2 Preliminary lemmata

In this section, we shall prove two simple results which will be the main tools used for our proof of (1.3) in the following section. First, we have the following generalisation of the square sieve of Heath-Brown [Reference Heath-Brown7].

Lemma 2.1. Suppose that $f:{\mathbb {N}}\hookrightarrow {\mathbb {N}}$ is a strictly increasing polynomial of degree two, with leading coefficient A and discriminant $\Delta _f$ . Moreover, suppose that $\mathscr {P}$ is a set of $P\geqslant 1$ primes and that $\phi : {\mathbb {N}}\to \mathbb {R}_+$ satisfies $\phi (n)=0$ whenever n is such that $4Af(n)+\Delta _f>e^P$ . Then, we have the majorisation

$$ \begin{align*} \sum_n\phi\circ f(n)\ll P^{-1}\sum_n\phi(n)+P^{-2}\sum_{\substack{p,p'\in \mathscr{P}\\p\neq p'}}\bigg|\!\sum_n\phi(n)\bigg(\frac{4An+\Delta_f}{pp'}\bigg)\bigg|, \end{align*} $$

where the implied constant is absolute.

Proof. Suppose that n is a natural number satisfying $4Af(n)+\Delta _f\leqslant e^P$ . Since $4Af(n)+\Delta _f=f'(n)^2$ for all natural numbers n,

$$ \begin{align*} \sum_{p\in\mathscr{P}}\bigg(\frac{4Af(n)+\Delta_f}{p}\bigg)=\sum_{\substack{p\in\mathscr{P}\\p\nmid4Af(n)+\Delta_f}}1\geqslant P-\omega(4Af(n)+\Delta_f)\gg P, \end{align*} $$

by virtue of the fact that $\omega (q)=o(\log q)$ . Hence,

$$ \begin{align*} \sum_n\phi\circ f(n)&\ll P^{-2}\sum_n\phi(n)\bigg(\sum_{p\in\mathscr{P}}\bigg(\frac{4An+\Delta_f}{p}\bigg)\bigg)^2\\ &=P^{-2}\sum_{p,p'\in\mathscr{P}}\sum_n\phi(n)\bigg(\frac{4An+\Delta_f}{pp'}\bigg)\\ &=P^{-2}\bigg(\sum_{p\in\mathscr{P}}\sum_{\substack{n\\p\nmid 4An+\Delta_f}}\phi(n)+\sum_{\substack{p,p'\in\mathscr{P}\\p\neq p'}}\sum_n\phi(n)\bigg(\frac{4An+\Delta_f}{pp'}\bigg)\bigg), \end{align*} $$

from which the assertion follows immediately.

Second, we have the following result pertaining to sums of values of the Jacobi symbol in an arithmetic progression, twisted by additive characters.

Lemma 2.2. Suppose that $M,N\geqslant 3$ are coprime and let $\chi $ be the Jacobi symbol modulo M. Moreover, suppose that B and C each belong to one of the primitive residue classes modulo M. Then, for any integer $\Delta $ ,

$$ \begin{align*} \sum_{r\leqslant MN}\chi(Br+\Delta)\boldsymbol{\mathrm{e}}\bigg(\frac{Cr}{MN}\bigg)=\delta_{N|C}\boldsymbol{\mathrm{e}}\bigg(\frac{-C\bar{B}\Delta}{MN}\bigg)\chi(CBN)\tau(\chi)N, \end{align*} $$

where $\boldsymbol {\mathrm {e}}(B\bar {B}/M)=\boldsymbol {\mathrm {e}}(1/M)$ , and $\tau (\chi )$ denotes the Gauß sum of the character $\chi $ .

Proof. It suffices to consider the case where $\Delta =0$ , for from this, the remaining cases follow by applying the translation $r\mapsto r-\bar {B}\Delta $ . So, on noting that

$$ \begin{align*} \{mN+nM:m\leqslant M,n\leqslant N\}/(MN {\mathbb{Z}})=\{r\leqslant MN\}/(MN {\mathbb{Z}}) \end{align*} $$

whenever $(M,N)=1$ , we see that

$$ \begin{align*} \sum_{r\leqslant MN}\chi(Br)\boldsymbol{\mathrm{e}}\bigg(\frac{Cr}{MN}\bigg)&=\sum_{m\leqslant M}\sum_{n\leqslant N}\chi(BmN)\boldsymbol{\mathrm{e}}\bigg(\frac{C(mN+nM)}{MN}\bigg)\\ &=\chi(CBN)\sum_{m\leqslant M}\chi(Cm)\boldsymbol{\mathrm{e}}\bigg(\frac{Cm}{M}\bigg)\sum_{n\leqslant N}\boldsymbol{\mathrm{e}}\bigg(\frac{Cn}{N}\bigg), \end{align*} $$

from which the assertion immediately follows.

Having established our two main preliminaries, we shall now move on to the demonstration of (1.3). Our approach will closely follow the work of Baier [Reference Baier2], so we will keep brief our treatment of the lesser details.

3 Demonstration

Similarly to [Reference Corrigan5], we start by breaking the sum over $q\leqslant Q$ in (1.3) into $O(\log f(Q))$ intervals of the form $\mathscr {Q}_f(M)=\{q\leqslant Q:f(q)\sim M\}$ , where $1\ll M\ll f(Q)$ . Let $\mathscr {A}$ be the set of Farey fractions with denominator in $\mathscr {Q}_f(M)$ and, for any real $\alpha $ and any small $\delta>0$ , define

$$ \begin{align*} P_\delta(\alpha)=\#\{\alpha'\in\mathscr{A}:|\alpha-\alpha'|\leqslant\delta\}. \end{align*} $$

Following the standard procedure laid down by Wolke [Reference Wolke8], we see that to establish (1.3), it suffices to show that the bound

(3.1) $$ \begin{align} \sup_{\alpha\in\mathbb{R}}P_\delta(\alpha)\mathop{\:\ll\:}_\varepsilon f(Q)^\varepsilon M\sqrt{\delta} \end{align} $$

holds whenever $M^{-2}\leqslant \delta \leqslant M^{-1}$ . As in [Reference Corrigan5], we note here that, in the case where $\delta $ is outside of the aforementioned range, (1.3) is contained in the trivial bound (1.1). Now, in the following, we may assume that $\alpha $ does not belong to the set

$$ \begin{align*} \mathfrak{M}=\bigcup_{r\leqslant1/(128M\delta)}\bigcup_{\substack{b\leqslant r\\(b,r)=1}}B(b/r,1/(8Mr)) \end{align*} $$

of major arcs, for if $\alpha \in B(b/r,1/(8Mr))$ , then since $\delta \leqslant 1/(128Mr)$ , we must have

$$ \begin{align*} P_\delta(\alpha)\leqslant\#\{\alpha\in\mathscr{A}:|b/r-\alpha|\leqslant1/(4Mr)\}\leqslant\#\{a/f(q)\in\mathscr{A}:|bf(q)-ar|\leqslant\tfrac12\}, \end{align*} $$

which is clearly zero, since $r\leqslant ({1}/{128})M<f(q)$ . Hence, in the remainder of this section, we shall assume that $\alpha $ belongs to the set $\mathfrak {m}=[0,1]\backslash \mathfrak {M}$ of minor arcs.

By Dirichlet’s approximation theorem, we see that for all $\alpha \in [0,1]$ , there exists a Farey fraction $b/r$ with $r\leqslant 128M$ such that $|b/r-\alpha |\leqslant 1/(128Mr)$ . If $r\leqslant 1/(128M\delta )$ , we must have $\alpha \in \mathfrak {M}$ , so we may assume that this is not the case. Hence, for all ${\alpha \in \mathfrak {m}}$ , there exists a Farey fraction $b/r$ such that $|b/r-\alpha |<\delta $ , and thus, we must have $P_\delta (\alpha )\leqslant P_{2\delta }(b/r)$ .

We now fix a Farey fraction $b/r$ , and suppose that $\phi ,\psi :\mathbb {R}\to \mathbb {R}_+$ are two infinitely differentiable functions having support on $[\tfrac 12,\tfrac 52]$ and $[-\tfrac 92,\tfrac 92]$ , respectively. Moreover, we suppose that $\phi $ and $\psi $ are bounded below by 1 on the intervals $[1,2]$ and $[-4,4]$ , respectively. Then, we have the majorisation

(3.2) $$ \begin{align} P_{2\delta}(b/r)\ll\sum_{q\in{\mathbb{Z}}}\sum_{a\in{\mathbb{Z}}}\phi\bigg(\frac{f(q)}{M}\bigg)\psi\bigg(\frac{ar-bf(q)}{Mr\delta}\bigg). \end{align} $$

Now, supposing that $R>f(Q)^\varepsilon $ , we note that the set $\mathscr {P}=\{p\sim R:p\nmid 4rA\}$ has cardinality $P\sim R/\log R$ , and thus applying Lemma 2.1 to (3.2) yields

(3.3) $$ \begin{align} P_{2\delta}(b/r)&\ll P^{-1}\sum_{q\in \mathbb{Z} }\sum_{a\in \mathbb{Z} }\phi\bigg(\frac{q}{M}\bigg)\psi\bigg(\frac{ar-bq}{Mr\delta}\bigg)\notag\\ &\quad+P^{-2}\sum_{\substack{p,p'\in\mathscr{P}\\p\neq p'}}\bigg|\sum_{q\in \mathbb{Z} }\sum_{a\in \mathbb{Z} }\bigg(\frac{4Aq+\Delta_f}{pp'}\bigg)\phi\bigg(\frac{q}{M}\bigg)\psi\bigg(\frac{ar-bq}{Mr\delta}\bigg)\bigg|. \end{align} $$

For the sake of brevity, we shall denote by $\Sigma _1$ and $\Sigma _2$ the inner double sums of the first and second terms, respectively, on the right-hand side of (3.3).

To treat the sum $\Sigma _1$ , we first note that if $a\in {\mathbb {Z}}$ is such that $|ar-bq|\leqslant \tfrac 92Mr\delta $ for some $q\in {\mathbb {Z}}$ satisfying $\tfrac 12M\leqslant q\leqslant \tfrac 52M$ , then $|a/q-\alpha |\leqslant |b/r-\alpha |+9\delta \leqslant 11\delta $ . If, moreover, $(a,q)>320M^2\delta $ , then $q/(a,q)<1/(128M\delta )$ and thus $11\delta <(a,q)/(8Mq)$ . In this case, we clearly have $\alpha \in B(a/q,(a,q)/(8Mq))\subset \mathfrak {M}$ , which contradicts our assumption that $\alpha \in \mathfrak {m}$ . Hence, we see that the $a,q$ in the double sum $\Sigma _1$ must all satisfy $(a,q)\leqslant 320M^2\delta $ , and thus

$$ \begin{align*} \Sigma_1\leqslant\sum_{\frac12M\leqslant q\leqslant\frac52M}\sum_{qb/r-\frac92M\delta\leqslant a\leqslant qb/r+\frac92M\delta}1\leqslant\sum_{m\leqslant320M^2\delta}\sum_{\frac12M/m\leqslant q\leqslant\frac52M/m}\sum_{\substack{qb/r-\frac92M\delta\leqslant a\leqslant qb/r+\frac92M\delta\\(a,q)=1}}1. \end{align*} $$

On noting that $|a/q-a'/q'|\geqslant 1/(qq')\asymp m^2/M^2$ for any distinct Farey fractions $a/q$ and $a'/q'$ satisfying $q,q'\asymp M/m$ , we derive the bound

(3.4) $$ \begin{align} \Sigma_1\ll\sum_{m\leqslant320M^2\delta}\bigg(1+\frac{M^2\delta}{m^2}\bigg)\ll M^2\delta, \end{align} $$

which completes our treatment of $\Sigma _1$ .

To treat the double sum $\Sigma _2$ , we first split the outer sum into subsums over the residue classes modulo $pp'r$ , and twice apply the Poisson summation formula to obtain the transformation

$$ \begin{align*} \Sigma_2&=\sum_{m\leqslant pp'r}\bigg(\frac{4Am+\Delta_f}{pp'}\bigg)\sum_{q\in\mathbb{Z} }\sum_{a\in\mathbb{Z} }\phi\bigg(\frac{m+pp'rq}{M}\bigg)\psi\bigg(\frac{ar-b(m+pp'rq)}{Mr\delta}\bigg)\\ &=\frac{M^2\delta}{pp'r}\sum_{m\leqslant pp'r}\bigg(\frac{4Am+\Delta_f}{pp'}\bigg)\sum_{q\in\mathbb{Z} }\sum_{a\in\mathbb{Z} }\hat{\phi}\bigg(\frac{M(q+bapp')}{pp'r}\bigg)\hat{\psi}(M\delta a)\boldsymbol{\mathrm{e}}\bigg(\frac{mq}{pp'r}\bigg)\\ &=\frac{M^2\delta}{pp'r}\sum_{q\in\mathbb{Z} }\sum_{a\in\mathbb{Z} }\hat{\phi}\bigg(\frac{Mq}{pp'r}\bigg)\hat{\psi}(M\delta a)\sum_{m\leqslant pp'r}\bigg(\frac{4Am+\Delta_f}{pp'}\bigg)\boldsymbol{\mathrm{e}}\bigg(\frac{m(q-bapp')}{pp'r}\bigg). \end{align*} $$

Hence, by Lemma 2.2 and the standard bound for Gauß sums,

(3.5) $$ \begin{align} \Sigma_2\ll\frac{M^2\delta}{\sqrt{pp'}}\sum_{q\in\mathbb{Z} }\sum_{a\in\mathbb{Z} }\delta_{r|(q-bapp')}\hat{\phi}\bigg(\frac{Mq}{pp'r}\bigg)\hat{\psi}(M\delta a), \end{align} $$

which, when averaged over distinct primes $p,p'\in \mathscr {P}$ , will suffice for our argument. So, first, on combining (3.4) and (3.5) with (3.3), we obtain the majorisation

(3.6) $$ \begin{align} P_{2\delta}(b/r)\ll\frac{M^2\delta}{P}+\frac{M^2\delta}{P^2R}\sum_{\substack{p,p'\in\mathscr{P}\\p\neq p'}}\mathop{\sum_{q\in\mathbb{Z} }\sum_{a\in\mathbb{Z}}}_{r|(q-bapp')}\hat{\phi}\bigg(\frac{Mq}{pp'r}\bigg)\hat{\psi}(M\delta a). \end{align} $$

Now, since $\phi $ and $\psi $ are supported only on singular closed intervals, we see that their respective Fourier transforms must satisfy the property that $\hat {\phi }(x)$ and $\hat {\psi }(x)$ are, for any $C>0$ , majorised by $(1+|x|)^{-C}$ . Consequently, we may truncate the triple sum on the right-hand side of (3.6) to derive the bound

$$ \begin{align*} \sum_{\substack{p,p'\in\mathscr{P}\\p\neq p'}}\mathop{\sum_{q\in \mathbb{Z} }\sum_{a\in \mathbb{Z} }}_{r|(q-bapp')}\hat{\phi}\bigg(\frac{Mq}{pp'r}\bigg)\hat{\psi}(M\delta a)\mathop{\:\ll\:}_\varepsilon f(Q)^{-8}+\sum_{R^2<m\leqslant4R^2}\mathop{\sum\sum}_{\substack{|q|<R^2rf(Q)^\varepsilon/M\\|a|<f(Q)^\varepsilon/(M\delta)\\r|(q-bam)}}1, \end{align*} $$

by virtue of the fact that $M\ll f(Q)$ . Hence, on noting that

$$ \begin{align*} \sum_{R^2<m\leqslant4R^2}\mathop{\sum\sum}_{\substack{|q|<R^2rf(Q)^\varepsilon/(M)\\|a|<f(Q)^\varepsilon/(M\delta)\\r|(q-bam)}}1&\ll\bigg(R^2+\frac{R^4f(Q)^\varepsilon}{M}\bigg)+\sum_{|q|<R^2rf(Q)^\varepsilon/M}\mathop{\sum\sum}_{\substack{R^2<m\leqslant4R^2\\a<f(Q)^\varepsilon/(M\delta)\\r|(q-bam)}}1\\ &\ll\bigg(R^2+\frac{R^4f(Q)^\varepsilon}{M}\bigg)+\sum_{|q|<R^2rf(Q)^\varepsilon/M}\sum_{\substack{m<R^2f(Q)^\varepsilon/(M\delta)\\r|(q-bm)}}\tau(m)\\ &\mathop{\:\ll\:}_\varepsilon\bigg(R^2+\frac{R^4f(Q)^\varepsilon}{M}\bigg)+R^\varepsilon f(Q)^{\varepsilon}\bigg(1+\frac{R^2r}{M}\bigg)\bigg(1+\frac{R^2}{Mr\delta}\bigg)\\ &\mathop{\:\ll\:}_\varepsilon R^\varepsilon f(Q)^{\varepsilon}\bigg(R^2+\frac{R^4}{M^2\delta}\bigg), \end{align*} $$

we obtain from (3.6) the majorisation

(3.7) $$ \begin{align} \sup_{\alpha\in\mathfrak{m}}P_\delta(\alpha)\mathop{\:\ll\:}_\varepsilon R^\varepsilon f(Q)^{\varepsilon}\bigg(\frac{M^2\delta}{R}+R\bigg). \end{align} $$

Taking $R=M\sqrt {\delta }$ in (3.7) yields (3.1), and thus the assertion (1.3) follows.