Fractional parts of polynomials over the primes. II

TL;DR

This paper improves the understanding of how closely quadratic polynomials with irrational leading coefficients evaluated at prime numbers approximate integers, demonstrating that the fractional parts are very small for infinitely many primes.

Contribution

It provides a 14% improvement in the exponent measuring the proximity of polynomial values to integers at prime arguments compared to previous results.

Findings

Fractional parts of quadratic polynomials at primes are very small infinitely often.

Achieved a 14% enhancement over previous bounds.

Results contribute to understanding polynomial behavior over primes.

Abstract

We consider the distance to the nearest integer of f(p), where f is a quadratic polynomial with irrational leading coefficient. This distance is very small as a function of p, for infinitely many primes p. We give a 14% improvement in the exponent that measures the distance, compared with the most recent result in the literature.

Equations

$$\Vert \alpha p^2+\beta p+\gamma \Vert <p^{-37/210}$$

$$\Vert f_k(p)\Vert <p^{-{\rho }_k}$$

$$\sum _{\ell \le L}{c}_{\ell }\sum _{R<r\le 2R}{a}_r\sum _{\frac{N}{2}<sr\le N}{b}_{s}e(\ell g(rs))$$

$${\ell }_1{y}_1^2-{\ell }_2{y}_2^2=n({\ell }_1,{\ell }_2\le L;y_1,y_2\in [Y,2Y)).$$

$$\sum _{\begin{array}{c}(2N{)}^{\frac{1}{2}-9\sigma }\le p<(2N{)}^{\frac{3}{8}+\frac{33\sigma }{4}}\\ p{p}^{\prime }\in \left(\frac{N}{2},N\right],\Vert g(p{p}^{\prime })\Vert \le \delta \end{array}}1$$

$$L_1^{1/2}N\ll q\ll L_1^{1/2}N\text{ with }{L}_1=2N^{\rho -\frac{\epsilon }{2}}$$

$$L=N^{\rho -\epsilon /3}.$$

$$\sigma =\rho -1/6.$$

$$\mathcal{A}=\left\{n:n\sim \frac{N}{2},\Vert g(n)\Vert <\delta \right\},\mathcal{B}=\left\{n:n\sim \frac{N}{2}\right\}.$$

$$V^3{W}^2\ll N^{2-3\rho },V{W}^3\ll N^{3-15\rho /2},VW\ll N^{2-8\rho }.$$

$$T:=\sum _{\ell =1}^L\sum _{v\sim V}\sum _{w\sim W}\left|\sum _{nϵI(vw)}e(\ell g(vwn))\right|.$$

$$T\ll N^{1-10\eta }.$$

$$T(y):=\sum _{\ell =1}^L\left|\sum _{n\in I(y)}e(\ell g(yn))\right|>N^{1-10\eta }{Y}^{-1}.$$

$$\sum _{y\in \mathcal{S}}T(y)\ll L^{1/2}{N}^{10\eta }{Z}^{1/2}\left|{\mathcal{S}}^{\ast }\right|;$$

$$\left|\frac{sa{y}^2}{q}-u\right|<Z^{-1}$$

$$s\sim S\ll L^2{N}^{\eta };$$

$$(N/Y{)}^2{L}^{-1}{N}^{-\eta }\ll Z\ll L{S}^{-1}(N/Y{)}^2.$$

$$\left|\theta x-z\right|<\zeta ,0<|x|<X,$$

$$\Vert \frac{sa{y}^2}{q}\Vert <\frac{1}{Z}$$

$$\ll N^{\eta }\left(\frac{Y+q^{1/2}}{Z^{1/2}}\right).$$

$$\sum _{(v,w)\in \mathcal{C}}T(v,w)\ll N^{1-10\eta }.$$

$$VW>N^{1-5\rho /2-\eta }.$$

$$\left|\mathcal{C}\right|\ll \frac{N^{2\eta }S(VW+q^{1/2})}{Z^{1/2}}$$

$$\left|\mathcal{C}\right|\ll \frac{N^{2\eta }SVW}{Z^{1/2}},$$

$$S<N^{1-23\eta }(VW{)}^{-1}{L}^{-1/2};$$

$$\sum _{(v,w)\in \mathcal{C}}T(v,w)$$

$$\ll N^{1-10\eta }$$

$$S\ge N^{1-23\eta }(VW{)}^{-1}{L}^{-1/2}.$$

$$Z\le {\left(\frac{N}{VW}\right)}^{2}L\frac{VW{L}^{1/2}}{N^{1-23\eta }}=\frac{N^{1+23\eta }}{VW}{L}^{3/2}.$$

$$\mathcal{C}(w)=\{v\sim V:(v,w)\in \mathcal{C}\}$$

$$\mathcal{E}(K)=\{w\sim W:K\le |\mathcal{C}(w)|<2K\}.$$

Full text

[TABLE]

has infinitely many solutions in primes $p$, sharpening a result due to Harman (1996) in the case $\beta=0$ and Baker (2017) in the general case.

*†*Research supported in part by Collaboration Grant 412557 from the Simons Foundation

1. Introduction

Let $f_{k}(x)=\alpha x^{k}+\cdots+\beta x+\gamma$ be a polynomial of degree $k>1$ with irrational leading coefficient. Inequalities of the form

[TABLE]

for infinitely many primes $p$ were studied by Vinogradov [11]; see [1] for the strongest available results. The present paper gives a new result for $k=2$.

Theorem 1**.**

Let $\rho_{2}=37/210=0.1761\ldots$; then (1.1) holds for infinitely many primes $p$.

The values $\rho_{2}=1/8-\varepsilon$, $3/20$, $2/13=0.1538\ldots$ were given by Ghosh [6], Baker and Harman [2], and Harman [7] in the case $\beta=0$; Baker [1] extended the last result to general $\beta$.

As in [1], [2], [7] we use the Harman sieve. We make progress in the present paper by giving new bounds for sums of the shape

[TABLE]

where $g$ is the approximating polynomial to $f_{2}$ in [1]. Type I sums (in which $b_{s}\equiv 1$) are treated in Section 2, and general (Type II) sums in Section 3. In the Type I case, $(a_{r})_{R<r\leq 2R}$ is restricted to convolutions of shorter sequences; a lemma of Birch and Davenport [4] on Diophantine approximation plays a key role. For Type II sums, a subsidiary task is the study of the average behavior, as $n$ varies, of the number of solutions of

[TABLE]

One sum that eludes the Harman sieve takes the form

[TABLE]

(where $\rho=\frac{37}{210}+\varepsilon$, $\sigma=\rho-\frac{1}{6}$ and $\delta=\frac{1}{2}\,N^{-\rho+\frac{\varepsilon}{2}}$). We bound it above using the form of the linear sieve given by Iwaniec [10]. This sum is treated in Section 5; the sums accessible via the Harman sieve are in Section 4. Section 6 contains the sieve decomposition of $S(\mathcal{A},(2N)^{1/2})$ (defined below), and the calculations leading to Theorem 1. Integrals that appear here and in earlier drafts were calculated by Andreas Weingartner; thanks, Andreas, for your generosity.

The following notations will be used:

$\|\theta\|=\min\limits_{n\in\mathbb{Z}}|\theta-n|$.

$|\mathcal{E}|=\sum\limits_{n\in\mathcal{E}}1\qquad(\mathcal{E}\subset[1,N])$.

$\chi_{\mathcal{E}}=$ indicator function of $\mathcal{E}$.

$C\ldots$ absolute constant, not the same at each occurrence.

$\lambda\ldots$ real number with $|\lambda|\leq C$, not the same at each occurrence.

$\varepsilon\ldots$ sufficiently small positive number; $\eta=\varepsilon^{9}$.

$\gg,\ll\ldots$ indicate implied constants that may depend on $\varepsilon$.

$y\sim Y\ldots$ indicates $Y<y\leq 2Y$.

$e(\theta)=e^{2\pi i\theta}$.

$\frac{a}{q}\ldots$ fraction in lowest terms with $|\alpha_{2}-\frac{a}{q}|<\frac{1}{q^{2}}$, with $q$ sufficiently large; $g(x)=\frac{a}{q}\,x^{2}+\beta x+\gamma$.

We choose $N$ so that

[TABLE]

and write

[TABLE]

Here $\rho$ will ultimately be $\frac{37}{210}+\varepsilon$; earlier in the paper we restrict $\rho$ somewhat less. We do suppose $\rho>\frac{1}{6}$, and write

[TABLE]

We reserve the symbols $p,p_{1},p^{\prime},\ldots$ for prime numbers.

Let $\delta=L_{1}^{-1}$ and

[TABLE]

We write $I(m)$ for an arbitrary subinterval of $\left(\frac{N}{2m},\frac{N}{m}\right]$.

2. Type I sums

The object of this section is to prove

Theorem 2**.**

Let $\frac{1}{6}<\rho<\frac{2}{11}$. Let $V\geq 1,W\geq 1$,

[TABLE]

Let

[TABLE]

Then

[TABLE]

We require several lemmas.

Lemma 1**.**

Let $1\leq Y\ll N^{1-2\rho}$. Let $\mathcal{S}$ be the set of $y\in(Y,4Y]$ with $\langle y,q\rangle\leq N^{\rho}$ and

[TABLE]

There are a set $\mathcal{S}^{*}\subset\mathcal{S}$ and positive numbers $S$, $Z$ with the following properties.

1. (i)

We have

[TABLE] 2. (ii)

for $y\in\mathcal{S}^{*}$, we have

[TABLE]

for some $s=s(y)$ in $\mathbb{N},u=u(y)$ in $\mathbb{Z},(u,q)=1$, and

[TABLE] 3. (iii)

$Z$* satisfies*

[TABLE]

Proof.

This can readily be extracted from the proof of [1, Lemma 8], with $5\eta$ in place of $\eta$. ∎

Lemma 2**.**

Let $\theta$ be a real number and suppose there exist $R$ distinct integer pairs $x,z$ satisfying

[TABLE]

where $R\geq 24\zeta X>0$. Then all integer pairs $x$, $z$ satisfying (2.6) have the same ratio $z/x$.

Proof.

Birch and Davenport[4]. ∎

Lemma 3**.**

Suppose that $s\geq 1$, $D\geq 1$, $sD^{2}<q$, $Z\geq 2$. The number of solutions $y\in(Y,4Y]$, with $\langle y,q\rangle\leq D$, of the inequality

[TABLE]

is

[TABLE]

Proof.

[1, Lemma 3]. ∎

Proof of Theorem 2.

In the notation of Lemma 1, with $Y=VW$, let $\mathcal{C}$ be the set of pairs $(v,w)$ for which $v\sim V$, $w\sim W$, $vw\in\mathcal{S}^{*}$. As in [1], proof of Lemma 8, it suffices to prove that

[TABLE]

In view of the Type I result obtained in [1, Lemma 8], we may suppose that

[TABLE]

We note that

[TABLE]

as a consequence of Lemma 3; indeed

[TABLE]

since $VW>N^{1-5\rho/2-\eta}>N^{1/2+\rho/4}\gg q^{1/2}$ for $\varepsilon$ sufficiently small.

Suppose for a moment that

[TABLE]

then (2.2) gives (with a divisor argument)

[TABLE]

(from (2.8),(2.9)), giving (2.7). So we may suppose that

[TABLE]

It now follows from (2.5) that

[TABLE]

Now let

[TABLE]

and for $K\geq$ 1, let

[TABLE]

Then

[TABLE]

We choose $K,1\leq K\leq V$, so that

[TABLE]

Suppose for a moment that

[TABLE]

Then arguing as above,

[TABLE]

Thus we may suppose that

[TABLE]

For the next stage of the argument, let $w$ be a fixed integer in $\mathcal{E}(K)$. We apply Lemma 2, taking

[TABLE]

since (by (2.3) and the definition of $\mathcal{C}$)

[TABLE]

for every $v$ in $\mathcal{C}(w)$. By a divisor argument, the number of distinct $sv^{2}$ as $v$ varies over $\mathcal{C}(w)$ is $\gg KN^{-\eta}$. Thus in the notation of Lemma 2,

[TABLE]

To see this,

[TABLE]

from the hypothesis of the theorem.

Accordingly, all $\frac{u}{sv^{2}}$ with $v\in\mathcal{C}(w)$ can be written in the form

[TABLE]

for a certain $t\in\mathbb{Z}$, $r\in\mathbb{N}$ independent of $v$.

We record a lower bound for $K$ that does not contain $Z$. From (2.12), (2.10), we have

[TABLE]

We now select a divisor $z$ of $r$ such that the set

[TABLE]

satisfies

[TABLE]

For each $v$ in $\mathcal{C}(w,z)$, we have

[TABLE]

It is convenient to write $z=bc^{2}$ where $b$ is square-free, and define $k=v^{2}/bc^{2}$. Then

[TABLE]

This leads to the upper bound

[TABLE]

from which we infer, using (2.15), that

[TABLE]

Now we re-examine our rational approximation

[TABLE]

(see (2.3) and the definition of $\mathcal{C}$). We observe that

[TABLE]

for $v\in\mathcal{C}(w,z)$, using (2.16).

Let $\mathcal{F}$ be the set of natural numbers $b_{0}c^{2}_{0}m$, $(b_{0},c_{0},m)\in\mathbb{N}^{3}$, $b_{0}c_{0}<\frac{VN^{2\eta}}{K}$, $m<L^{2}N^{\eta}$. The number of possibilities for $b_{0}c_{0}^{2}$ here is $\ll\frac{VN^{3\eta}}{K}$, since $b_{0}c_{0}^{2}$ is a divisor of $(b_{0}c_{0})^{2}$. Hence

[TABLE]

All the integers $r$ occurring in (2.17) are in $\mathcal{F}$. Hence

[TABLE]

on bounding the number of $w$ in $\mathcal{E}(K)$ with $r(w)=r$ via Lemma 3. Combining (2.18), (2.19), and recalling (2.11),

[TABLE]

We now use the lower bound (2.14) for $K$ and obtain

[TABLE]

Now (2.7) follows on applying the bounds for $VW^{3}$ and $VW$ in the hypothesis of Theorem 2. This completes the proof of Theorem 2. ∎

3. Type II sums

Lemma 4**.**

Let $Y\geq 1$. For $n\in\mathbb{N}$, let $R(n)$ denote the number of quadruples $(\ell_{1},\ell_{2},y_{1},y_{2})$ with $1\leq\ell_{i}\leq L$, $y_{i}\sim Y$ such that

[TABLE]

Proof.

For (3.2), fix $\ell_{2}$ and $y_{2}$, then the equation

[TABLE]

(with $\ell_{1}\neq 0$, $y_{1}\neq 0$) determines $\ell_{1}$, $y_{1}$ up to $O((LY)^{\eta})$ possibilities.

For (3.3), we observe that

[TABLE]

is the number of tuples $\ell_{1}$, $\ell_{2}$, $\ell_{3}$, $\ell_{4}$, $y_{1}$, $y_{2}$, $y_{3}$, $y_{4}$ with $1\leq\ell_{i}\leq L$, $y_{i}\sim Y$ and

[TABLE]

This may be expressed as an integral:

[TABLE]

by the Cauchy-Schwarz inequality, where

[TABLE]

Now $V$ is the number of solution of

[TABLE]

with $1\leq\ell$, $\ell_{3}$, $\ell_{4}\leq L$ and $y_{i}\sim Y(1\leq i\leq 4)$.

We first consider $V_{1}$, the number of solutions of (3.5) with

[TABLE]

If (3.5) and (3.6) hold, then $y_{1}=y_{2}$. There are $O((LY)^{1+\eta})$ possibilities for $\ell_{3}$, $\ell_{4}$, $y_{3}$, $y_{4}$ and for each of these, at most $LY$ possibilities for $\ell$, $y_{1}$, $y_{2}$. Thus

[TABLE]

Now consider $V_{2}$, the number of solutions of (3.5) for which (3.6) is violated. There are $O(L^{2}Y^{2})$ possibilities for $\ell_{3}$, $y_{3}$, $\ell_{4}$, $y_{4}$. For each of these, there are $O((LY)^{\eta})$ possibilities for $\ell$, $y_{1}-y_{2}$ and $y_{1}+y_{2}$, hence $O((LY)^{\eta})$ possibilities for $\ell$, $y_{1}$, $y_{2}$. Thus

[TABLE]

Now(3.3) now follows on combining(3.4) and (3.7). ∎

Theorem 3**.**

For $\frac{1}{6}<\rho<\frac{1}{5}$, and $N^{\rho}\ll Y\ll N^{1-4\rho}$, $|c_{\ell}|\leq 1$, $|a_{x}|\leq 1$, $|b_{y}|\leq 1$, we have

[TABLE]

Proof.

Just as in [1, proof of Lemma 9] we need only show that

[TABLE]

Again arguing as in that proof,

[TABLE]

where $\boldsymbol{\ell}=(\ell_{1},\ell_{2},y_{1},y_{2})$, $\mathcal{L}=\{\boldsymbol{\ell}:\ell_{1},\ell_{2}\leq L,y_{1},y_{2}\sim Y\}$ and

[TABLE]

The contribution to the right-hand side of (3.9) from those $\boldsymbol{\ell}$ with

[TABLE]

is

[TABLE]

since $Y\gg N^{\rho}$. The contribution from those $\boldsymbol{\ell}$ with

[TABLE]

It remains to consider $\mathcal{M}$, the set of $\boldsymbol{\ell}$ in $\mathcal{L}$ with

[TABLE]

and

[TABLE]

We apply [1, Lemma 5] with $M=1$, $X=NY^{-1}$, $P=\frac{N^{1-22\eta}}{YL^{2}}$. We require

[TABLE]

which holds since

[TABLE]

Thus for each $\boldsymbol{\ell}\in\mathcal{M}$ there exists a natural number $s$,

[TABLE]

with $u_{1}$, $u_{2}\in\mathbb{Z}$, $(u_{2},s)=1$,

[TABLE]

Now $s\leq N/Y$ and, writing $\gamma_{2}=(\ell_{1}y_{1}^{2}-\ell_{2}y_{2}^{2})a/q$, $\gamma_{1}=(\ell_{1}y_{1}-\ell_{2}y_{2})\alpha_{1},$

[TABLE]

Thus we can appeal to [1, Lemma 7] with $k=2$ and $L$ replaced by 1. Let

[TABLE]

We obtain

[TABLE]

Now $N^{3\eta}L^{2}$ is of smaller order than $\frac{N^{1-22\eta}}{Y}\,L^{-2}$, so that

[TABLE]

Moreover, by standard bounds, the left-hand side of (3.12) is

[TABLE]

We now use a standard splitting-up argument to choose a subset $Q$ of $\mathcal{M}$ such that

[TABLE]

where $(N/Y)^{2}L^{-4}N^{-\eta}\leq Z\leq Z_{0}$, with

[TABLE]

and moreover

[TABLE]

while

[TABLE]

Compare e.g. the argument in [1, proof of Lemma 8]. In order to obtain (3.8) it remains to show that

[TABLE]

Using (3.11)–(3.14), we find that

[TABLE]

and we must show that

[TABLE]

For each $\mathcal{\ell}$ in $Q$ there is an $s\sim S$ with

[TABLE]

where

[TABLE]

Clearly

[TABLE]

Given $m\in\mathcal{C}$, let $h_{1}(m),\ldots,h_{j}(m)$, $j=j(m)\ll N^{\eta}$, be the divisors $h$ of $m$ with $h\sim S$. Each element of $Q$ satisfies

[TABLE]

for some $m\in\mathcal{C}$ and some $i$, $1\leq i\leq j(m)$. Let

[TABLE]

Then

[TABLE]

while

[TABLE]

in the notation of Lemma 4. Applying Cauchy’s inequality,

[TABLE]

by Lemma 4 and (3.17).

Alternatively, (3.2) yields

[TABLE]

We now find that, depending in the value of $Y$, either (3.18) or (3.19) yields the desired bound (3.15). Suppose first that

[TABLE]

In view of (3.16), (3.18), we need to verify the four bounds

[TABLE]

First of all, (3.21) holds since

[TABLE]

Next, (3.22) holds since

[TABLE]

Next, (3.23) holds since

[TABLE]

Finally, (3.24) holds since

[TABLE]

from (3.20).

Now suppose that

[TABLE]

we employ (3.16) and (3.19). We need to verify the bounds

[TABLE]

First of all, (3.26) holds since

[TABLE]

from (3.25). Next, (3.27) holds since

[TABLE]

Next, (3.28) holds since

[TABLE]

Finally, (3.29) holds since

[TABLE]

In order to complete the proof, we show that the ranges of $Y$ in (3.20) and (3.25) overlap. We have

[TABLE]

that is

[TABLE]

since $\rho<1/5$. This completes the proof of (3.15), and Theorem 3 follows. ∎

4. Asymptotic formulae via Harman sieve and generalized Vaughan identity

In the present section and the next, we suppose that $\sigma=\rho-1/6$ satisfies

[TABLE]

We write $b=\frac{1}{6}-5\sigma$, $f=\frac{1}{3}-4\sigma$, $z=N^{b}$ and

[TABLE]

For a finite set $\mathcal{E}\subset\mathbb{N}$, let

[TABLE]

As in [1], our claim in Theorem 1 is a corollary of the lower bound

[TABLE]

for $\rho=\rho_{2}$.

We introduce some ‘comparison’ results for the pair $S(\mathcal{A},w)$ and $2\delta\,S(\mathcal{B},w)$, and similar pairs, that will be needed in Section 6. First of all, we have

[TABLE]

whenever $\tau$ is a positive constant and some subproduct $R$ of

$P_{1}\ldots P_{\ell}$ $Q_{1}\ldots Q_{t}$ satisfies

[TABLE]

This is a consequence of Theorem 2; compare the discussion in [8, Sections 3.2 and 3.5]. Additional inequalities such as ‘$p_{j}\leq K$’ may be included in the summation in (4.3) without affecting its validity, as explained in [8, Section 3.2].

The following lemma is essentially the special case $M=2X^{\alpha}$, $S=1$ of [3, Lemma 14], and is a variant of [8, Theorem 3.1].

Lemma 5**.**

Let $w$ be a complex function with support in $[1,N]$, $|w(n)|\leq N^{1/\eta}$ $(n\geq 1)$. Let $0<\theta<\theta+\psi<1/2$. Let

[TABLE]

Suppose that, for some $Y>1$ we have, (for any coefficients $a_{m},|a_{m}|\leq 1$, $c_{n}$, $|c_{n}|\leq 1$ and $b_{n}$, $|b_{n}|\leq\tau(n))$

[TABLE]

Let $u_{r}$ $(r<N^{\theta})$ be complex numbers with $|u_{r}|\leq 1$, $u_{r}=0$ for $(r,P(N^{\eta}))>1$. Then

[TABLE]

We can deduce the following ‘bilinear’ lemma.

Lemma 6**.**

Let $w$, $\theta$, $\psi$, $S(r,v)$ be as in Lemma 5. Suppose that we have the hypothesis (4.6) and in addition, for some $T\in[1,N)$,

[TABLE]

for any $a_{m}$, $c_{t}$ with $|a_{m}|\leq 1$, $|c_{t}|\leq 1$. Then for any $u_{r}$ $(r<N^{\theta})$, $v_{t}$ $(t\leq T)$ with $|u_{r}|\leq 1$, $|c_{t}|\leq 1$, $u_{r}=0$ for $(r,P(N^{\eta}))>1$, we have

[TABLE]

Proof.

We apply Lemma 4 with $w$ replaced by $w^{*}$,

[TABLE]

so that $S(r,v)$ is replaced by

[TABLE]

From (4.8), (4.9) the hypotheses of Lemma 5 are satisfied with $Y$ replaced by $YN^{\eta}$: for example,

[TABLE]

(we may group the product $mt$ as a single variable and apply (4.6)). The conclusion (4.7) with $S$ replaced with $S^{*}$ gives the desired bound (4.9). ∎

We now apply Lemma 6 with

[TABLE]

$\theta=\rho$, $\theta+\psi=f$, where $T=(2N)^{\nu}$ and the non-negative number $\nu$ satisfies

[TABLE]

Lemma 7**.**

Suppose that (4.10)–(4.12) hold. Then

[TABLE]

whenever $|u_{r}|\leq 1$, $(r,P(N^{\eta}))=1$ for $u_{r}\neq 0$, and $|v_{t}|\leq 1$.

Proof.

We take $Y=\delta N^{1-3\eta}$. The hypothesis (4.8) is a consequence of Theorem 2 because of (4.10)–(4.12). The hypothesis (4.6) is a consequence of Theorem 2. Now the conclusion (4.9) may be written in the form (4.13). ∎

Lemma 8**.**

Let $0<g\leq\frac{1}{6}-2\sigma$, $\frac{1}{3}-4\sigma<\gamma<\frac{2}{3}-8\sigma$. Let $\rho_{1}\geq\cdots\geq\rho_{t}\geq 0$ with $\rho_{1}+\cdots+\rho_{t}=\gamma$, $\rho_{1}\leq\gamma-g$. There is a set $\mathcal{C}\subset\{1,\ldots,t\}$ with $\sum\limits_{i\in\mathcal{C}}\rho_{i}\in\left[g,\frac{1}{3}-4\sigma\right]$.

Proof.

Suppose that no such $\mathcal{C}$ exists. Now suppose first that $\rho_{1}\leq g$. Since $2g\leq\frac{1}{3}-4\sigma$ we can prove successively that $\rho_{1}+\rho_{2},\ldots,\rho_{1}+\cdots+\rho_{t}$ are in $[0,g]$. This is absurd.

Thus we must have $\rho_{1}>\frac{1}{3}-4\sigma$. But now $\rho_{2}+\cdots+\rho_{t}=\gamma-\rho_{1}<\gamma-\left(\frac{1}{3}-4\sigma\right)<\frac{1}{3}-4\sigma$, and $\rho_{2}+\cdots+\rho_{t}\geq g$ since $\rho_{1}\leq\gamma-g$. This is absurd. ∎

Lemma 9**.**

*Let $F$ be a complex function on $[1,N]$. The sum

$\sum\limits_{k\leq N}\Lambda(k)$ $F(k)$ may be decomposed into at most $C(\log N)^{8}$ sums of the form*

[TABLE]

where $I_{i}=(N_{i},2N_{i}]$, $\prod N_{i}<N$ and $2N_{i}\leq N^{1/4}$ if $i>4$.

Proof.

This is a case of Heath-Brown’s ‘generalized Vaughan identity’ [9]. ∎

Lemma 10**.**

Let

[TABLE]

We have

[TABLE]

Proof.

Arguing as in the proof of (4.3), it will suffice to show that

[TABLE]

for $|c_{\ell}|\leq 1$. By a partial summation argument, it suffices to obtain

[TABLE]

Applying Lemma 9, we need only show that

[TABLE]

whenever $\prod\limits_{i}N_{i}<N$ and $2N_{i}\leq N^{1/4}$ for $i>4$. Thus $Q\ll N_{1}\ldots N_{8}\ll Q$. We write $N_{1}\ldots N_{8}=N^{\gamma}$.

We may assume in view of Theorem 3 that no subproduct $X$ of $N_{1},\ldots,N_{8}$ satisfies

[TABLE]

We now reorder $N_{1},\ldots,N_{k}$ as $(N_{1}\ldots N_{k})^{\rho_{j}}$ $(1\leq j\leq k)$ with $\rho_{1}\geq\cdots\geq\rho_{k}\geq 0$. Let $g=\min\left(3\gamma-\frac{5}{4}-\frac{15\sigma}{2},\gamma-\frac{1}{3}-8\sigma\right)$, so that $0<g\leq\gamma-\frac{1}{3}-8\sigma$ and

[TABLE]

We divide the argument into two cases.

Case 1**.**

We have $\rho_{1}\geq\gamma-g>\frac{1}{4}$. Thus $i\leq 4$ and (after a partial summation if necessary) we can apply Theorem 2 to the sum in (4.15) with

[TABLE]

We verify the hypotheses of Theorem 2. First,

[TABLE]

since

[TABLE]

Next,

[TABLE]

and

[TABLE]

from the definition of $g$. Now (4.15) follows from Theorem 2.

Case 2**.**

We have $\rho_{1}<\gamma-g$. By Lemma 8 and the absence of a product $X$ satisfying (4.17), there is a subsum $u=\sum\limits_{i\in\mathcal{C}}\rho_{i}$ such that

[TABLE]

We are now in a position to apply Lemma 6 with $\theta=\rho$, $\theta+\psi=f$,

[TABLE]

We need to verify (4.6), (4.8). Clearly (4.8) is a consequence of Theorem 3. As for (4.8), we need to verify the hypotheses of Theorem 2 with $V\leq 2N^{1/6+\sigma}$, $W\ll N^{\gamma-u}$. Suppose first that $\gamma>\frac{11}{24}-\frac{\sigma}{4}$. Then $\gamma-u\leq\frac{1}{3}+8\sigma$,

[TABLE]

Now suppose that $\gamma\leq\frac{11}{24}-\frac{\sigma}{4}$. Then $\gamma-u\leq\frac{5}{4}+\frac{15\sigma}{2}-2\gamma$,

[TABLE]

[TABLE]

and

[TABLE]

Thus Theorem 2 yields the desired estimate (4.8). Now the lemma follows from Lemma 6.∎

There is a short interval in which we can use Lemma 9 directly to obtain a conclusion stronger than (4.15).

Lemma 11**.**

We have

[TABLE]

whenever

[TABLE]

Proof.

In this range of $Q$ we have

[TABLE]

similarly with $\mathcal{B}$ in place of $\mathcal{A}$. We apply Lemma 9 to decompose the sum over $p^{\prime}$ (with $\Lambda(m)$ in place of $p^{\prime}$). Clearly it will be enough to show that, for $|c_{\ell}|\leq 1$,

[TABLE]

where $N\ll Q\prod\limits_{i=1}^{8}N_{i}\ll N$ and $N_{i}\ll(N/Q)^{1/4}$ for $i\geq 5$. As in the preceding proof we may suppose that no subproduct $X$ of $N_{1}\ldots N_{8}$ satisfies (4.17). Writing $Q=N^{\gamma}$, we have

[TABLE]

Thus it is clear by a ‘reflection’ argument that no such $X$ can satisfy

[TABLE]

Moreover,

[TABLE]

It follows that

[TABLE]

There cannot be two indices $i$ with $N_{i}>N^{\frac{5}{6}-\gamma-\sigma}$, since

[TABLE]

Hence there is a $j$ with

[TABLE]

We are now in a position to apply Theorem 2 with

[TABLE]

We make the usual verification:

[TABLE]

Thus Theorem 2 yields (4.18). This completes the proof of Lemma 11. ∎

5. Application of the linear sieve

In order to obtain an upper bound for

[TABLE]

whenever

[TABLE]

we apply Theorem 4 of Iwaniec [10], which we state in a form sufficient for our needs. The quantity estimated will actually exceed that in (5.1), which will be exploited in Section 6.

Let $\mathcal{E}$ be a set of integers in $[1,N]$. Fix an approximation $X$ to $|\mathcal{E}|$ and write

[TABLE]

Let $F(s)$ be the upper bound function for the linear sieve [10, p. 309]. In the following lemma, let $D\geq Z\geq 2$. Let

[TABLE]

Lemma 12**.**

With the above notations, we have

[TABLE]

with some coefficients $\phi_{d}^{+}(D^{\varepsilon})$ and $\Lambda_{d}^{+}(\varepsilon,\boldsymbol{D})$ bounded by $1$ in absolute value. Here

[TABLE]

In our application, we shall take

[TABLE]

for any $p\in[P,P^{\prime})$, and

[TABLE]

We write $P=N^{\gamma}$. We shall take $D=N^{\frac{2}{3}-8\sigma-\gamma-2\varepsilon}$ and $Z=N^{\frac{1}{12}+\frac{\sigma}{2}}$. It is easily verified that $D^{\frac{1}{3}}\leq Z\leq D^{1/2}$, so that

[TABLE]

We apply Lemma 11 and sum over $p$, obtaining (on noting that $P>D^{\delta}Z$)

[TABLE]

Here

[TABLE]

We shall show that

[TABLE]

the proof that $E_{1}\ll\delta N^{1-\eta}$ is similar but simpler.

Reducing the task of bounding $Z$ to estimating exponential sums as in previous sections, it suffices to prove that for $|c_{\ell}|\leq 1$, and a fixed $\boldsymbol{D}\in\mathcal{H}$, we have

[TABLE]

This is a consequence of Theorem 2, grouping the variables as $v=dp_{1}\ldots p_{r}$ and $p$,

[TABLE]

We make the usual verifications:

[TABLE]

The desired bound (5.5), and the same bound for $E_{2}$, now follows. Now (5.3), (5.4) yield

[TABLE]

6. The sieve decomposition

For the final stage of our work we take $\rho=\frac{37}{210}+\varepsilon$ so that $\sigma=\frac{1}{105}+\varepsilon$. In this section, each $S_{j}$ $(j\geq 0)$ that occurs takes the form

[TABLE]

where the asterisk indicates a restriction of $p_{1}\ldots p_{r}$ and $p_{1}^{\prime}\ldots p_{t}^{\prime}$ to certain subsets of $[1,N]$ depending on $j$; $S_{j}^{\prime}$ is obtained from $S_{j}$ on replacing $\mathcal{A}$ by $\mathcal{B}$. For some of these values of $j$ we write

[TABLE]

where $K_{j}$ is defined by the following additional condition of summation within $S_{j}$: a subproduct $R$ of $p_{1}\ldots p_{r}$ $p_{1}^{\prime}\ldots p_{t}^{\prime}$ satisfies

[TABLE]

We split up $S_{j}^{\prime}$ as $S_{j}^{\prime}=K_{j}^{\prime}+D_{j}^{\prime}$ in the same way. As noted in Section 5,

[TABLE]

whenever (6.1) is used.

Conversion of sums into integrals, with an acceptable error, in the following is along familiar lines (see [8]). Concerning Buchstab’s function $\omega(t)$, we note that

[TABLE]

provided that $\kappa\geq 0.5672$, and that

[TABLE]

see Cheer and Goldston [5].

Let $S_{0}=S(\mathcal{A},(2N)^{1/2})$. Using Buchstab’s identity and writing $p_{i}=(2N)^{\alpha_{i}}$, we have

[TABLE]

where

[TABLE]

with $I_{2}=\left[\frac{1}{6}-5\sigma,\frac{1}{6}+\sigma\right)$,$I_{3}=\left(\frac{1}{6}+\sigma,\frac{1}{3}-4\sigma\right)$, $I_{4}=\left[\frac{1}{3}-4\sigma,\frac{1}{3}+8\sigma\right)$, $I_{5}=\left[\frac{1}{3}+8\sigma,\frac{1}{2}-9\sigma\right)$, $I_{6}=\left[\frac{1}{2}-9\sigma,\frac{3}{8}+\frac{33\sigma}{4}\right)$ and $I_{7}=\left[\frac{3}{8}+\frac{33\sigma}{4},\frac{1}{2}\right]$.

Recalling (4.3) and Lemmas 7 (with $\nu=0$) and 11, we have

[TABLE]

for $j=1,3,5$. For $j=2$, we apply Buchstab’s identity three further times to obtain

[TABLE]

where

[TABLE]

We can apply Lemma 7 to $S_{8}$, $S_{9}$, $S_{10}$ to obtain (6.7): for example, in $S_{10}$,

[TABLE]

We treat $S_{11}$ via (6.3).

For $j=4$ we apply Buchstab’s identity once. Iterating once more for part of the sum over $p_{1}$, $p_{2}$, this gives

[TABLE]

where

[TABLE]

We have (6.7) for $S_{12}$, $S_{14}$ since Lemma 9 is applicable. For example, for $S_{14}$ we have

[TABLE]

We have (6.7) also for $S_{15}$, this time using (4.3), since $\rho\leq 2b\leq\alpha_{2}+\alpha_{3}<f$ in $S_{15}$. For $S_{13}$, we use the lower bound (6.3).

We also apply Buchstab once more to $S_{7}$,

[TABLE]

where

[TABLE]

satisfies (6.7) by Lemma 10, and

[TABLE]

is bounded below as in (6.3).

For $S_{6}$, we proceed differently. We have

[TABLE]

where

[TABLE]

is treated as in (6.3).

By (5.7),

[TABLE]

Moreover,

[TABLE]

Combining (6.11)–(6.13) and (6.3) with $j=18$,

[TABLE]

Our sieve decomposition, obtained by combining (6.6), (6.8), (6.9) and (6.10), is

[TABLE]

and also holds if $S_{j}$ is replaced by $S_{j}^{\prime}$. Combining all applications of (6.7) and (6.3) with (6.14), we end up with

[TABLE]

Our next task is to evaluate $D_{11}^{\prime}$, $D_{13}^{\prime}$, $D_{17}^{\prime}$, and $D_{18}^{\prime}$ with sufficient accuracy. Using $2b>c$ we find that in $D_{11}$, $\alpha_{3}+\alpha_{4}\notin[c,f]$ implies $\alpha_{3}+\alpha_{4}>f$. With a little thought we find that

[TABLE]

It is simplest to replace the $\omega$ factor by $0.59775$ using (6.4). Carrying out the $x$ and $w$ integrations (and using $z>f/2$) leads to

[TABLE]

For $D_{13}^{\prime}$, $D_{17}^{\prime}$ we write the numbers $p_{1},p_{2},p_{1}^{\prime},\ldots,p_{t}^{\prime}$ that appear in $D_{j}^{\prime}$ as

[TABLE]

where $k\geq 0$, $\ell\geq 0$, $k+\ell\geq 1$,

[TABLE]

We now consider $D_{13}^{\prime}$. First we treat the contribution $D^{(1)}$ from $\alpha_{2}>f$. Thus $k=0$. We cannot have $\ell\geq 2$, since

[TABLE]

If we consider $\ell=1$, we have $\alpha_{1}+2\alpha_{2}<1$ and obtain

[TABLE]

Let $D^{(2)}$ be the contribution to $D_{13}$ from $\alpha_{2}<\rho$. We have $\alpha_{2}+\beta_{1}\geq f$, $\beta_{1}\geq f/2$, by an argument used above. We cannot have $\ell\geq 2$ since

[TABLE]

If $\ell=1$, we must have $k\leq 1$ (use (6.20) again). If $\ell=0$, we must have $k\leq 3$ similarly; however, $k>2$, since

[TABLE]

The three remaining cases lead to

[TABLE]

where

[TABLE]

(The integral was bounded above by an integral in $x$, $y$, $z$, using $1-x-y-z-w\geq w$, before carrying out the $w$ integration.)

In $D_{17}^{\prime}$, we cannot have $\alpha_{2}\geq\rho$, since then

[TABLE]

which is absurd. Thus $\alpha_{2}<\rho$; we cannot have $\ell\geq 2$ or $\ell=1$, $k\geq 1$ or $\ell=0$, $k\geq 3$ since

[TABLE]

For $\ell=0$, we have $k>1$ since

[TABLE]

Thus

[TABLE]

where

[TABLE]

In $D_{18}^{\prime}$, we write the numbers that appear as $p_{1},p_{1}^{\prime},\ldots,p_{r}^{\prime}$ as

[TABLE]

where $k+\ell=r\geq 2$, $k\geq 0$, $\ell\geq 0$, $\beta_{k}<\rho$, $\gamma_{1}>f$,

[TABLE]

We observe that

[TABLE]

Thus we cannot have $\ell\geq 2$ or $\ell=1$, $k\geq 2$ or $\ell=0$, $k\geq 4$. We cannot have $\ell=0$, $k\leq 3$ since

[TABLE]

Since $\ell+k\geq 2$, the only remaining possibility is $\ell=1$, $k=1$, so that

[TABLE]

where

[TABLE]

Combining this with (6.14) we obtain

[TABLE]

with

[TABLE]

(we obtain this using (6.5)). Since $S_{0}^{\prime}=(1+\lambda\varepsilon)$ $\frac{N}{2\log N}$, we only need to add up our integrals to obtain Theorem 1 from (6.15)–(6.28):

[TABLE]