Variants of the Selberg sieve, and bounded intervals containing many primes

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1 Polymath Research in the Mathematical Sciences 4, : RESEARCH ARTICLE Variants of the Selberg sieve, and bounded intervals containing many primes DHJ Polymath Open Access Correspondence: tao@math.ucla.edu index.php Abstract For any m, let H m denote the quantity lim inf n p n+m p n ). A celebrated recent result of Zhang showed the finiteness of H, with the explicit bound H 7,,. This was then improved by us the Polymath8 project) to H 468, and then by Maynard to H 6, who also established for the first time a finiteness result for H m for m, and specifically that H m m 3 e 4m. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H, improving upon the previous bound H 6 of Goldston, Pintz, and Yıldırım, as well as the bound H m m 3 e m. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H 46 unconditionally and H 6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple h, h, h 3 ),thereare infinitely many n for which at least two of n + h, n + h, n + h 3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most, or both. We also modify the parity problem argument of Selberg to show that the H 6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by ) our project to obtain the unconditional asymptotic bound H m me m or H m me m under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for H m when m =, 3, 4, 5. Keywords: Selberg sieve; Elliott-Halberstam conjecture; Prime gaps Bacground For any natural number m,leth m denote the quantity H m := lim inf p n+m p n ), n where p n denotes the nth prime. The twin prime conjecture asserts that H = ; more generally, the Hardy-Littlewood prime tuples conjecture [] implies that H m = Hm + ) for all m, where H) is the diameter of the narrowest admissible -tuple see the Outline of the ey ingredients section for a definition of this term). Asymptotically, one has the bounds ) + o)) log H) + o) log 4 Polymath; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly credited.

2 Polymath Research in the Mathematical Sciences 4, : Page of 83 as see Theorem 7 below); thus, the prime tuples conjecture implies that H m is comparable to m log m as m. Until very recently, it was not nown if any of the H m were finite, even in the easiest case m =. In the breathrough wor of Goldston et al. [], several results in this direction were established, including the following conditional result assuming the Elliott- Halberstam conjecture EH[ϑ] see Claim 8 below) concerning the distribution of the prime numbers in arithmetic progressions: Theorem GPY theorem). Assume the Elliott-Halberstam conjecture EH[ϑ] for all <ϑ<.then,h 6. Furthermore, it was shown in [] that any result of the form EH [ + ϖ ] for some fixed <ϖ </4 would imply an explicit finite upper bound on H with this bound equal to 6 for ϖ>.9855). Unfortunately, the only results of the type EH[ϑ] thatarenown come from the Bombieri-Vinogradov theorem Theorem 9), which only establishes EH[ϑ] for <ϑ</. The first unconditional bound on H was established in a breathrough wor of Zhang [3]: Theorem Zhang s theorem). H 7,,. Zhang s argument followed the general strategy from [] on finding small gaps between primes, with the major new ingredient being a proof of a weaer version of EH [ + ϖ ], which we call MPZ[ϖ, δ] see Claim ) below. It was quicly realized that Zhang s numerical bound on H could be improved. By optimizing many of the components in Zhang s argument, we were able Polymath, DHJ: New equidistribution estimates of Zhang type, submitted), [4] to improve Zhang s bound to H 4, 68. Very shortly afterwards, a further breathrough was obtained by Maynard [5] with related wor obtained independently in an unpublished wor of Tao), who developed a more flexible multidimensional version of the Selberg sieve to obtain stronger bounds on H m. This argument wored without using any equidistribution results on primes beyond the Bombieri-Vinogradov theorem, and among other things was able to establish finiteness of H m for all m,notjustform =. More precisely, Maynard established the following results. Theorem 3 Maynard s theorem). Unconditionally, we have the following bounds: i) H 6 ii) H m Cm 3 e 4m for all m and an absolute and effective) constant C Assuming the Elliott-Halberstam conjecture EH[ϑ] for all < ϑ <, we have the following improvements: iii) H iv) H 6 v) H m Cm 3 e m for all m and an absolute and effective) constant C

3 Polymath Research in the Mathematical Sciences 4, : Page 3 of 83 For a survey of these recent developments, see [6]. In this paper, we refine Maynard s methods to obtain the following further improvements. Theorem 4. Unconditionally, we have the following bounds: i) H 46 ii) H 398, 3 iii) H 3 4, 797, 84 iv) H 4, 43, 556, 7 v) H 5 8, 55,, 48 vi) H m Cm exp ) m ) for all m and an absolute and effective) constant C Assume the Elliott-Halberstam conjecture EH[ϑ] for all <ϑ<. Then, we have the following improvements: vii) H 7 viii) H 3 5, 6 ix) H 4 474, 66. x) H 5 4, 37, 854. xi) H m Cme m for all m and an absolute and effective) constant C Finally, assume the generalized Elliott-Halberstam conjecture GEH[ϑ] see Claim below) for all <ϑ<.then, xii) H 6 xiii) H 5 In the Outline of the ey ingredients section, we will describe the ey propositions that will be combined together to prove the various components of Theorem 4. As with Theorem, the results in vii)-xiii) do not require EH[ϑ] orgeh[ϑ] for all <ϑ<, but only for a single explicitly computable ϑ that is sufficiently close to. Of these results, the bound in xii) is perhaps the most interesting, as the parity problem [7] prohibits one from achieving any better bound on H than 6 from purely sievetheoretic methods; we review this obstruction in the The parity problem section. If one only assumes the Elliott-Halberstam conjecture EH[ϑ] instead of its generalization GEH[ϑ], we were unable to improve upon Maynard s bound H ; however, the parity obstruction does not exclude the possibility that one could achieve xii) just assuming EH[ϑ] rather than GEH[ϑ], by some further refinement of the sieve-theoretic arguments e.g. by finding a way to establish Theorem ii) below using only EH[ϑ] instead of GEH[ϑ]). The bounds ii)-vi) rely on the equidistribution results on primes established in our previous paper. However, the bound i) uses only the Bombieri-Vinogradov theorem, and the remaining bounds vii)-xiii) of course use either the Elliott-Halberstam conjecture or a generalization thereof. A variant of the proof of Theorem 4xii), which we give in Additional remars section, also gives the following conditional near miss to a disjunction of) the twin prime conjecture and the even Goldbach conjecture:

4 Polymath Research in the Mathematical Sciences 4, : Page 4 of 83 Theorem 5 Disjunction). Assume the generalized Elliott-Halberstam conjecture GEH[ϑ] for all <ϑ<. Then, at least one of the following statements is true: a) Twin prime conjecture) H =. b) near-miss to even Goldbach conjecture) If n is a sufficiently large multiple of 6, then at least one of n and n is expressible as the sum of two primes, similarly with n replaced by n +. In particular, every sufficiently large even number lies withinofthesumoftwoprimes.) We remar that a disjunction in a similar spirit was obtained in [8], which established prior to the appearance of Theorem ) that either H was finite or that every interval [ x, x + x ε ] contained the sum of two primes if x was sufficiently large depending on ε>. There are two main technical innovations in this paper. The first is a further generalization of the multidimensional Selberg sieve introduced by Maynard and Tao, in which the support of a certain cutoff function F is permitted to extend into a larger domain than was previously permitted particularly under the assumption of the generalized Elliott- Halberstam conjecture). As in [5], this largely reduces the tas of bounding H m to that of efficiently solving a certain multidimensional variational problem involving the cutoff function F. Our second main technical innovation is to obtain efficient numerical methods for solving this variational problem for small values of the dimension, aswellas sharpenedasymptoticsinthecaseoflargevaluesof. The methods of Maynard and Tao have been used in a number of subsequent applications [9-]. The techniques in this paper should be able to be used to obtain slight numerical improvements to such results, although we did not pursue these matters here. Organization of the paper The paper is organized as follows. After some notational preliminaries, we recall in the Distribution estimates on arithmetic functions section the nown or conjectured) distributional estimates on primes in arithmetic progressions that we will need to prove Theorem 4. Then, in the section Outline of the ey ingredients, we give the ey propositions that will be combined together to establish this theorem. One of these propositions, Lemma 8, is an easy application of the pigeonhole principle. Two further propositions, Theorem 9 and Theorem, use the prime distribution results from the Distribution estimates on arithmetic functions section to give asymptotics for certain sums involving sieve weights and the von Mangoldt function; they are established in the Multidimensional Selberg sieves section. Theorems, 4, 6, and 8 use the asymptotics established in Theorems 9 and, in combination with Lemma 8, to give various criteria for bounding H m, which all involve finding sufficiently strong candidates for a variety of multidimensional variational problems; these theorems are proven in the Reduction to a variational problem section. These variational problems are analysedintheasymptoticregimeoflarge in the Asymptotic analysis section, and for small and medium in the The case of small and medium dimension section, with the results collected in Theorems 3, 5, 7, and 9. Combining these results with the previous propositions gives Theorem 6, which, when combined with the bounds on narrow admissible tuples in Theorem 7 that are established in the Narrow admissible tuples

5 Polymath Research in the Mathematical Sciences 4, : Page 5 of 83 section, will give Theorem 4. See also Table for more details of the logical dependencies between the ey propositions.) Finally, in the The parity problem section, we modify an argument of Selberg to show that the bound H 6 may not be improved using purely sieve-theoretic methods, and in the Additional remars section, we establish Theorem 5 and mae some miscellaneous remars. Notation The notation used here closely follows the notation in our previous paper. We use E to denote the cardinality of a finite set E, and E to denote the indicator function of a set E;thus, E n) = whenn E and E n) = otherwise. All sums and products will be over the natural numbers N := {,, 3,...} unless otherwise specified, with the exceptions of sums and products over the variable p, which will be understood to be over primes. The following important asymptotic notation will be in use throughout the paper. Definition 6 Asymptotic notation). We use x to denote a large real parameter, which one should thin of as going off to infinity; in particular, we will implicitly assume that it is larger than any specified fixed constant. Some mathematical objects will be independent of x and referred to as fixed; but unless otherwise specified, we allow all mathematical objects under consideration to depend on x or to vary within a range that depends on x, e.g. the summation parameter n in the sum x n x f n)). If X and Y are two quantities depending on x, wesaythatx = OY ) or X Y if one has X CY for some fixed C which we refer to as the implied constant), and X = oy ) if one has X cx)y for some function cx) of x and of any fixed parameters present) that goes to zero as x for each choice of fixed parameters). We use X Y to denote the estimate X x o) Y, X Y to denote the estimate Y X Y,andX Y to denote the estimate Y X Y.Finally,wesaythataquantityn is of polynomial size if one has n = O x O)). If asymptotic notation such as O) or appears on the left-hand side of a statement, this means that the assertion holds true for any specific interpretation of that notation. For instance, the assertion n=on) αn) N means that for each fixed constant C >, one has n CN αn) N. If q and a are integers, we write a q if a divides q.ifq is a natural number and a Z,we use a q) to denote the residue class a q) := { a + nq : n Z } Table Results used to prove various components of Theorem 6 Theorem 6 Results used i) Theorems 9, 6, and 7 ii)-vi) Theorems, 4, and 5 vii)-xi) Theorems and 3 xii) Theorems 8 and 9 xiii) Theorems 6 and 7 Note that Theorems, 4, 6, and 8 are in turn proven using Theorems 9 and and Lemma 8.

6 Polymath Research in the Mathematical Sciences 4, : Page 6 of 83 and let Z/qZ denote the ring of all such residue classes aq). The notation b = a q) is synonymous to b a q). Weusea, q) to denote the greatest common divisor of a and q,and[a, q] to denote the least common multiple a.wealsolet Z/qZ) := { a q) : a, q) = } denote the primitive residue classes of Z/qZ. We use the following standard arithmetic functions: i) ϕq) := Z/qZ) denotes the Euler totient function of q. ii) τq) := d q denotes the divisor function of q. iii) q) denotes the von Mangoldt function of q;thus, q) = log p if q is a power of a prime p,and q) = otherwise. iv) θq) is defined to equal log q when q is a prime, and θq) = otherwise. v) μq) denotes the Möbius function of q;thus,μq) = ) if q is the product of distinct primes for some,andμq) = otherwise. vi) q) denotes the number of prime factors of q counting multiplicity). We recall the elementary divisor bound τn) ) whenever n x O), as well as the related estimate τn) C log O) x ) n n x for any fixed C > see, e.g. [Lemma.5]). The Dirichlet convolution α β: N C of two arithmetic functions α, β : N C is defined in the usual fashion as α βn) := n ) αd)β = αa)βb). d d n ab=n Distribution estimates on arithmetic functions As mentioned in the introduction, a ey ingredient in the Goldston-Pintz-Yıldırım approach to small gaps between primes comes from distributional estimates on the primes, or more precisely on the von Mangoldt function, which serves as a proxy for the primes. In this wor, we will also need to consider distributional estimates on more general arithmetic functions, although we will not prove any new such estimates in this paper, relying instead on estimates that are already in the literature. More precisely, we will need averaged information on the following quantity: Definition 7 Discrepancy). For any function α : N C with finite support that is, α is non-zero only on a finite set) and any primitive residue class a q), we define the signed) discrepancy α; a q)) to be the quantity α; a q)) := n=a q) αn) ϕq) n,q)= For any fixed <ϑ<, let EH[ϑ] denote the following claim: αn). 3)

7 Polymath Research in the Mathematical Sciences 4, : Page 7 of 83 Claim 8 Elliott-Halberstam conjecture, EH[ϑ]). If Q x ϑ and A is fixed, then sup [x,x] ; a q) ) x log A x. 4) a Z/qZ) q Q In [], it was conjectured that EH[ϑ] held for all <ϑ<. Theconjecture fails at the endpoint case ϑ = ; see [3,4] for a more precise statement.) The following classical result of Bombieri [5] and Vinogradov [6] remains the best partial result of the form EH[ϑ]: Theorem 9 Bombieri-Vinogradov theorem). [5,6] EH[ϑ] holds for every fixed < ϑ</. In [], it wasshownthat anyestimate ofthe form EH[ϑ]withsomefixedϑ>/ would imply the finiteness of H. While such an estimate remains unproven, it was observed by Motohashi and Pintz [7] and by Zhang [3] that a certain weaened version of EH[ϑ] would still suffice for this purpose. More precisely and following the notation of our previous paper), let ϖ, δ> be fixed, and let MPZ[ϖ, δ] be the following claim: Claim Motohashi-Pintz-Zhang estimate, MPZ[ϖ, δ]). Let I [, x δ] and Q x /+ϖ.letp I denote the product of all the primes in I, and let S I denote the square-free natural numbers whose prime factors lie in I. If the residue class a P I ) is primitive and is allowed to depend on x), and A is fixed, then q Q q S I [x,x] ; a q) ) x log A x, 5) where the implied constant depends only on the fixed quantities A, ϖ, δ),butnotona. It is clear that EH [ + ϖ ] implies MPZ[ϖ, δ]wheneverϖ, δ. The first non-trivial estimate of the form MPZ[ϖ, δ] was established by Zhang [3], who essentially) obtained MPZ[ϖ, δ]whenever ϖ, δ<,68. In [Theorem.7], we improved this result to the following. Theorem. MPZ[ϖ, δ] holds for every fixed ϖ, δ with 6ϖ + 8δ <7. In fact, a stronger result was established, in which the moduli q were assumed to be densely divisible rather than smooth, but we will not exploit such improvements here. For our application, the most important thing is to get ϖ as large as possible; in particular, Theoremallowsonetogetϖ arbitrarily close to In this paper, we will also study the following generalization of the Elliott-Halberstam conjecture: Claim Generalized Elliott-Halberstam conjecture, GEH[ϑ]). Let ε> and A be fixed. Let N, M be quantities such that x ε N, M x ε with NM x, and let

8 Polymath Research in the Mathematical Sciences 4, : Page 8 of 83 α, β : N R be sequences supported on [N,N] and [M,M], respectively, such that one has the pointwise bound αn) τn) O) log O) x; βm) τm) O) log O) x 6) for all natural numbers n, m. Suppose also that β obeys the Siegel-Walfisz type bound β,r)= ; a q) ) τqr) O) M log A x 7) for any q, r, anyfixeda,andanyprimitiveresidueclassaq).thenforanyq x ϑ, we have sup α β; a q)) x log A x. 8) a Z/qZ) q Q In [8, Conjecture ], it was essentially conjectured b that GEH[ϑ] was true for all < ϑ<. This is stronger than the Elliott-Halberstam conjecture: Proposition 3. For any fixed <ϑ<, GEH[ϑ] implies EH[ϑ]. Proof. Setch) As this argument is standard, we give only a brief setch. Let A > be fixed. For n [x,x], we have Vaughan s identity c [9] n) = μ < Ln) μ < < n) + μ n), where Ln) := logn) and n) := n) n x /3, < n) := n) n<x /3 9) μ n) := μn) n x /3, μ < n) := μn) n<x /3. ) ) By decomposing each of the functions μ <, μ,, <, into O log A+ x functions supported on intervals of the form [N, + log A x)n], and discarding those contributions which meet the boundary of [x,x] cf. [3,8,3,3]), and using GEH[ϑ] with A replaced by a much larger fixed constant A ) to control all remaining contributions, we obtain the claim using the Siegel-Walfisz theorem; see, e.g. [3, Satz 4] or [33, Th. 5.9]). By modifying the proof of the Bombieri-Vinogradov theorem, Motohashi [34] established the following generalization of that theorem: Theorem 4 Generalized Bombieri-Vinogradov theorem). [34] GEH[ϑ] holds for every fixed <ϑ</. One could similarly describe a generalization of the Motohashi-Pintz-Zhang estimate MPZ[ϖ, δ], but unfortunately, the arguments in [3] or Theorem do not extend to this setting unless one is in the Type I/Type II case in which N,M are constrained to be somewhat close to x /, or if one has Type III structure to the convolution α β,inthesense that it can refactored as a convolution involving several smooth sequences. In any event, our analysis would not be able to mae much use of such incremental improvements to GEH[ϑ], as we only use this hypothesis effectively in the case when ϑ is very close to. In particular, we will not directly use Theorem 4 in this paper.

9 Polymath Research in the Mathematical Sciences 4, : Page 9 of 83 Outline of the ey ingredients In this section, we describe the ey subtheorems used in the proof of Theorem 4, with the proofs of these subtheorems mostly being deferred to later sections. We begin with a wea version of the Dicson-Hardy-Littlewood prime tuples conjecture [], which following Pintz [35]) we refer to as [, j]. Recall that for any N, an admissible -tuple is a tuple H = h,..., h ) of increasing integers h <...<h which avoids at least one residue class a p p) := {a p + np : n Z} for every p. For instance,,, 6) is an admissible 3-tuple, but,, 4) is not. For any j, we let DHL[; j] denote the following claim: Claim 5 Wea Dicson-Hardy-Littlewood conjecture, DHL[; j]). For any admissible -tuple H = h,..., h ),thereexistinfinitelymanytranslatesn+h = n+h,..., n+h ) of H which contain at least j primes. The full Dicson-Hardy-Littlewood conjecture is then the assertion that DHL[; ] holds for all. In our analysis, we will focus on the case when j is much smaller than ;infact,j will be of the order of log. For any,leth) denote the minimal diameter h h of an admissible -tuple; thus for instance, H3) = 6. It is clear that for any natural numbers m and m +, the claim DHL[ ; m + ] implies that H m H) and the claim DHL[; ] would imply that H = H)). We will therefore deduce Theorem 4 from a number of claims of the form DHL[; j]. More precisely, we have Theorem 6. Unconditionally, we have the following claims: i) DHL[5; ]. ii) DHL[35, 4; 3]. iii) DHL[, 649, 8; 4]. iv) DHL[75, 845, 77; 5]. v) DHL[3, 473, 955, 98; 6]. vi) DHL[; m + ] whenever m and C exp ) m ) for some sufficiently large absolute and effective) constant C. Assume the Elliott-Halberstam conjecture EH[θ] for all <θ<. Then, we have the following improvements: vii) DHL[54; 3]. viii) DHL[5, 5; 4]. ix) DHL[4, 588; 5]. x) DHL[39, 66; 6]. xi) DHL[; m + ] whenever m and C expm) for some sufficiently large absolute and effective) constant C. Assume the generalized Elliott-Halberstam conjecture GEH[θ] for all <θ<.then xii) DHL[3; ]. xiii) DHL[5; 3]. Theorem 4 then follows from Theorem 6 and the following bounds on H) ordered by increasing value of ):

10 Polymath Research in the Mathematical Sciences 4, : Page of 83 Theorem 7 Bounds on H)). xii) H3) = 6. i) H5) = 46. xiii) H5) = 5. vii) H54) = 7. viii) H5, 5) 5, 6. ii) H35, 4) 398, 3. ix) H4, 588) 474, 66. x) H39, 66) 4, 37, 854. iii) H, 649, 8) 4, 797, 84. iv) H75, 845, 77), 43, 556, 7. v) H3, 473, 955, 98) 8, 55,, 48. vi), xi) In the asymptotic limit,onehash) log + log log + o), with the bounds on the decay rate o) being effective. We prove Theorem 7 in the Narrow admissible tuples section. In the opposite direction, an application of the Brun-Titchmarsh theorem gives H) + o) ) log as see [4, 3.9] for this bound, as well as with some slight refinements). The proof of Theorem 6 follows the Goldston-Pintz-Yıldırım strategy that was also used in all previous progress on this problem e.g. [,3,5,7]), namely that of constructing a sieve function adapted to an admissible -tuple with good properties. More precisely, we set and w := log log log x W := p w p, and observe the crude bound W log log O) x. ) We have the following simple pigeonhole principle criterion for DHL[ ; m + ] cf. [Lemma 4.], though the normalization here is slightly different): Lemma 8 Criterion for DHL). Let and m be fixed integers and define the normalization constant B := ϕw) log x. ) W Suppose that for each fixed admissible -tuple h,..., h ) and each residue class b W) such that b + h i is coprime to W for all i =,...,, one can find a non-negative weight function ν : N R + and fixed quantities α>and β,..., β, such that one has the asymptotic upper bound νn) α + o)) B x W, 3) x n x n=b W)

11 Polymath Research in the Mathematical Sciences 4, : Page of 83 the asymptotic lower bound νn)θn + h i ) β i o))b x ϕw) x n x n=b W) 4) for all i =,...,, and the ey inequality β + +β > m. 5) α Then, DHL[; m + ] holds. Proof. Let h,..., h ) be a fixed admissible -tuple. Since it is admissible, there is at least one residue class b W) such that b + h i, W) = for all h i H. For an arithmetic function ν as in the lemma, we consider the quantity N := νn) θn + h i ) m log 3x. x n x n=b W) i= Combining 3) and 4), we obtain the lower bound N β + +β o))b x ϕw) mα + x o))b log 3x. W From ) and the crucial condition 5), it follows that N > ifxis sufficiently large. On the other hand, the sum θn + h i ) m log 3x i= can be positive only if n + h i is prime for at least m + indices i =,...,. We conclude that, for all sufficiently large x, there exists some integer n [x,x] suchthatn + h i is prime for at least m + values of i =,...,. Since h,..., h ) is an arbitrary admissible -tuple, DHL[; m + ] follows. The objective is then to construct non-negative weights ν whose associated ratio has provable lower bounds that are as large as possible. Our sieve majorants will β + +β α be a variant of the multidimensional Selberg sieves used in [5]. As with all Selberg sieves, the ν are constructed as the square of certain signed) divisor sums. The divisor sums we will use will be finite linear combinations of products of one-dimensional divisor sums. More precisely, for any fixed smooth compactly supported function F :[, + ) R, define the divisor sum λ F : Z R by the formula λ F n) := d n μd)flog x d) 6) where log x denotes the base x logarithm log x n := log n log x. 7) One should thin of λ F as a smoothed out version of the indicator function to numbers n which are almost prime in the sense that they have no prime factors less than x ε for some small fixed ε> see Proposition 4 for a more rigorous version of this heuristic).

12 Polymath Research in the Mathematical Sciences 4, : Page of 83 The functions ν we will use will tae the form J νn) = c j λ Fj, n + h )...λ Fj, n + h ) j= for some fixed natural number J, fixed coefficients c,..., c J R and fixed smooth compactly supported functions F j,i :[,+ ) R with j =,..., J and i =,...,. One can of course absorb the constant c j into one of the F j,i if one wishes.) Informally, ν is a smooth restriction to those n for which n + h,..., n + h are all almost prime. Clearly, ν is a positive-definite) linear combination of functions of the form n λ Fi n + h i )λ Gi n + h i ) i= for various smooth functions F,..., F, G,..., G :[,+ ) R. Thesumappearing in 3) can thus be decomposed into linear combinations of sums of the form x n x i= n=b W) 8) λ Fi n + h i )λ Gi n + h i ). 9) Also, since from 6) we clearly have λ F n) = F) ) when n x is prime and F is supported on [, ], the sum appearing in 4) can be similarly decomposed into linear combinations of sums of the form θn + h i ) λ Fi n + h i )λ Gi n + h i ). ) x n x n=b W) i ;i =i To estimate the sums ), we use the following asymptotic, proven in the Multidimensional Selberg sieves section. For each compactly supported F :[,+ ) R,let SF) := sup{x :Fx) = } ) denote the upper range of the support of F with the convention that S) = ). Theorem 9 Asymptotic for prime sums). Let be fixed, let h,..., h ) be a fixed admissible -tuple, and let b W) be such that b + h i is coprime to W for each i =,...,. Let i befixed,andforeach i distinctfromi,letf i, G i :[,+ ) R be fixed smooth compactly supported functions. Assume one of the following hypotheses: i) Elliott-Halberstam) There exists a fixed <ϑ< such that EH[ϑ] holds and such that i ;i =i SF i ) + SG i )) < ϑ. 3) ii) Motohashi-Pintz-Zhang) There exists fixed ϖ</4andδ > such that MPZ[ ϖ, δ] holds and such that SF i ) + SG i )) < + ϖ 4) i ;i =i

13 Polymath Research in the Mathematical Sciences 4, : Page 3 of 83 and max {SF i ), SG i )} <δ. i ;i =i 5) Then, we have θn + h i ) x 6) where x n x n=b W) c := i ;i =i λ Fi n + h i )λ Gi n + h i ) = c + o))b ϕw) i ;i =i F i t i)g i t i) dt i ). Here of course F denotes the derivative of F. To estimate the sums 9), we use the following asymptotic, also proven in the Multidimensional Selberg sieves section. Theorem Asymptotic for non-prime sums). Let be fixed, let h,..., h ) be a fixed admissible -tuple, and let b W) be such that b + h i is coprime to W for each i =,...,. For each fixed i, let F i, G i :[,+ ) R be fixed smooth compactly supported functions. Assume one of the following hypotheses: i) Trivial case) One has SF i ) + SG i )) <. 7) i= ii) Generalized Elliott-Halberstam) There exists a fixed <ϑ<and i {,..., } such that GEH[ϑ] holds, and SF i ) + SG i )) < ϑ. 8) i ;i =i Then, we have λ Fi n + h i )λ Gi n + h i ) = c + o))b x W, 9) where x n x i= n=b W) c := ) F i t i)g i t i) dt i. 3) i= A ey point in ii) is that no upper bound on SF i ) or SG i ) is required although, as we will see in the The generalized Elliott-Halberstam case section, the result is a little easier to prove when one has SF i ) + SG i )<). This flexibility in the F i, G i functions will be particularly crucial to obtain part xii) of Theorem 6 and Theorem 4. Remar. Theorems 9 and can be viewed as probabilistic assertions of the following form: if n is chosen uniformly at random from the set {x n x : n = b W)}, then the random variables θn + h i ) and λ Fj n + h j )λ Gj n + h j ) for i, j =,..., have ) mean + o)) ϕw) W and F j t)g j t) dt + o) B, respectively, and furthermore,

14 Polymath Research in the Mathematical Sciences 4, : Page 4 of 83 these random variables enjoy a limited amount of independence, except for the fact as can be seen from )) that θn + h i ) and λ Fi n + h i )λ Gi n + h i ) are highly correlated. Note though that we do not have asymptotics for any sum which involves two or more factors of θ, as such estimates are of a difficulty at least as great as that of the twin prime conjecture which is equivalent to the divergence of the sum n θn)θn + )). Theorems 9 and may be combined with Lemma 8 to reduce the tas of establishing estimates of the form DHL[; m + ] to that of establishing certain variational problems. For instance, in the Proof of Theorem section, we reprove the following result of Maynard [5, Proposition 4.]): Theorem Sieving on the standard simplex). Let and m be fixed integers. For any fixed compactly supported square-integrable function F :[,+ ) R, define the functionals IF) := Ft,..., t ) dt...t 3) [,+ ) and ) J i F) := Ft,..., t ) dt i dt...dt i dt i+...dt 3) [,+ ) for i =,...,, and let M be the supremum i= J i F) M := sup IF) over all square integrable functions F that are supported on the simplex { } R := t,..., t ) [, + ) : t + +t and are not identically zero up to almost everywhere equivalence, of course). Suppose that there is a fixed <ϑ< such that EH[ϑ] holds and such that M > m ϑ. Then, DHL[; m + ] holds. 33) Parts vii)-xi) of Theorem 6 and hence Theorem 4) are then immediate from the following results, proven in the Asymptotic analysis and The case of small and medium dimension sections, and ordered by increasing value of : Theorem 3 Lower bounds on M ). vii) M 54 > viii) M 5,5 > 6. ix) M 4,588 > 8. x) M 39,66 >. xi) One has M log C for all C, where C is an absolute and effective) constant. For the sae of comparison, in [5, Proposition 4.3]), it was shown that M 5 >, M 5 > 4, and M log loglog for all sufficiently large. As remared in that paper, the sieves used on the bounded gap problem prior to the wor in [5] would

15 Polymath Research in the Mathematical Sciences 4, : Page 5 of 83 essentially correspond, in this notation, to the choice of functions F of the special form Ft,..., t ) := f t + + t ), which severely limits the size of the ratio in 33) in particular, the analogue of M in this special case cannot exceed 4, as shown in [36]). In the converse direction, in Corollary 37, we will also show the upper bound M log for all, which shows in particular that the bounds in vii) and xi) of the above theorem cannot be significantly improved. We remar that Theorem 3vii) and the Bombieri-Vinogradov theorem also give a weaer version DHL[54; ] of Theorem 6i). We also have a variant of Theorem which can accept inputs of the form MPZ[ϖ, δ]: Theorem 4 Sieving on a truncated simplex). Let and m be fixed integers. Let <ϖ </4 and <δ</ be such that MPZ[ϖ, δ] holds. For any α>, let M [α] be defined as in 33), but where the supremum now ranges over all square-integrable F supported in the truncated simplex { } t,..., t ) [, α] : t + +t and are not identically zero. If M [ ] δ /4+ϖ > m /4 + ϖ, then DHL[; m + ] holds. 34) In the Asymptotic analysis section, we will establish the following variant of Theorem 3, which when combined with Theorem, allows one to use Theorem 4 to establish parts ii)-vi) of Theorem 6 and hence Theorem 4): Theorem 5 Lower bounds on M [α] ). ii) There exist δ, ϖ> with 6ϖ + 8δ <7 and M iii) There exist δ, ϖ> with 6ϖ + 8δ <7 and M iv) There exist δ, ϖ> with 6ϖ + 8δ <7 and M [ ] δ /4+ϖ 35 4 > /4+ϖ. [ ] δ /4+ϖ > /4+ϖ 3. [ ] δ /4+ϖ > /4+ϖ 4. [ ] δ /4+ϖ > /4+ϖ 5. 6 C log,and v) There exist δ, ϖ>with 6ϖ + 8δ <7 and M vi) For all C, there exist δ, ϖ>with 6ϖ + 8δ <7, ϖ 7 [ δ /4+ϖ ] M log C for some absolute and effective) constant C. The implication is clear for ii)-v). For vi), observe that from Theorem 5vi), Theorem, and Theorem 4, we see that DHL[; m + ] holds whenever is sufficiently large and m log C) C ) log which is in particular implied by m log C for some absolute constant C, giving Theorem 6vi).

16 Polymath Research in the Mathematical Sciences 4, : Page 6 of 83 Now we give a more flexible variant of Theorem, in which the support of F is enlarged, at the cost of reducing the range of integration of the J i. Theorem 6 Sieving on an epsilon-enlarged simplex). Let and m be fixed integers, and let < ε < be fixed also. For any fixed compactly supported squareintegrable function F :[,+ ) R, define the functionals ) J i, ε F) := Ft,..., t ) dt i dt...dt i dt i+...dt ε) R for i =,...,, and let M,ε be the supremum i= J i, ε F) M,ε := sup IF) over all square-integrable functions F that are supported on the simplex { } + ε) R = t,..., t ) [, + ) : t + +t + ε and are not identically zero. Suppose that there is a fixed <ϑ<, such that one of the following two hypotheses hold: i) EH[ϑ] holds, and + ε< ϑ. ii) GEH[ϑ] holds, and ε<. If M,ε > m ϑ then DHL[; m + ] holds. We prove this theorem in the Proof of Theorem 6 section. We remar that due to the continuity of M,ε in ε, the strict inequalities in i) and ii) of this theorem may be replaced by non-strict inequalities. Parts i) and xiii) of Theorem 6, and a weaer version DHL[4; ] of part xii), then follow from Theorem 9 and the following computations, proven in the Bounding M,ε for medium and Bounding M 4,ε sections: Theorem 7 Lower bounds on M,ε ). i) M 5,/5 > xii ) M 4,.68 >.558. xiii) M 5,/5 > We remar that computations in the proof of Theorem 7xii ) are simple enough that the bound may be checed by hand, without use of a computer. The computations used to establish the full strength of Theorem 6xii) are however significantly more complicated. Infact,wemayenlargethesupportofF further. We give a version corresponding to part ii) of Theorem 6; there is also a version corresponding to part i), but we will not give it here as we will not have any use for it. Theorem 8 Going beyond the epsilon enlargement). Let and m be fixed integers, let <ϑ<beafixedquantity such that GEH[ϑ] holds, and let < ε< be fixed also. Suppose that there is a fixed non-zero square-integrable function

17 Polymath Research in the Mathematical Sciences 4, : Page 7 of 83 F :[,+ ) R supported in R, such that for i =,...,,onehasthevanishing marginal condition Ft,..., t ) dt i = 35) whenever t,..., t i, t i+,..., t are such that t + +t i + t i+ + +t > + ε. Supposethatwealsohavetheinequality i= J i,ε F) > m IF) ϑ. Then DHL[; m + ] holds. This theorem is proven in the Proof of Theorem 8 section. Theorem 6xii) is then an immediate consequence of Theorem 8 and the following numerical fact, established in the Three-dimensional cutoffs section. Theorem 9 A piecewise polynomial cutoff). Set ε := 4.Then,thereexistsapiecewise polynomial function F :[,+ ) 3 R supported on the simplex { 3 R 3 = t, t, t 3 ) [, + ) 3 : t + t + t 3 3 } and symmetric in the t, t, t 3 variables, such that F is not identically zero and obeys the vanishing marginal condition Ft, t, t 3 ) dt 3 = whenever t, t with t + t > + ε and such that 3 ) t +t ε Ft, t, t 3 ) dt 3 dt dt [, ) 3 Ft, t, t 3 ) >. dt dt dt 3 There are several other ways to combine Theorems 9 and with equidistribution theorems on the primes to obtain results of the form DHL[ ; m+], but all of our attempts to do so either did not improve the numerology or else were numerically infeasible to implement. Multidimensional Selberg sieves In this section, we prove Theorems 9 and. A ey asymptotic used in both theorems is the following: Lemma 3 Asymptotic). Let be) a fixed integer, and let N be a natural number coprime to W with log N = O log O) x.letf,..., F, G,..., G :[,+ ) R be fixed smooth compactly supported functions. Then, μ ) ) d j μ d j ) ) F j logx d j Gj log x d j [ ] = c+o))b N j= d j, d j ϕn) d,...,d,d,...,d [d,d ],..., [ d,d ],W,Ncoprime 36)

18 Polymath Research in the Mathematical Sciences 4, : Page 8 of 83 where B was defined in ), and c := j= F j t j)g j t j) dt j. The same claim holds if the denominators [ ] [ ]) d j, d j are replaced by ϕ d j, d j. Such asymptotics are standard in the literature see, e.g. [37] for some similar computations). In older literature, it is common to establish these asymptotics via contour integration e.g. via Perron s formula), but we will use the Fourier analytic approach here. Of course, both approaches ultimately use the same input, namely the simple pole of the Riemann zeta function at s =. Proof. We begin with the first claim. For j =,...,, the functions t e t F j t), t e t G j t) may be extended to smooth compactly supported functions on all of R,andsowe have Fourier expansions e t F j t) = e itξ f j ξ) dξ 37) R and e t G j t) = e itξ g j ξ) dξ R for some fixed functions f j, g j : R C that are smooth and rapidly decreasing in the sense that f j ξ), g j ξ) = O + ξ ) A) for any fixed A > andallξ R here the implied constant is independent of ξ and depends only on A). We may thus write ) f j ξ j ) F j logx d j = dξ j and R ) G j log x d j = R +iξ j log x dj ) g j ξ j ) +iξ j d j log x dξ j for all d j, d j. We note that μ ) ) d j μ d j [ ] d j,d j d j, d j / log x / log x = d d j) p j + p +/ log x + p+/ log x ) expolog log x)). Therefore, if we substitute the Fourier expansions into the left-hand side of 36), the resulting expression is absolutely convergent. Thus, we can apply Fubini s theorem, and the left-hand side of 36) can thus be rewritten as where R... K ξ,..., ξ, ξ,..., ξ ) R Kξ,..., ξ, ξ,..., ξ ) := ) f j ξj gj j= ξ j d,...,d,d,...,d [d,d ],..., [ j= d,d ],W,Ncoprime ) dξ j dξ j, 38) [ ] d j, d j μ d j ) μ d j ) +iξ j ) +iξ j log x dj d j log x.

19 Polymath Research in the Mathematical Sciences 4, : Page 9 of 83 This latter expression factorizes as an Euler product K = K p, p WN where the local factors K p are given by K p ξ,..., ξ, ξ,..., ξ ) := + p We can estimate each Euler factor as Since d,...,d,d [,...,d d,...,d,d ],...,d =p [d,d ],..., [ d,d ] coprime )) K p ξ,..., ξ, ξ,..., ξ ) = + O p j= p:p>w )) + O p = + o), j= μ d j ) μ d j ) +iξ j ) +iξ j log x dj d j log x ) ) p +iξ j log x p +iξ j log x p +iξ j+iξ j log x.. 39) 4) we have K ξ,..., ξ, ξ,..., ξ ) = + o)) j= ) ζ WN + +iξ j+iξ j ζ WN + +iξ j log x where the modified zeta function ζ WN is defined by the formula ζ WN s) := p WN p s ) for Rs) >. For Rs) + log x,wehavethecrudebounds + ζ WN s), ζ WN s) + O p+/ log x p exp p+/ log x p explog log x + O)) log x. Thus, K ξ,..., ξ, ξ,..., ξ ) ) = O log 3 x. )) p log x ) ζ WN + +iξ j log x Combining this with { the rapid decrease of f j, g j, we see that the contribution to 38) outside of the cube max ξ,..., ξ, ξ,..., ξ ) } log x say) is negligible. Thus, it will suffice to show that log x log x... K ξ,..., ξ, ξ,..., ξ ) ) ) f j ξj gj ξ j dξ j dξ j = c + o))b N log x log x ϕn). j= )

20 Polymath Research in the Mathematical Sciences 4, : Page of 83 When ξ j log x, we see from the simple pole of the Riemann zeta function ζs) = ) p at s = that s p ζ + + iξ ) j = + o)) log x. log x + iξ j For log x ξ j log x,weseethat ) = p + O log p p. log x p WN p + +iξ j log x Since log WN log O) x, this gives = ϕwn) WN exp O p + +iξ j log x p WN log p p = + o)) ϕwn) log x WN, since the sum is maximized when WN is composed only of primes p log O) x.thus, ζ WN + + iξ ) j + o))bϕn) =, log x + iξ j )N similarly with + iξ j replaced by + iξ j or + iξ j + iξ j. We conclude that K ξ,..., ξ, ξ,..., ξ ) = + o))b N ϕn) Therefore, it will suffice to show that ) ) + iξj + iξ j... R R + iξ j + iξ j f j ξ j )g j j= ξ j ) j= dξ j dξ j = c, ) + iξj + iξ j + iξ j + iξ j ). 4) since the errors caused by the + o) multiplicative factor in 4) or the truncation ξ j, ξ j log x canbeseentobenegligibleusingtherapiddecayoff j, g j.byfubini s theorem, it suffices to show that R R + iξ) + iξ ) + iξ + iξ f j ξ)g j ξ ) dξdξ = + F j t)g j t) dt for each j =,...,. But from dividing 37) by e t and differentiating under the integral sign, we have F j t) = + iξ)e t+iξ) f j ξ) dξ, R and the claim then follows from Fubini s [ theorem. ] [ ]) Finally, suppose that we replace d j, d j with ϕ d j, d j. An inspection of the above argument shows that the only change that occurs is that the p term in 39) is replaced by ) p ; but this modification may be absorbed into the + O factor in 4), and the p rest of the argument continues as before. The trivial case We can now prove the easiest case of the two theorems, namely case i) of Theorem ; a closely related estimate also appears in [5, Lemma 6.]). We may assume that x is suf-

21 Polymath Research in the Mathematical Sciences 4, : Page of 83 ficiently large depending on all fixed quantities. By 6), the left-hand side of 9) may be expanded as μd i )μ d i) ) Fi logx d i Gi logx d i ) S d,..., d, d,..., ) d d,...,d,d,...,d i= where S d,..., d, d,..., ) d := x n x n=b W) n+h i = [d i,d i ]) i By hypothesis, b + h i is coprime to W for all i =,...,,and h i h j < w for all distinct i, j. Thus,S d,..., d, d,..., [ ) d vanishes unless the di, d i] are coprime to each other and to W. Inthiscase,S d,..., d, d,..., ) d is summing the constant function over an arithmetic progression in [x,x]ofspacingw [ d, d ] [... d, d ],andso S d,..., d, d,..., ) x d = W [ d, d ] [... d, d ] + O). x By Lemma 3, the contribution of the main term W[d,d ]... [ d,d ] to 9) is c + o))b W x ; note that the restriction of the integrals in 3) to [, ] instead of [, + ) is harmless since SF i ), SG i )<for all i. Meanwhile, the contribution of the O) error is then bounded by O F i log x d i ) G i log x d i ). d,...,d,d,...,d i= By the hypothesis in Theorem i), we see that for d,..., d, d,..., d contributing a non-zero term here, one has [ d, d ] [... d, d ] x ε. 4) for some fixed ε >. From the divisor bound ), we see that each choice of [ d, d ] [... d, d ] arises from choices of d,..., d, d,..., d. We conclude that the net contribution of the O) error to 9) is x ε, and the claim follows. The Elliott-Halberstam case Now we show case i) of Theorem 9. For the sae of notation, we tae i =, as the other cases are similar. We use 6) to rewrite the left-hand side of 6) as μd i )μ d i) ) Fi logx d i Gi logx d i ) S d,..., d, d,..., ) d d,...,d,d,...,d i= where S d,..., d, d,..., ) d := θn + h ). x n x n=b W) n+h i = [d i,d i]) i=,..., 43)

22 Polymath Research in the Mathematical Sciences 4, : Page of 83 ) As in the previous case, S d,..., d, d,..., d vanishes unless the [ d i, d i] are coprime to each other and to W, and so the summand in 43) vanishes unless the modulus q W,d,...,d defined by q W,d,...,d := W [ d, d ] [... d, d ] 44) is square-free. In that case, we may use the Chinese remainder theorem to concatenate the congruence conditions on n into a single primitive congruence condition ) n + h = a W,d,...,d q W,d,...,d for some a W,d,...,d depending on W, d,..., d, d,..., d, and conclude using 3) that S d,..., d, d,..., ) d = ) d,...,d,d,...,d i= ϕ + q W,d,...,d x+h n x+h θn) [x+h,x+h ]θ; a W,d,...,d )) q W,d,...,d. 45) From the prime number theorem, we have θn) = + o))x x+h n x+h and this expression is clearly independent of d,..., d. Thus, by Lemma 3, the contribution of the main term in 45) is c + o))b ϕw) x. By ) and ), it thus suffices to show that for any fixed A we have ) F i logx d i G i logx d i ) [x+h,x+h ]θ; a q) ) x log A x, where a = a W,d,...,d and q = q W,d,...,d. For future reference, we note that we may restrict the summation here to those d,..., d for which q W,d,...,d is square-free. From the hypotheses of Theorem 9i), we have q W,d,...,d xϑ whenever the summand in 43) is non-zero, and each choice q of q W,d,...,d is associated to O τq) O)) choices of d,..., d, d,..., d.thus,thiscontributionis τq) O) sup [x+h,x+h ]θ; a q) ). q x ϑ a Z/qZ) Using the crude bound [x+h,x+h ]θ; a q) ) x q logo) x and ), we have τq) C sup [x+h,x+h ]θ; a q) ) x log O) x q x ϑ a Z/qZ) for any fixed C >. By the Cauchy-Schwarz inequality, it suffices to show that q x ϑ sup [x+h,x+h ]θ; a q) ) x log A x a Z/qZ) 46)

23 Polymath Research in the Mathematical Sciences 4, : Page 3 of 83 for any fixed A >. However, since θ only differs from on powers p j of primes with j >, it is not difficult to show that [x+h,x+h ]θ; a q) ) [x+h,x+h ] ; a q) ) x q, sotheneterrorinreplacingθ here by is x ϑ)/, which is certainly acceptable. The claim now follows from the hypothesis EH[ϑ], thans to Claim 8. The Motohashi-Pintz-Zhang case Now we show case ii) of Theorem 9. We repeat the arguments from the The Elliott- Halberstam case section, with the only difference being in the derivation of 46). As observed previously, we may restrict q W,d,...,d to be square-free. From the hypotheses in Theorem 9ii), we also see that q W,d,...,d xϑ and that all the prime factors of q W,d,...,d are at most xδ.thus,ifweseti := [, x δ ], we see using the notation from Claim ) that q W,d,...,d lies in S I and is thus a factor of P I. If we then let A Z/P I Z denote all the primitive residue classes a P I ) with the property that a = b W), and such that for each prime w < p x δ,onehasa + h i = p) for some i =,...,, thenweseethata W,d,...,d lies in the projection of A to Z/q W,d,...,d Z. Each q S I is equal to q W,d,...,d for O τq) O)) choices of d,..., d.thus,the left-hand side of 46) is q S I :q x ϑ τq) O) sup a A [x+h,x+h ]θ; a q) ). Note from the Chinese remainder theorem that for any given q, ifoneletsa range uniformly in A, thena q) is uniformly distributed among O τq) O)) different moduli. Thus, we have sup [x+h,x+h ]θ; a q) ) τq)o) [x+h,x+h a A A ]θ; a q) ), a A and so it suffices to show that τq) O) [x+h,x+h A ]θ; a q)) x log A x q S I :q x ϑ a A for any fixed A >. We see it suffices to show that [x+h,x+h ]θ; a q)) x log A x q S I :q x ϑ τq) O) for any given a A. But this follows from the hypothesis MPZ[ϖ, δ] by repeating the arguments of the The Elliott-Halberstam case section. Crude estimates on divisor sums To proceed further, we will need some additional information on the divisor sums λ F defined in 6)), namely that these sums are concentrated on almost primes ; results of this type have also appeared in [38]. Proposition 4 Almost primality). Let be fixed, let h,..., h ) be a fixed admissible -tuple, and let b W) be such that b + h i is coprime to W for each i =,...,.

24 Polymath Research in the Mathematical Sciences 4, : Page 4 of 83 Let F,..., F :[,+ ) R be fixed smooth compactly supported functions, and let m,..., m and a,..., a be fixed natural numbers. Then, λfj n + h j ) a j τn + h j ) m ) j B x W. 47) x n x:n=b W) j= Furthermore, if j isfixedandp is a prime with p x, then we have the variant x n x:n=b W) j= As a consequence, we have λfj n + h j ) a j τn + h j ) m j ) p n+h j log x p p B x W. 48) λfj n + h j ) a j τn + h j ) m ) j pn+hj ) xε x εb W, 49) x n x:n=b W) j= for any ε>,wherepn) denotes the least prime factor of n. The exponent can certainly be improved here, but for our purposes, any fixed positive exponent depending only on will suffice. Proof. The strategy is to estimate the alternating divisor sums λ Fj n + h j ) by nonnegative expressions involving prime factors of n + h j, which can then be bounded combinatorially using standard tools. We first prove 47). As in the proof of Proposition 3, we can use Fourier expansion to write f j ξ) F j logx d ) = R d +iξ log x dξ for some rapidly decreasing f j : R C and all natural numbers d.thus, λ Fj n) = μd) f R d n d +iξ j ξ) dξ, log x which factorizes using Euler products as λ Fj n) = f j ξ) dξ. R p n The function s p log s x Rs) >, and thus p +iξ log x p +iξ log x has a magnitude of O) and a derivative of O log x p ) when = O min + ξ ) log x p,) ). From the rapid decrease of f j and the triangle inequality, we conclude that λ Fj n) O min + ξ ) log x p,) ) dξ R + ξ ) A p n

25 Polymath Research in the Mathematical Sciences 4, : Page 5 of 83 for any fixed A >. Thus, noting that p n O) τn)o),wehave λ Fj n) a j τn) O)... a j min + ξ l ) log x p,) R R p n l= dξ...dξ aj + ξ ) A... + ξ aj ) A for any fixed a j, A. However, we have a j min + ξ i ) log x p, ) min + ξ + + ξ aj ) log x p, ), i= and so λ Fj n) a j τn) O) R p n min + ξ + + ξ aj ) log x p, )) dξ...dξ aj... R + ξ + + ξ aj ). A Maing the change of variables σ := + ξ + + ξ aj,weobtain λ Fj n) a j τn) O) minσ log x p,) dσ p n for any fixed A >. In view of this bound and the Fubini-Tonelli theorem, it suffices to show that τn + h j ) O) minσ j log x p,) B x W σ + +σ ) O) p n x n x:n=b W) j= for all σ,..., σ. By setting σ := σ + +σ, it suffices to show that τ ) O) n + h j σ logx p, ) B x W σ O) 5) x n x:n=b W) j= p n+h j min for any σ. To proceed further, we factorize n + h j as a product n + h j = p...p r of primes p p r in increasing order and then write n + h j = d j m j where d j := p...p ij and i j is the largest index for which p...p ij < x,andm j := p ij +...p r. By construction, we see that i j < r, d j x.also,wehave p ij + ) i p...p j ij + + i x j +). Since n x,thisimpliesthat and so r = Oi j + ) τn + h j ) O+ d j)), σ A