A HIGHER RANK SELBERG SIEVE WITH AN ADDITIVE TWIST AND APPLICATIONS M. Ram Murty, Akshaa Vatwani. 1. Introduction

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1 Functiones et Approximatio ** 201*, 1 34 doi: /facm/1622 A HIGHER RANK SELBERG SIEVE WITH AN ADDITIVE TWIST AND APPLICATIONS M. Ram Murty, Akshaa Vatwani To the memory of Atle Selberg, with respect and admiration Abstract: We develop an axiomatic formulation of the higher rank version of the classical Selberg sieve with an additive twist and provide asymptotic formulas for the same. As an application of this higher rank sieve, we obtain improvements of results of Heath-Brown and Ho-Tsang on almost prime k-tuples. Keywords: the Selberg sieve, almost prime k-tuples. 1. Introduction Atle Selberg [16] was the first to suggest the existence of a higher rank version of his celebrated sieve method see page 351 of [16] and page 245 of [15]. This has been applied by Maynard [9] and the Polymath project [14] to give, among other results, a simplified proof of the Yitang Zhang s ground-breaking result [18] regarding bounded gaps between primes. In [17], the second author gives an axiomatic formulation of the higher rank sieve as well as applications to various problems. In this paper, we develop the theory of the Selberg sieve with an additive twist, the general theory of which seems to be new even in the rank one case, though the ideas are nascent in some of Selberg s work. Thus, our main theorems, which can be seen as a natural extension of this progression of ideas, are contained in Theorems 3.3 and 4.3 below. Though the theory can be employed to a number of problems, in this paper we restrict our attention to the well-known prime k-tuples conjecture. The remarkable advantage of the general formulation leads us naturally to realize the culmination of Selberg s idea hinted in the two papers cited earlier, namely [15] and [16]. Research of the first author partially supported by an NSERC Discovery grant and a Simons fellowship. Research of the second author partially supported by an Ontario Graduate Scholarship Mathematics Subject Classification: primary: 11N05; secondary: 11N35, 11N36

2 2 M. Ram Murty, Akshaa Vatwani As an application of our general theory, we prove Theorem 5.13, which we state below. Let τ denote the divisor function and H = {h 1,..., h k } be an admissible set, by which we mean that h i s are distinct non-negative integers and for every prime p, there is a residue class b p mod p such that b p / H mod p. Theorem. There exists ρ k such that there are xlog log x 1 log x k integers n x satisfying: the product k n + h i is square-free and τn + h i ρ k. For large k, we have ρ k 3 4 k2. This improves upon Theorem 1 of Heath-Brown [5], which gives ρ k 3 2 k2. Ho and Tsang [6] obtained the above result with ρ k k 2. Similar computations have also been performed by Maynard [10]. It is worth noting that the above mentioned results do not use the higher rank sieve and rely on the combinatorial version of the classical sieve. Note that for k = 2, the best result is due to Chen [1]. As a consequence, we obtain bounds on the number of distinct prime divisors of the product k n+h i as well as for each n+h i individually, which are stated as Corollaries 5.14 and Our derivation of the above results uses the k-rank Selberg sieve along with the divisor function analogue of the Bombieri-Vinogradov theorem. More precisely, let a, q = 1, and set Ex, q, a = n x n a mod q τn 1 φq n x n,q=1 τn, 1.1 where φ is Euler s function. Then for any A > 0 and any θ < 2/3, max Ex, q, a x a,q=1 log x A. 1.2 q x θ In fact, the sharper result that Ex, q, a q 1/4 x 1/2+ɛ for any ɛ > 0 was known to Selberg see p.237 of [15] as well as Hooley [7] and Linnik unpublished. It is conjectured that 1.2 is valid for any θ < 1. Fouvry and Iwaniec [2] have investigated this conjecture in a variety of directions. In particular, they showed that if a is fixed, then for any ɛ, A > 0, we have Ex, q, a x log x A, x 2 3 +ɛ <q<x 1 ɛ a,q=1 where the implied constant depends on ɛ, A and a. For other variants of the Bombieri-Vinogradov theorem, we refer the reader to [11].

3 A higher rank Selberg sieve with an additive twist and applications 3 2. Notation We repeat much of the notation and terminology set up in [17] for convenience of the reader. A k-tuple of integers d := d 1,..., d k is said to be square-free if the product of its components is square-free. For a real number R, the inequality d R means that i d i R. The notion of divisibility among tuples is defined component-wise, that is, d n d i n i for all 1 i k. It follows that the notion of congruence among tuples, modulo a tuple, is also defined component-wise. On the other hand, we say a scalar q divides the tuple d if q divides the product i d i. However when we explicitly write the congruence relation d e mod q, we mean that it holds for each component. When we say that a tuple d divides a scalar q, we mean that i d i divides q. For a square-free tuple, this is equivalent to each component dividing q. We do not invoke any special notation for vector functions, that is, functions acting on k-tuples. It will be evident from its argument whether a function is a vector or scalar function. Most of the functions that we deal with are multiplicative. A vector function is said to be multiplicative if all its component functions are multiplicative. In this context, we define the function fd to mean the product of its component multiplicative functions acting on the corresponding components of the tuple, that is, k fd = f i d i. The identity function acting on a tuple d is denoted by d itself. In this case, d would represent the product k d i. It will be clear from the context whether we mean the above product or the vector tuple itself. Similarly, when we write a tuple raised to some power, we interpret it as the appropriate function acting on the tuple. For example, d 2 = k d2 i. Similarly, we define for k-tuples d and α, d α = k dαi i. Some more vector functions that will be used by us are the Euler phi function, as well as the lcm and gcd functions. For example, [d, e] := k [d i, e i ]. We also use the notation [d, e] n to mean [d i, e i ] n i for 1 i k. When written as the argument of a vector function, [d, e] will denote the tuple whose components are [d i, e i ]. The meaning of the use will be clear from the context. Similarly, a vector function νd is called additive if all its components ν i are additive, in which case, we define νd = ν i d i.

4 4 M. Ram Murty, Akshaa Vatwani We use the convention n N to denote N n < 2N. Alternatively, fx fx gx may also denote that lim x gx = 1. The meaning will be clear from the context. Moreover, if we have an expression of the form fx = 1 + o1cgx, where c is a constant independent of x, it is understood that the case c = 0 implies that fx = ogx. Furthermore, we let τn denote the number of divisors of the integer n and ωn denote the number of distinct prime factors of n. The greatest integer less than or equal to x is denoted as x. Throughout this paper, δ denotes a positive quantity which can be made as small as needed. We employ the following multi-index notation to denote mixed derivatives of a function on k-tuples, Ft. for any k-tuple α with α := k α j. 3. The higher rank Selberg sieve revisited F α t := α Ft 1,..., t k t 1 α1... t k α, 2.1 k In this section, we summarize the salient features of the higher rank Selberg sieve discussed in [17]. Our exposition is concise for the sake of brevity and the reader is encouraged to peruse Section 3.2 of the above mentioned paper. Given a set S of k-tuples, S = {n = n 1,..., n k }, in [17], we undertook a systematic study of sums of the form 2 λ d, 3.1 n S w n d n where w n is a weight attached to the tuples n and λ d s are parameters to be chosen. Throughout this section, the condition n S is understood to hold without being explicitly stated. We impose the following hypotheses on this sum: H1. If a prime p divides a tuple n such that p divides n i and n j, with i j, then p must lie in some fixed finite set of primes P 0. This hypothesis allows us to perform the W trick. We fix some W = p<d 0 p, with D 0 depending on S, such that p P 0 implies that p W. We then fix some tuple of residue classes b mod W, with b i, W = 1 for all i and restrict n to be congruent to b in the above sum. H2. With W, b as in H1, the function w n satisfies d n n b mod W w n = X fd + r d, for some multiplicative function f and some quantity X depending on the set S.

5 H3. The components of f satisfy A higher rank Selberg sieve with an additive twist and applications 5 f j p = p α j + Op t, with t < 1, for some fixed α j N, α j independent of X, k. We denote the tuple α 1,..., α k as α and the sum of the components k α j as α. H4. There exists θ > 0 and Y X such that for any A > 0, as Y. [d,e]<y θ r [d,e] Y log Y A, Henceforth, we assume D 0 and hence W as X. The big oh and little oh notation used henceforth is understood to be with respect to X, unless stated otherwise. Moreover, the implied constants may depend on those parameters which are independent of X such as the function f, parameters A, α j, β j etc but not those quantities which do depend on X such as D 0, W, R, Y With all this in place, we state the main results of the higher rank sieve obtained in [17]. Lemma 3.1. Set R to be some fixed power of X and let D 0 = olog. Let f be a multiplicative function satisfying H3 and G, H : [0, k R be smooth functions with compact support. We denote log d log d1 G := G,, log d k and similarly for H. Let the dash over the sum mean that we sum over k-tuples d and e with [d, e] square-free and co-prime to W. Then, d,e µdµe f[d, e] G log d log e H = 1 + o1cg, H α cw α, where CG, H α = 0 cw := p W 0 p α φp α. k t αj 1 j Gt α Ht α dt, α j 1! Here Gt α and Ht α are as in the notation of 2.1.

6 6 M. Ram Murty, Akshaa Vatwani Let η G, η H be shifted Fourier transforms of G and H respectively. More precisely, let η G u = Gt expt expiu tdt, R k where expt = n etj and the dot denotes dot product of tuples. We have a similar expression for η H u. Lemma 3.2. Let a denote the tuple a 1,..., a k and let a = j a j. We follow the same notation for b and c and the notation of 2.1 for the relevant mixed derivatives. Then the integral is given by R k R k η G uη H v k CG, H a,b,c := 1 a+b iu j aj 1 + iv j bj dudv cj 1 + iu j iv j 0 k t cj 1 j Gt a Ht b dt. c j 1! We now choose λ d in terms of a fixed symmetric smooth function F : [0, k R, supported on the simplex k 1 := {t 1,..., t k [0, k : t t k 1}, as follows log d1 log d1 λ d = µdf := µd 1... µd k F,..., log d k. 3.2 Theorem 3.3. Let λ d s be as chosen above. Suppose hypotheses H1 to H3 hold and H4 holds with Y = X. Set R = X θ/2 δ for small δ > 0 and let D 0 = olog. Then, with CF, F α = 2 w n λ d = 1 + o1cf, F α X cw α, n b mod W d n 0 0 k t αj 1 j 2 F α W α t dt, cw := α j 1! φw α. 4. The higher rank sieve with an additive function In [17], one could have considered a more general setting for the sieve, in which the weights w n satisfy:

7 H2. A higher rank Selberg sieve with an additive twist and applications 7 d n n b mod W w n = X fd + X f d νd + r d, where f and f are multiplicative functions and ν is an additive function. In other words, each component ν j d j of νd is an additive function. H2 used earlier can be thought of as H2 with X = 0. However, there do arise situations when the expressions we need to analyze are of this more general form, as we shall see. One could consider even more general situations as hinted by Selberg [16], but we do not do so here. The reader may consult p.37 of [3] for a cursory discussion of this kind of setting. This expression motivates the analysis of sums of the following form. Lemma 4.1. Let f, g, h be multiplicative vector functions and ν be an additive vector function. Let d and e denote k-tuples as usual. We define by S the twisted sum d,e µdµegdhe ν[d, e], f[d, e] where the dash over the initial sum means that we are summing over tuples d and e such that [d, e] is square-free. Assuming that all the sums involved are absolutely convergent, we have S = S q Sq ν j qg j q f j q + ν jqh j q f j q ν jqg j qh j q, f j q where the summation in the second term runs over all primes q, and S = d,e µdµegdhe, Sq = f[d, e] d,e q [d,e] Proof. Fix a prime q. We denote by S q the sum d,e q [d,e] µdµegdhe. f[d, e] µdµegdhe ν[d, e] 4.1 f[d, e] Then as [d i, e i ], [d j, e j ] are co-prime for all i j, q can divide only one of the [d j, e j ] s if it divides the tuple [d, e]. Hence, S = S q + d,e q [d j,e j] µdµegdhe ν[d, e]. f[d, e] Again as before, for each j, the condition q [d j, e j ] leads to three cases:

8 8 M. Ram Murty, Akshaa Vatwani a q d j, q e j b q d j, q e j c q d j, q e j. For each case, the dash over the sum indicates that q cannot divide [d i, e i ] for any i j. The sum for case a is given by, d,e q d j,q e j µdµegdhe ν[d, e] f[d, e] We write d j = qd j, so that [d j, e j ] = q[d j, e j] and hence ν[d j, e j ] = ν j q + ν j [d j, e j]. Then with obvious notation, the sum for case a is equal to g j qν j q µd µegd he f j q f[d, e] that is, d,e q [d,e] g j q f j q g j qν j q Sq f j q d,e q [d,e] µd µegd he f[d ν j [d, e],, e] g j q f j q S q. Due to the additive function, we have obtained an extra term compared to the expression for case a in Lemma 3.2 of [17]. Cases b and c yield exactly the same expressions, with g j replaced by h j and g j h j respectively. Thus S = 1 gj q f j q + h jq f j q g jqh j q S q 4.2 f j q gj qν j q f j q + h jqν j q f j q g jqh j qν j q Sq. f j q By Lemma 3.2 of [17], we know that S has an Euler product. Note that the first term in the parenthesis is in fact the Euler factor in this Euler product, corresponding to the prime q. Let us denote it by P q. Now, fixing a prime q q and repeating this process for S q, we obtain S gj q q = 1 f j q + h jq f j q g jq h j q f j q S q, q 4.3 gj q ν j q f j q + h jq ν j q f j q g jq h j q ν j q f j q Sq, q, where S q, q and Sq, q are the respective sums with the primes q and q eliminated in the sense of 4.1.

9 A higher rank Selberg sieve with an additive twist and applications 9 Note that the Euler product for Sq, q is the Euler product for S with P q and P q the Euler factors corresponding to q and q, removed. We plug the above expression for S q into the expression for S preceding it, to obtain S = P q P q S q, q gj q ν j q f j q gj qν j q f j q + h jq ν j q f j q + h jqν j q f j q g jq h j q ν j q f j q P q Sq, q g jqh j qν j q Sq. f j q Noting that P q Sq, q = Sq and continuing this process over all primes gives S equal to P p gj qν j q + h jqν j q g jqh j qν j q Sq. f p q j q f j q f j q The product p P p is simply the Euler product for S. Thus, S is expressible in terms of S and Sq with q running over all primes, both of whose Euler products are known. We proceed to the general sieve as in the previous section, with hypotheses H1, H3 and H4. However, the hypothesis H2 is now replaced by H2. We also impose an additional hypothesis H5 on our additive function ν, which is akin to H3 for the multiplicative function f. H5. For each j, we have p ν j p p 1+δ = β j δ + O1, p ν j p 1 p 1+δ δ δ as δ 0. We will be concerned with this sum for p > D 0. Then, ν j p p 1+δ = β j δ + O ν j p p 1+δ. p>d 0 p<d 0 Moreover, ν j p p 1+δ e ν j p p 1+δ e log p log D 0 e p<d 0 p<d 0 p From 4.4, as D 0, we have p ν j p p 1+ 1 log D 0 = Olog D 0. ν j p p log D +δ 0 4.4

10 10 M. Ram Murty, Akshaa Vatwani Hence, ν j p p 1+δ p>d 0 = β j δ + Olog D 0. In practice one usually finds that for a fixed j, ν j p is of the same sign for all primes p, so that the two conditions of 4.4 can be reconciled into a single condition. The following result is the analogue of Lemma 3.1 in the additive function case. Let α = k α j, W = p<d 0 p and D 0 as X, as before. Lemma 4.2. Set R to be some fixed power of X and let D 0 = olog. Let f be a multiplicative function and ν be an additive function satisfying H3 and H5 respectively. Let G, H be smooth functions with compact support. Let all notation be as in Lemma 3.2 and Theorem 3.3. Then, the sum d,e is given by as R where, cw 1 + o1 α 1 µdµe ν[d, e]g f[d, e] log d H log e log β j α j Cj G, H α D0 + O α, C j G, H α = CG, H α,α,α+ej CG, H α ej,α,α CG, H α,α ej,α and the tuple α ± e j is α 1,..., α j ± 1,..., α k. Proof. Following the proof of Lemma 3.4 of [17], we extend the functions G and H to smooth compactly supported functions on R k. Then by Fourier inversion, we have Gt = η G u exp 1 + iu t du, R k 4.5 Ht = η H v exp 1 + iv t dv, R k where η G and η H are shifted Fourier transforms of G and H respectively and the dot denotes dot product of tuples. η G and η H are Fourier transforms of smooth functions with compact support and are hence rapidly decaying smooth functions, satisfying the bounds for any A 1, A 2 > 0. η G t 1 + t A1, η H t 1 + t A2, 4.6

11 where A higher rank Selberg sieve with an additive twist and applications 11 The required sum can be written as R k Z u, v = d,e p>d 0 R k η G uη H vz u, vdudv, 4.7 µdµe f[d, e] ν[d, e] 1 1 d 1+iu/ e 1+iv/. Consider the Euler product for Zu, v given by equation 3.7 of [17]: Zu, v = 1 + o1 α j p p 1+iu j p 1+iv j p 1+iu j + 1+iv j. 4.8 Let Zu, v q be Zu, v with the Euler factor corresponding to some prime q W removed. Applying Lemma 4.1 gives the following expression for Z u, v in terms of Zu, v and Zu, v q : Zu, v q q W ν j q f j q 1 q 1+iu j + 1 q 1+iv j 1 q 1+iu j +1+iv j + Zu, v Noting from equation 3.7 of [17], the Euler factor for the prime q in the Euler product for Zu, v, we have, Zu, v = 1 + Oq 1 Zu, v q. As 1/1 + Oq 1 = 1 + Oq 1, one obtains Thus, Z u, v equals, Zu, v q = 1 + Oq 1 Zu, v. ν j q 1 Zu, v + 1 f j q q W q 1+iu j q 1+iv j + Zu, v O ν j q + Zu, v. q f j q q>d 0 1 q 1+iu j +1+iv j The hypothesis H3 on f gives ν j q f j q = O νj q q, q W ν j q f j q = α j ν j q q q>d 0 + O ν j q q 2 t, q>d 0

12 12 M. Ram Murty, Akshaa Vatwani with t < 1. As done in the comments following H5, we can show that ν j q q q<d 0 q ν j q Keeping this in mind, we apply partial summation on ν j q 1 q q 1 t, q>d 0 q 1+ 1 log D 0 log D to get ν j q q2 t log D0 D 1 t q>d 0 0 Since 0 t < 1, the above term is o1 as D 0. We conclude that as D 0, Z u, v is given by 1 + o1zu, v ν j qα j q q>d 0 q 1+iu j q 1+iv j log D0 + Zu, v + Zu, v O. D 0 By partial summation and the bound 4.9, we have 1 q 1+iu j +1+iv j ν j q q q>d 0 1 q 1+u j log D 0 D 1+uj 0 log D 0, as R. Similar bounds can be obtained for each summand in the first term above. This shows that Z u, v log D 0 Zu, v. Recalling the bound Zu, v O1 given by 3.9 of [17], we see that Z u, v O Using the same argument as in Lemma 3.4 of [17], one can show that the contribution from the region u or v ɛ to the integral 4.7 is O A for any A > 0. In the region u, v ɛ, we apply H5 with δ being terms of the type 1 + iu j /, to obtain that Z u, v = 1 + o1zu, v β j α j 1 + iu j iv j 1 + iu j 1 + iv j + Olog D 0 Zu, v.

13 A higher rank Selberg sieve with an additive twist and applications 13 We then use the known result cf. 3.10, [17] 1 Zu, v = 1 + o1cw α k to simplify the above expression, finally obtaining 1 + iu j αj 1 + iv j αj, 4.12 αj 1 + iu j iv j Z 1 u, v = 1 + o1cw α 1 L 2u, v L 1 u, v L 1 v, u + Olog D 0 Zu, v, 4.13 with L 1 u, v = = L 2 u, v = = L 1 u, v j 4.14 β j α j 1 + iu j αj iv j αj 1 + iu j iv j αj L 2 u, v j l j β j α j 1 + iu j αj 1 + iv j αj 1 + iu j iv j αj+1 l j 1 + iu l α l 1 + iv l α l 1 + iu l iv l α, l 1 + iu l α l 1 + iv l α l 1 + iu l iv l α. l Plugging all this into 4.7, we are led to evaluate new integrals of the form η G uη H vl 1 u, vdudv, η G uη H vl 2 u, vdudv. R k R k R k R k Note that we have extended the integrals to be over all of R k. This can be done since the O A contribution from the complementary region gets absorbed into the o1 term of Now, using Lemma 3.2, we have η G uη H vl 1 u, v j dudv = β j α j CG, H α ej,α,α, R k R k where α is as before and α e j = α 1,..., α j 1,..., α k. Similarly, η G uη H vl 1 v, u j dudv = β j α j CG, H α,α ej,α, R k R k and R k R k η G uη H vl 2 u, v j dudv = β j α j CG, H α,α,α+ej, where α + e j = α 1,..., α j + 1,..., α k.

14 14 M. Ram Murty, Akshaa Vatwani Putting together all these evaluated integrals gives for the main term of 4.13, the term involving Cj G, Hα in the lemma. We have not considered yet the error term of In the region u, v < ɛ, the expression 4.12 gives 1 k Zu, v = 1 + o1cw 1 + iu j αj 1 + iv j αj α 1 + iu j iv j αj, 4.15 Plugging Zu, v into 4.7 leads to 1 + o1 cw k α η G uη H v R k R k 1 + iu j αj 1 + iv j αj 1 + iu j iv j αj dudv. As this integral is absolutely convergent due to rapid decay of the integrand, in this region we obtain Zu, v Olog D 0 cw log D 0 α = o1 In the region u or v ɛ, the bound 4.11 as well as rapid decay bounds 4.6 mean that one can pull out any negative power of out of the integrand, and hence the error obtained in the former region dominates. We find that the required sum is as required. cw 1 + o1 α 1 β j α j Cj G, H α, Theorem 4.3. Let λ d s be chosen as in 3.2. We assume hypotheses H1, H2, H4 and H5. We also assume that both functions, f and f arising from H2 satisfy H3 with α j s and α j s respectively. Choose R = Xθ/2 δ and assume that D 0 = olog. Then, 2 cw X w n λ d = 1 + o1 CF, Fα α n b mod W d n o1 c W X β α 1 j αj Cj F, F α where C j F, Fα denotes the quantity and CF, F α,α,α +e j CF, F α e j,α,α CF, F α,α e j,α, cw = W α φw α, c W = W α. φw α All notation is as in Lemmas 3.1, 3.2, 4.2 and α := j α j.

15 A higher rank Selberg sieve with an additive twist and applications 15 Proof. We proceed exactly as in the proof of Theorem 3.6 of [17], by expanding the square, interchanging the order of summation, applying the W -trick and finally using H2. This gives us the following expression for the above sum, X d,e<r λ d λ e f[d, e] + X d,e<r λ d λ e ν[d, e] + O f [d, e] d,e<r λ d λ e r [d,e]. Now, we have two main terms. Using the given choice of λ d s, the first term can be analyzed as in Theorem 3.6 of [17] to obtain The second term yields X d,e<r 1 + o1cf, F α X cw α. µdµe ν[d, e]f f [d, e] log d log e F. By Lemma 4.2, this is given by 1 + o1 β j αj Cj F, F α c X W α 1. To complete the proof, we note that the choice of R along with H4 ensures that the error term is negligible. Note that a general version of the above theorem holds, without the condition log D 0 / = o1. In that case, in addition to the two terms already obtained in the theorem, we would get another term, namely X Olog D 0 cw α. 5. Application to almost prime k-tuples We apply the higher rank sieve to the well-known prime k-tuples problem, by making use of the divisor function τ to sieve out primes. A set H of distinct non-negative integers is said to be admissible if for every prime p, there is a residue class b p mod p such that b p / H mod p. We will work with a fixed admissible set of size k, H = {h 1,..., h k }. We use the W trick to remove the effect of small primes, that is we restrict n to be in a fixed residue class b modulo W, where W = p<d 0 p and b is chosen so that b + h i is co-prime to W for each h i. This choice of b is possible because of admissibility of the set H.

16 16 M. Ram Murty, Akshaa Vatwani One can choose D 0 = log log log N, so that W log log N 1+o1 by an application of the prime number theorem. We consider the expressions, S 1 = S 2 = n N n b mod W n N n b mod W d j n+h j j λ d 2, τn + h j For ρ positive, if we denote by SN, ρ the quantity ρs 1 S 2, then we have the following key observation. Proposition 5.1. Given a positive number ρ, if ρs 1 S 2 > 0 d j n+h j j λ d 2. for all large N, then there are infinitely many integers n such that τn + h j ρ, where ρ denotes the greatest integer less than or equal to ρ. Proof. As λ d s are non-negative, we see that if SN, ρ > 0, there must exist n N such that ρ τn + h j > 0. As this happens for all large N, this inequality holds for infinitely many integers n. As each τn + h j is an integer, this completes the proof. The asymptotic formula for S 1 was derived in Lemma 4.2 of [17]. We proceed to derive an asymptotic formula for S 2. Recall the definition of Ex, q, a in Section 1. Given θ > 0, we say that θ is permissible for the divisor function if for any A > 0, a, q = 1, we have max max a,q=1 y x q<x θ Ey, q, a x log x A. 5.1 We define the level of distribution θ 0 of the divisor function to be the supremum of all the permissible values of θ. Henceforth we will work with a fixed permissible θ and set R = N θ/2 δ, for some small δ > 0. In the context of our problem, θ is assumed to be less than 1.

17 A higher rank Selberg sieve with an additive twist and applications Asymptotic formula for S 2 We write S 2 = m=1 S m 2, S m 2 = n N n b mod W τn + h m d j n+h j j λ d 2, and proceed to obtain an asymptotic formula for S m 2. This calls upon the theory of the sieve with an additive function, as we will see. We begin with some preliminary propositions. Proposition 5.2. Let q 1 and a, q = 1. Then, n x n a mod q τn = φq q 2 x log x + c + 2 p q with c = 2γ 1, where γ is Euler s constant. Proof. We have, n x n a mod q τn = 1 φq Lemma 16 on p. 234 of [15] gives 1 φq n x n,q=1 n x n,q=1 τn = φq q 2 x log x + c + 2 p q log p + Ex, q, a + O q ɛ 1 x, p 1 τn + Ex, q, a. x log p + O p 1 φq p q p 2 Now, p q p 2 3 ωq q ɛ, where ωq denotes the number of distinct prime factors of q. Hence this error term is Oq ɛ 1 x. This completes the proof. The function τ is not completely multiplicative, but it is not very far from being so. We state some results that will make this more precise. Proposition 5.3. We have, τmτn = d m,n mn τ d 2

18 18 M. Ram Murty, Akshaa Vatwani Proof. The result is trivial when m, n = 1. It is enough to prove it in the case m = p a, n = p b for some prime p. Assume without loss of generality that a b. Then the left hand side of the result gives τp a τp b = a + 1b + 1. The right hand side gives a τ p a+b 2j = j=0 a a + b + 1 2j = a + b + 1a + 1 aa + 1 j=0 = a + 1b + 1, completing the proof. We now state the following two variable version of Möbius inversion c.f. Lemma 2.1 of [13], which can be proved in the usual manner. Lemma 5.4. Let Then, and conversely. F m, n = Gm, n = d m,n d m,n Gm/d, n/d. µdf m/d, n/d Applying this lemma to Proposition 5.3 gives the following expression for τmn in terms of τm and τn. It can also be proved directly following the method of Proposition 5.3. Proposition 5.5. We have, τmn = d m,n µdτm/dτn/d. The following proposition allows us to obtain a more general form of Proposition 5.2, without the condition a, q = 1. Proposition 5.6. Let q 1 be square-free, a, q = δ. Then, τn = µdτδ/d τn d δ n x n a mod q n x/δd n a d mod q, where q = q/δ and a d a 1 δd mod q. In particular, a d, q = 1. Proof. We write n = n 1 δ and q = q δ. As q is square-free, we have q, δ = 1. Letting δ denote the inverse of δ modulo q and a 1 aδ mod q, we have, τn = τn 1 δ. n x n a mod q n 1 x/δ n 1 a 1 mod q

19 A higher rank Selberg sieve with an additive twist and applications 19 Note that q, a 1 = 1. Using Proposition 5.5 and then interchanging summation, we obtain that the above sum equals µdτδ/dτn 1 /d = µdτδ/d τn 1 /d n 1 x/δ d δ,n 1 d δ n 1 a 1 mod q = d δ µdτδ/d n 1 x/δ n 1 a 1 mod q n 1=dn n x/dδ n a d mod q τn, where a d a 1 d mod q, which exists as δ, q = 1 implies d, q = 1. Here, a d, q = 1 as needed. Proposition 5.7. Let q 1 be square-free, a, q = δ. Let q = q/δ. Then, n x n a mod q τn δτδ µd φq = φδ dτd q 2 d δ + E x, q, a + O δ 1/2 q ɛ 1 x. x log x log δ + p δ log p 2p 1 + c + 2 p q log p p 1 Here, E x, q, a = τδ d δ µd τd Ex/δd, q, a d, with a d aδd mod q. Proof. Combining Propositions 5.2 and 5.6 for the sum τn, we obtain the following main term d δ µdτδ/d φq q 2 n x n a mod q x δd log x logδd + c + 2 p q log p. p 1 Here as q is square-free, so are δ and each d dividing δ. Hence, τδ/d = τδ/τd. Similarly, φq = φq/φδ. With some simplification, this gives δτδ φδ d δ µd dτd φq q 2 x log x log δ log d + c + 2 p q log p p 1

20 20 M. Ram Murty, Akshaa Vatwani We examine the term involving log d. We would like to express the relevant series as a product, namely, µd dτd log d = µd hδ, dτd d δ d δ for some suitable function h. Consider the Dirichlet series f δ s := d δ µd d s τd. Then f δs = d δ µd log d d s τd = f δ s p δ log p 2p s, 1 by logarithmic differentiation of the Euler product of f δ s. Thus f δ1 =: µd log d = µd log p. dτd dτd 2p 1 d δ d δ p δ Plugging this into the obtained expression gives the desired main term. The error terms are given by τδ µd τδ τd Ex/δd, 1 q, a d + O d δ δ τd d q ɛ 1 x. d δ We denote the first term by E x, q, a. As d δ the second error term gives O δ ɛ 1/2 q ɛ 1 x 1 τd d τδ δɛ/2, = O δ 1/2 q ɛ 1 x. We now seek to show that the error term E x, q, a appearing in the proposition above satisfies an average bound of the type 5.1, as this will be needed in our estimation of S 2. In order to do this, let us define some notation. We define E x, q := max a,q=1 max y x Ey, q, a, Ẽ x, q := max a mod q max y x E y, q, a. Then 5.1 can be rewritten as q<x θ E x, q x log x A. 5.2

21 We wish to prove that A higher rank Selberg sieve with an additive twist and applications 21 q<x θ Ẽ x, q := max max a mod q q<x θ y x E y, q, a x log x A. 5.3 We begin with the following result giving a bound for each individual E y, q, a, y x. Proposition 5.8. Let q 1 be square-free, δ = a, q and q = q/δ. Then, for any y x, E y, q, a τδ 2 E x/δ, q. Proof. By definition of E y, q, a, E y, q, a τδ d δ 1 τd Ey/δd, q, a d, with a d aδd mod q. We know that a d, q = 1. Notice that Ey/δd, q, a d max max Ez, a,q =1 z x/δ q, a The right hand side of the above inequality is simply E x/δ, q, giving E y, q, a E x/δ, q τδ d δ 1 τd. It is clear that d δ 1 τd d δ 1 τδ, thereby completing the proof. Theorem 5.9. Let q 1 be square-free. For any A > 0, we have for any permissible θ. q<x θ max y x max a,q=1 E y, q, a x log x A, Proof. Using the result of the previous proposition with the same notation, we have Ẽ x, q := max max a mod q y x E y, q, a max a mod q τδ2 max y x E x/δ, q/δ. Clearly, the condition max y x is redundant. As δ := a, q is the only parameter in the right hand side that depends on the choice of residue class a mod q, the condition max a mod q can be replaced with max δ q, to give Ẽ x, q max δ q τδ2 max y x E x/δ, q/δ.

22 22 M. Ram Murty, Akshaa Vatwani Then, Ẽ x, q q x θ max δ q q x θ δ x θ τδ 2 τδ2 max y x E x/δ, q/δ q x θ /δ E x/δ, q, q x θ δ q τδ 2 max y x E x/δ, q after interchanging summation. Using x θ /δ x/δ θ, for θ < 1, and 5.2 gives Ẽ x, q τδ 2 x 1 δ logx/δ A, q x θ δ x θ for any A > 0. In the given range of δ, logx/δ log x. One can use the Tauberian theorem cf. Ex of [12] or even elementary estimations to obtain n x τn2 cxlog x 3 for some constant c, followed by partial summation to get τn 2 log x 4. n n x Thus, Ẽ x, q q x θ τδ2 δ δ x θ x xlog x4 log x A log x A, This completes the proof of the theorem. We are ready to derive an asymptotic expression for S m 2. The following can be compared to Theorem 2.1 of Li-Pan [8]. It must be noted that the expression for S m 2 obtained by Li-Pan contains a sign error for the term β 2. This sign is crucial; if the expression obtained by [8] were correct, one could show that it leads to infinitude of twin primes for any θ > 2/3. Moreover, it would violate the analogue of the Elliott-Halberstam conjecture for the divisor function. Namely, one could then prove that the divisor function cannot have level of distribution greater than 4/5. This is contrary to expected heuristic reasoning. Lemma With λ d s as chosen in 3.2 in terms of F, and R = N θ/2 δ, we have as N, S m 2 := n N n b mod W = 1 + o1 W k 1 φw k τn + h m d j n+h j j λ d 2 N log N k αm β m 1 4β m 2,

23 A higher rank Selberg sieve with an additive twist and applications 23 with and α m := β m 1 := k 1 t 2 m k 1 t m F 1+em t 2 dt1... dt k, F 1+em t 2 dt1... dt k, β m 2 := t m F 1+em tf 1 tdt 1... dt k. k 1 We use the usual notation 2.1 as before. Proof. We begin by establishing the setting of the sieve. The tuple n in this case is n + h 1,..., n + h k, where h i are elements of the fixed set H. The set S is given by S = {n = n + h 1,..., n + h k : n N}. The choice of W and b was stated at the beginning of this section. The weights in this case are given by w n = τn + h m. Clearly, hypothesis H1 holds as H is a fixed set. We try to show H2. For this, consider the sum w n = τn + h m d n n b mod W n N d j n+h j j n b mod W For the above expression, we can use the same argument as in Lemma 4.2 of [17] to conclude that d i, d j = 1 for all i j, and d j, W = 1 for all j. We rewrite the above sum as τn, N+h m n <2N+h m n h m h j mod d j j n b+h m mod W where n = n + h m. Note that if a prime p divides h m h j, for j m, then p divides W. Hence d j, W = 1 implies that h m h j must be co-prime to d j for j m. It is also clear that b + h m is co-prime to W. Then, using the Chinese remainder theorem, the above sum can be written as the sum over a single residue class a modulo q = W j d j, with a, q = d m to obtain N+h m n <2N+h m n a mod q τn.

24 24 M. Ram Murty, Akshaa Vatwani Letting q denote q/d m, we apply Proposition 5.7 to write this sum as d m τd m φd m t d m µt tτt φq q 2 N log N log d m + p d m t d m log p 2p 1 + c + 2 log p p 1 p q + E N, q, a + O q ɛ 1 N p d m d 1/2 m. 5.4 We first deal with the main term the first term of this sum. Further simplification using the definition of q along with µt tτt = 1 1, 2p gives, φw W 2 N d mτd m φd m p d m + φw W 2 N d mτd m φd m 1 1 k φd j 2p d 2 log N j 5.5 k p d m φw W 2 N log d d m τd m m φd m 1 1 2p p d m φd j d 2 c + 2 j p q 1 1 k 2p φd j d 2. j log p p 1 log p 2p 1 p d m As all the d j s are co-prime to W as well as mutually co-prime to each other, we can rewrite 2 log p p 1 = 2 log p p log p p 1. p q p W j m p d j It is clear that the latter sum, being the sum of component additive functions, is an additive function of the k-tuple d. Thus H2 is satisfied, with X = φw W 2 N log N + c + p W fd = f d = φd m d m τd m log p p d m νd = log d m + 2p 1 p d m j m Moreover, the error terms in 5.4 give r d = E N, q, a + O 2 log p, X = φw p 1 W 2 N, 2p k 2p 1 2 log p p 1. p d j d 1/2 m d 2 j φd j, q ɛ 1 N.

25 A higher rank Selberg sieve with an additive twist and applications 25 It is clear that H3 holds for f and f with { α j = αj 1 if j = 1,..., k, j m = 2 if j = m. This brings us to H5 for the additive function ν, given by ν j p = 2 log p p 1 for j m, ν m p = log p log p 2p 1. Noting that ν m p > 0 for all primes p, we see that if one fixes a component j then the sign of ν j is fixed, which means that the absolute value sign in the statement of H5 is irrelevant. The behaviour of ζ s/ζs as s 1 + as well as the absolute convergence of the series p β j = log p p 2 shows that H5 holds for ν with { 0 if j = 1,..., k, j m 1 if j = m. Recall that we are using θ to denote the level of distribution of the divisor function see 5.1. To verify H4 it suffices to show that for any A > 0, E N, q, a + O [d,e]<n θ Denoting j m [d j, e j ] as [d, e], we have [d,e]<n θ [d m, e m ] 1/2 q ɛ 1 W ɛ 1 [d,e]<n θ [d m, e m ] 1/2 q ɛ 1 [d m,e m]<n θ [d m, e m ] ɛ 1/2 N N log N A, 5.6 [d,e] <N θ [d, e] ɛ 1 Using Proposition 3.1 of [17] and partial summation along with the fact that the average order of τ 3 n is log n 2, we get [d m,e m]<n θ [d m, e m ] ɛ 1/2 r<n θ r ɛ 1/2 τ 3 r N θ ɛ+1/2 log N 2. Similarly, [d, e] ɛ 1 r ɛ 1 τ 3k 1 r N θ ɛ log N 3k. [d,e] <N θ r<n θ As ɛ can be made arbitrarily small and W log log N 2, we obtain for the second term of 5.6, the estimate [d m, e m ] 1/2 q ɛ 1 N N ɛ +θ/2 N, [d,e]<n θ

26 26 M. Ram Murty, Akshaa Vatwani for any ɛ > 0. As θ < 1, this term is indeed of the order of N/log N A for any A > 0 as required. The first term of 5.6 can be written as q<w N θ E N, q, a q<n θ+ɛ E N, q, a for any ɛ > 0, due to the choice of W. For small enough ɛ > 0, θ + ɛ is permissible being less than the level of distribution θ 0, and one can apply Theorem 5.9, to obtain that the first term of 5.6 is ON/log N A as well. Thus H4 holds. As the choice of D 0 gives log D 0 = o1, we are now in a position to apply Theorem 4.3 with X, X, α, α, β as given above, α = α = k + 1 and cw = c W = W k+1 /φw k+1. After some simplification, this gives the following asymptotic formula as R for the required sum 1 + o1 W k 1 φw k N k log N + c + 2 log p p W p o1 W k 1 N φw k k 2CmF, F α. CF, F α Here α is the tuple 1,..., 1, 2, 1,..., 1, with 2 in the m-th place. As W = p<d 0 p, p W 2 log p p 1 1 D 0 p W By the choice of D 0 made in the beginning of this section, it is clear that D 0 = o. Hence, the asymptotic formula for the required sum as R, becomes 1 + o1 W k 1 N log N φw k k CF, Fα 2CmF, F α. Using Lemma 3.2 keeping in mind the notation 2.1, we obtain that CF, F α = α m F. Similarly, by definition see Theorem 4.3, C mf, F α equals CF, F α,α,α+em CF, F α em,α,α CF, F α,α em,α, which equals β m β m 2, after applying Lemma 3.2. This completes the proof.

27 A higher rank Selberg sieve with an additive twist and applications 27 Noting that condition H4 holds for both S 1 and S 2 when θ is permissible for the divisor function, we put together the above lemma with the asymptotic formula for S 1 obtained in Lemma 4.2 of [17] to obtain Lemma The quantity SN, ρ := ρs 1 k m=1 Sm with 1 + o1 W k 1 φw k α = N k ρif α m, β1 = 5.2. Choice of the test function 2 as N is given by, α θ/2 δ + β 1 + 4β 2 β m 1, β 2 = β m 2. Inspired by the choice of the function in [4], let us choose the function F to be Ft 1,..., t k = 1 k l! 1 k + l! when k t j 1, and zero otherwise. Then by 2.1, F 1 t 1,..., t k = 1 l+k t i 5.7 l t i, 5.8 We wish to compute all the integrals appearing in Lemma 5.11 with this choice of function. In order to compute the integral IF, we begin by considering the integral Then I l,k = 1 0 = 1 l + 1 I l,k = k 1 1 k i=2 ti = 1 l + 1 I l+1,k 1 1 k i=3 ti 0 1 l t i dt. 5.9 l t i dt 1 dt 2... dt k 1 This gives a recursion formula for I l,k. Moreover, I l+k 1,1 = 1 0 l+1 t i dt 2 dt 3... dt k i=2 1 t k l+k 1 dt k = 1 l + k.

28 28 M. Ram Murty, Akshaa Vatwani Thus, I l,k = l!/l + k!. By definition of IF in Lemma 4.2 of [17], one has We have from 5.8, IF = I 2l,k = F 1+em t 1,..., t k = l 2l! 2l + k! l 1 t i, 5.11 Let us compute the integrals α and β 2. Consider first the integral J l,k = k 1 t i 1 l t i dt. Then I l,k J l,k = 1 t i 1 k 1 l t i dt = I l+1,k Using known values for the integrals I l,k and I l+1,k, l! J l,k = k l + k + 1!. Using definitions in Lemmas 5.10 and 5.11, we have α F = l 2 k 1 β2f = l k 1 t i 1 t i 1 2l 2 t i dt, 2l 1 t i dt. From 5.8 and 5.11 for our choice of function, this gives α = l 2 J 2l 2,k = l 2 2l 2! k 2l + k 1!, 2l 1! β 2 = lj 2l 1,k = l k 2l + k! We are now left to compute the integral β1. In order to do this, we first consider the integral β m 1 defined in Lemma For our choice of function, using 5.11, we have 2l 2 β m 1 = l 2 t 2 m 1 t i dt k 1

29 A higher rank Selberg sieve with an additive twist and applications 29 Let σ m denote i m t i. Integration by parts with respect to t m gives us 1 σm t 2 m 0 1 2l 2 t i dt m = t2 m 1 2l 1 2l 1 t i σm t m 1 2l 1 0 t m=0 t m=1 σ m 2l 1 t i dt m The first term evaluated over the given limits is zero. Let t m denote the k-tuple t with the mth component removed. Then, β m 1 = l = 2l2 2l 1 1 i m ti t 2 m = 2l2 t m 1 2l 1 k 1 2l 2 t i dt m dt m 1 i m ti t m 1 0 2l 1 t i dt m dt m 2l 1 t i dt Thus, β 1 = 2l2 2l 1 J 2 2l 2! 2l 1,k = 2kl 2l + k! 5.13 Plugging obtained values for the required integrals into Lemma 5.11 yields the following result. Theorem SN, ρ = 1 + o1 W k 1 φw k N 2l! k 2l + k! S k,l,θ, where S k,l,θ = kl2l + k ρ 2l 1θ 2δ + kl 2l 1 2k 5.3. Improvement of a result of Heath-Brown We obtain the following improvement of a result of Heath-Brown [5], who obtained the theorem below with ρ k = 3k k.

30 30 M. Ram Murty, Akshaa Vatwani Theorem There exists ρ k such that there are xlog log x 1 log x k integers n x satisfying: the product k n + h i is square-free and For large k, we have ρ k 3 4 k2. τn + h i ρ k. Proof. It can be seen from the expression for SN, ρ in Theorem 5.12, that ρ > kl2l + k 2l 1θ 2δ kl + 2k, l 1 implies that S k,l,θ > 0 and hence SN, ρ > 0 for all large N. Thus, with λ d s chosen as in 3.2, we have that the quantity 2 ρs 1 S m 2 = ρ τn + h i λ d 5.15 m=1 n N n b mod W = 1 + o1 W k 1 φw k N k d j n+h j j 2l! 2l + k! S k,l,θ is positive for all large N. As explained in Proposition 5.1, this means that the inequality τn + h j < ρ, 5.16 holds infinitely often. In fact, one can estimate the number of integers n x for which 5.16 holds as follows. Let all n N, n b mod W for which 5.16 holds be denoted by n. Then the positive contribution to the first term in parenthesis in 5.15 can come only from these integers. As a consequence, n N n b mod W ρ τn + h j d j n +h j j 2 λ d W k 1 N φw k k For each n, the absolute value of the first term in parenthesis is bounded above by ρ. With the λ d s as chosen in 3.2, we have that each λ d is bounded in absolute value. Moreover, arguing using the W -trick as done in the proof of Lemma 4.2 of [17] shows that the n + h j s are pairwise co-prime. Thus, for some constant c, independent of n, k k λ d c 1 = c τ n + h j = c τn + h j. d j n +h j j d j n +h j j

31 A higher rank Selberg sieve with an additive twist and applications 31 However, by construction, all the n satisfy 5.16; in particular, each n + h j can have at most ρ divisors. Hence the above product of the divisor functions is bounded above by some constant M say independent of n. This gives 2 ρ τn + h i λ d 1, 5.18 n N n b mod W d j n +h j j n N n b mod W with both summations running only over integers n satisfying Combining 5.17 and 5.18, we get n N n b mod W 1 W k 1 φw k N k, which means that the number of integers x that satisfy the conditions of the theorem is xlog log x 1 log x k. We now wish to optimize the ρ in 5.16 as a function of k. Let θ 2δ = 2/3 ɛ in Putting l = 1 gives the theorem with ρ k = k 3k Notice that this is precisely Theorem 1 of Heath-Brown [5]. Letting l = 2 gives the theorem with ρ k = k k/3, which is an improvement for k > 2. In order to optimize ρ k, the right hand side of 5.14 should be viewed as a function hl and minimized with respect to l. Then we obtain l2l + k θ 2δl hl = k + 2k θ 2δ2l 1 Differentiating this with respect to l, we have h k 4l 2 4l k l = 2l 1 2 θ 2δ + 1. From this and h l, it can be seen that the hl is minimum when 4ll 1 = k θ 2δ. For large k, this means l k/2. Plugging this into hl gives hl k 2 2θ 2δ Thus, for large k, S k,l,θ can be made positive with l k/2, and ρ > Choosing θ 2δ = 2 3 ɛ completes the proof. k2 2θ 2δ.

32 32 M. Ram Murty, Akshaa Vatwani We remark that Heath-Brown discusses what seems to be a more general case, with n + h i replaced by linear functions of the type L i n = a i n + h i satisfying certain hypotheses on the a i s, namely that each a i is composed of the same set of primes. It is easy to see that our method can be adapted to the same setting, though in the interest of simplicity and elegance, we do not do so here. Remark. Our quantitative result is only slightly weaker than that of Heath-Brown by a factor of log log x 1. This can be improved further by a finer estimate of the constants depending on W. The above proof gives that for n satisfying the conditions of the theorem, n N n b mod W 1 1 W k W N φw k. By Merten s theorem, W φw = p<d 0 where γ is Euler s constant. This gives n x n b mod W p 1 e γ log D 0, log log log log xk log log x x log x k, thereby improving the quantitative estimate of Theorem It is also worth noting that if we assume the divisor function analogue of the Elliott-Halberstam conjecture, more precisely, if we assume that the divisor function has a level of distribution θ = 1, then the statement of Theorem 5.13 holds with ρ k k2 2. Interestingly, the above theorem leads to some bounds on the number of prime factors of k n + h i as well as the number of prime factors of each n + h i individually. We state these results as corollaries. It is worth noting that better bounds have been obtained through different means in the literature. The first corollary is as stated in [5] and follows immediately from the fact that τn + h i is at least 2 for each i. Corollary There exists R k such that there are xlog log x 1 log x k integers n x satisfying: the product k n + h i is square-free and max ωn + h i R k = log 2 ρ k 2k 1. i {1,...,k} For large k, we have R k 2 log k log 2.

33 A higher rank Selberg sieve with an additive twist and applications 33 To control the number of prime factors of the product k n + h i, observe that k ωn + h i k log 2ωn+hi ρk 2 k log k 2, 5.19 k from the AM-GM inequality and Theorem Equality occurs everywhere iff all the ωn + h i s are equal to the quantity log ρk 2 k, which we denote as c k. However c k may not be an integer, in which case this maximum is not attained in the context of our problem. We optimize by taking the ωn + h i s to be close to c k and to each other. More precisely, let k m where m k of the ωn + h i s be equal to c k and the remaining m of them be c k + 1. Then one has 2 ωn+hi = 2 ck k + m ρ k, from Theorem 5.13, giving m ρk k 2 c k As k ωn + h i = k c k + m, we have the following bound on the number of distinct prime factors of the product k n + h i. Corollary There exists r k such that there are xlog log x 1 log x k integers n x satisfying: the product k n + h i is square-free and ωn + h i r k = k c k 1 + ρk. 2 c k For large k, we have ρ k 3 4 k2 and c k = log ρk 2 k, giving r k k log k log Concluding remarks We remark that the quantitative results of this paper are by no means optimal and can be improved by various means. For instance, one natural device that suggests itself is to optimize the choice of W. Acknowledgements. We would like to thank V. Kumar Murty and John Friedlander for some helpful discussion and inputs. We also thank the referee for detailed comments on a previous version of this paper

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