Small gaps in coefficients of L-functions and B-free numbers in short intervals

Transcription

1 Rev. Mat. Iberoamericana 23 (2007), no., Small gaps in coefficients of L-functions and B-free numbers in short intervals Emmanuel Kowalski, Olivier Robert and Jie Wu Abstract We discuss questions related to the non-existence of gaps in the series defining modular forms and other arithmetic functions of various types, and improve results of Serre, Balog & Ono and Alkan using new results about exponential sums and the distribution of B-free numbers.. Introduction The motivation of this paper is a result of Serre ([43, Th. 5]) and the questions he subsequently raises. Let f be a primitive holomorphic cusp form (i.e. a newform in the Atkin-Lehner terminology) of weight k, with conductor N and nebentypus χ. Write (.) f(z) = n λ f (n)e(nz) its Fourier expansion at infinity, where e(z) =exp(2πiz), so that λ f (n) is also the Hecke eigenvalue of f for the Hecke operator T n. Serre s result is that (.2) {p x λ f (p) =0} x(log x) δ, for x 2andanyδ<, the implied constant depending on f and δ, from 2 which he deduces that the series (.), or equivalently the L-function (.3) L(f,s) = n λ f (n)n s is not lacunary, i.e. the set of indices n where λ f (n) 0, has a positive density Mathematics Subject Classification: F2, F30, F66, L5, N25. Keywords: B-free numbers, Fourier coefficients of modular forms, Rankin-Selberg convolution, exponential sums.

2 282 E. Kowalski, O. Robert and J. Wu Serre asked ([43, p. 83]) for more precise statements, in particular for bounding non-trivially the function i f (n) defined by (.4) i f (n) :=max{k λ f (n + j) =0for0<j k}, where non-trivial means an estimate of type i f (n) n θ for some θ<and all n. A stronger form of the problem is to find y as small as possible (as a function of x, sayy = x θ with θ<) such that (.5) {x <n x + y λ f (n) 0} y (where the implied constant can depend on f). Non-lacunarity means y = x is permitted, and one wishes to improve this. Note i f (n) y so this generalizes the first question. The history of this problem is somewhat confused. First, Serre could have solved it quite simply in (at least) two ways available at the time. The first is to argue that by multiplicativity λ f (n) 0ifnis squarefree and not divisible by primes p for which λ f (p) = 0. The latter have density zero by (.), so estimating i f (n) becomes a special case of a problem in multiplicative number theory, that of counting so-called B-free numbers in small intervals, where for a set B = {b i } of integers with (b i,b j )=if i j and < +, b i i one says that n isb-free if it is not divisible by any element in B. Erdös [] already showed in 966 that with no further condition there exists a constant θ< (absolute) such that the interval (x, x + x θ ]containsabfree number for x large enough, thereby solving Serre s first question in the affirmative. A quantitative result proving the analogue of (.5) for general B-free numbers was also obtained Szemerédi [44] as early as 973. This was apparently first noticed by Balog and Ono [2]. By this time results about B-free numbers had been refined a number of times, and they deduced from a result of Wu [45] that i f (n) n 7/4+ε for n andany ε>0, the implied constant depending on f and ε. Using this idea and other results (such as a version of the Chebotarev density theorem in small intervals and the Shimura correspondence), they also get weaker results for modular forms of weight or half-integral weight. The latter is noteworthy in this respect since the Fourier coefficients of half-integral weight forms are highly non-multiplicative (see [7] for a strong quantitative expression of this fact). Alkan [] has developed and improved the results of [2], tailoring some arguments to the specific instance of B-free numbers involved for the problem at hand.

3 Small gaps and B-free numbers 283 A second method of estimating i f (n) available to Serre was a direct appeal to the properties of the Rankin-Selberg L-function L(f f,s). Specifically this proves [36, 42] (for f any cusp form of integral weight k and level N) λ f (n) 2 n k = c f x + O(x 3/5 ) n x for some c f > 0, and x, the implied constant depending only on f. Trivially this implies i f (n) n 3/5, and incidentally this fact is implicit in [27] (which Serre quotes as one source for his problems!). It turns out however that there are still a number of things which seem to have been overlooked. For instance we will show that it is not necessary to sieve by squarefree numbers, and we will explain the applications of the Rankin-Selberg L-functions (in particular to non-congruence subgroups, another of the questions in [43]). We also look at lacunarity in some other Dirichlet series coming from arithmetic or analysis, including one which is really neither fish nor fowl (see Proposition 4). On the other hand (this is our main new contribution), we will improve quite significantly the B-free number results that can be used. Some of our tools are new estimates for exponential sums and bilinear forms which are of independent interest in analytic number theory. We of course welcome any further corrections to the picture thus produced about this problem. Acknowledgement. The authors would like to thank Emmanuel Royer for helpful comments on an earlier version of this paper. Notation. For any k, N and any character χ modulo N, we denote S k (N,χ) the vector space of cusp forms of weight k for the group Γ 0 (N), with nebentypus χ. Ifχ is the trivial character modulo N, wesimply write S k (N). We also denote by Sk (N,χ), or S k (N), the set of primitive forms in S k (N,χ) ors k (N), i.e. those forms which are eigenfunctions of all Hecke operators T n and of the Fricke involution and are normalized by λ f () =, where λ f (n) isthen-thfourier coefficient, which is then equal to the n-th Hecke eigenvalue. See e.g. [23] for basic analytic facts about modular forms. For s a complex number, we denote σ its real part and t its imaginary part. Also, we use f(x) =O(g(x)) and f(x) g(x) forx in some set X as synonyms, meaning f(x) Cg(x) for all x X, C 0 being called the implied constant.

4 284 E. Kowalski, O. Robert and J. Wu 2. Algebraic aspects We start by noticing that the restriction to squarefree numbers present in [2] and [] is in fact unnecessary, because the set of primes for which λ f (p ν )=0 for any ν still satisfies an estimate similar to (.2). This is partly implicit in [43, p ]. Lemma 2.. Let f Sk (N,χ) be a primitive holomorphic cusp form. There exists an integer ν f depending only on f such that for any prime p N, either λ f (p ν ) 0for all ν 0, or there exists ν ν f such that λ f (p ν )=0. Proof. Let p N. By multiplicativity we have the power series expansion (2.) λ f (p ν )X ν = ν 0 λ f (p)x + χ(p)p k X 2. Let α p and β p be the complex numbers such that (2.2) λ f (p)x + χ(p)p k X 2 =( α p X)( β p X). Thus (2.3) α p + β p = λ f (p) and α p β p = χ(p)p k 0. Expanding (2.) using (2.2) by geometric series gives the well-known expressions (2.4) λ f (p ν )= αν+ p βp ν+ α p β p if α p β p, and the simpler (2.5) λ f (p ν )=(ν +)α ν p =(ν +)τpν(k )/2 0 if α p = β p,whereτ 2 = χ(p). So we can assume α p β p.inthiscaseweget by (2.4) λ f (p ν ) = 0 if and only if (α p /β p ) ν+ = so that there exists ν 0forwhichλ f (p ν ) = 0 if and only if α p /β p is a root of unity, and if this ratio is a primitive root of unity of order d, then λ f (p d )=0. Now we input some more algebraic properties of the Fourier coefficients. The field K f = Q((λ f (n)) n, (χ(n)) n ) generated by all Fourier coefficients and values of χ is known to be a number field. By (2.2), the roots α p and β p lie in a quadratic extension of K f. This extension (say K p ) depends on p, but it has degree [K p : Q] 2[K f : Q] for all p.

5 Small gaps and B-free numbers 285 Now we combine both remarks and the fact that a number field L/Q can only contain a primitive d-th root of unity if ϕ(d) [L : Q]. It follows that if p N and λ f (p ν )=0forsomeν 0, α p /β p is a primitive root of unity of some order d such that ϕ(d) 2[K f : Q], and then λ f (p d ) = 0. Since ϕ(d) d/ log log(3d), this proves the lemma. It is clear that ν f is effectively computable. Here are some simple cases. Lemma 2.2. Let k be even and f Sk (N,χ). There exists M such that for any p M, eitherλ f (p) =0or λ f (p ν ) 0for any ν. If χ is trivial and f has integer coefficients, one can take M = N. Proof. If p N, the condition λ f (p ν )=0isequivalenttoλ f (p) =0by total multiplicativity, so we can assume that p N. Let p be such a prime with λ f (p ν )=0forsomeν 2, but λ f (p) 0. Using the same notation as the proof of Lemma 2., we have α p = ξβ p forsomerootofunityξ of order d +, andξ. We derive from the second relation of (2.3) that α 2 p = ξχ(p)pk, hence α p = ±τp (k )/2,whereτ 2 = ξχ(p). By the second relation of (2.3), we get λ f (p) =(+ ξ)α p = ±τ( + ξ)p (k )/2 0. In particular, since k is even, Q(τ( + ξ) p) K f. As K f is a number field, this can happen only for finitely many p, and one can take as M the product of those primes and those p N with λ f (p) =0. Furthermore, if χ is trivial and f has integer coefficients, then for p N, α p /β p = ξ is a root of unity in a quadratic extension of Q (see (2.2)), hence ξ {, ±j, ±j 2, ±i} (with ν {, 2, 3, 5}). All those except ξ = contradict the fact that f has integer coefficients by simple considerations such as the following, for ξ = j say: we have αp 2 = jp k, α p = ±j 2 p (k )/2 and λ f (p) =(+ j)α p = ±( + j)j 2 p (k )/2 = ±(j 2 + j)p (k )/2 / Z (compare [43, p ]). We now prove the analogue of (.2) for primes p such that λ f (p ν )=0 for some ν. Lemma 2.3. Let f Sk (N,χ) be a primitive cusp form not of CM type, in particular with k 2. Forν, let (2.6) P f,ν = {p N λ f (p ν )=0}.

6 286 E. Kowalski, O. Robert and J. Wu For any ν we have (2.7) Pf,ν [,x] x (log x) +δ for x 2 and any δ<, the implied constant depending on f and δ. Let 2 P f be the union of P f,ν. We have (2.8) P f [,x] x (log x) +δ for x 2 and any δ<, the implied constant depending on f and δ. 2 Proof. All the tools needed to prove (2.7), if not the exact statements, can be gathered from [43], in particular Section 7.2. By Lemma 2., we need only prove (2.7), so let ν befixed. Fix a prime number l totally split in the field K f = Q(λ f (n),χ(n)) already considered. Thus K f Q l. There exists an l-adic Galois representation ρ f,l :Gal( Q/Q) GL(2, Q l ) constructed by Deligne, such that for p Nl we have Trρ f,l (σ p )=λ f (p) and det ρ f,l (σ p )=χ(p)p k, where σ p is a Frobenius at p. LetG l be the image of ρ f,l. As explained by Serre [43, Prop. 7], it is an open subgroup of GL(2, Q l ), hence an l-adic group of dimension 4. By symmetry, there exists a polynomial P ν Z[X, Y ] such that the identity X ν+ Y ν+ = P ν (X + Y,XY ) X Y holds. Consider the set C G l defined by C = {s G l P ν (Tr(s), det(s)) = 0}. Note the following facts about C: it is a closed l-adic subvariety of G l, stable by conjugation, and of dimension 3. Moreover, C is stable by multiplication by H l = {homotheties in G l }, and therefore C = π (C )for a certain subvariety C G l /H l,whereπ : G l G l /H l is the projection. The set C is an l-adic variety of dimension 2 and all its elements are regular ([43, Section 5.2]), since they have distinct eigenvalues α, ξα for some root of unity ξ of order ν +. Now remark that if p P f,ν and p Nl,wehaveπ(σ p ) C (going back to the proof of Lemma 2. if necessary). Hence our result (2.7) follows from Theorem2of[43],asintheproofofthecaseh =0ofTheorem5ofloc. cit., p. 77.

7 Small gaps and B-free numbers 287 For ease of reference we recall the lemma which allows the extension of the results for i f (n) to general cusp forms from that of newforms. Lemma 2.4. Let f S k (N,χ) be a cusp form not in the space spanned by CM forms. There exist: (i) an integer s and algebraic numbers β j and positive rational numbers γ j for j s; (ii) a divisor δ N such that χ is induced by χ modulo N/δ and a divisor δ δ; (iii) a primitive form g Sk (N/δ, χ ), not of CM type; such that λ g (n) = j s β j λ f (γ j δ n) for n. By convention, we put λ f (x) =0if x Q is not a positive integer. This is just a formal restatement of the computations in [2, p. 362], or follows from [43, 7.6]. We now discuss briefly the possibility of extending the results above to higher rank situations. From the proof of Lemma 2.3, it is natural to start from an l-adic representation ρ :Gal( Q/Q) GL(V ) where V Q r l for some r. We assume it is sufficiently geometric, namely that it is unramified outside a finite set of primes S, and that the L-function of ρ, defined as usual by the Euler product (2.9) L(ρ, s) = det ( ρ(σ p )p s V Ip) = λ ρ (n)n s, p (where σ p is a Frobenius element at p and I p the inertia group at p) has coefficients in a number field K ρ Q l. Note that we view this here as a formal Dirichlet series. If the image of ρ is fairly big, one can use the methods of Serre to get n {p x p/ S and λ ρ (p) =Trρ(σ p )=0} x(log x) δ for some δ > 0, see Proposition below. On the other hand, it is not clear if the analogue of Lemma 2. holds, and this seems a hard question in general. The analogue of (2.4) does not provide an equation easily solvable

8 288 E. Kowalski, O. Robert and J. Wu to characterize the values of ν for which λ ρ (p ν ) = 0. The best that seems doable is to notice that, for fixed (unramified) p, u ν = λ ρ (p ν )isgivenbya linear recurrence relation of degree r with companion polynomial given by det(x ρ(σ p )) = X r λ ρ (p)x r + +( ) r det ρ(σ p )= (X α p,i ) so that u ν = λ ρ (p ν )= i r γ p,i α ν p,i i r for some γ p,i. The Skolem-Mahler-Lech theorem (see e.g. [6, p. 88]) says that for any linear recurrence sequence (u ν ), either u ν = 0 for only finitely many values of ν, or there exists an arithmetic progression a + tv with v 0 such that u a+tv = 0 for all t 0. In the latter case, spelling this out yields a Vandermonde type linear system for powers of the αp,i v, hence it implies that αp,i v = αv p,j for some i j. Coming back to ρ, this case implies that an extension of degree r[k ρ : Q] containsav-th root of unity. As in Lemma 2., this bounds v, and an analogue of Lemma 2.3 is possible, given ξ a root of unity, to get {p x p/ S and two roots α i,p, α j,p = ξα i,p with i j} x (log x) +δ for some δ>0. However, in the first case where u ν = 0 has only finitely many solutions, despite the remarkable fact that there exists a uniform bound for the number of solutions depending only on r (see [2]), this is insufficient because only the number of solutions, not the value of ν, is bounded, so that an integer ν 0 (independent of p) for which the smallest solution is ν ν 0 is not known to exist. The question amounts to asking for a bound for the height of the solutions to the relevant linear equations in multiplicative groups [2, p. 820], and is thus in full generality of the same type as asking for effective versions of Roth s theorem, or of Schmidt s Subspace Theorem. (Note that by replacing u ν by p (k )/2 u ν one gets a linear recurrence relation with companion polynomial having height absolutely bounded, by the Ramanujan-Petersson conjecture proved by Deligne). The theory of B-free numbers does however still apply; although we use results proved later, we include a sample application here for consistency. Proposition. Let ρ be an l-adic representation of Gal( Q/Q) on GL(r,Q l ). Assume that ρ is unramified for p outside a finite set S and that its L- function has coefficients in a number field K ρ.letg =Imρ, C = G {s GL(r, Q l ) Tr s =0}.

9 Small gaps and B-free numbers 289 Assume that, as l-adic varieties, we have dim C < dim G. Then for any ε>0, x x 0 (ρ, ε) and y x 7/7+ε we have {x <n x + y λ ρ (n) 0} y. In particular, with obvious notation, we have i ρ (n) n 7/7+ε. Proof. One can argue as for modular forms using B-free numbers (see Proposition 6) with B = {p p S or λ ρ (p) =0} {p 2 λ ρ (p) 0}, after applying Theorem 0 of [43] to G and C, withe = Q ker ρ,toderive {p x p/ S or λ ρ (p) =0} x(log x) δ for some δ>0 depending on the dimensions of G and C (for instance, any δ< dim C/ dim G). Strictly speaking, to apply this theorem as stated we must also treat separately the case where G is finite. One can then see ρ as a linear representation of the finite group G =Gal(E/Q) intogl(n, Q), or into GL(n, C). In that case the condition dim C < dim G means that the character of ρ does not vanish. By a well-known fact about linear representations of finite groups (see e.g. [7, Ex. 2.39]), this means that the representation ρ is a one-dimensional character of Gal( Q/Q), which by the Kronecker-Weber theorem corresponds to (i.e. has the same L-function as) a Dirichlet character χ, of conductor N say (divisible only by primes in S). Then λ ρ (n) 0ifandonlyif(n, N) =. This is also implicit in [43, 6.4, 6.5]. 3. Maass forms, cofinite groups and the Rankin-Selberg method In this section, we describe what results follow from the Rankin-Selberg method. Although, for fixed f Sk (N,χ), they are weaker than those obtained by means of B-free numbers, this method has the advantage of yielding quite easily estimates uniform in terms of f, i.e. with explicit dependency on k and N. Those are by no means obvious from the l-adic point of view leading to (.2). Moreover, the Rankin-Selberg method applies, at least as far as bounding i f (n), to non-congruence subgroups, as shown by Good [9], Sarnak [4] and Petridis [34]. This answers the last question in [43, p. 83].

10 290 E. Kowalski, O. Robert and J. Wu Proposition 2. Let Γ SL(2, R) be a discrete subgroup such that the quotient Γ\H has finite hyperbolic volume and Γ contains the integral translation matrices acting by z z + n. Let f be either a holomorphic cusp form of weight k 2 or a Maass cusp form with eigenvalue λ /4. Define i f (n) by (.4) where λ f (n) are the Fourier coefficients in the expansion of f at the cusp of Γ. Then for some θ< we have i f (n) n θ for n where the implied constant depends on f. Specifically, one can take θ =2/3 if f is holomorphic and any θ>4/5 if f is non-holomorphic. Proof. The non-holomorphic case follows from [34] as the holomorphic case follows from [9], so we describe only the latter. Good shows that (3.) λ f (n) 2 = n x 2/3<s j (4πx) s j+k Γ(k + s j ) r j,y k f 2 + O(x k +2/3 ) for x, where = s 0 >s s r are the finitely many poles of the Eisenstein series E(z, s) for Γ in the interval [2/3, ] (those with s j 2/3 go to the error term), r j (z) is the residue of E(z, s) ats j and, is the inner product on L 2 (Γ\H).Thepoleats 0 = with residue V contributes (4πx) k f 2 k!v where f is the Petersson norm of f and V the volume of Γ\H. Comparing (3.) at x = n and x = n+cn 2/3,whereC is some large constant, shows that i f (n) Cn 2/3. Remark. As for half-integral weight forms, it is not expected that the coefficients of a cusp form for a non-arithmetic group satisfy any multiplicativity properties. In fact, it would be quite interesting to express this in a quantitative manner as done by Duke and Iwaniec [7] for half-integral forms using bilinear forms in the Fourier coefficients. In the case of congruence subgroups the methods using B-free numbers yield stronger results such as (.5) for y quite small. Those however are not uniform in terms of f (i.e. in terms of N and k for holomorphic forms). The Rankin-Selberg method can quite easily yield some uniform estimates. Here are sample statements; note that we have not tried to get the best possible results. Since this paper was written, A. Venkatesh has proved this in a qualitative form using techniques of ergodic theory.

11 Small gaps and B-free numbers 29 Proposition 3. () Let N and f a primitive Maass form of conductor N with eigenvalue λ /4 and trivial nebentypus. Let Λ=λ + and let λ f (n) be the Fourier coefficients of f. For any ε>0, there exists c>0 depending only on ε such that when (3.2) y>x 37/40 Λ 45/32 N 9/8 (xλn) ε, we have {x <n x + y λ f (n) 2 c(λn) ε } y(log x) (ΛN) ε, the implied constants depending only on ε. (2) Let N and f a primitive holomorphic cusp form of conductor N, weight k and nebentypus χ, not of CM type. There exists an absolute constant c>0 such that for any ε>0 and we have y>x 4/5 (kn) /2 (xkn) ε {x <n x + y λ f (n) 2 n k c(log kn) } y(log x) 4 (log kn) 3 the implied constant depending only on ε. Proof. We prove () and only give some indications for the easier (2) at the end. It turns out to be simpler to reduce to squarefree numbers (so in fact we could impose this condition on n). The result will follow by Cauchy s inequality from the two asymptotic formulas (3.3) n x λ f (n) 2 = c f x + O((ΛN) 25/64 x 2/46 (xλn) ε ) and (3.4) n x λ f (n) 4 = d f x log x + e f x + O(Λ 45/32 N 9/8 x 37/40 (xλn) ε ) where restricts n to squarefree integers coprime with N. Both hold for any ε>0, with the implied constant depending only on ε, andc f, d f, e f are real numbers with c f, d f > 0and (3.5) c f (ΛN) ε, d f, e f (ΛN) ε for any ε>0, the implied constant depending only on ε.

12 292 E. Kowalski, O. Robert and J. Wu Indeed, let ε > 0 and η > 0 be any positive numbers, and put the integers n with x<n x+y in two sets L and S if, respectively, λ f (n) 2 >η or λ f (n) 2 η. Ify satisfies (3.2) we have by (3.3) and (3.5) λ f (n) 2 c f y Cy(ΛN) ε/4 where C depends only on ε, whereas by positivity and Cauchy s inequality λ f (n) 2 η S + ( λ f (n) 2 ηy + L /2 ) /2. λ f (n) 4 x L If η< C 2 (ΛN) ε/4 we derive by (3.4) and (3.5) L y 2 (ΛN) ε y (log x), as desired. We give the proof of (3.4) and the upper bounds on d f, e f, since (3.3) is easier. The lower bound for c f is deeper, and follows immediately from the bound L(F, ) (ΛN) ε of Hoffstein and Lockhart [22] for the adjoint square F of f (which is also its symmetric square since the nebentypus is trivial). Since f is primitive and has trivial nebentypus, hence real coefficients, we have λ f (p) 4 = λ f (p) 4 =(+λ f (p 2 )) 2 =+2λ f (p 2 )+λ f (p 2 ) 2 for p N and thus we find that L(s) := λ f (n) 4 n s = ( + λ f (p) 4 p s ) n p N = ζ (s)l (Sym 2 f,s) 2 L (Sym 2 f Sym 2 f,s)h(s) where ζ (s) = ( + p s ), p N L (Sym 2 f,s) = ( + λ f (p 2 )p s ), p N L (Sym 2 f Sym 2 f,s) = ( + λ f (p 2 ) 2 p s ) p N and H(s) is an Euler product which converges absolutely for σ> 23 by the 32 estimate λ f (p) 2p 7/64 of Kim and Sarnak [26] (any estimate λ f (p) 2p θ with θ < /4 would do, at the cost of worsening the exponent), and is moreover uniformly bounded on any line σ = σ 0 with σ 0 >

13 Small gaps and B-free numbers 293 To see this, define H as the obvious ratio for σ large enough, and check on the Euler factors individually that H(s) ζ(2σ 4/32) B for some absolute constant B>0 (for a similar argument, see e.g. [9, Prop. 2]). Each of the three L-functions is obtained by removing non-squarefree coefficients (and those not coprime with N) from an L-function which has analytic continuation and a functional equation of the standard type: the first one is the zeta function, the second one is the adjoint square F =Sym 2 f of Shimura and Gelbart-Jacquet [8], and the third is the Rankin-Selberg square F F of the latter (which exists as a special case of convolution of cusp forms on GL(3)). The same bound and reasoning already used shows that ζ (s)l (F, s) 2 L (F F, s) =ζ(s)l(f, s) 2 L(F F, s)h (s) where H (s) has the same properties as H(s) above. In particular we see that L(s) has a pole of order 2 at s = (by [3] since λ /4 sothatf is a cusp form on GL(3)). We can now proceed along classical lines: let U> (to be chosen later) and let ψ be a C function on [0, + ) such that 0 ψ and { if 0 x U, ψ(x) = 0 if x +U. The Mellin transform ˆψ(s) is holomorphic for σ > 0, it satisfies ˆψ(s) = s + O( σ U ) and by integration by parts (3.6) ˆψ(s) U A ( + t ) A for σ /2 andforanya>0, the implied constant depending on A and ψ only. For suitable choices (say ψ + and ψ )ofψ we get n λ f (n) 4 ψ (n/x) λ f (n) 4 λ f (n) 4 ψ + (n/x). n x Thus it is enough to prove (3.4) for a sum weighted by ψ(n/x). We have λ f (n) 4 ψ(n/x) = L(s)x s ˆψ(s)ds 2πi (3) n = ζ(s)l(f, s) 2 L(F F, s)h(s)h (s) 2πi ˆψ(s)x s ds. (3) n

14 294 E. Kowalski, O. Robert and J. Wu For any fixed α > 23 we can move the line of integration (the three L- 32 functions are polynomially bounded in vertical strips and ˆψ decays rapidly) to σ = α. We pass the double pole at s = with residue of the form d f x log x + e f x + O(x(log x)(λn) ε U ) with d f = L(F, )H()H ()res s= L(F F, s) > 0, d f and e f being estimated by [32] to get d f (ΛN) ε, e f (ΛN) ε for any ε>0, the implied constant depending only on ε. Now the integral on σ = α< is estimated using H(s)H (s), the uniform convexity bound for automorphic L-functions (see e.g. [24, 5.2]) yielding L(s) ( + t ) 8( α)+ε Λ 5( α)+ε N 8( α)+ε for the product of the three L-functions, the implied constant depends only on α and ε. Then (3.6) with A =8( α) +/2 (to get an absolutely convergent integral) yields n λ f (n) 4 ψ(n/x) =d f x log x + e f x + O(x(log x)(λn) ε U ) + O(x α Λ 5( α)+ε N 8( α)+ε U 8( α)+/2+ε ). Without trying to optimize, we take U so that xu = x α U 8( α)+/2,which gives n λ f (n) 4 ψ(n/x) =d f x log x + e f x + O(Λ 5( α)+ε N 8( α)+ε x β+ε ) with 2( α) β = 6( α)+3. Taking α = 23 + ε, we get (3.4), up to renaming ε. 32 For holomorphic forms, we proceed in the a slightly different manner. First since we have a nebentypus we use the adjoint square instead of the symmetric square in proving the analogue of (3.3), namely (3.7) n x λ f (n) 2 = c f x k + O(x k /5+ε (kn) /2+ε ) with c f (log kn) (by Goldfeld, Hoffstein and Lieman, see the Appendix to [22]).

15 Small gaps and B-free numbers 295 Secondly we can avoid proving the analogue of (3.4), for which we require only an upper bound, by means of the Ramanujan-Petersson bound (proved by Deligne) λ f (n) d(n)n (k )/2, where d(n) is the divisor function. In fact it is more efficient then to argue with the third power moment, and use Hölder s Inequality with (p, q) = (3, 3/2) for the final estimates: (3.8) λ f (n) 2 η S + λ f (n) 2 x L ( ηy + L /3 ) 2/3. λ f (n) 3 We have λ f (n) 3 d(n) 3 n 3(k )/2, and estimating this is classical. Here are the main steps for completeness. The generating Dirichlet series for d(n) 3 is L (s) := n d(n) 3 n s = ζ(s) 4 p ( + 4p s + p 2s )=ζ(s) 8 H 2 (s) where H 2 is absolutely convergent, hence holomorphic, for σ>.sayithas 2 coefficients α(n), and ζ(s) 8 has coefficients d 8 (n). By [2, Th. 2] we have d 8 (n) =xp (log x)+o(x 5/8+ε ) n x where P is some polynomial of degree 7. Hence d(n) 3 = α(b) d 8 (a) n x b x a x/b = xp (log x)+o(x 5/8+ε ) for some polynomial P of degree 7 since H 2 (5/8) is absolutely convergent. By partial summation we get d(n) 3 n 3(k )/2 = x 3(k )/2+ P 2 (log x)+o(x 3(k )/2+5/8+ε ), n x hence the result follows using (3.7), (3.8) since 5/8 < 4/5.

16 296 E. Kowalski, O. Robert and J. Wu Remark 2. We see that this method provides n where a lower bound for λ f (n) holds, and this also seems very hard to get by purely algebraic techniques. In applications to analytic number theory, this can be of crucial importance; see for instance [8], [9]. In these papers the question is somewhat different: one needs to find very small n, compared to some large parameter x (say n x ε ), such that λ f (n) isnottoosmall,andthisissolvedbyusing the trick of Iwaniec that for any prime p N, wehaveλ f (p) 2 λ f (p 2 )=, so one of λ f (p), λ f (p 2 )isatleast/ 2 in absolute value, and p 2 remains small enough for the application in mind. There is a strong contrast between the proof of Proposition 3, which depends on quite deep analytic properties of L-functions, and the algebraic approach of the previous section, where not even convergence mattered! It is clear that one can extend Proposition 2 to any cuspidal automorphic form on GL(n)/Q using its Rankin-Selberg convolution (compare [9]), but Proposition 3 requires either that f satisfies the Ramanujan-Petersson conjecture, or that the adjoint square be automorphic (in which case there is also a bound of the type α p p θ with θ< for the local parameters of f at 4 unramified primes). This is not known for n 3. It is natural to ask if the property in Lemma 2. holds for primitive Maass forms. If the eigenvalue is λ = /4, conjecturally the Fourier coefficients still generate a number field, and in this case the proof goes through without change. If λ /4, the field K f = Q((λ f (n)) n, (χ(n)) n )isnot expected to be a number field. However we still see that if Lemma 2. is false for f, thenq((α p ) p, (β p ) p, (χ(p)) p ) Q ab is an infinite extension of Q, where Q ab is the cyclotomic field generated by all roots of unity. This does not sound very likely, as the field generated by the local roots α p, β p could be expected to be mostly transcendental, but it is certainly beyond proof or disproof today! (The corresponding fact is true, however, for the field K t = Q(2 it, 3 it,...,p it,...) generated by the local roots of the Eisenstein series E(z, + it), for SL(2, Z) say, for all t R except maybe those in a 2 countable set; it doesn t seem easy to decide if the latter is really empty, but this would follow from Schanuel s Conjecture, as observed by B. Poonen). One is tempted to confront this with the famous optimistic question of Katz ([25, p.5]): is L(s) = p ( S(, ; p)p s + p 2s) = n λ S (n)n s the L-function of a (primitive) Maass form (of weight 2), even up to finitely many factors, where S(, ;p) denotes the usual Kloosterman sums? Note that S(, ; p) generates the maximal real subfield of the field of p-th roots

17 Small gaps and B-free numbers 297 of unity, so in this case the field generated by λ S (p) is an infinite algebraic extension of Q. However we can prove the analogue of Lehmer s conjecture for this Dirichlet series! (Of course, the answer to Katz s question is widely expected to be No, see [5] for some strong evidence). Proposition 4. For any n, we have λ S (n) 0. Proof. We give two proofs (suggested by Katz and simpler than our original argument). We need to show that λ S (p ν ) 0forp prime and ν 0. For the first argument, consider the Euler factor at p as a rational function of X = p s with coefficients in the cyclotomic field Q(e(/p)). It is congruent (modulo the ideal generated by p) to S(, ; p)x = ν S(, ; p) ν X ν. Thus the result follows from the well-known fact that S(, ; p) is non-zero modulo p, in fact we have S(, ; p) (mod ( e(/p))), and the prime ideal e(/p) divides p. For the other argument, notice that since the form of the Euler product is the same as for a holomorphic form of weight 2, we must show that α p /β p is not a root of unity, where α p and β p satisfy α p + β p = S(, ; p) and α p β p = p. Hence the product α b β p is divisible by e(/p), whereas by the congruence above, the sum is invertible modulo e(/p). This means one of α p, β p must also be invertible while the other is not, which implies that the ratio α p /β p is not a p-unit, hence not a root of unity. It is probably possible to derive a fancy proof of this proposition (more amenable to generalizations, if desired) using ideas as in [4], Lemma 4.9, applied to some Kloosterman/Gauss sum sheaves on G m /F p with traces of Frobenius at α G m (F p ) given by both sides of (3.9). Note also that if ν andp is odd we do have (see e.g. [23, Lemma 4.]) S(, ; p 2ν )=p ν( e ( 2/p 2ν) + e ( 2/p 2ν)), so Proposition 4 is special to Kloosterman sums with prime modulus.

18 298 E. Kowalski, O. Robert and J. Wu 4. Applications of B-free numbers We now come to the technical heart of this paper where we consider the original question of proving (.5) for a cusp form f S k (N,χ), not in the space spanned by CM forms. Recall that Balog and Ono [2] proved (.5) for y = x 7/4+ε, ε>0 being arbitrary. It is interesting to look for smaller exponents, in particular since it is natural to expect that y = x ε should be sufficient. (By a result of Plaksin [35] on B-free numbers, this is true for almost all n). For one very natural f, namely the Ramanujan function with coefficients τ(n), a famous conjecture of Lehmer [29] says that τ(n) 0 for any n. Since this problem seems very difficult, approaching it by means of conditional statements based on solid conjectures is also desirable. Very recently Alkan [] gave two such results: he showed that the exponent 7/4 can be reduced to 69/69and /3 ([], Theorems 3 and 4) under the generalised Riemann hypothesis (GRH) for Dedekind zeta-function and the Lang-Trotter conjecture [28], respectively. We will prove a number of results improving the previously known statements, both conditional and unconditional. The following is a general bound, where we recall that P f, is defined in (2.6): Theorem. Suppose that k 2 and f Sk (N,χ) is a primitive form not of CM type such that (4.) Pf, [,x] ρ (log log x)ψρ f x (x 2), (log x) Θρ where ρ [0, ] and Θ ρ, Ψ ρ are real constants such that Θ >. Define if 0 ρ, 4 3 0ρ if <ρ 9, 9ρ ρ 9 5 if <ρ 4ρ+3 7 θ(ρ) =, 28 5 if 5 <ρ 5, ρ if 5 24ρ+29 8 <ρ 9, 0 7ρ <ρ. 9 if 9ρ+8 0 For every ε>0, x x 0 (f,ε) and y x θ(ρ)+ε, we have (4.2) {x <n x + y λ f (n) 0} f,ε y. In particular for any ε>0 and all n, we have (4.3) i f (n) f,ε n θ(ρ)+ε.

19 Small gaps and B-free numbers 299 Theorem follows immediately by multiplicativity from Corollary 0 below which gives a more effective treatment for B-free numbers in short intervals, applied with P = {p p N or λ f (p) =0}. The new ideas and new ingredients will be explained in 5. According to (.2), the hypothesis (4.) holds with (ρ, Θ ρ, Ψ ρ )=(, +δ, 0) for any δ< 2. Thus, applying this result and Lemma 2.4, we immediately obtain an improvement of the result of Balog and Ono. Corollary. Suppose that k 2 and f S k (N,χ) is not in the space spanned by CM forms. Then for any ε>0, x x 0 (f,ε) and y x 7/7+ε, we have {x <n x + y λ f (n) 0} f,ε y. In particular i f (n) f,ε n 7/7+ε. In proving this we do not exploit Lemma 2. (so we could claim that we obtain the correct proportion of squarefree numbers if f is primitive). It can be used to simplify the proof, as we will see, but it does not influence the strength of the exponent. This is mainly due to the fact that we have ρ =, and when ρ is close to we do not succeed in getting better results by not imposing the numbers to be squarefree. However, if one can get ρ quite small, e.g. smaller than the current best results about squarefree numbers in short intervals (see [3]), it is clear that using Lemma 2. will yield an improvement. So consider the set of prime numbers P f := {p p N} ν= P f,ν, where as before P f,ν = {p p N and λ f (p ν )=0}. Clearly λ f (n) 0(for(n, N) = ) if and only if n is P f-free. We then have the following result: Theorem 2. Assume that k 2, f Sk (N,χ) is not a CM form, and that (4.4) P f [,x] ρ (log log x)ψρ f x (x 2), (log x) Θρ where ρ [0, ] and Θ ρ, Ψ ρ are real constants such that Θ >. Then the inequalities (4.2) and (4.3) hold with θ(ρ) =ρ/( + ρ) for 0 ρ.

20 300 E. Kowalski, O. Robert and J. Wu This theorem gives a better exponent than Theorem when ρ under 3 a slightly stronger hypothesis than (4.). However recall from Lemma 2.2 that the hypotheses (4.) and (4.4) are in fact equivalent when k is even. It is of course particularly interesting that this new exponent tends towards 0 when ρ 0. As for Theorem, this result follows directly by multiplicativity from the corresponding result for B-free numbers, Proposition 9 below, where this time P = P f. Another consequence of Lemma 2.3 and Corollary 0 is an extension to all symmetric powers: Corollary 2. Let k 2 and f Sk (N,χ) which is not a CM form. Let m and define the unramified m-th symmetric power L-function of f by L nr (Sym m f,s) = ( αp j βm j p p s ) = λ (m) f (n)n s. p N 0 j m n Then for any ε>0, x x 0 (f,ε) and y x 7/7+ε, we have {x <n x + y λ (m) f (n) 0} f,m,ε y, and in particular i Sym m f(n) f,ε,m n 7/7+ε for n. Proof. For p N prime, we have λ (m) f (p) =λ f (p m ). Hence by Lemma 2.3 we derive {p x p N and λ (m) f (p) =0} x(log x) δ for any δ<. By multiplicativity and Corollary 0 below, the result follows. 2 Note we do not need the automorphy of Sym m f (which is known in general only for m 4). The hypothesis (4.) is known only with ρ =, with the one exception of primitive forms f S2 (N) with integral coefficients. Those are associated to elliptic curves over Q, and Elkies [0] has proved that (4.) (or (4.4)) holds with ρ =3/4, Θ = Ψ = 0. Theorem is still better for this value of ρ than Theorem 2 and we get: Corollary 3. Let E/Q be an elliptic curve without complex multiplication and let f be the associated primitive form. Then for every ε > 0, x x 0 (E,ε) and y x 33/94+ε, we have {x <n x + y λ f (n) 0} E,ε y. In particular for any ε>0 and all n, we have i f (n) E,ε n 33/94+ε. This improves Theorem 2 of [], which requires 69/69 in place of 33/94.

21 Small gaps and B-free numbers 30 Some well-known conjectures imply that (4.) holds for smaller values of ρ. For example, Serre ([43, (82) R ]) showed that the GRH for Dedekind zeta-functions implies (4.) with (ρ, Θ ρ, Ψ ρ ) = ( 3, 0, 0). Lang and Trotter [28] formulated a conjecture for the size of the set P f,, in the case 4 where f is associated to an elliptic curve over Q. This, if true, implies for these forms an estimate (4.) with (ρ, Θ ρ, Ψ ρ )=(,, 0). Generalizations of 2 the Lang-Trotter conjecture (see e.g. Murty s version [33], especially Conjecture 3.4) imply that if k 2andf Sk (N,χ) is not of CM type, then we have (4.) with (,, 0) if k =2and[F 2 f : Q] =2, (0, 0, ) if k =2and[F f : Q] =3 (4.5) (ρ, Θ ρ, Ψ ρ )= or k =3and[F f : Q] =2, (0, 0, 0) otherwise, where F f is the stable trace field (see 2and 3 of [33]). Applying Theorem, we get the following conditional result, which improves Theorem of []. Corollary 4. Suppose that k 2 and f S k (N,χ) is not in the space spanned by CM forms. (i) Under the GRH for Dedekind zeta-function, the exponent 7/7 of Corollary can be further improved to 33/94. (ii) Under the generalized Lang-Trotter conjecture, the exponent 7/7 can be further improved to 0/33 if k =[F f : Q] =2,andto/4 otherwise. We can apply Theorem 2 instead if f satisfies the assumptions of Lemma 2.2, but it is just as simple to extend the Lang-Trotter type conjectures to deal with the sets P f,ν for any ν. The heuristics which lead to these conjectures, based on Deligne s estimate α p = β p = p (k )/2, suggest the following: Conjecture. Let ν be any integer. If k 2 and f Sk (N,χ) is not of CM type, then Pf,ν [,x] ρ (log log x)ψρ f x (x 2) (log x) Θρ with (ρ, Θ ρ, Ψ ρ )=(,, 0) if k =2, (0, 0, ) if k =3and (0, 0, 0) if k 4. 2 We only state upper bounds, but one could propose a more precise statement, which involves looking at the possibility of f having extra twists and eliminating the all but finitely many ν for which P f,ν is empty. About

22 302 E. Kowalski, O. Robert and J. Wu this conjecture, recall that even under GRH, one can not get a better general result towards the Lang-Trotter conjecture than Pf, [,x] f x 3/4 for f of weight k 2. The exponent is the same for all weights, so this gets worse (compared to what we expect) as k grows. In particular, this conjecture for k 3 seems hopeless for the time being. Lemma 2. implies: Corollary 5. Let k 2 and f Sk (N,χ) not of CM type. Assuming Conjecture for f, the inequality (4.4) holds with (ρ, Θ ρ, Ψ ρ ) given by (ρ, Θ ρ, Ψ ρ )=(,, 0) if k =2, (0, 0, ) if k =3and (0, 0, 0) if k 4. 2 As applications (or cautionary tale...), here are some very impressivelooking results. Corollary 6. Suppose that k 3 and f S k (N,χ) is not in the space spanned by CM type. If Conjecture holds for all primitive forms, then the exponent 7/7 of Corollary can be improved to 0. If k 4 and f is primitive, then there exists M such that (n, M) =implies λ f (n) 0. Specializing to the Ramanujan τ-function, which is integer valued, Lemma 2.2 allows us to deduce the following result (implicit in [43]): Corollary 7. Assume Conjecture, or equivalently the generalized Lang- Trotter conjecture, for f = S2 (). There exists P such that τ(n) =0if and only if (n, P ) is a square, i.e. if and only if v p (n) is even for p P.Inparticulari (n) P for n, andforallx 2 and y, we have {x <n x + y τ(n) 0} = y p P p p + + O( ω(p (log(x + y)) )) ϕ(p ) P y + O() where the implied constant is absolute and ω(p ) is the number of prime divisors of P. Proof. The first statement is the rephrasing of Lemma 2.2 and Conjecture in this case. Notice that τ(n) 0if(n, P )=soi (n) P follows (an interval of length P contains elements prime to P ) as does the last inequality by trivial counting. For the asymptotic, write {n x τ(n) 0} = λ(d), d P,d x n x/d

23 Small gaps and B-free numbers 303 where λ(n) is the Liouville function, i.e. λ(p k )=( ) k.since λ(d) = d d P p P p p +, we get the result after elementary estimates. This is of course trivial and of little practical significance towards the Lehmer conjecture. 5. Multiple exponential sums and bilinear forms This section is devoted to the study of multiple exponential sums and bilinear forms, which will be used in the proofs of our results on B-free numbers in the next sections, but are also of independent interest. Recall that exponential sums in analytic number theory are often classified as of type I or type II, meaning roughly bilinear forms with at least one smooth variable (type I) and general bilinear forms (type II). We begin by investigating a double exponential sum of type II: S(M,N) := ) ϕ m ψ n e (X mα n β, M α N β m M n N where e(t) :=exp{2πit}, X>0, M, N, ϕ m, ψ n, α, β R and m M means M m<2m. Such a sum occurs in many arithmetic problems and is studied by many authors (for example, [5] and [40]). We shall estimate this sum by the method of Fouvry and Iwaniec [5] together with the refinement of Robert and Sargos [39]. When X<N 2, we need to use an idea in [40]. The following result is an improvement of Theorem 4 in [5] and Theorem 0 in [40]. Proposition 5. If α, β R {0, }, then for any ε>0 we have S(M,N) { (XM 6 N 6 ) /8 + M /2 N + MN 3/4 + X /2 MN } (MN) ε. Proof. We shall distinguish two cases. A. The case of X N 2 By applying twice the Cauchy-Schwarz inequality, it follows that S(M,N) 4 (MN) ( 2 e X (mα mα 2 )(nβ nβ 2 ) ). M α N β n N n 2 N m M m 2 M

24 304 E. Kowalski, O. Robert and J. Wu The double large sieve inequality ([5, Proposition ]) with the choice of X = { (m α mα 2 )/M α} m,m 2 M and Y = { (n β nβ 2 )/N β} n,n 2 N leads to the following estimate (5.) S(M,N) 8 X(MN) 4 N (M,/X)N (N,/X), where N (M, ) is the number of quadruplets (m,m 2,m 3,m 4 ) {M +,...,2M} 4 satisfying m α + m α 2 m α 3 m α 4 M α. According to Theorem 2 of [39], we have N (M,/X) ( M 2 + X M 4) M ε. Inserting this into (5.) and simplifying the estimate obtained by using the hypothesis X N 2, we find that S(M,N) { (XM 6 N 6 ) /8 + MN 3/4} (MN) ε. B. The case of X N 2 By Lemma 2. of [40], we deduce that, for any Q [,M ε ], (5.2) S(M,N) 2 (MN) 2 Q + MNQ (log M) max Q Q S(Q ), where (5.3) S(Q ):= and q Q m M ϕ m,q n N ) t(m, q)nβ e (X TN β t(m, q) :=(m + q) α m α, T := M α Q, X := XM Q. If N := X /N, applying Lemma 2.2 of [40] to the sum over n yields 2 ) t(m, q)nβ e (X X /2 N ( w TN β n e βx u(m, q)n β ) (5.4) UN β where n N n I(m,q) + R + R 2 +logn, I(m, q) := [ c Xt(m, q)m α N,c 2 Xt(m, q)m α N ], R j := min { X /2 N, / c jxm α N t(m, q) }, u(m, q) :=t(m, q) /( β), U := T /( β), β := β/(β ), β := β β β, w n, and c j = c j (β), c j = c j (β) are some suitable constants.

25 Small gaps and B-free numbers 305 Inserting this into (5.3), using Lemma 2.5 of [40] to eliminate multiplicative restrictions and using Lemma 2.3 of [40] with n = m to estimate the related error terms, we find S(Q ) X /2 N + Ξ(r)S(Q,r)dr + { (XM Q 3 ) /2 + MQ } (MN) ε, where Ξ(r) :=max{m, (πr), (πr) 2 }, ψ n (r) :=w n e(rn )and S(Q,r):= ( ψ n (r) e βx u(m, q)n β ). UN β m M n N q Q If X /N, the Kusmin-Landau inequality (see e.g. [20, Theorem 2.]) 2 implies S(Q ) X MNQ. Thus we always have (5.5) S(Q )S(Q ) X /2 N + Ξ(r)S(Q,r)dr + { (XM Q 3 )/2 + MQ + X MNQ } (MN) ε, Now by applying Cauchy-Schwarz inequality, it follows that S(Q,r) 2 MQ ψ n (r)ψ n 2 (r) q Q m M n N n 2 N ( e βx u(m, q)( n β n β )) 2. UN β The double large sieve inequality with the choice of X = { u(m, q)/u } m M,q Q and Y = { (n β n β 2 )/N β } n,n 2 N allows us to deduce (5.6) S(Q,r) 4 (MQ ) 2 X N (u, U;/X )N (N ;/X ) uniformly for r R, wheren (u, U; ) denotes the number of quadruplets (m + q,m 2 + q 2,m,m 2 ) such that m,m 2 M, q,q 2 Q and u(m,q ) u(m 2,q 2 ) U, and N (N; ) is the number of quadruplets (n,...,n 4 ) {N +,...,2N} 4 satisfying n β + n β 2 n β 3 n β 4 N β.

26 306 E. Kowalski, O. Robert and J. Wu Since u(m,q ) u(m 2,q 2 ) t(m,q ) t(m 2,q 2 ) T β/( β), we have, for some suitable constant C>0, N (u, U; )=N (t, T ; C ). Noticing that t(m,q ) t(m 2,q 2 ) = (m + q ) α m α (m 2 + q 2 ) α m α 2 and (m + q,m,m 2 + q 2,m 2 ) {M +,...,3M} 4, clearly we have Thus Theorem 2 of [39] implies that N (t, T ; C ) N(M; C ). N (u, U;/X ) ( M 2 + X M 4) M ε, N (N, /X ) ( N 2 + X N 4) N ε. Inserting these into (5.6), we obtain uniformly for r R, S(Q,r) 4 { X M 4 (N Q ) 2 +(MN ) 4 Q 2 + M 6 (N Q ) 2 + X M 6 N 4 Q 2 } (MN) ε. Combining this with (5.5), we find that S(Q ) { (XM 3 N 2 Q 3 ) /4 +(XMQ 2 ) /2 +(M 3 NQ ) /2 +(XM 5 Q 3 ) /4 +(XM Q 3 )/2 + MQ + X MNQ } (MN) ε. Since Q M ε, the fifth and sixth terms on the right-hand side are superfluous. Inserting the simplified estimate into (5.2) and taking Q = M ε, we find (5.7) S(M,N) 2 { MN 2 +(XM 6 N 6 ) /4 +(XM 3 N 2 ) /2 +(M 4 N 3 ) /2 +(XM 8 N 4 ) /4 + X (MN) 2} (MN) ε. Similarly by interchanging the role of M and N, wealsohave (5.8) S(M,N) 2 { M 2 N +(XM 6 N 6 ) /4 +(XM 2 N 3 ) /2 +(M 3 N 4 ) /2 +(XM 4 N 8 ) /4 + X (MN) 2} (MN) ε. Now the required estimate follows from (5.7) if X N 2 and M N, and from (5.8) when X N 2 and M>N. This completes the proof.

27 Small gaps and B-free numbers 307 Next as an application of Proposition 5, we consider a particular triple exponential sum of type I: S I (H, M, N) := ) ξ h ψ n e (X hβ m β n α, H β M β N α h H m I n N where X > 0, H, M, N, ξ h, ψ n andi is a subinterval of [M,2M]. Corollary 8. Let α, β R satisfy β, 0 and α/( + β) 0,. For any ε>0, we have (5.9) S I (H, M, N) { (X 3 H 6 M 2 N 6 ) /8 +(XH 2 N) /2 + HN +(XH 3 M) /4 N + X HMN } (HMN) ε, and (5.0) S I (H, M, N) { (X κ+λ H +κ+λ M +κ λ N 2+κ ) /(2+2κ) +(XH 2 N) /2 +(HM) /2 N + HN + X HMN } (HMN) ε, where (κ, λ) is an exponent pair. Proof. If X/M, the Kusmin-Landau inequality implies 2 S I (H, M, N) X HMN. When X/M >, applying Lemma 2.2 of [40] to the sum over m and 2 using Lemma 2.3 of [40] with n = n to estimate the related error terms, we find S I (H, M, N) X /2 MS +(HN + X /2 H)logM, where S := ) m ψ n ξh ϕ m e ( αx hβ β n α, H β M β N α n N h H m I (h,n) where I (h, n) is a subinterval of [M, 2M ]withm := X/M, β := β/(+β), α := α/( + β), α := +β β β, ξ h, ϕ m and ψ n. Noticing that the exponents of h and m are equal, we can express this new triple sum as a double exponential sum over (h,n)withh = hm hi (h, n). We use Lemma 2.5 of [40] to relax the condition h = hm hi (h, n) toh H := HM = XH/M. Finally applying Proposition 5 with (M,N) =(N,H ) yields the desired estimate (5.9). The last inequality follows from (3.) of [30] with the choice of (H, M, N) =(H,,N). This completes the proof.

28 308 E. Kowalski, O. Robert and J. Wu Finally we study bilinear form of type I: (5.) ψ n r mn (x, y), m M n N where ψ n and (5.2) r d (x, y) := d n y d. In the sequel, ε denotes an arbitrarily small positive number and ε a constant multiple of ε, which may be different in each occurrence. Corollary 9. Let y := x θ and ψ n. Then for any ε>0 we have (5.3) ψ n r mn (x, y) ε yx ε m M n N provided one of the following two conditions holds <θ 5, 3 (5.4) N y 9/4 x 3/4 ε, MN x ε, or (5.5) (κ + λ)/( + 2κ +2λ) <θ (κ + λ)/(2κ + λ), N y (+2κ+2λ)/(+λ) x (κ+λ)/(+λ) ε, MN x ε. Proof. Without loss of generality, we can suppose that MN yx ε. By applying (5.9) of Corollary 8, we see that ( ) xh ψ n e MNx 2ε, mn provided h H m M n N <θ 5, H 3 MNy x 3ε, N y 9/4 x 3/4 ε, MN x ε. Combining this with Lemma 9 of [46] with the choice of ϕ m, we deduce (5.3) provided (5.4) holds. The other one can be proved by using (5.0) of Corollary 8. A particular case of (5.) linear forms (with N = ) will be needed in the proof of Corollary 0.

29 Small gaps and B-free numbers 309 Lemma 5.. Let y := x θ. Then for any ε>0 we have (5.6) r m (x, y) ε yx ε m M provided one of the following two conditions holds (5.7) <θ and M y 9/7 x 3/7 ε ; 9 (5.8) <θ 29 2 and M y 4/3 x ε. Proof. Without loss of generality, we can suppose that M yx ε. Theorem of [38] allows us to write ( ) xh e { (x 3 H 9 M 6 ) /8 +(xh 6 M) /5 m h H m M Mx 2ε, provided one of the following two conditions holds + HM 3/4 +(x H 2 M 4 ) /3} M ε <θ 9, H 4 29 My x 3ε, M y 9/7 x 3/7 ε or 9 <θ, H 29 2 My x 3ε, M y 4/3 x ε. This implies (5.6) if (5.7) or (5.8) holds. 6. B-free numbers in short intervals In this section we explain our new results about B-free numbers. Let B = {b k <b <b 2 < } be an infinite sequence of integers such that (6.) k= b k < and (b j,b k )= (j k). As already mentioned in the introduction, the B-free numbers are the integers that are divisible by no element of B. We also already explained that the existence of B-free numbers in short intervals was proved by Erdős [], who showed that there is a constant θ (0, ) such that the short interval (x, x + x θ ]withxsufficiently large contains B-free numbers. Szemerédi [44] showed that θ = + ε is admissible. 2

30 30 E. Kowalski, O. Robert and J. Wu This result was further improved to θ = 9 + ε 20 by Bantle & Grupp [3], θ = 5 + ε 2 by Wu [45], θ = 7 4 θ = θ = ε by Wu [46], + ε by Wu [47] and by Zhai [48] (independently), + ε by Sargos & Wu [40]. Inserting our new result on bilinear form ((5.4) of Corollary 9) into the argument of [46], we immediately obtain a slightly better exponent. Proposition 6. For any ε>0, x x 0 (B,ε) and y x 7/7+ε, we have B,ε y. b n ( b B) Next we shall consider special sets B, of the type which occurs in the applications to modular forms (Theorems and 2). Let P be the set of all prime numbers and P be a subset of P for which there is a constant ρ [0, ] such that (6.2) P [,x] x ρ (log x) Θρ (x 2), where Θ ρ is a real constant such that Θ >. Define B P := P { p 2 p P P } = {b k b <b 2 < }. Clearly the hypothesis (6.2) guarantees that B P satisfies the condition (6.). One can hope to obtain a smaller exponent for this special set of integers B P than in the general case. In this direction, Alkan ([], Theorems 2.2 and 2.3) proved, by exploiting the structure of the first component P of B P, the following result: If y x θ with (6.3) θ = θ(ρ) = then (6.4) { + ε if ρ =, 3 2 max { 7, } 23ρ 9 35ρ+6 + ε if <ρ, 2 b n ( b B P ) P,ε y. His proof is based on the method of Bantle & Grupp [3], using the weight of the form w(n) :=, where P i := { p P x δ i <p i x } δ i+ε. p P p 2 P 2 p p 2 n