Abstract
For any m≥1, let H_{ m } denote the quantity ${liminf}_{n\to \infty}({p}_{n+m}{p}_{n})$. A celebrated recent result of Zhang showed the finiteness of H_{1}, with the explicit bound H_{1}≤70,000,000. This was then improved by us (the Polymath8 project) to H_{1}≤4680, and then by Maynard to H_{1}≤600, who also established for the first time a finiteness result for H_{ m } for m≥2, and specifically that H_{ m }≪m^{3}e^{4m}. If one also assumes the ElliottHalberstam conjecture, Maynard obtained the bound H_{1}≤12, improving upon the previous bound H_{1}≤16 of Goldston, Pintz, and Yıldırım, as well as the bound H_{ m }≪m^{3}e^{2m}.
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_{1}≤246 unconditionally and H_{1}≤6 under the assumption of the generalized ElliottHalberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h_{1},h_{2},h_{3}), there are infinitely many n for which at least two of n+h_{1},n+h_{2},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 2, or both. We also modify the ‘parity problem’ argument of Selberg to show that the H_{1}≤6 bound is the best possible that one can obtain from purely sievetheoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound ${H}_{m}\ll m{e}^{\left(4\frac{28}{157}\right)m}$ or H_{ m }≪m e^{2m} under the assumption of the ElliottHalberstam conjecture. We also obtain explicit upper bounds for H_{ m } when m=2,3,4,5.
Background
For any natural number m, let H_{ m } denote the quantity
where p_{ n } denotes the n th prime. The twin prime conjecture asserts that H_{1}=2; more generally, the HardyLittlewood prime tuples conjecture [1] implies that H_{ m }=H(m+1) for all m≥1, where H(k) is the diameter of the narrowest admissible ktuple (see the ‘Outline of the key ingredients’ section for a definition of this term). Asymptotically, one has the bounds
as k→∞ (see Theorem 17 below); thus, the prime tuples conjecture implies that H_{ m } is comparable to m logm as m→∞.
Until very recently, it was not known if any of the H_{ m } were finite, even in the easiest case m=1. In the breakthrough work of Goldston et al. [2], several results in this direction were established, including the following conditional result assuming the ElliottHalberstam conjecture EH[ 𝜗] (see Claim 8 below) concerning the distribution of the prime numbers in arithmetic progressions:
Theorem 1(GPY theorem).
Assume the ElliottHalberstam conjecture EH[ 𝜗] for all 0<𝜗<1. Then, H_{1}≤16.
Furthermore, it was shown in [2] that any result of the form $\text{EH}\left[\phantom{\rule{0.3em}{0ex}}\frac{1}{2}+2\varpi \right]$ for some fixed 0<ϖ<1/4 would imply an explicit finite upper bound on H_{1} (with this bound equal to 16 for ϖ>0.229855). Unfortunately, the only results of the type EH[ 𝜗] that are known come from the BombieriVinogradov theorem (Theorem 9), which only establishes EH[ 𝜗] for 0<𝜗<1/2.
The first unconditional bound on H_{1} was established in a breakthrough work of Zhang [3]:
Theorem 2(Zhang’s theorem).
H_{1}≤70,000,000.
Zhang’s argument followed the general strategy from [2] on finding small gaps between primes, with the major new ingredient being a proof of a weaker version of $\text{EH}\left[\frac{1}{2}+2\varpi \right]$, which we call MPZ[ ϖ,δ] (see Claim 10) below. It was quickly realized that Zhang’s numerical bound on H_{1} 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
Very shortly afterwards, a further breakthrough was obtained by Maynard [5] (with related work obtained independently in an unpublished work of Tao), who developed a more flexible ‘multidimensional’ version of the Selberg sieve to obtain stronger bounds on H_{ m }. This argument worked without using any equidistribution results on primes beyond the BombieriVinogradov theorem, and among other things was able to establish finiteness of H_{ m } for all m, not just for m=1. More precisely, Maynard established the following results.
Theorem 3(Maynard’s theorem).
Unconditionally, we have the following bounds:
(i) H_{1}≤600
(ii) H_{ m }≤C m^{3}e^{4m} for all m≥1 and an absolute (and effective) constant C
Assuming the ElliottHalberstam conjecture EH[ 𝜗] for all 0<𝜗<1, we have the following improvements:
(iii) H_{1}≤12
(iv) H_{2}≤600
(v) H_{ m }≤C m^{3}e^{2m} for all m≥1 and an absolute (and effective) constant C
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_{1}≤246
(ii) H_{2}≤398,130
(iii) H_{3}≤24,797,814
(iv) H_{4}≤1,431,556,072
(v) H_{5}≤80,550,202,480
(vi)${H}_{m}\le \mathit{\text{Cm}}exp\left(\left(4\frac{28}{157}\right)m\right)$ for all m≥1 and an absolute (and effective) constant C
Assume the ElliottHalberstam conjecture EH[ 𝜗] for all 0<𝜗<1. Then, we have the following improvements:
(vii) H_{2}≤270
(viii) H_{3}≤52,116
(ix) H_{4}≤474,266.
(x) H_{5}≤4,137,854.
(xi) H_{ m }≤C m e^{2m} for all m≥1 and an absolute (and effective) constant C
Finally, assume the generalized ElliottHalberstam conjecture GEH[ 𝜗] (see Claim 12 below) for all 0<𝜗<1. Then,
(xii) H_{1}≤6
(xiii) H_{2}≤252
In the ‘Outline of the key ingredients’ section, we will describe the key propositions that will be combined together to prove the various components of Theorem 4. As with Theorem 1, the results in (vii)(xiii) do not require EH[ 𝜗] or GEH[ 𝜗] for all 0<𝜗<1, but only for a single explicitly computable 𝜗 that is sufficiently close to 1.
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_{1} than 6 from purely sievetheoretic methods; we review this obstruction in the ‘The parity problem’ section. If one only assumes the ElliottHalberstam conjecture EH[ 𝜗] instead of its generalization GEH[ 𝜗], we were unable to improve upon Maynard’s bound H_{1}≤12; 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 sievetheoretic arguments (e.g. by finding a way to establish Theorem 20(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 BombieriVinogradov theorem, and the remaining bounds (vii)(xiii) of course use either the ElliottHalberstam conjecture or a generalization thereof.
A variant of the proof of Theorem 4(xii), which we give in ‘Additional remarks’ section, also gives the following conditional ‘near miss’ to (a disjunction of) the twin prime conjecture and the even Goldbach conjecture:
Theorem 5(Disjunction).
Assume the generalized ElliottHalberstam conjecture GEH[ 𝜗] for all 0<𝜗<1. Then, at least one of the following statements is true:
(a) (Twin prime conjecture) H_{1}=2.
(b) (nearmiss to even Goldbach conjecture) If n is a sufficiently large multiple of 6, then at least one of n and n−2 is expressible as the sum of two primes, similarly with n−2 replaced by n+2. (In particular, every sufficiently large even number lies within 2 of the sum of two primes.)
We remark that a disjunction in a similar spirit was obtained in [8], which established (prior to the appearance of Theorem 2) that either H_{1} was finite or that every interval [x,x+x^{ε}] contained the sum of two primes if x was sufficiently large depending on ε>0.
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 ElliottHalberstam conjecture). As in [5], this largely reduces the task 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 k, as well as sharpened asymptotics in the case of large values of k.
The methods of Maynard and Tao have been used in a number of subsequent applications [9][21]. 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.
1.1 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 known (or conjectured) distributional estimates on primes in arithmetic progressions that we will need to prove Theorem 4. Then, in the section ‘Outline of the key ingredients’, we give the key propositions that will be combined together to establish this theorem. One of these propositions, Lemma 18, is an easy application of the pigeonhole principle. Two further propositions, Theorem 19 and Theorem 20, 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 22, 24, 26, and 28 use the asymptotics established in Theorems 19 and 20, in combination with Lemma 18, 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 analysed in the asymptotic regime of large k in the ‘Asymptotic analysis’ section, and for small and medium k in the ‘The case of small and medium dimension’ section, with the results collected in Theorems 23, 25, 27, and 29. Combining these results with the previous propositions gives Theorem 16, which, when combined with the bounds on narrow admissible tuples in Theorem 17 that are established in the ‘Narrow admissible tuples’ section, will give Theorem 4. (See also Table 1 for more details of the logical dependencies between the key propositions.)
Finally, in the ‘The parity problem’ section, we modify an argument of Selberg to show that the bound H_{1}≤6 may not be improved using purely sievetheoretic methods, and in the ‘Additional remarks’ section, we establish Theorem 5 and make some miscellaneous remarks.
1.2 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 1_{ E } to denote the indicator function of a set E; thus, 1_{ E }(n)=1 when n∈E and 1_{ E }(n)=0 otherwise.
All sums and products will be over the natural numbers $\mathbb{N}:=\{1,2,3,\dots \}$ 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 think 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 $\sum _{x\le n\le 2x}f(n)$). If X and Y are two quantities depending on x, we say that X=O(Y) or X≪Y if one has X≤C Y for some fixed C (which we refer to as the implied constant), and X=o(Y) if one has X≤c(x)Y for some function c(x) 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(1)}Y, X∼Y to denote the estimate Y≪X≪Y, and X≈Y to denote the estimate Y⪻ ⪻X⪻ ⪻Y. Finally, we say that a quantity n is of polynomial size if one has n=O(x^{O(1)}).
If asymptotic notation such as O() or ⪻ ⪻ appears on the lefthand side of a statement, this means that the assertion holds true for any specific interpretation of that notation. For instance, the assertion $\sum _{n=O(N)}\alpha (n)\u2abb\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\u2abbN$ means that for each fixed constant C>0, one has $\sum _{n\le \mathit{\text{CN}}}\alpha (n)\u2abb\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\u2abbN$.
If q and a are integers, we write aq if a divides q. If q is a natural number and $a\in \mathbb{Z}$, we use a (q) to denote the residue class
and let $\mathbb{Z}/\mathrm{q\mathbb{Z}}$ denote the ring of all such residue classes a(q). The notation b=a (q) is synonymous to b∈ a (q). We use (a,q) to denote the greatest common divisor of a and q, and [ a,q] to denote the least common multiple^{a}. We also let
denote the primitive residue classes of $\mathbb{Z}/\mathrm{q\mathbb{Z}}$.
We use the following standard arithmetic functions:

(i)
$$\phi (q):={(\mathbb{Z}/\mathrm{q\mathbb{Z}})}^{\times}$$
denotes the Euler totient function of q.

(ii)
$$\tau (q):=\sum _{dq}1$$
denotes the divisor function of q.

(iii)
Λ(q) denotes the von Mangoldt function of q; thus, Λ(q)= logp if q is a power of a prime p, and Λ(q)=0 otherwise.

(iv)
θ(q) is defined to equal logq when q is a prime, and θ(q)=0 otherwise.

(v)
μ(q) denotes the Möbius function of q; thus, μ(q)=(−1)^{k} if q is the product of k distinct primes for some k≥0, and μ(q)=0 otherwise.

(vi)
Ω(q) denotes the number of prime factors of q (counting multiplicity).
We recall the elementary divisor bound
whenever n≪x^{O(1)}, as well as the related estimate
for any fixed C>0 (see, e.g. [Lemma 1.5]).
The Dirichlet convolution$\alpha \star \beta :\mathbb{N}\to \u2102$ of two arithmetic functions $\alpha ,\beta :\mathbb{N}\to \u2102$ is defined in the usual fashion as
Distribution estimates on arithmetic functions
As mentioned in the introduction, a key ingredient in the GoldstonPintzYıldırım approach to small gaps between primes comes from distributional estimates on the primes, or more precisely on the von Mangoldt function Λ, which serves as a proxy for the primes. In this work, 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 $\alpha :\mathbb{N}\to \u2102$ with finite support (that is, α is nonzero only on a finite set) and any primitive residue class a (q), we define the (signed) discrepancy Δ(α;a (q)) to be the quantity
For any fixed 0<𝜗<1, let EH[ 𝜗] denote the following claim:
Claim 8(ElliottHalberstam conjecture, EH[ 𝜗]).
If Q⪻ ⪻x^{𝜗} and A≥1 is fixed, then
In [22], it was conjectured that EH[ 𝜗] held for all 0<𝜗<1. (The conjecture fails at the endpoint case 𝜗=1; see [23],[24] for a more precise statement.) The following classical result of Bombieri [25] and Vinogradov [26] remains the best partial result of the form EH[ 𝜗]:
Theorem 9(BombieriVinogradov theorem).
[25],[26] EH[ 𝜗] holds for every fixed 0<𝜗<1/2.
In [2], it was shown that any estimate of the form EH[ 𝜗] with some fixed 𝜗>1/2 would imply the finiteness of H_{1}. While such an estimate remains unproven, it was observed by Motohashi and Pintz [27] and by Zhang [3] that a certain weakened version of EH[ 𝜗] would still suffice for this purpose. More precisely (and following the notation of our previous paper), let ϖ,δ>0 be fixed, and let MPZ[ ϖ,δ] be the following claim:
Claim 10(MotohashiPintzZhang estimate, MPZ[ ϖ,δ]).
Let I⊂[1,x^{δ}] and Q⪻ ⪻x^{1/2+2ϖ}. Let P_{ I } denote the product of all the primes in I, and let S_{ I } denote the squarefree 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≥1 is fixed, then
where the implied constant depends only on the fixed quantities (A,ϖ,δ), but not on a.
It is clear that $\text{EH}\left[\frac{1}{2}+2\varpi \right]$ implies MPZ[ ϖ,δ] whenever ϖ,δ≥0. The first nontrivial estimate of the form MPZ[ ϖ,δ] was established by Zhang [3], who (essentially) obtained MPZ[ ϖ,δ] whenever $0\le \varpi ,\delta <\frac{1}{1,168}$. In [Theorem 2.17], we improved this result to the following.
Theorem 11.
MPZ[ ϖ,δ] holds for every fixed ϖ,δ≥0 with 600ϖ+180δ<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, Theorem 11 allows one to get ϖ arbitrarily close to $\frac{7}{600}\approx 0.01167$.
In this paper, we will also study the following generalization of the ElliottHalberstam conjecture:
Claim 12(Generalized ElliottHalberstam conjecture, GEH[ 𝜗]).
Let ε>0 and A≥1 be fixed. Let N,M be quantities such that x^{ε}⪻ ⪻N,M⪻ ⪻x^{1−ε} with N M≍x, and let $\alpha ,\beta :\mathbb{N}\to \mathbb{R}$ be sequences supported on [ N,2N] and [ M,2M], respectively, such that one has the pointwise bound
for all natural numbers n,m. Suppose also that β obeys the SiegelWalfisz type bound
for any q,r≥1, any fixed A, and any primitive residue class a (q). Then for any Q⪻ ⪻x^{𝜗}, we have
In [28], Conjecture 1], it was essentially conjectured^{b} that GEH[ 𝜗] was true for all 0<𝜗<1. This is stronger than the ElliottHalberstam conjecture:
Proposition 13.
For any fixed 0<𝜗<1, GEH[ 𝜗] implies EH[ 𝜗].
Proof.
(Sketch) As this argument is standard, we give only a brief sketch. Let A>0 be fixed. For n∈[ x,2x], we have Vaughan’s identity^{c}[29]
where L(n):= log(n) and
By decomposing each of the functions μ_{<}, μ_{≥}, 1, Λ_{<}, Λ_{≥} into O(logA+1x) functions supported on intervals of the form [ N,(1+ log−A x)N], and discarding those contributions which meet the boundary of [ x,2x] (cf. [3],[28],[30],[31]), 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 SiegelWalfisz theorem; see, e.g. [32], Satz 4] or [33], Th. 5.29]).
By modifying the proof of the BombieriVinogradov theorem, Motohashi [34] established the following generalization of that theorem:
Theorem 14(Generalized BombieriVinogradov theorem).
[34] GEH[ 𝜗] holds for every fixed 0<𝜗<1/2.
One could similarly describe a generalization of the MotohashiPintzZhang estimate MPZ[ ϖ,δ], but unfortunately, the arguments in [3] or Theorem 11 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^{1/2}, or if one has ‘Type III’ structure to the convolution α⋆β, in the sense that it can refactored as a convolution involving several ‘smooth’ sequences. In any event, our analysis would not be able to make 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 1. In particular, we will not directly use Theorem 14 in this paper.
Outline of the key ingredients
In this section, we describe the key subtheorems used in the proof of Theorem 4, with the proofs of these subtheorems mostly being deferred to later sections.
We begin with a weak version of the DicksonHardyLittlewood prime tuples conjecture [1], which (following Pintz [35]) we refer to as [ k,j]. Recall that for any $k\in \mathbb{N}$, an admissible ktuple is a tuple $\mathcal{\mathscr{H}}=({h}_{1},\dots ,{h}_{k})$ of k increasing integers h_{1}<…<h_{ k } which avoids at least one residue class ${a}_{p}\phantom{\rule{1em}{0ex}}(p):=\{{a}_{p}+\mathit{\text{np}}:n\in \mathbb{Z}\}$ for every p. For instance, (0,2,6) is an admissible 3tuple, but (0,2,4) is not.
For any k≥j≥2, we let DHL[ k;j] denote the following claim:
Claim 15(Weak DicksonHardyLittlewood conjecture, DHL[ k;j]).
For any admissible ktuple $\mathcal{\mathscr{H}}=({h}_{1},\dots ,{h}_{k})$, there exist infinitely many translates $n+\mathcal{\mathscr{H}}=(n+{h}_{1},\dots ,n+{h}_{k})$ of which contain at least j primes.
The full DicksonHardyLittlewood conjecture is then the assertion that DHL[ k;k] holds for all k≥2. In our analysis, we will focus on the case when j is much smaller than k; in fact, j will be of the order of logk.
For any k, let H(k) denote the minimal diameter h_{ k }−h_{1} of an admissible ktuple; thus for instance, H(3)=6. It is clear that for any natural numbers m≥1 and k≥m+1, the claim DHL[k;m+1] implies that H_{ m }≤H(k) (and the claim DHL[ k;k] would imply that H_{k−1}=H(k)). We will therefore deduce Theorem 4 from a number of claims of the form DHL[ k;j]. More precisely, we have
Theorem 16.
Unconditionally, we have the following claims:
(i) DHL[50;2].
(ii) DHL[35,410;3].
(iii) DHL[1,649,821;4].
(iv) DHL[75,845,707;5].
(v) DHL[3,473,955,908;6].
(vi) DHL[k;m+1] whenever m≥1 and $k\ge Cexp\left(\left(4\frac{28}{157}\right)m\right)$ for some sufficiently large absolute (and effective) constant C.
Assume the ElliottHalberstam conjecture EH[ θ] for all 0<θ<1. Then, we have the following improvements:
(vii) DHL[54;3].
(viii) DHL[5,511;4].
(ix) DHL[41,588;5].
(x) DHL[309,661;6].
(xi) DHL[k;m+1] whenever m≥1 and k≥C exp(2m) for some sufficiently large absolute (and effective) constant C.
Assume the generalized ElliottHalberstam conjecture GEH[ θ] for all 0<θ<1. Then
(xii) DHL[3;2].
(xiii) DHL[51;3].
Theorem 4 then follows from Theorem 16 and the following bounds on H(k) (ordered by increasing value of k):
Theorem 17(Bounds on H(k)).
(xii) H(3)=6.
(i) H(50)=246.
(xiii) H(51)=252.
(vii) H(54)=270.
(viii) H(5,511)≤52,116.
(ii) H(35,410)≤398,130.
(ix) H(41,588)≤474,266.
(x) H(309,661)≤4,137,854.
(iii) H(1,649,821)≤24,797,814.
(iv) H(75,845,707)≤1,431,556,072.
(v) H(3,473,955,908)≤80,550,202,480.
(vi), (xi) In the asymptotic limit k→∞, one has H(k)≤k logk+k log logk−k+o(k), with the bounds on the decay rate o(k) being effective.
We prove Theorem 17 in the ‘Narrow admissible tuples’ section. In the opposite direction, an application of the BrunTitchmarsh theorem gives $H(k)\ge \left(\frac{1}{2}+o(1)\right)klogk$ as k→∞ (see [4], §3.9] for this bound, as well as with some slight refinements).
The proof of Theorem 16 follows the GoldstonPintzYıldırım strategy that was also used in all previous progress on this problem (e.g. [2],[3],[5],[27]), namely that of constructing a sieve function adapted to an admissible ktuple with good properties. More precisely, we set
and
and observe the crude bound
We have the following simple ‘pigeonhole principle’ criterion for DHL[k;m+1] (cf. [Lemma 4.1], though the normalization here is slightly different):
Lemma 18(Criterion for DHL).
Let k≥2 and m≥1 be fixed integers and define the normalization constant
Suppose that for each fixed admissible ktuple (h_{1},…,h_{ k }) and each residue class b (W)such that b+h_{ i } is coprime to W for all i=1,…,k, one can find a nonnegative weight function $\nu :\mathbb{N}\to {\mathbb{R}}^{+}$ and fixed quantities α>0 and β_{1},…,β_{ k }≥0, such that one has the asymptotic upper bound
the asymptotic lower bound
for all i=1,…,k, and the key inequality
Then, DHL[ k;m+1] holds.
Proof.
Let (h_{1},…,h_{ k }) be a fixed admissible ktuple. Since it is admissible, there is at least one residue class b (W) such that (b+h_{ i },W)=1 for all ${h}_{i}\in \mathcal{\mathscr{H}}$. For an arithmetic function ν as in the lemma, we consider the quantity
Combining (13) and (14), we obtain the lower bound
From (12) and the crucial condition (15), it follows that N>0 if x is sufficiently large.
On the other hand, the sum
can be positive only if n+h_{ i } is prime for at least m+1 indices i=1,…,k. We conclude that, for all sufficiently large x, there exists some integer n∈[ x,2x] such that n+h_{ i } is prime for at least m+1 values of i=1,…,k.
Since (h_{1},…,h_{ k }) is an arbitrary admissible ktuple, DHL[ k;m+1] follows.
The objective is then to construct nonnegative weights ν whose associated ratio $\frac{{\beta}_{1}+\cdots +{\beta}_{k}}{\alpha}$ has provable lower bounds that are as large as possible. Our sieve majorants will 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 ‘onedimensional’ divisor sums. More precisely, for any fixed smooth compactly supported function $F:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty )\to \mathbb{R}$, define the divisor sum ${\lambda}_{F}:\mathbb{Z}\to \mathbb{R}$ by the formula
where logx denotes the base x logarithm
One should think 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>0 (see Proposition 14 for a more rigorous version of this heuristic).
The functions ν we will use will take the form
for some fixed natural number J, fixed coefficients ${c}_{1},\dots ,{c}_{J}\in \mathbb{R}$ and fixed smooth compactly supported functions ${F}_{j,i}:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty )\to \mathbb{R}$ with j=1,…,J and i=1,…,k. (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_{1},…,n+h_{ k } are all almost prime.
Clearly, ν is a (positivedefinite) linear combination of functions of the form
for various smooth functions ${F}_{1},\dots ,{F}_{k},{G}_{1},\dots ,{G}_{k}:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty )\to \mathbb{R}$. The sum appearing in (13) can thus be decomposed into linear combinations of sums of the form
Also, since from (16) we clearly have
when n≥x is prime and F is supported on [ 0,1], the sum appearing in (14) can be similarly decomposed into linear combinations of sums of the form
To estimate the sums (21), we use the following asymptotic, proven in the ‘Multidimensional Selberg sieves’ section. For each compactly supported $F:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty )\to \mathbb{R}$, let
denote the upper range of the support of F (with the convention that S(0)=0).
Theorem 19(Asymptotic for prime sums).
Let k≥2 be fixed, let (h_{1},…,h_{ k }) be a fixed admissible ktuple, and let b (W) be such that b+h_{ i } is coprime to W for each i=1,…,k. Let 1≤i_{0}≤k be fixed, and for each 1≤i≤k distinct from i_{0}, let ${F}_{i},{G}_{i}:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty )\to \mathbb{R}$ be fixed smooth compactly supported functions. Assume one of the following hypotheses:
(i) (ElliottHalberstam) There exists a fixed 0<𝜗<1 such that EH[ 𝜗] holds and such that
(ii) (MotohashiPintzZhang) There exists fixed 0≤ϖ<1/4 and δ>0 such that MPZ[ϖ,δ] holds and such that
and
Then, we have
where
Here of course F^{′} denotes the derivative of F.
To estimate the sums (19), we use the following asymptotic, also proven in the ‘Multidimensional Selberg sieves’ section.
Theorem 20(Asymptotic for nonprime sums).
Let k≥1 be fixed, let (h_{1},…,h_{ k }) be a fixed admissible ktuple, and let b (W) be such that b+h_{ i } is coprime to W for each i=1,…,k. For each fixed 1≤i≤k, let ${F}_{i},{G}_{i}:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty )\to \mathbb{R}$ be fixed smooth compactly supported functions. Assume one of the following hypotheses:
(i) (Trivial case) One has
(ii) (Generalized ElliottHalberstam) There exists a fixed 0<𝜗<1 and i_{0}∈{1,…,k} such that GEH[ 𝜗] holds, and
Then, we have
where
A key point in (ii) is that no upper bound on $S({F}_{{i}_{0}})$ or $S({G}_{{i}_{0}})$ is required (although, as we will see in the ‘The generalized ElliottHalberstam case’ section, the result is a little easier to prove when one has $S({F}_{{i}_{0}})+S({G}_{{i}_{0}})<1$). This flexibility in the ${F}_{{i}_{0}},{G}_{{i}_{0}}$ functions will be particularly crucial to obtain part (xii) of Theorem 16 and Theorem 4.
Remark 21.
Theorems 19 and 20 can be viewed as probabilistic assertions of the following form: if n is chosen uniformly at random from the set {x≤n≤2x:n=b (W)}, then the random variables θ(n+h_{ i }) and ${\lambda}_{{F}_{j}}(n+{h}_{j}){\lambda}_{{G}_{j}}(n+{h}_{j})$ for i,j=1,…,k have mean $(1+o(1))\frac{W}{\phi (W)}$ and $\left(\underset{0}{\overset{1}{\int}}{F}_{j}^{\prime}(t){G}_{j}^{\prime}(t)\phantom{\rule{1em}{0ex}}\mathit{\text{dt}}+o(1)\right){B}^{1}$, respectively, and furthermore, these random variables enjoy a limited amount of independence, except for the fact (as can be seen from (20)) that θ(n+h_{ i }) and ${\lambda}_{{F}_{i}}(n+{h}_{i}){\lambda}_{{G}_{i}}(n+{h}_{i})$ 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 $\sum _{n}\theta (n)\theta (n+2)$).
Theorems 19 and 20 may be combined with Lemma 18 to reduce the task of establishing estimates of the form DHL[ k;m+1] to that of establishing certain variational problems. For instance, in the ‘Proof of Theorem 22’ section, we reprove the following result of Maynard ([5], Proposition 4.2]):
Theorem 22(Sieving on the standard simplex).
Let k≥2 and m≥1 be fixed integers. For any fixed compactly supported squareintegrable function $F:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty {)}^{k}\to \mathbb{R}$, define the functionals
and
for i=1,…,k, and let M_{ k } be the supremum
over all square integrable functions F that are supported on the simplex
and are not identically zero (up to almost everywhere equivalence, of course). Suppose that there is a fixed 0<𝜗<1 such that EH[ 𝜗] holds and such that
Then, DHL[ k;m+1] holds.
Parts (vii)(xi) of Theorem 16 (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 k:
Theorem 23(Lower bounds on M_{ k }).
(vii) M_{54}>4.00238.
(viii) M_{5,511}>6.
(ix) M_{41,588}>8.
(x) M_{309,661}>10.
(xi) One has M_{ k }≥ logk−C for all k≥C, where C is an absolute (and effective) constant.
For the sake of comparison, in ([5], Proposition 4.3]), it was shown that M_{5}>2, M_{105}>4, and M_{ k }≥ logk−2 log logk−2 for all sufficiently large k. As remarked in that paper, the sieves used on the bounded gap problem prior to the work in [5] would essentially correspond, in this notation, to the choice of functions F of the special form F(t_{1},…,t_{ k }):=f(t_{1}+⋯+t_{ k }), which severely limits the size of the ratio in (33) (in particular, the analogue of M_{ k } 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 ${M}_{k}\le \frac{k}{k1}logk$ for all k≥2, which shows in particular that the bounds in (vii) and (xi) of the above theorem cannot be significantly improved. We remark that Theorem 23(vii) and the BombieriVinogradov theorem also give a weaker version DHL[ 54;2] of Theorem 16(i).
We also have a variant of Theorem 22 which can accept inputs of the form MPZ[ ϖ,δ]:
Theorem 24(Sieving on a truncated simplex).
Let k≥2 and m≥1 be fixed integers. Let 0<ϖ<1/4 and 0<δ<1/2 be such that MPZ[ ϖ,δ] holds. For any α>0, let ${M}_{k}^{\left[\alpha \right]}$ be defined as in (33), but where the supremum now ranges over all squareintegrable F supported in the truncated simplex
and are not identically zero. If
then DHL[ k;m+1] holds.
In the ‘Asymptotic analysis’ section, we will establish the following variant of Theorem 23, which when combined with Theorem 11, allows one to use Theorem 24 to establish parts (ii)(vi) of Theorem 16 (and hence Theorem 4):
Theorem 25(Lower bounds on ${M}_{k}^{\left[\alpha \right]}$).
(ii) There exist δ,ϖ>0 with 600ϖ+180δ<7 and ${M}_{35\phantom{\rule{1em}{0ex}}410}^{\left[\frac{\delta}{1/4+\varpi}\right]}>\frac{2}{1/4+\varpi}$.
(iii) There exist δ,ϖ>0 with 600ϖ+180δ<7 and ${M}_{1\phantom{\rule{1em}{0ex}}649\phantom{\rule{1em}{0ex}}821}^{\left[\frac{\delta}{1/4+\varpi}\right]}>\frac{3}{1/4+\varpi}$.
(iv) There exist δ,ϖ>0 with 600ϖ+180δ<7 and ${M}_{75\phantom{\rule{1em}{0ex}}845\phantom{\rule{1em}{0ex}}707}^{\left[\frac{\delta}{1/4+\varpi}\right]}>\frac{4}{1/4+\varpi}$.
(v) There exist δ,ϖ>0 with 600ϖ+180δ<7 and ${M}_{3\phantom{\rule{1em}{0ex}}473\phantom{\rule{1em}{0ex}}955\phantom{\rule{1em}{0ex}}908}^{\left[\frac{\delta}{1/4+\varpi}\right]}>\frac{5}{1/4+\varpi}$.
(vi) For all k≥C, there exist δ,ϖ>0 with 600ϖ+180δ<7, $\varpi \ge \frac{7}{600}\frac{C}{logk}$, and ${M}_{k}^{\left[\frac{\delta}{1/4+\varpi}\right]}\ge logkC$ for some absolute (and effective) constant C.
The implication is clear for (ii)(v). For (vi), observe that from Theorem 25(vi), Theorem 11, and Theorem 24, we see that DHL[ k;m+1] holds whenever k is sufficiently large and
which is in particular implied by
for some absolute constant C^{′}, giving Theorem 16(vi).
Now we give a more flexible variant of Theorem 22, in which the support of F is enlarged, at the cost of reducing the range of integration of the J_{ i }.
Theorem 26(Sieving on an epsilonenlarged simplex).
Let k≥2 and m≥1 be fixed integers, and let 0<ε<1 be fixed also. For any fixed compactly supported squareintegrable function $F:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty {)}^{k}\to \mathbb{R}$, define the functionals
for i=1,…,k, and let M_{k,ε} be the supremum
over all squareintegrable functions F that are supported on the simplex
and are not identically zero. Suppose that there is a fixed 0<𝜗<1, such that one of the following two hypotheses hold:
(i) EH[𝜗] holds, and $1+\epsilon <\frac{1}{\mathit{\vartheta}}$.
(ii) GEH[𝜗] holds, and $\epsilon <\frac{1}{k1}$.
If
then DHL[ k;m+1] holds.
We prove this theorem in the ‘Proof of Theorem 26’ section. We remark that due to the continuity of M_{k,ε} in ε, the strict inequalities in (i) and (ii) of this theorem may be replaced by nonstrict inequalities. Parts (i) and (xiii) of Theorem 16, and a weaker version DHL[ 4;2] of part (xii), then follow from Theorem 9 and the following computations, proven in the ‘Bounding M_{k,ε} for medium k’ and ‘Bounding M_{4,ε}’ sections:
Theorem 27(Lower bounds on M_{k,ε}).
(i) M_{50,1/25}>4.0043.
(xii’) M_{4,0.168}>2.00558.
(xiii) M_{51,1/50}>4.00156.
We remark that computations in the proof of Theorem 27(xii’) are simple enough that the bound may be checked by hand, without use of a computer. The computations used to establish the full strength of Theorem 16(xii) are however significantly more complicated.
In fact, we may enlarge the support of F further. We give a version corresponding to part (ii) of Theorem 26; 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 28(Going beyond the epsilon enlargement).
Let k≥2 and m≥1 be fixed integers, let 0<𝜗<1 be a fixed quantity such that GEH[ 𝜗] holds, and let $0<\epsilon <\frac{1}{k1}$ be fixed also. Suppose that there is a fixed nonzero squareintegrable function $F:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty {)}^{k}\to \mathbb{R}$ supported in $\frac{k}{k1}\xb7{\mathcal{R}}_{k}$, such that for i=1,…,k, one has the vanishing marginal condition
whenever t_{1},…,t_{i−1},t_{i+1},…,t_{ k }≥0 are such that
Suppose that we also have the inequality
Then DHL[ k;m+1] holds.
This theorem is proven in the ‘Proof of Theorem 28’ section. Theorem 16(xii) is then an immediate consequence of Theorem 28 and the following numerical fact, established in the ‘Threedimensional cutoffs’ section.
Theorem 29(A piecewise polynomial cutoff).
Set $\epsilon :=\frac{1}{4}$. Then, there exists a piecewise polynomial function $F:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty {)}^{3}\to \mathbb{R}$ supported on the simplex
and symmetric in the t_{1},t_{2},t_{3} variables, such that F is not identically zero and obeys the vanishing marginal condition
whenever t_{1},t_{2}≥0 with t_{1}+t_{2}>1+ε and such that
There are several other ways to combine Theorems 19 and 20 with equidistribution theorems on the primes to obtain results of the form DHL[k;m+1], 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 19 and 20. A key asymptotic used in both theorems is the following:
Lemma 30(Asymptotic).
Let k≥1 be a fixed integer, and let N be a natural number coprime to W with logN=O(logO(1)x). Let ${F}_{1},\dots ,{F}_{k},{G}_{1},\dots ,{G}_{k}:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty )\to \mathbb{R}$ be fixed smooth compactly supported functions. Then,
where B was defined in (12), and
The same claim holds if the denominators $\left[\phantom{\rule{0.3em}{0ex}}{d}_{j},{d}_{j}^{\prime}\right]$ are replaced by $\phi \left(\left[\phantom{\rule{0.3em}{0ex}}{d}_{j},{d}_{j}^{\prime}\right]\right)$.
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=1.
Proof.
We begin with the first claim. For j=1,…,k, 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 , and so we have Fourier expansions
and
for some fixed functions ${f}_{j},{g}_{j}:\mathbb{R}\to \u2102$ that are smooth and rapidly decreasing in the sense that f_{ j }(ξ),g_{ j }(ξ)=O((1+ξ)^{−A}) for any fixed A>0 and all $\xi \in \mathbb{R}$ (here the implied constant is independent of ξ and depends only on A).
We may thus write
and
for all ${d}_{j},{d}_{j}^{\prime}\ge 1$. We note that
Therefore, if we substitute the Fourier expansions into the lefthand side of (36), the resulting expression is absolutely convergent. Thus, we can apply Fubini’s theorem, and the lefthand side of (36) can thus be rewritten as
where
This latter expression factorizes as an Euler product
where the local factors K_{ p } are given by
We can estimate each Euler factor as
Since
we have
where the modified zeta function ζ_{ WN } is defined by the formula
for ℜ(s)>1.
For $\Re (s)\ge 1+\frac{1}{logx}$, we have the crude bounds
Thus,
Combining this with the rapid decrease of f_{ j },g_{ j }, we see that the contribution to (38) outside of the cube $\left\{max\left({\xi}_{1},\dots ,{\xi}_{k},{\xi}_{1}^{\prime},\dots ,{\xi}_{k}^{\prime}\right)\le \sqrt{logx}\right\}$ (say) is negligible. Thus, it will suffice to show that
When ${\xi}_{j}\le \sqrt{logx}$, we see from the simple pole of the Riemann zeta function $\zeta (s)=\prod _{p}{\left(1\frac{1}{{p}^{s}}\right)}^{1}$ at s=1 that
For $\sqrt{logx}\le {\xi}_{j}\le \sqrt{logx}$, we see that
Since logW N≪ logO(1)x, this gives
since the sum is maximized when WN is composed only of primes p≪ logO(1)x. Thus,
similarly with 1+i ξ_{ j } replaced by $1+i{\xi}_{j}^{\prime}$ or $2+i{\xi}_{j}+i{\xi}_{j}^{\prime}$. We conclude that
Therefore, it will suffice to show that
since the errors caused by the 1+o(1) multiplicative factor in (41) or the truncation ${\xi}_{j},{\xi}_{j}^{\prime}\le \sqrt{logx}$ can be seen to be negligible using the rapid decay of f_{ j },g_{ j }. By Fubini’s theorem, it suffices to show that
for each j=1,…,k. But from dividing (37) by e^{t} and differentiating under the integral sign, we have
and the claim then follows from Fubini’s theorem.
Finally, suppose that we replace $\left[\phantom{\rule{0.3em}{0ex}}{d}_{j},{d}_{j}^{\prime}\right]$ with $\phi \left(\left[\phantom{\rule{0.3em}{0ex}}{d}_{j},{d}_{j}^{\prime}\right]\right)$. An inspection of the above argument shows that the only change that occurs is that the $\frac{1}{p}$ term in (39) is replaced by $\frac{1}{p1}$; but this modification may be absorbed into the $1+O\left(\frac{1}{{p}^{2}}\right)$ factor in (40), and the rest of the argument continues as before.
4.1 The trivial case
We can now prove the easiest case of the two theorems, namely case (i) of Theorem 20; a closely related estimate also appears in ([5], Lemma 6.2]). We may assume that x is sufficiently large depending on all fixed quantities. By (16), the lefthand side of (29) may be expanded as
where
By hypothesis, b+h_{ i } is coprime to W for all i=1,…,k, and h_{ i }−h_{ j }<w for all distinct i,j. Thus, $S\left({d}_{1},\dots ,{d}_{k},{d}_{1}^{\prime},\dots ,{d}_{k}^{\prime}\right)$ vanishes unless the $\left[\phantom{\rule{0.3em}{0ex}}{d}_{i},{d}_{i}^{\prime}\right]$ are coprime to each other and to W. In this case, $S\left({d}_{1},\dots ,{d}_{k},{d}_{1}^{\prime},\dots ,{d}_{k}^{\prime}\right)$ is summing the constant function 1 over an arithmetic progression in [ x,2x] of spacing $W\left[\phantom{\rule{1.0pt}{0ex}}{d}_{1},{d}_{1}^{\prime}\right]\dots \left[\phantom{\rule{1.0pt}{0ex}}{d}_{k},{d}_{k}^{\prime}\right]$, and so
By Lemma 30, the contribution of the main term $\frac{x}{W\left[{d}_{1},{d}_{1}^{\prime}\right]\dots \left[{d}_{k},{d}_{k}^{\prime}\right]}$ to (29) is $(c+o(1)){B}^{k}\frac{x}{W}$; note that the restriction of the integrals in (30) to [ 0,1] instead of [ 0,+∞) is harmless since S(F_{ i }),S(G_{ i })<1 for all i. Meanwhile, the contribution of the O(1) error is then bounded by
By the hypothesis in Theorem 20(i), we see that for ${d}_{1},\dots ,{d}_{k},{d}_{1}^{\prime},\dots ,{d}_{k}^{\prime}$ contributing a nonzero term here, one has
for some fixed ε>0. From the divisor bound (1), we see that each choice of $\left[{d}_{1},{d}_{1}^{\prime}\right]\dots \left[{d}_{k},{d}_{k}^{\prime}\right]$ arises from ⪻ ⪻1 choices of ${d}_{1},\dots ,{d}_{k},{d}_{1}^{\prime},\dots ,{d}_{k}^{\prime}$. We conclude that the net contribution of the O(1) error to (29) is ⪻ ⪻x^{1−ε}, and the claim follows.
4.2 The ElliottHalberstam case
Now we show case (i) of Theorem 19. For the sake of notation, we take i_{0}=k, as the other cases are similar. We use (16) to rewrite the lefthand side of (26) as
where
As in the previous case, $\stackrel{~}{S}\left({d}_{1},\dots ,{d}_{k1},{d}_{1}^{\prime},\dots ,{d}_{k1}^{\prime}\right)$ vanishes unless the $\left[{d}_{i},{d}_{i}^{\prime}\right]$ are coprime to each other and to W, and so the summand in (43) vanishes unless the modulus ${q}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ defined by
is squarefree. In that case, we may use the Chinese remainder theorem to concatenate the congruence conditions on n into a single primitive congruence condition
for some ${a}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ depending on $W,{d}_{1},\dots ,{d}_{k1},{d}_{1}^{\prime},\dots ,{d}_{k1}^{\prime}$, and conclude using (3) that
From the prime number theorem, we have
and this expression is clearly independent of ${d}_{1},\dots ,{d}_{k1}^{\prime}$. Thus, by Lemma 30, the contribution of the main term in (45) is $(c+o(1)){B}^{1k}\frac{x}{\phi (W)}$. By (11) and (12), it thus suffices to show that for any fixed A we have
where $a={a}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ and $q={q}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$. For future reference, we note that we may restrict the summation here to those ${d}_{1},\dots ,{d}_{k1}^{\prime}$ for which ${q}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ is squarefree.
From the hypotheses of Theorem 19(i), we have
whenever the summand in (43) is nonzero, and each choice q of ${q}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ is associated to O(τ(q)^{O(1)}) choices of ${d}_{1},\dots ,{d}_{k1},{d}_{1}^{\prime},\dots ,{d}_{k1}^{\prime}$. Thus, this contribution is
Using the crude bound
and (2), we have
for any fixed C>0. By the CauchySchwarz inequality, it suffices to show that
for any fixed A>0. However, since θ only differs from Λ on powers p^{j} of primes with j>1, it is not difficult to show that
so the net error in replacing θ here by Λ is ⪻ ⪻x^{1−(1−𝜗)/2}, which is certainly acceptable. The claim now follows from the hypothesis EH[ 𝜗], thanks to Claim 8.
4.3 The MotohashiPintzZhang case
Now we show case (ii) of Theorem 19. We repeat the arguments from the ‘The ElliottHalberstam case’ section, with the only difference being in the derivation of (46). As observed previously, we may restrict ${q}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ to be squarefree. From the hypotheses in Theorem 19(ii), we also see that
and that all the prime factors of ${q}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ are at most x^{δ}. Thus, if we set I:= [ 1,x^{δ}], we see (using the notation from Claim 10) that ${q}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ lies in ${\mathcal{S}}_{I}$ and is thus a factor of P_{ I }. If we then let $\mathcal{A}\subset \mathbb{Z}/{P}_{I}\mathbb{Z}$ 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}, one has a+h_{ i }=0 (p) for some i=1,…,k, then we see that ${a}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ lies in the projection of to $\mathbb{Z}/{q}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}\mathbb{Z}$. Each $q\in {\mathcal{S}}_{I}$ is equal to ${q}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ for O(τ(q)^{O(1)}) choices of ${d}_{1},\dots ,{d}_{k1}^{\prime}$. Thus, the lefthand side of (46) is
Note from the Chinese remainder theorem that for any given q, if one lets a range uniformly in , then a (q) is uniformly distributed among O(τ(q)^{O(1)}) different moduli. Thus, we have
and so it suffices to show that
for any fixed A>0. We see it suffices to show that
for any given $a\in \mathcal{A}$. But this follows from the hypothesis MPZ[ ϖ,δ] by repeating the arguments of the ‘The ElliottHalberstam case’ section.
4.4 Crude estimates on divisor sums
To proceed further, we will need some additional information on the divisor sums λ_{ F } (defined in (16)), namely that these sums are concentrated on ‘almost primes’; results of this type have also appeared in [38].
Proposition 14(Almost primality).
Let k≥1 be fixed, let (h_{1},…,h_{ k }) be a fixed admissible ktuple, and let b (W)be such that b+h_{ i } is coprime to W for each i=1,…,k. Let ${F}_{1},\dots ,{F}_{k}:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty )\to \mathbb{R}$ be fixed smooth compactly supported functions, and let m_{1},…,m_{ k }≥0 and a_{1},…,a_{ k }≥1 be fixed natural numbers. Then,
Furthermore, if 1≤j_{0}≤k is fixed and p_{0} is a prime with ${p}_{0}\le {x}^{\frac{1}{10k}}$, then we have the variant
As a consequence, we have
for any ε>0, where p(n) denotes the least prime factor of n.
The exponent $\frac{1}{10k}$ can certainly be improved here, but for our purposes, any fixed positive exponent depending only on k will suffice.
Proof.
The strategy is to estimate the alternating divisor sums ${\lambda}_{{F}_{j}}(n+{h}_{j})$ 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 30, we can use Fourier expansion to write
for some rapidly decreasing ${f}_{j}:\mathbb{R}\to \u2102$ and all natural numbers d. Thus,
which factorizes using Euler products as
The function $s\mapsto {p}^{\frac{s}{logx}}$ has a magnitude of O(1) and a derivative of O(logx p) when ℜ(s)>1, and thus
From the rapid decrease of f_{ j } and the triangle inequality, we conclude that
for any fixed A>0. Thus, noting that $\prod _{pn}O(1)\ll \tau {(n)}^{O(1)}$, we have
for any fixed a_{ j },A. However, we have
and so
Making the change of variables $\sigma :=1+{\xi}_{1}+\cdots +{\xi}_{{a}_{j}}$, we obtain
for any fixed A>0. In view of this bound and the FubiniTonelli theorem, it suffices to show that
for all σ_{1},…,σ_{ k }≥1. By setting σ:=σ_{1}+⋯+σ_{ k }, it suffices to show that
for any σ≥1.
To proceed further, we factorize n+h_{ j } as a product
of primes p_{1}≤⋯≤p_{ r } in increasing order and then write
where ${d}_{j}:={p}_{1}\dots {p}_{{i}_{j}}$ and i_{ j } is the largest index for which ${p}_{1}\dots {p}_{{i}_{j}}<{x}^{\frac{1}{10k}}$, and ${m}_{j}:={p}_{{i}_{j}+1}\dots {p}_{r}$. By construction, we see that 0≤i_{ j }<r, ${d}_{j}\le {x}^{\frac{1}{10k}}$. Also, we have
Since n≤2x, this implies that
and so
where we recall that Ω(d_{ j })=i_{ j } denotes the number of prime factors of d_{ j }, counting multiplicity. We also see that
where p(n) denotes the least prime factor of n. Finally, we have that
and we see that the d_{1},…,d_{ k },W are coprime. We may thus estimate the lefthand side of (50) by
where the outer sum $\sum _{\ast}$ is over ${d}_{1},\dots ,{d}_{k}\le {x}^{\frac{1}{10k}}$ with d_{1},…,d_{ k },W coprime, and the inner sum $\sum _{\ast \ast}$ is over x≤n≤2x with n=b (W) and n+h_{ j }=0 (d_{ j }) for each j, with $p\left(\frac{n+{h}_{j}}{{d}_{j}}\right)\ge R$ for each j.
We bound the inner sum $\sum _{\ast \ast}1$ using a Selberg sieve upper bound. Let G be a smooth function supported on [ 0,1] with G(0)=1, and let d=d_{1}…d_{ k }. We see that
since the product is G(0)^{2k}=1 if $p\left(\frac{n+{h}_{j}}{{d}_{j}}\right)\ge R$, and nonnegative otherwise. The righthand side may be expanded as
As in the ‘The trivial case’ section, the inner sum vanishes unless the ${e}_{i}{e}_{i}^{\prime}$ are coprime to each other and dW, in which case it is
The O(1) term contributes ⪻ ⪻R^{k}⪻ ⪻x^{1/10}, which is negligible. By Lemma 30, if Ω(d)≪ log1/2x, then the main term contributes
We see that this final bound applies trivially if Ω(d)≫ log1/2x. The bound (50) thus reduces to
Ignoring the coprimality conditions on the d_{ j } for an upper bound, we see this is bounded by
But from Mertens’ theorem, we have
and the claim (47) follows.
The proof of (48) is a minor modification of the argument above used to prove (47). Namely, the variable ${d}_{{j}_{0}}$ is now replaced by [ d_{0},p_{0}]<x^{1/5k}, which upon factoring out p_{0} has the effect of multiplying the upper bound for (51) by $O\left(\frac{\sigma \underset{x}{log}{p}_{0}}{{p}_{0}}\right)$ (at the negligible cost of deleting the prime p_{0} from the sum $\left.\sum _{p\le x}\right)$, giving the claim; we omit the details.
Finally, (49) follows immediately from (47) when $\epsilon >\frac{1}{10k}$, and from (48) and Mertens’ theorem when $\epsilon \le \frac{1}{10k}$.
Remark 32.
As in [38], one can use Proposition 14, together with the observation that the quantity λ_{ F }(n) is bounded whenever n=O(x) and p(n)≥x^{ε}, to conclude that whenever the hypotheses of Lemma 18 are obeyed for some ν of the form (18), then there exists a fixed ε>0 such that for all sufficiently large x, there are $\gg \frac{x}{\stackrel{k}{log}x}$ elements n of [x,2x] such that n+h_{1},…,n+h_{ k } have no prime factor less than x^{ε}, and that at least m of the n+h_{1},…,n+h_{ k } are prime.
4.5 The generalized ElliottHalberstam case
Now we show case (ii) of Theorem 20. For the sake of notation, we shall take i_{0}=k, as the other cases are similar; thus, we have
The basic idea is to view the sum (29) as a variant of (26), with the role of the function θ now being played by the product divisor sum ${\lambda}_{{F}_{k}}{\lambda}_{{G}_{k}}$, and to repeat the arguments in the ‘The ElliottHalberstam case’ section. To do this, we rely on Proposition 14 to restrict n+h_{ i } to the almost primes.
We turn to the details. Let ε>0 be an arbitrary fixed quantity. From (49) and CauchySchwarz, one has
with the implied constant uniform in ε, so by the triangle inequality and a limiting argument as ε→0, it suffices to show that
where c_{ ε } is a quantity depending on ε but not on x, such that
We use (16) to expand out ${\lambda}_{{F}_{i}},{\lambda}_{{G}_{i}}$ for i=1,…,k−1, but not for i=k, so that the lefthand side of (29) becomes
where
As before, the summand in (54) vanishes unless the modulus^{d}${q}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ defined in (44) is squarefree, in which case we have the analogue
of (45). Here we have put $q={q}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ and $a={a}_{W,{d}_{1},\dots ,{d}_{k1}^{\prime}}$ for convenience. We thus split
where,