisors in short intervals, Quart. J. Math. Oxford , ,(1985), 57—69.[9] H. Iwaniec and E. Kowalski, Analytic number theory , American Mathematical Society Colloquium Pub-lications, , American Mathematical Society, Providence, RI, 2004.[10] K. F. Roth, On the large sieves of Linnik and R´enyi, Mathematika , , (1965), 1—9. SD: Aix Marseille Universit´e, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Mar-seille, France E-mail address : [email protected] AG: D´epartement de math´ematiques et de statistique, Universit´e de Montr´eal, CP 6128succ. Centre-Ville, Montr´eal, QC H3C 3J7, Canada; and Department of Mathematics,University College London, Gower Street, London WC1E 6BT, England. E-mail address : [email protected] XS: Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, OxfordOX2 6GG, United Kingdom E-mail address ::