# Representing an integer as the sum of a prime and the product of two small
factors

ROGER BAKER and GLYN HARMAN

_Abstract._ Let $\epsilon>0$. We show that every large integer $n$ may be
written in the form

$n=p+ab\,,$

where $a,b\leq n^{\frac{1}{2}-\delta}$ for a positive absolute constant
$\delta$, and $ab\leq n^{0.55+\epsilon}$. This sharpens a result of Heath-
Brown [10]. The improvement depends on a lower bound version of Bombieri’s
theorem in short intervals. In establishing such a result we shall need to
“intersect” two lower bound prime-detecting sieves, and we give a more general
discussion on this point which may have further applications.

MSC (2020): 11N13 (Primary).

##  1 Introduction

Answering a question raised by M. Car, Heath-Brown [10] showed that a large
integer $n$ can be written as

$n=p+ab$ (1.1)

where $p$ is prime and $a,b$ are positive integers,

$\max(a,b)\leq p^{\frac{1}{2}-\delta}$ (1.2)

(where $\delta$ denotes a positive absolute constant) with

$ab\leq n^{\theta}$ (1.3)

where $\theta<1$. In [10] it is shown that $\theta$ may be assigned any value
$>\frac{3}{4}$.

Recently Lü and Ren [13] claimed a sharpening of the above result, requiring
only $\theta>\frac{3}{5}$ in (1.3). Unfortunately, there is a substantial
error in the last paragraph of [13].

In the present paper the following theorem is proved.

######  Theorem 1.1.

Given $\epsilon>0$, every large integer $n$ can be written in the form (1.1)
with $a,b$ positive integers subject to (1.2) and

$ab\leq n^{0.55+\epsilon}.$

The key auxiliary result that we use is given below as Theorem 1.2. We require
some notation. Let $x$ be a large positive number. Write ${\cal L}=\log x$ and
$\rho(n)$ for the indicator function of the primes. Given an arithmetic
function $f(n)$, let

$E_{f}(y,h;q,a)=\sum_{\begin{subarray}{c}{\scriptstyle y-h<n\leq y}\\\
{\scriptstyle n\equiv
a\bmod{q}}\end{subarray}}f(n)\,-\,\frac{h}{\phi(q)h_{0}}\sum_{y-h_{0}<n\leq
y}f(n),$

where $\frac{1}{2}x<y\leq x$ and $h_{0}=x\exp\left(-3{\cal
L}^{\frac{1}{3}}\right)$. Constants implied by the $\ll$ notation will depend
at most on two parameters $A$ and $\epsilon$ unless otherwise indicated. We
write $B$ for a positive absolute constant, not the same at each occurrence.
We comment on one consequence of this convention: when we put a hypothesis
such as $\max(M,N)\ll x^{\theta}{\cal L}^{-2A-B}$ in a result (see (2.3)
below, for example), we mean the value $B$ is chosen to cancel out all ${\cal
L}^{B}$ factors entering in the proof. We write $n\asymp N$ to indicate
$n\in[N/B,BN]$ and $n\sim N$ for $n\in(N/2,N]$.

######  Theorem 1.2.

Let $\theta=0.55+\epsilon$. There is an arithmetic function $\lambda(n)$ with
the following properties.

(i) We have

$\lambda(n)\leq\rho(n)\quad(2\leq n\leq x),$

and

$\text{\rm if}\ \lambda(n)\neq 0\ \ \text{\rm then}\ \ p|n\Rightarrow p\geq
x^{\frac{1}{10}}.$

(ii) If $\frac{1}{2}x\leq y<x$, then

$\sum_{y-h_{0}<n\leq y}\lambda(n)\gg h_{0}{\cal L}^{-1}.$

(iii) Let $A>0$, $Q=x^{\theta-\frac{1}{2}}{\cal L}^{-C}$ where C=C(A).

Then

$\sum_{q\leq Q}\max_{(a,q)=1}\max_{h\leq x^{\theta}}\max_{y\sim
x}\left|E_{\lambda}(y,h;q,a)\right|\ll x^{\theta}{\cal L}^{-A}.$ (1.4)

This result is closely related to work of Huxley and Iwaniec [11], Perelli,
Pintz and Salerno [15, 16], Timofeev [17], Kumchev [12] and Harman, Watt and
Wong [9], but not comparable to any result in these papers.

We require some further notation. The symbol $p$, with or without subscript,
is reserved for primes. Let

$P(z)=\prod_{p<z}p\qquad\text{and}\quad\psi(n,z)=\begin{cases}1&\text{if}\
(n,P(z))=1\\\ 0&\text{otherwise.}\end{cases}$

As usual, we attack Theorem 1.2 by using Dirichlet characters. For a character
$\chi$, let $\delta(\chi)=1$ if $\chi$ is principal and $\delta(\chi)=0$
otherwise. We note that principal characters are not primitive, except in the
trivial case of modulus $1$. All the new work we carry out in this paper only
involves primitive characters with modulus $q\geq 2$, so we will tacitly
assume $\delta(\chi)=0$ in any results we quote from the literature. The
notation

${\sum_{\chi\bmod{q}}\kern-6.0pt}^{*}$

denotes a sum restricted to the primitive characters $\bmod{\,q}$. Let

$E_{f}(y,h;\chi)=\sum_{y-h<k\leq
y}f(k)\chi(k)-\delta(\chi)\frac{h}{h_{0}}\sum_{y-h_{0}<k\leq y}f(k)$

for an arithmetic function $f$. The following lemma reduces the proof of
Theorem 1.2 to establishing a mean value result for $E_{f}(y,h;\chi)$.

######  Lemma 1.1.

Suppose that $f$ satisfies

$\sum_{q\sim Q}{\sum_{\chi\bmod{q}}\kern-6.0pt}^{*}\max_{h\leq
x^{\theta}}\max_{y\sim x}\left|E_{f}(y,h;\chi)\right|\ll Qx^{\theta}{\cal
L}^{-A}$ (1.5)

for all $A>0$ and $Q\leq Q_{0}=Q(A)$. Suppose further that $|f(k)|\leq
d(k)^{B}$ and that

$f(k)=0\ \text{\rm if}\ \ k\ \ \text{\rm has a prime factor}\ <x^{\epsilon}.$
(1.6)

Then, for all $A>0$ and the value $Q_{0}$ as above, we have

$\sum_{q\leq Q_{0}}\max_{(a,q)=1}\max_{h\leq x^{\theta}}\max_{y\sim
x}\left|E_{f}(y,h;q,a)\right|\ll x^{\theta}{\cal L}^{2-A}.$

###### Proof.

This is established in [12, §4.1]. ∎

We already have in [2] results of the type (1.5) suitable for use in Theorem
1.2 when $Q\leq{\cal L}^{A+B}$. We develop similar results for $Q>{\cal
L}^{A+B}$ in sections 2 and 3. To make these results work in tandem, we
develop the idea of intersecting two prime-detecting sieves in sections 4 and
5. As we do this we will prove a result of a more general nature than needed
here which may have other applications. Then, in section 6, we prove Theorem
1.2 using this approach. In section 7 we prove Theorem 1.1 by introducing the
result of Theorem 1.2 into the method of Heath-Brown [10]. We remark that the
idea of intersecting two prime-detecting sieves is particularly useful in the
situation we have here, when a lot of hard work has already been done to get a
strong result for one of the sieves (see [2]).

We would like to thank Andreas Weingartner for computer calculations used in
an earlier version of this paper.

##  2 Mean-Values of Dirichlet Polynomials

Throughout sections 2 and 3 we suppose that

${\cal L}^{A+B}<Q\leq Q_{0}.$ (2.1)

For $N,M,K\geq 1$, a character $\chi(\cdot)$ and a complex variable $s$, we
write

$N(s,\chi)=\sum_{n\asymp N}b_{n}n^{-s}\chi(n),\quad M(s,\chi)=\sum_{m\asymp
M}a_{m}m^{-s}\chi(m),$ $K(s,\chi)=\sum_{k\asymp K}k^{-s}\chi(k)$

for Dirichlet polynomials, where

$a_{m}\ll d(m)^{B},\qquad b_{n}\ll d(n)^{B}.$ (2.2)

For a fixed $Q>{\cal L}^{A+B}$ and $x^{2}\geq T\geq 1$ we write, for $1\leq
u<\infty$,

$||N||_{u}=\left(\sum_{q\sim
Q}{\sum_{\chi\bmod{q}}\kern-6.0pt}^{*}\int_{(T-1)/2}^{T}\left|N\left(\tfrac{1}{2}+it,\chi\right)\right|^{u}dt\right)^{1/u}.$

We also write

$||N||_{\infty}=\sup_{(t,q,\chi)}\left|N\left(\tfrac{1}{2}+it,\chi\right)\right|$

where the supremum is taken over all primitive characters $\chi\pmod{q},q\sim
Q$ and $\frac{1}{2}(T-1)\leq t\leq T$.

It is convenient to write (see Lemma 3.2 below for the application)

$\Psi(T)=\min\left(x^{\theta-\frac{1}{2}},x^{\frac{1}{2}}T^{-1}\right).$

Here, and throughout the rest of the paper, $\theta=0.55+\epsilon$.

######  Lemma 2.1.

We have

$||N||_{2}^{2}\ll\left(N+Q^{2}T\right)\sum_{n\asymp N}|b_{n}|^{2}n^{-1}.$

###### Proof.

See [14, Theorem 7.1]. ∎

######  Lemma 2.2.

Let $M(s,\chi),N(s,\chi)$ be defined as above and $MN\asymp x$. Suppose that

$\max(M,N)\ll x^{\theta}{\cal L}^{-2A-B}.$ (2.3)

Then

$\Psi(T)||MN||_{1}\ll Qx^{\theta}{\cal L}^{-A}.$ (2.4)

###### Proof.

By the Cauchy-Schwarz inequality, Lemma 2.1, and (2.2) we have

$\Psi(T)||MN||_{1}\ll\Psi(T)\left(Q^{2}T+M\right)^{\frac{1}{2}}\left(Q^{2}T+N\right)^{\frac{1}{2}}{\cal
L}^{B}.$ (2.5)

For $0\leq\beta\leq 1$, we observe that

$\max_{T\geq
1}\Psi(T)T^{\beta}=x^{\theta-\frac{1}{2}}\left(x^{1-\theta}\right)^{\beta}.$

Thus, in estimating the right hand side of (2.5) we may suppose that
$T=x^{1-\theta}$. Now

$x^{1-\theta}(MN)^{\frac{1}{2}}{\cal L}^{B}\ll x^{\theta}{\cal L}^{B}\ll
Qx^{\theta}{\cal L}^{-A}$

from (2.1);

$\displaystyle
x^{\theta-\frac{1}{2}}QT^{\frac{1}{2}}\max(M,N)^{\frac{1}{2}}{\cal L}^{B}$
$\displaystyle\ll
x^{\theta-\frac{1}{2}}Qx^{\frac{1}{2}(1-\theta)}x^{\frac{1}{2}\theta}{\cal
L}^{-A-B}{\cal L}^{B}$ $\displaystyle=Qx^{\theta}{\cal L}^{-A};$

and

$\displaystyle x^{\theta-\frac{1}{2}}Q^{2}x^{1-\theta}{\cal L}^{B}$
$\displaystyle\ll x^{\theta-\frac{1}{2}}Qx^{\theta-\frac{1}{2}}{\cal
L}^{-A-B}x^{(1-\theta)}{\cal L}^{B}$ $\displaystyle=Qx^{\theta}{\cal L}^{-A}.$

This gives a satisfactory estimate for the right-hand side of (2.5) and proves
(2.4). ∎

######  Lemma 2.3.

Let $Q\leq x,1\leq T\leq x,K\leq\min(x,QT)$ and suppose $K(s,\chi)$ is as
above. Then

$||K||_{4}^{4}\ll Q^{2}T{\cal L}^{B}+Q^{4}T^{-1}{\cal L}^{B}.$

###### Proof.

This follows from [2, Lemma 9] on summing over $q$. ∎

######  Lemma 2.4.

Let $q\geq 2$ and $\chi$ be a primitive character $\pmod{q}$ with $K(s,\chi)$
as above. Writing $\tau=|t|+2$, we have

$K(\tfrac{1}{2}+it,\chi)\ll(q\tau/K)^{\frac{1}{2}}\log(q\tau).$

###### Proof.

This follows from [4, Theorem 1] by partial summation. ∎

######  Lemma 2.5.

Let $K(s,\chi),M(s,\chi)$ and $N(s,\chi)$ be defined as above. Suppose that
$KMN\asymp x,Q\leq x^{\theta-\frac{1}{2}}{\cal L}^{-A-B}$ and

$M\ll x^{0.55},\quad N\ll x^{0.275},\quad MN\ll x^{0.775}.$ (2.6)

Then

$\Psi(T)||KMN||_{1}\ll Qx^{\theta}{\cal L}^{-A}.$

###### Proof.

For small $Q$ this would follow from [2, Lemma 10] . We give a full proof of
the result here for completeness. We consider two cases.

_Case 1._ $K\leq QT$. We apply Hölder’s inequality, Lemmas 2.1 and 2.3 in
conjunction with (2.2) to give

$\begin{split}\Psi(T)||&KMN||_{1}\leq\Psi(T)||M||_{2}||N||_{4}||K||_{4}\\\
&\ll\Psi(T)\left(Q^{2}T+M\right)^{\frac{1}{2}}\left(Q^{2}T+N^{2}\right)^{\frac{1}{4}}\left(Q^{2}T+Q^{4}T^{-1}\right)^{\frac{1}{4}}{\cal
L}^{B}.\end{split}$ (2.7)

Write $S$ for the sum of all the terms obtained by multiplying out the last
expression, except for $\Psi(T)(MN)^{\frac{1}{2}}QT^{-\frac{1}{4}}$. In
estimating $S$ we may take $T=x^{1-\theta}$. Now

$S\ll{\cal
L}^{B}x^{\theta-\frac{1}{2}}\left(Q^{2}x^{1-\theta}+Qx^{\frac{1}{2}(1-\theta)}M^{\frac{1}{2}}+Q^{\frac{3}{2}}x^{\frac{3}{4}(1-\theta)}N^{\frac{1}{2}}+Q^{\frac{1}{2}}x^{\frac{1}{4}(1-\theta)}(MN)^{\frac{1}{2}}\right).$

Since $Q\leq x^{\theta-\frac{1}{2}}{\cal L}^{-A-B}$, we have

$\displaystyle{\cal L}^{B}x^{\theta-\frac{1}{2}}Q^{2}x^{1-\theta}\ll
Qx^{\theta}{\cal L}^{-A},$ $\displaystyle{\cal
L}^{B}x^{\theta-\frac{1}{2}}Qx^{\frac{1}{2}(1-\theta)}M^{\frac{1}{2}}\ll
Q{\cal L}^{B}x^{\theta-\frac{1}{2}\epsilon},$ $\displaystyle{\cal
L}^{B}x^{\theta-\frac{1}{2}}Q^{\frac{3}{2}}x^{\frac{3}{4}(1-\theta)}N^{\frac{1}{2}}\ll
Q{\cal L}^{B}x^{\theta-\frac{1}{4}\epsilon},$

and

${\cal
L}^{B}x^{\theta-\frac{1}{2}}Q^{\frac{1}{2}}x^{\frac{1}{4}(1-\theta)}(MN)^{\frac{1}{2}}\ll
Q{\cal L}^{B}x^{\theta-\frac{1}{4}\epsilon}.$

This gives a satisfactory estimate for $S$. For the term
$\Psi(T)(MN)^{\frac{1}{2}}QT^{-\frac{1}{4}}$ we use $T\geq 1$:

$\displaystyle\Psi(T)(MN)^{\frac{1}{2}}QT^{-\frac{1}{4}}$ $\displaystyle\ll
x^{\theta-\frac{1}{2}}x^{\frac{1}{4}(1+\theta)}Q$ $\displaystyle\ll
Qx^{\theta}{\cal L}^{-A}$

since $\frac{1}{4}(1+\theta)<\frac{1}{2}$. This covers Case 1.

_Case 2._ $K>QT$. From Lemma 2.4,

$\displaystyle\Psi(T)||KMN||_{1}$
$\displaystyle\leq\Psi(T)||M||_{2}||N||_{2}||K||_{\infty}$
$\displaystyle\ll\Psi(T)\left(Q^{2}T+M\right)^{\frac{1}{2}}\left(Q^{2}T+N\right)^{\frac{1}{2}}{\cal
L}^{B}.$

Since

$\left(Q^{2}T+N\right)^{\frac{1}{2}}\ll\left(Q^{2}T+N^{2}\right)^{\frac{1}{4}}\left(Q^{2}T\right)^{\frac{1}{4}}$

the result follows as in Case 1. This completes the proof of Lemma 2.5. ∎

##  3 Sieve Estimates

We first state Perron’s formula in a suitable manner.

######  Lemma 3.1.

Let $b>0,T>1$, and write

$E(u)=\begin{cases}0&\text{if}\ 0<u<1\\\ 1&\text{if}\ u>1.\end{cases}$

Then

$\frac{1}{2\pi
i}\int_{b-iT}^{b+iT}\frac{u^{s}}{s}\,ds=E(u)+O\left(\frac{u^{b}}{T|\log
u|}\right).$ (3.1)

###### Proof.

This is [8, Lemma A.1] in different notation. ∎

We note that, by standard procedures using the above, we can derive results of
the form (1.5) for certain $f$ using Lemma 2.2 or 2.5. We do this explicitly
for Lemma 2.2 giving the following result.

######  Lemma 3.2.

Suppose $x^{0.45}\leq M\leq x^{0.55}$ and $MN\asymp x$. Write

$f(k)=\sum_{\begin{subarray}{c}{\scriptstyle mn=k}\\\ {\scriptstyle m\asymp
M,n\asymp N}\end{subarray}}a_{m}b_{n}$

with $a_{m},b_{n}$ subject to (2.2). Then (1.5) holds for $f$.

###### Proof.

We first reduce the “edge effects” by noting that we can restrict $y$ and
$y-h$ to values $n+\frac{1}{2},n\in\mathbb{N}$ with a maximum error

$\sum_{q\sim Q}{\sum_{\chi\bmod{q}}\kern-6.0pt}^{*}\max_{n\ll x}\tau(n)^{B}\ll
Q^{2}x^{\epsilon},$

which is certainly a negligible error. It then follows from Lemma 3.1 that

$\displaystyle E_{f}(y,h;\chi)$ $\displaystyle=\sum_{y-h<k\leq y}f(k)\chi(k)$
$\displaystyle=\frac{1}{2\pi
i}\int_{\frac{1}{2}-iT}^{\frac{1}{2}+iT}M(s,\chi)N(s,\chi)\frac{y^{s}-(y-h)^{s}}{s}ds+O\left(V_{y,h}\right),$

where

$V_{y,h}=\sum_{k\asymp
x}\tau(k)^{B}\frac{1}{T\min(|\log(y/k)|,|\log((y-h)/k)|)}\,.$

A simple calculation gives

$\max_{\frac{1}{2}x\leq y\leq x}\,\max_{h\leq
x^{\theta}}V_{y,h}\ll\frac{x^{1+\frac{1}{2}\epsilon}}{T}\,.$

If we take $T=x$ the error is again negligible from the above.

Next we note that

$\frac{y^{s}-(y-h)^{s}}{s}=\int_{y-h}^{y}u^{s-1}du\ll hx^{-\frac{1}{2}},$

and

$\frac{y^{s}-(y-h)^{s}}{s}\ll\frac{x^{\frac{1}{2}}}{|s|}.$

It follows that, for $t\sim T$, we have

$\frac{y^{s}-(y-h)^{s}}{s}\ll\Psi(T).$

This demonstrates the significance of our choice for $\Psi(T)$. It therefore
remains to prove that, with $s=\frac{1}{2}+it$,

$\sum_{q\sim Q}{\sum_{\chi\bmod{q}}\kern-6.0pt}^{*}\max_{1\leq T\leq
x}\Psi(T){\cal L}\int_{\frac{1}{2}(T-1)}^{T}|M(s,\chi)N(s,\chi)|dt\ll
Qx^{\theta}{\cal L}^{-A}.$

This follows immediately from Lemma 2.2 and completes the proof. ∎

The result we need to cover the case $Q>{\cal L}^{A+B}$ is as follows. The
result essentially follows for $Q\leq{\cal L}^{A+B}$ by the work done in [2].
The result for $q=1$ goes back in principle at least as far as [1] (see Lemma
4 and following there). The proof of the result we need is actually simpler in
principle since, in the case $Q>{\cal L}^{A+B}$, we use Lemma 2.2 which is a
straightforward Type II estimate that does not require one of

$||M||_{\infty}\ll M^{\frac{1}{2}}{\cal L}^{-A}\quad\text{or}\
||N||_{\infty}\ll N^{\frac{1}{2}}{\cal L}^{-A}$

to hold.

######  Lemma 3.3.

Suppose $M,N$ satisfy (2.6). Write

$f(k)=\sum_{\begin{subarray}{c}{\scriptstyle mn\ell=k}\\\ {\scriptstyle
m\asymp M,n\asymp N}\end{subarray}}a_{m}b_{n}\psi(\ell,x^{0.1})$

with $a_{m},b_{n}$ subject to (2.2). Then (1.5) holds for $f$.

###### Proof.

We could say: use the results of Lemmas 2.2 and 2.5 in the method of proof of
[2, §4], also given in [8, §10.5]. The value $x^{0.1}$ comes from the “width”
of the Type sum in Lemma 2.2 (that is $x^{0.45}$ to $x^{055}$), and Lemma 2.5
is exactly analogous to the corresponding results in [1, 2, 8]. By
“analogous”, we mean here that our results are, as expected by the nature of
the problem, a factor $Q$ larger than those in the previously cited works.
However, as noted above, we are in a simpler situation (the preliminary proof
of the result with $x^{0.1}$ replaced by $\exp\left({\cal L}^{0.9}\right)$ is
unnecessary). We could therefore prove the result using a slight modification
to [8, Theorem 3.1]. ∎

By combining the two above lemmas it is possible to construct a lower bound
prime detecting sieve using the method first delineated in [6] and developed
in [7], with many developments chronicled in [8]. This will be done explicitly
in the following sections once we have outlined the general method and how to
“intersect” one sieve with another. The crucial additional result needed to
construct the sieve is Buchstab’s identity, which in the form we require,
states that, for any $2\leq w<z$, we have

$\psi(k,z)=\psi(k,w)-\sum_{\begin{subarray}{c}{\scriptstyle k=p\ell}\\\
{\scriptstyle w\leq p<z}\end{subarray}}\psi(\ell,p)\,.$

##  4 Some Further Notation

Given an arithmetic function $f(n)$ we write $\mathbb{f}$ to mean the sequence
$f(k),k\in[\tfrac{1}{2}x,x)\cap\mathbb{N}$. So, for example,
$\boldsymbol{\lambda}$ means the sequence $\lambda(k),\frac{1}{2}x\leq k<x$.
We recall the definition of a _sublinear functional_ ${\cal M}$ defined over
the set of all real sequences. That is, given sequences
$\mathbb{w},\mathbb{v}$, we have

${\cal M}(c\mathbb{w})=|c|{\cal M}(\mathbb{w})\ \ (c\in\mathbb{R}),\qquad{\cal
M}(\mathbb{w}+\mathbb{v})\leq{\cal M}(\mathbb{w})+{\cal M}(\mathbb{v})\,.$

We then define a sequence $\bf w$ to be _${\cal M}$ -regular_ if it satisfies

${\cal M}(\mathbb{w})\ll 1.$

Here we assume the parameter $x\rightarrow\infty$ and the implied constant is
independent of $x$. For a set $\Phi$ of sublinear functionals, a sequence
$\mathbb{w}$ is said to be _$\Phi$ -regular_ if $\mathbb{w}$ is ${\cal
M}$-regular for each ${\cal M}\in\Phi$.

Example 1. Let $A>0$; let $Q_{0}(A)=x^{\theta-\frac{1}{2}}{\cal
L}^{-A-B},\eta=\eta(Q)=x^{-\theta}Q^{-1}{\cal L}^{A}$ and

${\cal M}_{Q}({\bf w})=\eta\sum_{q\sim
Q}{\sum_{\chi\bmod{q}}\kern-6.0pt}^{*}\max_{h\leq x^{\theta}}\,\max_{y\sim
x}\left|E_{w}(y,h,\chi)\right|\,.$

We put

$\Phi_{1}=\left\\{{\cal M}_{Q}:{\cal L}^{A+B}\leq Q\leq
Q_{0}\right\\}\,,\quad\Phi_{2}=\left\\{{\cal M}_{Q}:1\leq Q\leq{\cal
L}^{A+B}\right\\}\,.$

Then (1.4) of Theorem 2 is the requirement that $\boldsymbol{\lambda}$ be both
$\Phi_{1}$-regular and $\Phi_{2}$-regular.

In the following we write $\mathcal{B}$ for the set of integers in the
interval $(y-h_{0},y)$ for some $y\sim x$, Example 2. Let ${\cal
A}\subset\mathcal{B}$ and let $\sigma$ be an approximation to $|{\cal
A}||\mathcal{B}|^{-1}$. We wish to show that ${\cal A}$ contains primes. Let

${\cal M}({\bf w})=\frac{{\cal
L}^{2}}{\sigma|\mathcal{B}|}\left|\sum_{k\in{\cal
A}}w(k)-\sigma\sum_{k\in\mathcal{B}}w(k)\right|\,.$

Suppose an ${\cal M}$-regular sequence $\boldsymbol{\lambda}$ satisfies
$\lambda(k)\leq\rho(k)$ and

$\sum_{k\in\mathcal{B}}\lambda(k)\gg\sum_{k\in\mathcal{B}}\rho(k)\,.$ (4.1)

Then we have

$\left|\sum_{k\in{\cal
A}}\lambda(k)-\sigma\sum_{k\in\mathcal{B}}\lambda(k)\,\right|\ll\frac{\sigma|\mathcal{B}|}{{\cal
L}^{2}},$

and, from the prime number theorem for the slightly shortened interval
$(y-h_{0},y)$,

$\sigma\sum_{k\in\mathcal{B}}\lambda(k)\gg\frac{\sigma|\mathcal{B}|}{{\cal
L}}\,.$

Consequently

$\sum_{k\in{\cal A}}\rho(k)\geq\sum_{k\in{\cal
A}}\lambda(k)\gg\sigma|\mathcal{B}|{\cal L}^{-1}\,,$

and so ${\cal A}$ contains primes.

By the sublinearity definition, we can thus establish lower bounds for the
number of primes in sets by constructing $\boldsymbol{\lambda}$ as above using
a linear combination of a bounded number of regular sequences. In particular,
to prove Theorem 1.2, we need to show that the $\boldsymbol{\lambda}$ we
construct is a combination of sequences that are regular for both $\Phi_{1}$
and $\Phi_{2}$. We now briefly sketch the process developed in [6, 7, 8] for
constructing a lower bound sieve $\boldsymbol{\lambda}$. Let us only consider
$k>\frac{1}{2}x$. Then, for any value $z_{0}$,

$\rho(k)=\psi(k,x^{\frac{1}{2}})=\psi(k,z_{0})-\sum_{\begin{subarray}{c}{\scriptstyle
z_{0}\leq p<x^{\frac{1}{2}}}\\\ {\scriptstyle
k=p\ell}\end{subarray}}\psi(\ell,p)\,,$

using Buchstab’s identity. We choose $z_{0}$ to be the largest value with
$\psi(k,z_{0})$ being regular. Now we may apply Buchstab’s identity a bounded
number of times – indeed we want an _even_ (say $K$) number of times – to give

$\rho(k)=\psi(k,z_{0})-f_{1}(k)-f_{3}(k)-\ldots-
f_{K-1}(k)+g_{2}(k)+\ldots+g_{K}(k)$

for certain functions $f_{j},g_{j}$ defined as sums of the $\psi$ function
over $j+1$ variables which multiply together to give $k$. By their definition,
these functions are non-negative. Now, suppose each $f_{j}(k)$ is regular, and
we split $g_{j}$ into a regular term and a (possibly non-existent) term which
is not regular: $g_{j}(k)=r_{j}(k)+s_{j}(k)$, say. Then the function

$\lambda(k)=\psi(k,z_{0})-f_{1}(k)-f_{3}(k)-\ldots-
f_{K-1}(k)+r_{2}(k)+\ldots+r_{K}(k)$ (4.2)

is certainly regular by the sublinearity definition. Since we have only
discarded non-negative terms, we have $\lambda(k)\leq\rho(k)$. The challenge
that remains is to carry out the above process in such a way that (4.1) holds.
When successful, we shall call $\boldsymbol{\lambda}$ a _non-trivial lower
bound prime-detecting sieve_ for whatever ${\cal M}$ or family of ${\cal M}$
is under consideration. Clearly our ultimate challenge is to find such a
$\boldsymbol{\lambda}$ for Example 1, regular for both families $\Phi_{1}$ and
$\Phi_{2}$ defined there.

Given a sublinear functional ${\cal M}$ (or family $\Phi$ of such), we call
the process leading to (4.2) a _permissible decomposition for ${\cal M}$
(respectively, $\Phi$)_. We write ${\mathfrak{D}}$ with or without a subscript
for a permissible decomposition. Now, if (4.1) holds with
$\lambda(k)\leq\rho(k)$, then clearly

$\sum_{k\in\mathcal{B}}\lambda(k)=(1-d)\sum_{k\in\mathcal{B}}\rho(k)\,$

for some $d\in(0,1)$. By what we have written so far, $d$ might vary somewhat
with $x$, but we shall show later that in practice $d=\delta+o(1)$ for a fixed
$\delta\in(0,1)$. We write this as $\delta({\mathfrak{D}})$ to show that it
depends essentially on the decomposition process (equivalently we could have
defined it as $\delta(\boldsymbol{\lambda})$).

##  5 Intersecting Two Prime-Detecting Sieves

When constructing a prime detecting sieve as outlined in [8, Chapters 3,
5–11], one first garners the necessary arithmetical information (what is
commonly know as Type I & II information, although there are sub-varieties of
these types that appear as well), and then one applies Buchstab’s identity to
decompose a sum over primes in a given set, ${\cal A}$, into multiple sums.
Alternatively, and equivalently, as in the previous section we use Buchstab’s
identity applied to $\rho(k)$ summed over the set of interest, which often
means we weight $\rho(k)$ with auxiliary functions like $k^{-s},\chi(k)$ or
$e(\alpha\ell k)$.

Suppose $K$ is an even integer, and $\zeta>0$ (but not arbitrarily small) is
given. Write $\xi=x^{\zeta}$. For $j\leq K<\zeta^{-1}$ write

$E_{j}=\\{\boldsymbol{\alpha}_{j}:\alpha_{i}\geq\zeta\ (1\leq i\leq
j),\alpha_{1}+\ldots+\alpha_{j}\leq 1\\}.$

We write $\mathbf{p}_{j}=(p_{1},\ldots,p_{j})$ where
$p_{i}=x^{\alpha_{i}},1\leq i\leq j,\boldsymbol{\alpha}_{j}\in E_{j}$, and

$\Pi_{j}=\prod_{1\leq i\leq j}p_{j}\,.$

Given a Jordan measurable subset ${\cal F}$ of $E_{j}$ we may thus write

$\sum_{\boldsymbol{\alpha}_{j}\in{\cal F}}$

to indicate the summation ranges for $\mathbf{p}_{j}$. In practice, ${\cal F}$
will always be the interior of a polyhedron (or union of finitely many
polyhedrons). When we apply Buchstab’s identity in a straightforward way we
will have summation ranges satisfying

$p_{j}<p_{j-1}<\ldots<p_{1},\qquad{p_{r}}^{2}(p_{1}\ldots p_{r-1})<x,\
\text{for}\ \ 2\leq r\leq j.$

We write $\mathcal{H}_{j}$ for the set of all possible values of
$\boldsymbol{\alpha}_{j}$ in this case.

When we sketched out a decomposition in the previous section we assumed we
would always decompose the “inner variable” counted with the $\psi$ notation
using Buchstab’s identity. This is not always the most efficient course of
action. Instead one can apply what is known as a rôle-reversal. We describe
the simplest case of this technique (see also [8, §3.5]). Suppose ${\cal F}$
is a Jordan measurable region of $\mathcal{H}_{2}$ and suppose we have a term
to consider of the form

$w(k)=\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{2}\in{\cal
F}}\\\ {\scriptstyle k=\Pi_{2}\ell}\end{subarray}}\psi(\ell,p_{2})\,.$

Instead of applying Buchstab’s identity to $\psi(\ell,p_{2})$, we rewrite
${\bf w}$ as

$w(k)=\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{2}\in{\cal
F}}\\\ {\scriptstyle
k=n_{1}p_{2}\ell}\end{subarray}}\psi(\ell,p_{2})\psi\left(n_{1},\left(\frac{x}{\ell
p_{2}}\right)^{\frac{1}{2}}\right)$ (5.1)

where $n_{1}=x^{\alpha_{1}}$, and apply Buchstab’s identity to

$\psi\left(n_{1},\left(\frac{x}{\ell p_{2}}\right)^{\frac{1}{2}}\right)\,.$

Clearly one could alternatively have applied this process to decompose $p_{2}$
as well or instead of $p_{1}$. The possibilities become rather complicated at
this point (consider the working in [2, pp.32-41]). We will first set up the
sieve machinery in the case where there are no rôle-reversals. We shall need
to take these into account later to prove our main theorems, but the general
result we prove initially is not valid when rôle-reversals are used. For the
purposes of clarity it may help the reader to see the simplest case first.

Our first task is to describe the decomposition process in a way that will
help develop the method for intersecting two sieves. Let $\Phi$ be a set of
sublinear functionals, and our desire is to construct $\boldsymbol{\lambda}$
which is $\Phi$-regular. Now, given any Jordan-measurable set $R_{j}\subset
E_{j}$ we write $R_{j}^{*}$ for the set of all $\boldsymbol{\alpha}_{t}\in
E_{t},t\geq j$ whose coordinates can be partitioned into $j$ sets (none of
which is empty) such that if we form the vector $\boldsymbol{\alpha}_{j}$ by
summing the $\alpha_{i}$ in the $j$ sets, then $\boldsymbol{\alpha}_{j}\in
R_{j}$. Suppose there are sets $R_{j}$ and functions
$z(\boldsymbol{\alpha}_{j})\geq\xi$ such that if we write

${\cal R}=\bigcup_{j}R_{j}^{*}$

then the sequence

$v_{k}=\sum_{\begin{subarray}{c}{\scriptstyle k=\Pi_{t}\ell}\\\
{\scriptstyle\boldsymbol{\alpha}_{t}\in V_{t}\cap{\cal
R}}\end{subarray}}\psi(\ell,z^{\prime}(\boldsymbol{\alpha}_{t}))$

is $\Phi$-regular for every Jordan-measurable subset $V_{t}\subset E_{t}$ and
all functions $z^{\prime}(\boldsymbol{\alpha}_{t})\leq
z(\boldsymbol{\alpha}_{t})$. If $U_{j}$ is a Jordan-measurable subset of
$E_{j}\cap{\cal R}$ then we say $U_{j}$ is a _Type I $\Phi$-regular domain_.
Let $U_{j}$ be such a Type I $\Phi$-regular domain with associated
$z(\boldsymbol{\alpha}_{j})$. Write

$w_{k}=\sum_{\begin{subarray}{c}{\scriptstyle k=\Pi_{j}\ell p}\\\
{\scriptstyle\boldsymbol{\alpha}_{j}\in
U_{j}}\end{subarray}}\sum_{z^{\prime}(\boldsymbol{\alpha}_{j})\leq
p<z(\boldsymbol{\alpha}_{j})}\psi(\ell,p)\,.$

Then, by Buchstab’s identity, for any
$z^{\prime}(\boldsymbol{\alpha}_{j})<z(\boldsymbol{\alpha}_{j})$ we have

$w_{k}=\sum_{\begin{subarray}{c}{\scriptstyle k=\Pi_{j}\ell}\\\
{\scriptstyle\boldsymbol{\alpha}_{j}\in
U_{j}}\end{subarray}}\psi(\ell,z(\boldsymbol{\alpha}_{j}))-\sum_{\begin{subarray}{c}{\scriptstyle
k=\Pi_{j}\ell}\\\ {\scriptstyle\boldsymbol{\alpha}_{j}\in
U_{j}}\end{subarray}}\psi(\ell,z^{\prime}(\boldsymbol{\alpha}_{j}))\,,$

and so ${\bf w}$ is $\Phi$-regular. For future reference we call this
_Property ${\cal R}$_.

Results such as these usually follow from what the second-named author has
called _The Fundamental Theorem_ : see [8, Theorems 3.1, 5.2]. By the method
of proof, if we get regularity with values $z(\boldsymbol{\alpha}_{t})$, we
also get regularity for any
$z^{\prime}(\boldsymbol{\alpha}_{t})\in[2,z(\boldsymbol{\alpha}_{t})]$. In our
present context, for $\Phi_{1}$-regularity, Lemma 3.3 gives

$R_{2}=\\{\boldsymbol{\alpha}_{2}\in E_{2}:\alpha_{1}\leq 0.55,\
\alpha_{2}\leq 0.275,\ \alpha_{1}+\alpha_{2}\leq 0.775\\},$

with $z(\boldsymbol{\alpha}_{j})\equiv x^{\frac{1}{10}}$. We would thus take
$\zeta={\frac{1}{10}}$ in this case. We write, in general,
$z(\boldsymbol{\alpha}_{j})=x^{\zeta(\boldsymbol{\alpha}_{j})}$, and note that
$\zeta(\boldsymbol{\alpha}_{j})$ will be piece-wise linear in each coordinate
by the methods used (for example, see [8, p.153, Diagram 7.3]) and $\zeta$
would be the infimum of all $\zeta(\boldsymbol{\alpha}_{j})$.

Now, if we use rôle-reversals, we might have in our sums instead of $p_{i}$ a
“sieved-variable” $n_{i}$ all of whose prime factors exceed
$g(\boldsymbol{\alpha}_{t})=x^{\gamma(\boldsymbol{\alpha}_{t})}$ where
$\gamma(\cdot)$ is piecewise linear in each variable. That is we are counting
$n_{i}$ with a weight $\psi(n_{i},g(\boldsymbol{\alpha}_{t}))$. We can do this
by breaking up the sum into multiple sums over appropriate domains. We remark
that we can replace as many prime variables $p_{i}$ as we wish by this
process. We illustrate this by considering (5.1). The variable $\ell$ there
has all its prime factors exceeding $p_{2}$ so, for some integer
$t<\zeta^{-1}$,

$w(k)=\sum_{3\leq h\leq
t}\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{2}\in{\cal F}}\\\
{\scriptstyle k=n_{1}p_{2}p_{3}\cdots
p_{h}}\end{subarray}}\psi\left(n_{1},\left(\frac{x}{p_{2}p_{3}\cdots
p_{h}}\right)^{\frac{1}{2}}\right)$

Here the variables $p_{j},j\geq 3$ satisfy $p_{h}>p_{h-1}>\ldots>p_{3}>p_{2}$.
We can rewrite $w(k)$ then as

$w(k)=\sum_{3\leq h\leq
t}\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{h}\in{\cal
F}_{h}}\\\ {\scriptstyle k=np_{2}p_{3}\cdots
p_{h}}\end{subarray}}\psi\left(n,\left(\frac{x}{p_{2}p_{3}\cdots
p_{h}}\right)^{\frac{1}{2}}\right)$

for certain regions ${\cal F}_{h}\in E_{h}$. For example,

${\cal
F}_{2}=\\{\boldsymbol{\alpha}_{2}:(1-\alpha_{1}-\alpha_{2},\alpha_{1})\in{\cal
F},\alpha_{2}>\alpha_{1}\\}\,,$

and

${\cal
F}_{3}=\\{\boldsymbol{\alpha}_{3}:(1-\alpha_{1}-\alpha_{2}-\alpha_{3},\alpha_{1})\in{\cal
F},\alpha_{3}>\alpha_{2}>\alpha_{1}\\}\,.$

Similarly, but with one crucial difference, given any Jordan-measurable set
$S_{j}\subset E_{j}$ we define $S_{j}^{*}$ to be the set of all
$\boldsymbol{\alpha}_{t}\in E_{t},t\geq j$ whose coordinates can be
partitioned into $j+1$ sets (at most one of which is empty) such that if we
form the vector $\boldsymbol{\alpha}_{j}$ by summing the $\alpha_{i}$ in $j$
of the (non-empty) sets, then $\boldsymbol{\alpha}_{j}\in S_{j}$. Suppose
there are sets $S_{j}$ such that if we write

${\cal S}=\bigcup_{j}S_{j}^{*}$

then the sequence

$v_{k}=\sum_{\begin{subarray}{c}{\scriptstyle k=\Pi_{t}}\\\
{\scriptstyle\boldsymbol{\alpha}_{t}\in V_{t}\cap{\cal S}}\end{subarray}}1$

is $\Phi$-regular for every Jordan-measurable subset $V_{t}\subset E_{t}$. If
$U_{j}$ is a Jordan-measurable subset of $E_{j}\cap{\cal S}$ then we say
$U_{j}$ is a _Type II $\Phi$-regular domain_. This corresponds to Type II (and
possible more complicated variants) information in the usual description of
this sieve method. For $\Phi_{1}$ we have just the one set $S_{1}=[0.45,0.55]$
by Lemma 3.2. So, in this case, ${\cal S}$ is comprised of all regions in $j$
dimensions, $1\leq j\leq\zeta^{-1}$, contained in $E_{j}$ where a sum of a
subset of the coordinates lies in $S_{1}$. This is just the union of a finite
number of polyhedra. It follows from our definition of ${\cal S}$ that the
sequence

$w_{k}=\sum_{\begin{subarray}{c}{\scriptstyle k=n_{1}\cdots n_{t}}\\\
{\scriptstyle\boldsymbol{\alpha}_{t}\in V_{t}\cap{\cal
S}}\end{subarray}}c_{1}(n_{1})\cdots c_{t}(n_{t})$

is $\Phi$-regular for any Jordan-measurable subset $V_{t}\subset E_{t}$, where
each $c_{j}(n)$ is either $\rho(n)$ or $\psi(n,g(\boldsymbol{\alpha}_{t}))$
for some function
$g(\boldsymbol{\alpha}_{t})=x^{\gamma(\boldsymbol{\alpha}_{t})}$ where
$\gamma(\cdot)$ is piecewise linear in each variable. Let us illustrate this
with the simplest case. Suppose we investigate

$\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{j}\in
V_{j}\cap{\cal S}}\\\ {\scriptstyle
k=\Pi_{j}\ell}\end{subarray}}\psi(\ell,g(\boldsymbol{\alpha}_{j})).$

This can be written as the sum of $<\zeta^{-1}$ sequences corresponding to
$k=\Pi_{j}p_{j+1}\cdots p_{t}$. Each sequence is $\Phi$-regular from the
definition of ${\cal S}$, because $\boldsymbol{\alpha}_{t}$ belongs to a
Jordan-measurable region (guaranteed by $\boldsymbol{\alpha}_{j}\in V_{j}$ and
the subsequent coordinates only have piecewise linear restrictions).

Consider now how the decomposition is framed to construct a lower bound sieve.
We always start with

$\rho(k)=\psi(k,z_{0})-\sum_{\begin{subarray}{c}{\scriptstyle z_{0}\leq
p_{1}<x^{\frac{1}{2}}}\\\ {\scriptstyle
k=p_{1}\ell}\end{subarray}}\psi(\ell,z(\alpha_{1}))+\sum_{\begin{subarray}{c}{\scriptstyle
z(\alpha_{1})\leq p_{2}<p_{1}<x^{\frac{1}{2}}}\\\ {\scriptstyle
k=\Pi_{2}\ell}\end{subarray}}\psi(\ell,p_{2}).$ (5.2)

The $z$ values are usually chosen to be the largest such that the first two
sequences on the right hand side above are $\Phi$-regular, and, writing
$z_{0}=x^{\zeta_{0}}$, we assume $[\zeta_{0},\tfrac{1}{2}]$ is a Type I
$\Phi$-regular domain. As above, we let $\xi$ be the minimum taken by any $z$
value. We write the second term on the right of (5.2) as $f_{1}(k)$ to conform
with (4.2). We note that the final sum, which for future reference we shall
call $\Sigma_{2}(k)$, has no terms with $p_{2}^{2}p_{1}>x$ (this corresponds
to the condition $\boldsymbol{\alpha}_{2}\in\mathcal{H}_{2}$). Let ${\cal
F}\subset\mathcal{H}_{2}$ correspond to the values of
$\boldsymbol{\alpha}_{2}$ in the final sum above. Let us write ${\cal
F}_{1}={\cal F}\cap{\cal S}$. So

$v_{2}(k)=\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{2}\in{\cal
F}_{1}}\\\ {\scriptstyle k=\Pi_{2}\ell}\end{subarray}}\psi(\ell,p_{2})$

is $\Phi$-regular. Now let ${\cal G}={\cal F}\setminus{\cal F}_{1}$, and write
${\cal F}_{3}$ for the subset of ${\cal G}$ for which two further applications
of Buchstab’s identity are permissible, that is

${\cal F}_{3}=\left\\{\boldsymbol{\alpha}_{2}\in{\cal G}\cap{\cal
R}:\\{\boldsymbol{\alpha}_{3}\in\mathcal{H}_{3}:\zeta(\boldsymbol{\alpha}_{2})\leq\alpha_{3}<\alpha_{2}\\}\subset{\cal
R}\right\\}\,.$

We have thus written

$\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{2}\in{\cal
F}_{3}}\\\ {\scriptstyle
k=\Pi_{2}\ell}\end{subarray}}\psi(\ell,p_{2})=\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{2}\in{\cal
F}_{3}}\\\ {\scriptstyle
k=\Pi_{2}\ell}\end{subarray}}\psi(\ell,z(\boldsymbol{\alpha}_{2}))-\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{2}\in{\cal
F}_{3}}\\\
{\scriptstyle\zeta(\boldsymbol{\alpha}_{2})\leq\alpha_{3}<\alpha_{2}}\\\
{\scriptstyle
k=\Pi_{3}\ell}\end{subarray}}\psi(\ell,z(\boldsymbol{\alpha}_{3}))+\Sigma_{4}(k)\,,$

say, where the first two sequences on the right hand side above are
$\Phi$-regular and we call the second of these $f_{3}(k)$.

Finally, we write ${\cal F}_{2}={\cal F}\setminus\left({\cal F}_{1}\cup{\cal
F}_{3}\right)$. We then have the term

$s_{2}^{\prime}(k)=\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{2}\in{\cal
F}_{2}}\\\ {\scriptstyle k=\Pi_{2}\ell}\end{subarray}}\psi(\ell,p_{2})$

which at first sight we might discard as a positive non-regular term. However,
it is possible to rewrite $s_{2}(k)$ as the combination of sequences written
as sums over two, three or more primes:

$s_{2}^{2}(k)+s_{2}^{3}(k)+s_{2}^{4}(k)+\ldots=\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{2}\in{\cal
F}_{2}}\\\ {\scriptstyle
k=\Pi_{2}p_{3}}\end{subarray}}1+\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{3}\in{\cal
J}_{3}}\\\ {\scriptstyle
k=\Pi_{3}p_{4}}\end{subarray}}1+\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{4}\in{\cal
J}_{4}}\\\ {\scriptstyle k=\Pi_{4}p_{5}}\end{subarray}}1\ldots,$

for certain sets ${\cal J}_{i}$. The sequence $s_{2}^{2}(k)$ must be
discarded. But we can split each ${\cal J}_{i}$ into ${\cal J}_{i}\cap{\cal
S}$ and ${\cal J}_{i}\setminus{\cal S}$ (one or other may be empty, of course)
to obtain another regular sequence and a sequence to be discarded. We may thus
write

$s_{2}^{\prime}(k)=s_{2}(k)+\sigma_{2}(k)$

where $s_{2}(k)$ is discarded and $\sigma_{2}(k)$ is regular. We then have

$r_{2}(k)=v_{2}(k)+\sigma_{2}(k)+\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{2}\in{\cal
F}_{3}}\\\ {\scriptstyle
k=\Pi_{2}\ell}\end{subarray}}\psi(\ell,z(\boldsymbol{\alpha}_{2}))$

in the notation of (4.2). We can present the regions ${\cal F}_{j}$ on a
diagram for $\Phi_{1}$-regularity as follows.

$\alpha_{1}$$\alpha_{2}$${\cal F}_{3}$${\cal F}_{1}$ ${\cal F}_{2}$${\cal
F}_{1}$$0$$0.1$$0.2$$0.25$$0.3$$0.35$$0.4$$0.45$$0.5$$0$$0.1$$0.2$$0.3$$0.35$

Clearly we can analyze $\Sigma_{4}(k)$ as we did $\Sigma_{2}(k)$ and continue
by induction to reach $\Sigma_{K}(k)$. For this last term we split into just
two sums – one corresponding to a region in ${\cal S}$, and the remainder is
discarded. We thus arrive at (4.2) with $\boldsymbol{\lambda}$ the sum of
$\Phi$-regular sequences, and so $\boldsymbol{\lambda}$ itself is
$\Phi$-regular. We have thus defined our $\Phi$-permissible decomposition
${\mathfrak{D}}$ leading to the lower bound sieve $\boldsymbol{\lambda}$ which
is $\Phi$-regular.

Now suppose that ${\cal J}_{i}\cap{\cal S}=\emptyset$ for every $i$ at each
stage of the decomposition (so $s_{j}^{\prime}(k)=s_{j}(k)$). Let ${\cal
F}_{2}(j)$ be the region corresponding to the discarded sum at each stage ($j$
even, $j\leq K$). Let

$\delta_{j}=\int_{{\cal
F}_{2}(j)}\omega\left(\frac{1-\alpha_{1}-\ldots-\alpha_{j}}{\alpha_{j}}\right)\frac{d\alpha_{1}\ldots
d\alpha_{j}}{\alpha_{1}\ldots\alpha_{j-1}\alpha_{j}^{2}}\,,$

and

$\delta({\mathfrak{D}})=\sum_{\begin{subarray}{c}{\scriptstyle j\
\text{even}}\\\ {\scriptstyle j\leq K}\end{subarray}}\delta_{j}.$

As explained in [8, pp 16, 56–62],

$\sum_{k\in\mathcal{B}}\lambda(k)=(1-\delta({\mathfrak{D}})+o(1))\sum_{k\in\mathcal{B}}\rho(k)\,.$

This value $\delta({\mathfrak{D}})$ is independent of $x$ and so we can call
it the _deficit_ of the lower bound sieve $\boldsymbol{\lambda}$. When ${\cal
J}_{i}\cap{\cal S}\neq\emptyset$ we can still define $\delta_{j}$ and hence
$\delta({\mathfrak{D}})$ independent of $x$, but the integrals concerned
become much more complicated. For example, if we write

$\tau(\boldsymbol{\alpha}_{j})=(\alpha_{1}\cdots\alpha_{j}(1-\alpha_{1}-\ldots-\alpha_{j}))^{-1},$

then

$\delta_{2}=\int_{\boldsymbol{\alpha}_{2}\in{\cal
F}_{2}(2)}\tau(\boldsymbol{\alpha}_{2})d\alpha_{2}d\alpha_{1}+\sum_{j\geq
3}\int_{\boldsymbol{\alpha}_{j}\in{\cal J}_{j}\setminus{\cal
S}}\tau(\boldsymbol{\alpha}_{j})d\alpha_{j}\ldots d\alpha_{2}d\alpha_{1}\,.$

We have now given enough notation and explanation to state and prove a general
result on intersecting lower bound prime-detecting sieves.

######  Theorem 5.1.

Let ${\mathfrak{D}}_{1},{\mathfrak{D}}_{2}$ be two decompositions (constructed
as above with no rôle-reversals) which are permissible for the families of
sublinear functionals $\Xi_{1},\Xi_{2}$ respectively. Then we can define a
decomposition ${\mathfrak{D}}_{3}={\mathfrak{D}}_{1}\cap{\mathfrak{D}}_{2}$
which is permissible for $\Xi_{1}\cup\Xi_{2}$, and which satisfies

$\delta({\mathfrak{D}}_{3})\leq\delta({\mathfrak{D}}_{1})+\delta({\mathfrak{D}}_{2}).$
(5.3)

######  Corollary.

Given the conditions of the theorem with
$\delta({\mathfrak{D}}_{1})+\delta({\mathfrak{D}}_{2})<1$, then we can
construct a non-trivial lower bound prime-detecting sieve
$\boldsymbol{\lambda}$ which is both $\Xi_{1}$ and $\Xi_{2}$ regular.

_Proof of Theorem 5.1._ For simplicity we suppose that ${\cal J}_{i}\cap{\cal
S}=\emptyset$ for every $i$ at each stage for both decompositions. This does
not change the principle involved in the proof – it just simplifies the
details of describing the discarded sums at each stage. Let ${\cal
R}_{1},{\cal R}_{2},{\cal S}_{1},{\cal S}_{2}$ correspond to
${\mathfrak{D}}_{1},{\mathfrak{D}}_{2}$ respectively. Our first task is to
define ${\mathfrak{D}}_{3}={\mathfrak{D}}_{1}\cap{\mathfrak{D}}_{2}$. We do
this inductively as follows. We let $z_{0,j}$ and $z_{j}(\alpha_{1})$
correspond to ${\mathfrak{D}}_{j}$. We then put

$z_{0,3}=\min_{j=1,2}z_{0,j},\qquad
z_{3}(\alpha_{1})=\min_{j=1,2}z_{j}(\alpha_{1})\,.$

Now modify ${\mathfrak{D}}_{1},{\mathfrak{D}}_{2}$ by replacing $z_{0,j}$ and
$z_{j}(\alpha_{1}),j=1,2$ in their definition with the values for $j=3$. Of
course, in some circumstances we may not have modified them at all, but in
general the regions ${\cal F}^{(j)}$ corresponding to our construction for
each decomposition will have increased in size to a new ${\cal F}^{(3)}$.
Consider the region ${\cal F}^{(3)}\setminus{\cal F}^{(1)}$. For the sake of
argument, suppose that $z_{1}(\alpha_{1})>z_{2}(\alpha_{1})$ for
$\alpha_{1}\in{\cal I}$. This means that a new region

$\\{(\alpha_{1},\alpha_{2}):\alpha_{1}\in{\cal
I},\zeta_{2}(\alpha_{1})<\alpha_{2}<\zeta_{1}(\alpha_{1})\\}$

has appeared in the extended ${\mathfrak{D}}_{1}$ decomposition. This
increases the range of summation for the variable $\alpha_{2}$. However, by
what we called _Property ${\cal R}$_, the sum over this new range is a
$\Xi_{1}$-regular sequence. We get a similar result for ${\cal
F}^{(3)}\setminus{\cal F}^{(2)}$. The crucial point is that increasing the
summation domain does not disrupt the regularity property of the resulting
sequence. So in the following we can redefine ${\cal F}_{1}^{(1)}$ to be
${\cal F}_{1}^{(1)}\cup({\cal F}^{(3)}\setminus{\cal F}^{(1)})$, and similarly
for ${\cal F}_{1}^{(2)}$.

In ${\mathfrak{D}}_{3}$ we discard sums over the region ${\cal
F}_{2}^{(1)}\cup{\cal F}_{2}^{(2)}$. The sum over ${\cal F}_{1}^{(1)}\cap{\cal
F}_{1}^{(2)}$ is clearly regular for both $\Xi_{1}$ and $\Xi_{2}$: call this
term $u_{2}(k)$. Buchstab’s identity can be applied twice more for ${\cal
F}_{3}^{(1)}\cap{\cal F}_{3}^{(2)}$ leading to terms
$v_{2}(k)-f_{3}(k)+\Sigma_{4}(k)$, say. Write $w_{2}(k)=u_{2}(k)+v_{2}(k)$.
That only leaves the cases ${\cal F}_{1}^{(i)}\cap{\cal F}_{3}^{(j)},\ i\neq
j$ to discuss. Without loss of generality consider ${\cal
F}_{1}^{(1)}\cap{\cal F}_{3}^{(2)}$. In this case we work in
${\mathfrak{D}}_{3}$ just as we would in ${\mathfrak{D}}_{2}$ (that is,
applying Buchstab’s identity twice more) giving the same shape sums. The
important point to observe is that each of these sums is still regular for
$\Xi_{1}$. To see this, simply note that in each of the new sums we still have
$\alpha_{1}$ and $\alpha_{2}$ unchanged. Here we have observed that, keeping
the first two coordinates fixed, $\boldsymbol{\alpha}_{2}\in{\cal
S}_{1}\Rightarrow\boldsymbol{\alpha}_{3}\in{\cal S}_{1}$ and
$\boldsymbol{\alpha}_{4}\in{\cal S}_{1}$. This is a trivial consequence of our
definition of a Type II $\Phi$-regular domain. We have thus started
constructing our lower bound sieve $\boldsymbol{\lambda}$ which is regular for
both $\Xi_{1}$ and $\Xi_{2}$. We have $\lambda(k)\leq\rho(k)$ with

$\lambda(k)=\psi(k,z_{0,3})-f_{1}(k)+w_{2}(k)-f_{3}(k)+\Sigma_{4}(k),$

where the first $4$ terms on the right hand side above are regular for both
$\Xi_{1}$ and $\Xi_{2}$. As in our previous discussion we can treat
$\Sigma_{4}$ similarly and continue to the case $K_{3}=\max(K_{1},K_{2})$ by
induction. At each stage
$\delta_{j}^{(3)}\leq\delta_{j}^{(1)}+\delta_{j}^{(2)}$. Summing over $j$ then
gives (5.3) as required. ∎Now, if rôle-reversals came into play, we can
instantly see a problem might arise over a new type of region, even proceeding
beyond $j=2$, where one applies Buchstab twice more in ${\mathfrak{D}}_{1}$ in
a straightforward manner but uses a rôle reversal in ${\mathfrak{D}}_{2}$,
before applying Buchstab twice. We are thus jumping straight from a two
dimensional region to a six dimensional one. Our philosophy would suggest we
discard those parts of sums which are non-negative and not regular in the
individual cases. This immediately gets complicated, because in mixing the two
decompositions we then have to remove several sums, not all of which are non-
negative. We illustrate this as follows.

Suppose at the $j=2$ stage we have the term

$\sum_{\begin{subarray}{c}{\scriptstyle\boldsymbol{\alpha}_{1}\in{\cal F}}\\\
{\scriptstyle k=p_{1}p_{2}\ell}\end{subarray}}\psi(\ell,p_{2})$

which is regular for neither $\Phi_{1}$ nor $\Phi_{2}$. Further suppose, that
for $\Phi_{1}$ we can apply Buchstab twice more, and for $\Phi_{2}$ we rewrite
it as per (5.1). This leads to the following two decompositions, where for
simplicity (and since we are demonstrating that the result cannot be true in
general) we shall assume that $z$ is fixed, for certain regions for the
variables:

$\sum_{k=\Pi_{2}\ell}\psi(\ell,z)-\sum_{k=\Pi_{3}\ell}\psi(\ell,z)+\sum_{k=\Pi_{4}\ell}\psi(\ell,p_{4})\,,$
(5.4)
$\sum_{k=np_{2}\ell}\psi(\ell,p_{2})\psi(n,z)-\sum_{k=np_{2}p_{5}\ell}\psi(\ell,p_{2})\psi(n,z)+\sum_{k=np_{2}p_{5}p_{6}\ell}\psi(\ell,p_{2})\psi(n,p_{6})\,.$
(5.5)

We would expect to split the final sum in each of (5.4), (5.5) into a regular
term and a term which must be discarded. Suppose that the discarded sum in
(5.4) corresponds to $\boldsymbol{\alpha}_{4}\in{\cal K}$. We would then have
to split the variable $\ell$ in each of the three terms in (5.5) into sums
matching those in (5.4) and discard terms such as a sum,say $\Sigma^{\prime}$,
over (we state the most complicated) $np_{2}p_{3}p_{4}p_{5}p_{6}\ell$ with
$np_{5}p_{6}=x^{\alpha_{1}}$ and $\boldsymbol{\alpha}_{4}\in{\cal K}$. Of
course, at this point we are discarding some sums which could be negative
(corresponding to the middle term in (5.5)). However, on reversing Buchstab’s
identity it is clear that overall the discarded term is non-negative (indeed,
it is $\delta_{2}^{(1)}$), and so the procedure is legitimate. It is a
different matter, though, when we come to discard that part of the final sum
in (5.5) over a region not belonging to ${\cal S}_{2}$, say we write this as
${\boldsymbol{\alpha}_{4}}^{\prime}\in{\cal K}^{\prime}$
(${\boldsymbol{\alpha}_{4}}^{\prime}$ corresponding to the variables
$p_{2},\ell,p_{5},p_{6}$). We split into three cases with the variable $\ell$
unchanged, or decomposed as $\ell p_{3}$ or $\ell p_{3}p_{4}$. If we had not
discarded $\Sigma^{\prime}$ we could reverse Buchstab’s identity and verify
that the new discarded sum is overall non-negative (indeed, it would have been
$\delta_{2}^{(2)}$). However, we cannot rule out that we are discarding
negative terms and so our proof collapses.

We can put this argument another way. Let

$\sigma(k)=\sum_{\begin{subarray}{c}{\scriptstyle
k=np_{2}p_{3}p_{4}p_{5}p_{6}\ell}\\\
{\scriptstyle\boldsymbol{\alpha}_{4}\in{\cal K}}\\\
{\scriptstyle{\boldsymbol{\alpha}_{4}}^{\prime}\in{\cal
K}^{\prime}}\end{subarray}}\psi(\ell,p_{2})\psi(n,p_{6})\,.$

Then, in general, we would have to assume that $\sigma(k)$ is regular for
neither $\Phi_{1}$ nor $\Phi_{2}$ (though for some examples there might be a
reason for it to be regular). Now, if we were constructing a lower bound
$\boldsymbol{\lambda}$ (regular for both $\Phi_{1}$ and $\Phi_{2}$) for
$\rho(k)+\sigma(k)$, our method would work. But $\boldsymbol{\lambda}$ is no
longer a lower bound prime-detecting sieve as it possibly gives a positive
count for the numbers for which $\sigma(k)$ is non-zero. We cannot retrieve
the situation by subtracting off this term, because, since $\sigma(k)$ is not
regular, $\lambda(k)-\sigma(k)$ cannot be regular.

##  6 Application to Theorem 1.2

Before proving Theorem 1.2 we first state a more general result which allows
us to have rôle-reversals in one of the decompositions. The trick, of course,
is to ensure the problem that we described at the end of the last section
cannot occur. To this end we establish the following result.

######  Theorem 6.1.

Let ${\mathfrak{D}}_{1},{\mathfrak{D}}_{2}$ be two decompositions which are
permissible for the families of sublinear functionals $\Xi_{1},\Xi_{2}$
respectively, and ${\mathfrak{D}}_{1}$ involves no rôle-reversals. Assume for
${\mathfrak{D}}_{1}$ we apply Buchstab no more than 4 times, and at the $K=2$
stage there is a region ${\cal F}_{1}$, sums over which are $\Xi_{1}$-regular,
a region ${\cal F}_{2}$ which must be discarded, and a region ${\cal F}_{3}$
where Buchstab can be applied twice more. Suppose further that ${\cal
F}_{3}={\cal F}_{4}\cup{\cal F}_{5}$ where sums over variables in ${\cal
F}_{4}$ are $\Xi_{2}$-regular (possibly using rôle-reversals) and, however the
sums are treated over ${\cal F}_{5}$ for ${\mathfrak{D}}_{2}$, there are no
rôle-reversals. Then (5.3) holds.

###### Proof.

We only need to consider the two new cases introduced not covered by Theorem
5.1. First of all, there may be a subset, say ${\cal G}$ of ${\cal F}_{1}$,
where rôle-reversals are applied in the ${\mathfrak{D}}_{2}$ decomposition.
This is easily dealt with, since, however the variables in $k=p_{1}p_{2}\ell$
are broken up, we can still combine them at any stage to give variables of the
same size as $p_{1},p_{2},\ell$. For ${\cal F}_{4}$ no obstacle arises since
we would only be discarding a four dimensional region in ${\mathfrak{D}}_{1}$
and we have all the sums remaining $\Xi_{2}$-regular. ∎

We can apply the above theorem directly to our problem with
$\Xi_{j}=\Phi_{j},j=1,2$. We have described the regions ${\cal F}_{j},1\leq
j\leq 3$ in the previous section. To cover ${\mathfrak{D}}_{2}$ and the
corresponding $\Phi_{2}$-regularity, consider the situation described in [2]
which gives $\delta({\mathfrak{D}}_{2})<0.01$ when $\theta=0.55+\epsilon$.
Here rôle-reversals are crucial. Looking at the proof, though, we notice that
sums with $0.4\leq\alpha_{1}+\alpha_{2}\leq 0.46$ are $\Phi_{2}$-regular [2,
Lemma 13]. Also, if $\alpha_{1}+\alpha_{2}<0.4$, no rôle-reversals are used
(see [2, p.37–38]). We can thus apply our theorem with
$\delta({\mathfrak{D}}_{2})<0.01$. Also, $\delta({\mathfrak{D}}_{1})$ is the
sum of the two integrals

$\int_{0.275}^{0.45}\int_{0.55-\alpha_{1}}^{\min(\alpha_{1},(1-\alpha_{1}-\alpha_{2})/2)}\omega\left(\frac{1-\alpha_{1}-\alpha_{2}}{\alpha_{2}}\right)\frac{d\alpha_{2}d\alpha_{1}}{\alpha_{2}^{2}\alpha_{1}}\,;$
$\int_{\boldsymbol{\alpha}_{4}\in{\cal
G}}\omega\left(\frac{1-\alpha_{1}-\ldots\alpha_{4}}{\alpha_{4}}\right)\frac{d\alpha_{4}\ldots
d\alpha_{1}}{\alpha_{4}^{2}\ldots\alpha_{1}}\,.$

Here ${\cal G}$ is the region with $\boldsymbol{\alpha}_{2}\in{\cal
F}_{3},\boldsymbol{\alpha}_{4}\in\mathcal{H}_{4},\alpha_{4}\geq 0.1$, and no
sum of three or all four of $\alpha_{1},\ldots,\alpha_{4}$ is between $0.45$
and $0.55$.

Calculations give the first integral as $<0.71$. The second integral would
have been $<0.16$ even before removing sums of variables between $0.45$ and
$0.55$. With those removals the integral is $<0.02$. So, we get a prime
detecting lower bound sieve with a deficit $<\frac{3}{4}$ for
$\Phi_{1}\cup\Phi_{2}$, which establishes Theorem 2.

##  7 Proof of Theorem 1.1

This is not very different from the proof of Heath-Brown’s result [10], so we
shall be brief.

######  Lemma 7.1.

We have

$\sum_{\begin{subarray}{c}{\scriptstyle n-2y<p\leq n-y}\\\ {\scriptstyle
v|n-p}\end{subarray}}1\ll\frac{y}{\phi(v)}\log{y/v}$

for $y\geq 3v,n\geq 2y$.

###### Proof.

See [5, Theorem 3.7]. ∎

######  Lemma 7.2.

When $(n,d)=1$ we have

$\sum_{\begin{subarray}{c}{\scriptstyle u\leq E}\\\ {\scriptstyle
d|u,(u,n)=1}\end{subarray}}\frac{u}{\phi(u)}=CE\frac{\omega(d)}{d}f(n)+O\left((ED)^{\frac{1}{2}}d(n)\phi(d)^{-1}\right)$

where

$C=\prod_{p}\left(1+\frac{1}{p(p-1)}\right)$

and $\omega,f$ are multiplicative functions given by

$\omega(p^{e})=\frac{p^{2}}{p^{2}-p+1},\quad
f(p^{e})=\frac{(p-1)^{2}}{p^{2}-p+1}\,.$

###### Proof.

This is [10, Lemma 3]. ∎

_Proof of Theorem 1.1._ Let $y=\frac{1}{2}n^{\theta}$ and
$Q=yn^{-\frac{1}{2}}(\log n)^{-2C}$ where $C$ is as in Theorem 1.2 with $A=4$.
We shall estimate

$\\#\\{(p,u,v):n=p+uv,y\leq uv<2y,Q<u\leq 2Q,(u,q)=1\\}$

in two different ways. We have

$N\geq\sum_{\begin{subarray}{c}{\scriptstyle n=k+uv}\\\
{\scriptstyle\rm{(7.1)}}\end{subarray}}\lambda(k)$

with $\lambda$ as in Theorem 1.2 with $n$ in place of $x$, where (7.1) denotes
the set of conditions

$y\leq uv<2y,\quad Q<u\leq 2Q,\quad(u,n)=1.$ (7.1)

It is an easy deduction from Theorem 1.2 that

$N\geq\sum_{\begin{subarray}{c}{\scriptstyle Q<u\leq 2Q}\\\
{\scriptstyle(u,n)=1}\end{subarray}}\ \sum_{\begin{subarray}{c}{\scriptstyle
n-2y<k\leq n-y}\\\ {\scriptstyle
u|n-k}\end{subarray}}\lambda(k)\gg\sum_{\begin{subarray}{c}{\scriptstyle
Q<u\leq 2Q}\\\ {\scriptstyle(u,n)=1}\end{subarray}}\frac{y}{\phi(u)\log n}\,.$

On taking $d=1$ in Lemma 7.2, we have

$\displaystyle\sum_{\begin{subarray}{c}{\scriptstyle Q<u\leq 2Q}\\\
{\scriptstyle(u,n)=1}\end{subarray}}\frac{1}{\phi(u)}$
$\displaystyle\gg\frac{1}{Q}\sum_{\begin{subarray}{c}{\scriptstyle Q<u\leq
2Q}\\\ {\scriptstyle(u,n)=1}\end{subarray}}\frac{u}{\phi(u)}$
$\displaystyle\gg f(n)+O\left(Q^{-\frac{1}{2}}d(n)\right)$ $\displaystyle\gg
f(n),$

since $f(n)\gg\phi(n)/n$ and $Q\gg n^{0.05}$. It follows that

$N\gg\frac{f(n)y}{\log n}\,.$ (7.2)

We now take $\delta$ to be a small positive constant, and assume there are no
solutions of $n=p+ab$ for which $1\leq a,b\leq n^{\frac{1}{2}-\delta}$ and
$ab<2y$, and estimate $N$ from above. Using the argument in [10], we find that

$N\ll\frac{Q}{\log n}\sum_{\begin{subarray}{c}{\scriptstyle F\leq v<4F}\\\
{\scriptstyle(v,nK)=1}\end{subarray}}\frac{v}{\phi(v)}$ (7.3)

where $F=\frac{1}{2}yQ^{-1}$ and

$K=\prod_{q_{1}\leq p<q_{2}}p,\quad q_{1}=2yn^{\delta-\frac{1}{2}}Q^{-1},\quad
q_{2}=\tfrac{1}{2}n^{\frac{1}{2}-\delta}Q^{-1}.$

Following Heath-Brown’s application of the Selberg sieve method in [10], we
obtain the bound

$\sum_{\begin{subarray}{c}{\scriptstyle F\leq v<4F}\\\
{\scriptstyle(v,nK)=1}\end{subarray}}\frac{v}{\phi(v)}\leq\frac{3CFf(n)}{G}+O\left(n^{\frac{3}{7}}\right)$
(7.4)

where, with

$g(m)=\prod_{p|m}(p-\omega(p))^{-1},$

we have

$G=\sum_{\begin{subarray}{c}{\scriptstyle m<n^{\frac{1}{6}}}\\\
{\scriptstyle(m,n)=1,m|K}\end{subarray}}\mu^{2}(m)g(m)\geq\sum_{\begin{subarray}{c}{\scriptstyle
q_{1}\leq p<n^{\frac{1}{6}}}\\\ {\scriptstyle p\nmid n}\end{subarray}}g(p).$

Thus

$\displaystyle G$ $\displaystyle\geq\sum_{q_{1}\leq
p<n^{\frac{1}{6}}}\frac{1}{p}-\sum_{p|n}\frac{1}{q_{1}}$ $\displaystyle=-\log
6+\log\log n-\log\log q_{1}+O\left((\log q_{1})^{-1}+q_{1}^{-1}\log n\right).$

Now

$\displaystyle q_{1}$ $\displaystyle=2(\log n)^{2C}n^{\delta},$
$\displaystyle\log q_{1}$ $\displaystyle=\delta\log n+O(\log\log n),$
$\displaystyle\log\log q_{1}$ $\displaystyle=\log\delta+\log\log
n+O\left((\log n)^{-\frac{1}{2}}\right).$

It follows that

$G\geq-\log(6\delta)+O\left((\log
n)^{-\frac{1}{2}}\right)\geq-\tfrac{1}{2}\log\delta$ (7.5)

if $\delta$ is sufficiently small. We now see from (7.3), (7.4), (7.5), that

$N\ll\frac{yf(n)}{\log n\log(1/\delta)},$

which contradicts (7.2) if $\delta$ is chosen to be sufficiently small. This
completes the proof of Theorem 1.1.

## References

  * [1] R. C. Baker and G. Harman, On the difference between consecutive primes. _Proc. London Math. Soc._ 72 (1996), 261-280.
  * [2] R. C. Baker, G. Harman, and J. Pintz, The exceptional set for Goldbach’s problem in short intervals. _Sieve Methods, Exponential Sums and Their Applications in Number Theory_ , Cambridge University Press, Cambridge, 1997, pp. 11–54.
  * [3] R. C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes II. _Proc. London Math. Soc._ 83 (2001), 532–562.
  * [4] A. Fujii, P.X. Gallagher, and H.L. Montgomery, Some hybrid bounds for character sums and Dirichlet L-functions. _Topics in Number Theory_ , 41-57, North Holland, Amsterdam, 1976.
  * [5] H. Halberstam and H. -E. Richert, _Sieve Methods,_ Academic Press, London, 1974.
  * [6] G. Harman. On the distribution of $\alpha p$ modulo one. _J. London Math. Soc._ (2) 27 (1983), 9–18.
  * [7] G. Harman. On the distribution of $\alpha p$ modulo one II. _Proc. London Math. Soc._ (3) 72 (1996), 241–260.
  * [8] G. Harman, _Prime-detecting Sieves_ , London Mathematical Society Monographs Series, 33, Princeton University Press (Princeton NJ, 2007).
  * [9] G. Harman, N. Watt, and K. C. Wong. A new mean-value result for Dirichlet L-functions and polynomials, _Quart. J. Math._ 55 (2004), 307–324.
  * [10] D.R. Heath-Brown, Representation of an integer as a prime plus the product of two small factors. _Math. Proc. Cambridge Phil. Soc._ 89 (1981), 29-33.
  * [11] M.N. Huxley and H. Iwaniec, Bombieri’s Theorem in short intervals. _Mathematika_ 22 (1975), 188-194.
  * [12] A. Kumchev. The difference between consecutive primes in an arithmetic progression. _Quart. J. Math._ 53 (2002), 479–501.
  * [13] X. Lü and Y. Ren, Representation of an integer as a prime plus the product of two small factors. _J. Number Theory_ , 176 (2017), 204-210.
  * [14] H. L. Montgomery, _Topics in Multiplicative Number Theory_ , Lecture Notes in Mathematics 227, Springer-Verlag, Berlin–New York, 1971.
  * [15] A. Perelli, J. Pintz, and S. Salerno. Bombieri’s theorem in short intervals. _Ann. Scuola Norm. Sup. Pisa_ 11 (1984), 529–538.
  * [16] A. Perelli, J. Pintz, and S. Salerno. Bombieri’s theorem in short intervals II. _Invent. Math._ 79 (1985), 1-9.
  * [17] N.M. Timofeev, Distribution of arithmetic functions in short intervals in the mean with respect to arithmetic progressions (Russian). _Akad. Nauk SSSR Ser. Mat._ 51 (1987), 341-362.

Roger Baker,
Department of Mathematics,
Brigham Young University,
Provo, UT 84602, USA
E-mail<EMAIL_ADDRESS>
Glyn Harman,
Department of Mathematics,
Royal Holloway, University of London,
Egham, Surrey TW20 0EX, UK
E-mail<EMAIL_ADDRESS>