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

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## 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 Elliott-Halberstam 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 Elliott-Halberstam 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 sieve-theoretic 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 Elliott-Halberstam conjecture. We also obtain explicit upper bounds for *H*_{m} when *m*=2,3,4,5.

### Keywords

Selberg sieve Elliott-Halberstam conjecture Prime gaps## Background

*m*, let

*H*

_{m}denote the quantity

*p*

_{n}denotes the

*n*th prime. The twin prime conjecture asserts that

*H*

_{1}=2; more generally, the Hardy-Littlewood 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

*k*-tuple (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* log*m* 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 Elliott-Halberstam conjecture EH[ *𝜗*] (see Claim 8 below) concerning the distribution of the prime numbers in arithmetic progressions:

**Theorem****1**(GPY theorem).

Assume the Elliott-Halberstam 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 Bombieri-Vinogradov 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.

*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 Bombieri-Vinogradov 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 Elliott-Halberstam 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 Elliott-Halberstam 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 Elliott-Halberstam 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 sieve-theoretic 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*_{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 sieve-theoretic 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 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 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 Elliott-Halberstam 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)* (near-miss 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 Elliott-Halberstam 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

*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.)

**Results used to prove various components of Theorem 16**

Theorem 16 | Results used |
---|---|

(i) | Theorems 9, 26, and 27 |

(ii)-(vi) | Theorems 11, 24, and 25 |

(vii)-(xi) | Theorems 22 and 23 |

(xii) | Theorems 28 and 29 |

(xiii) | Theorems 26 and 27 |

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 sieve-theoretic 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 left-hand 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 $\sum _{|n|\le \mathit{\text{CN}}}|\alpha (n)|\u2abb\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\u2abbN$.

*q*and

*a*are integers, we write

*a*|

*q*if

*a*divides

*q*. If

*q*is a natural number and $a\in \mathbb{Z}$, we use

*a*(

*q*) to denote the residue class

*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 $\mathbb{Z}/\mathrm{q\mathbb{Z}}$.

- (i)denotes the Euler totient function of$\phi (q):=|{(\mathbb{Z}/\mathrm{q\mathbb{Z}})}^{\times}|$
*q*. - (ii)denotes the divisor function of$\tau (q):=\sum _{d|q}1$
*q*. - (iii)
*Λ*(*q*) denotes the von Mangoldt function of*q*; thus,*Λ*(*q*)= log*p*if*q*is a power of a prime*p*, and*Λ*(*q*)=0 otherwise. - (iv)
*θ*(*q*) is defined to equal log*q*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).

*divisor bound*

*n*≪

*x*

^{O(1)}, as well as the related estimate

for any fixed *C*>0 (see, e.g. [Lemma 1.5]).

*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 Goldston-Pintz-Yıldırım 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).

*α*is non-zero 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**(Elliott-Halberstam conjecture, EH[ *𝜗*]).

*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**(Bombieri-Vinogradov 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**(Motohashi-Pintz-Zhang estimate, MPZ[ *ϖ*,*δ*]).

*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 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*≥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 non-trivial 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 600*ϖ*+180*δ*<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 Elliott-Halberstam conjecture:

**Claim****12**(Generalized Elliott-Halberstam conjecture, GEH[ *𝜗*]).

*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*,2

*N*] and [

*M*,2

*M*], respectively, such that one has the pointwise bound

*n*,

*m*. Suppose also that

*β*obeys the Siegel-Walfisz type bound

*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 Elliott-Halberstam conjecture:

**Proposition****13**.

For any fixed 0<*𝜗*<1, GEH[ *𝜗*] implies EH[ *𝜗*].

*Proof*.

*A*>0 be fixed. For

*n*∈[

*x*,2

*x*], we have Vaughan’s identity

^{c}[29]

*L*(

*n*):= log(

*n*) and

By decomposing each of the functions *μ*_{<}, *μ*_{≥}, 1, *Λ*_{<}, *Λ*_{≥} into *O*(log*A*+1*x*) functions supported on intervals of the form [ *N*,(1+ log−*A**x*)*N*], and discarding those contributions which meet the boundary of [ *x*,2*x*] (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 Siegel-Walfisz theorem; see, e.g. [32], Satz 4] or [33], Th. 5.29]).

By modifying the proof of the Bombieri-Vinogradov theorem, Motohashi [34] established the following generalization of that theorem:

**Theorem****14**(Generalized Bombieri-Vinogradov theorem).

[34] GEH[ *𝜗*] holds for every fixed 0<*𝜗*<1/2.

One could similarly describe a generalization of the Motohashi-Pintz-Zhang 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 Dickson-Hardy-Littlewood prime tuples conjecture [1], which (following Pintz [35]) we refer to as [ *k*,*j*]. Recall that for any $k\in \mathbb{N}$, an *admissible k-tuple* 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 3-tuple, but (0,2,4) is not.

For any *k*≥*j*≥2, we let DHL[ *k*;*j*] denote the following claim:

**Claim****15**(Weak Dickson-Hardy-Littlewood conjecture, DHL[ *k*;*j*]).

For any admissible *k*-tuple $\mathcal{\mathscr{H}}=({h}_{1},\dots ,{h}_{k})$, 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 Dickson-Hardy-Littlewood 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 log*k*.

For any *k*, let *H*(*k*) denote the minimal diameter *h*_{k}−*h*_{1} of an admissible *k*-tuple; 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 Elliott-Halberstam 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(2*m*) for some sufficiently large absolute (and effective) constant *C*.

Assume the generalized Elliott-Halberstam 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* log*k*+*k* log log*k*−*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 Brun-Titchmarsh 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).

*k*-tuple with good properties. More precisely, we set

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).

*k*≥2 and

*m*≥1 be fixed integers and define the normalization constant

*k*-tuple (

*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 non-negative 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

*i*=1,…,

*k*, and the key inequality

Then, DHL[ *k*;*m*+1] holds.

*Proof*.

*h*

_{1},…,

*h*

_{k}) be a fixed admissible

*k*-tuple. 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

From (12) and the crucial condition (15), it follows that *N*>0 if *x* is sufficiently large.

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*,2*x*] 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 *k*-tuple, DHL[ *k*;*m*+1] follows.

*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 ‘one-dimensional’ 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

*x*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).

*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.

*ν*is a (positive-definite) linear combination of functions of the form

*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

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 *k*-tuple, 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)*(Elliott-Halberstam) There exists a fixed 0<

*𝜗*<1 such that EH[

*𝜗*] holds and such that

*(ii)*(Motohashi-Pintz-Zhang) There exists fixed 0≤

*ϖ*<1/4 and

*δ*>0 such that MPZ[

*ϖ*,

*δ*] holds and such that

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 non-prime sums).

Let *k*≥1 be fixed, let (*h*_{1},…,*h*_{k}) be a fixed admissible *k*-tuple, 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 Elliott-Halberstam) There exists a fixed 0<

*𝜗*<1 and

*i*

_{0}∈{1,…,

*k*} such that GEH[

*𝜗*] holds, and

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 Elliott-Halberstam 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*≤2*x*:*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).

*k*≥2 and

*m*≥1 be fixed integers. For any fixed compactly supported square-integrable function $F:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty {)}^{k}\to \mathbb{R}$, define the functionals

*i*=1,…,

*k*, and let

*M*

_{k}be the supremum

*F*that are supported on the simplex

*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}≥ log*k*−*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}≥ log*k*−2 log log*k*−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 ${M}_{k}\le \frac{k}{k-1}logk$ 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 Bombieri-Vinogradov 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).

*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 square-integrable

*F*supported in the

*truncated*simplex

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 600*ϖ*+180*δ*<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 600*ϖ*+180*δ*<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 600*ϖ*+180*δ*<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 600*ϖ*+180*δ*<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 600*ϖ*+180*δ*<7, $\varpi \ge \frac{7}{600}-\frac{C}{logk}$, and ${M}_{k}^{\left[\frac{\delta}{1/4+\varpi}\right]}\ge logk-C$ for some absolute (and effective) constant *C*.

*k*;

*m*+1] holds whenever

*k*is sufficiently large and

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 epsilon-enlarged simplex).

*k*≥2 and

*m*≥1 be fixed integers, and let 0<

*ε*<1 be fixed also. For any fixed compactly supported square-integrable function $F:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty {)}^{k}\to \mathbb{R}$, define the functionals

*i*=1,…,

*k*, and let

*M*

_{k,ε}be the supremum

*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 $\epsilon <\frac{1}{k-1}$.

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 non-strict 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).

*k*≥2 and

*m*≥1 be fixed integers, let 0<

*𝜗*<1 be a fixed quantity such that GEH[

*𝜗*] holds, and let $0<\epsilon <\frac{1}{k-1}$ be fixed also. Suppose that there is a fixed non-zero square-integrable function $F:\phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}0,+\infty {)}^{k}\to \mathbb{R}$ supported in $\frac{k}{k-1}\xb7{\mathcal{R}}_{k}$, such that for

*i*=1,…,

*k*, one has the vanishing marginal condition

*t*

_{1},…,

*t*

_{i−1},

*t*

_{i+1},…,

*t*

_{k}≥0 are such that

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 ‘Three-dimensional cutoffs’ section.

**Theorem****29**(A piecewise polynomial cutoff).

*t*

_{1},

*t*

_{2},

*t*

_{3}variables, such that

*F*is not identically zero and obeys the vanishing marginal condition

*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).

*k*≥1 be a fixed integer, and let

*N*be a natural number coprime to

*W*with log

*N*=

*O*(log

*O*(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,

*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*.

*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

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*).

*K*

_{p}are given by

*ζ*

_{WN}is defined by the formula

for ℜ(*s*)>1.

*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

*s*=1 that

*W*

*N*≪ log

*O*(1)

*x*, this gives

*WN*is composed only of primes

*p*≪ log

*O*(1)

*x*. Thus,

*i*

*ξ*

_{j}replaced by $1+i{\xi}_{j}^{\prime}$ or $2+i{\xi}_{j}+i{\xi}_{j}^{\prime}$. We conclude that

*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

*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 $\frac{1}{p-1}$; 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

*x*is sufficiently large depending on all fixed quantities. By (16), the left-hand side of (29) may be expanded as

*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*,2

*x*] 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

*∞*) 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

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 Elliott-Halberstam case

*i*

_{0}=

*k*, as the other cases are similar. We use (16) to rewrite the left-hand side of (26) as

*W*, and so the summand in (43) vanishes unless the modulus ${q}_{W,{d}_{1},\dots ,{d}_{k-1}^{\prime}}$ defined by

*n*into a single primitive congruence condition

*A*we have

where $a={a}_{W,{d}_{1},\dots ,{d}_{k-1}^{\prime}}$ and $q={q}_{W,{d}_{1},\dots ,{d}_{k-1}^{\prime}}$. For future reference, we note that we may restrict the summation here to those ${d}_{1},\dots ,{d}_{k-1}^{\prime}$ for which ${q}_{W,{d}_{1},\dots ,{d}_{k-1}^{\prime}}$ is square-free.

*q*of ${q}_{W,{d}_{1},\dots ,{d}_{k-1}^{\prime}}$ is associated to

*O*(

*τ*(

*q*)

^{O(1)}) choices of ${d}_{1},\dots ,{d}_{k-1},{d}_{1}^{\prime},\dots ,{d}_{k-1}^{\prime}$. Thus, this contribution is

*C*>0. By the Cauchy-Schwarz inequality, it suffices to show that

*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 Motohashi-Pintz-Zhang case

*x*

^{δ}. Thus, if we set

*I*:= [ 1,

*x*

^{δ}], we see (using the notation from Claim 10) that ${q}_{W,{d}_{1},\dots ,{d}_{k-1}^{\prime}}$ 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 ${a}_{W,{d}_{1},\dots ,{d}_{k-1}^{\prime}}$ lies in the projection of to $\mathbb{Z}/{q}_{W,{d}_{1},\dots ,{d}_{k-1}^{\prime}}\mathbb{Z}$. Each $q\in {\mathcal{S}}_{I}$ is equal to ${q}_{W,{d}_{1},\dots ,{d}_{k-1}^{\prime}}$ for

*O*(

*τ*(

*q*)

^{O(1)}) choices of ${d}_{1},\dots ,{d}_{k-1}^{\prime}$. Thus, the left-hand side of (46) is

*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

*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 Elliott-Halberstam 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).

*k*≥1 be fixed, let (

*h*

_{1},…,

*h*

_{k}) be a fixed admissible

*k*-tuple, 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,

*j*

_{0}≤

*k*is fixed and

*p*

_{0}is a prime with ${p}_{0}\le {x}^{\frac{1}{10k}}$, then we have the variant

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 non-negative expressions involving prime factors of *n*+*h*_{j}, which can then be bounded combinatorially using standard tools.

*d*. Thus,

*O*(1) and a derivative of

*O*(log

*x*

*p*) when ℜ(

*s*)>1, and thus

*f*

_{j}and the triangle inequality, we conclude that

*A*>0. Thus, noting that $\prod _{p|n}O(1)\ll \tau {(n)}^{O(1)}$, we have

*a*

_{j},

*A*. However, we have

*A*>0. In view of this bound and the Fubini-Tonelli theorem, it suffices to show that

*σ*

_{1},…,

*σ*

_{k}≥1. By setting

*σ*:=

*σ*

_{1}+⋯+

*σ*

_{k}, it suffices to show that

for any *σ*≥1.

*n*+

*h*

_{j}as a product

*p*

_{1}≤⋯≤

*p*

_{r}in increasing order and then write

*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

*n*≤2

*x*, this implies that

*Ω*(

*d*

_{j})=

*i*

_{j}denotes the number of prime factors of

*d*

_{j}, counting multiplicity. We also see that

*p*(

*n*) denotes the least prime factor of

*n*. Finally, we have that

*d*

_{1},…,

*d*

_{k},

*W*are coprime. We may thus estimate the left-hand 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*≤2*x* 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*.

*G*be a smooth function supported on [ 0,1] with

*G*(0)=1, and let

*d*=

*d*

_{1}…

*d*

_{k}. We see that

*G*(0)

^{2k}=1 if $p\left(\frac{n+{h}_{j}}{{d}_{j}}\right)\ge R$, and non-negative otherwise. The right-hand side may be expanded as

*dW*, in which case it is

*O*(1) term contributes ⪻ ⪻

*R*

^{k}⪻ ⪻

*x*

^{1/10}, which is negligible. By Lemma 30, if

*Ω*(

*d*)≪ log1/2

*x*, then the main term contributes

*Ω*(

*d*)≫ log1/2

*x*. The bound (50) thus reduces to

*d*

_{j}for an upper bound, we see this is bounded by