4.1 Preparations

Owing to (1.3), there exists positive integers $z_{0}=z_{0}(\mathbf{f}),m=m(\mathbf{f})$ such that if we let

$$\begin{eqnarray}W:=\mathop{\prod }_{p\leqslant z_{0}}p^{m},\end{eqnarray}$$

then there exists $\mathbf{y}\in (\mathbb{N}\cap [1,W])^{n}$ fulfilling the following:

(4.1) $$\begin{eqnarray}\text{gcd}(y_{1}\cdots y_{n},W)=1\end{eqnarray}$$

and

(4.2) $$\begin{eqnarray}p\leqslant z_{0}\Rightarrow \unicode[STIX]{x1D70E}_{p}(\mathbf{f}(\mathbf{y}+W\mathbf{x}))>0.\end{eqnarray}$$

Define

(4.3) $$\begin{eqnarray}\mathscr{A}:=\{\mathbf{x}\in \mathbb{Z}^{n}:\mathbf{f}(\mathbf{x})=\mathbf{0},\mathbf{x}\equiv \mathbf{y}~(\text{mod}~W)\}.\end{eqnarray}$$

Let us now choose a non-singular point $\unicode[STIX]{x1D73B}\in V_{\mathbf{f}}(\mathbb{R})$ (whose existence is guaranteed by (1.3)) and we let $\unicode[STIX]{x1D702}\in (0,\min _{i}\{\min \{\unicode[STIX]{x1D701}_{i}/2,(1-\unicode[STIX]{x1D701}_{i})/2\}\})$ be arbitrary. Defining

(4.4) $$\begin{eqnarray}\mathscr{B}_{\unicode[STIX]{x1D702}}:=\bigg\{\mathbf{x}\in \mathbb{R}^{n}:\bigg|\mathbf{x}-{\displaystyle \frac{\unicode[STIX]{x1D73B}}{2|\unicode[STIX]{x1D73B}|}}\bigg|<\unicode[STIX]{x1D702}\bigg\},\end{eqnarray}$$

we see that for any such $\unicode[STIX]{x1D702}$ , one has $\mathscr{B}_{\unicode[STIX]{x1D702}}\subset (0,1)^{n}$ . Now we choose any smooth function $w:\mathbb{R}\rightarrow \mathbb{R}_{{\geqslant}0}$ of compact support in $[-\unicode[STIX]{x1D702}/2,\unicode[STIX]{x1D702}/2]$ and such that if $|t|\leqslant \unicode[STIX]{x1D702}/4$ then $w(t)>0$ . Letting $w_{0}:=\sup \{w(t):t\in \mathbb{R}\}$ we have $\mathbf{1}_{\{0<t\leqslant B\}}(t)\geqslant w_{0}^{-1}w(t/B-\unicode[STIX]{x1D701}_{i}/(2|\unicode[STIX]{x1D73B}|))$ and therefore for every $\mathbf{x}\in \mathbb{Z}^{n}$ ,

(4.5) $$\begin{eqnarray}\mathop{\prod }_{i=1}^{n}\mathbf{1}_{\{0<x_{i}\leqslant B\}}(\mathbf{x})\geqslant w_{0}^{-n}\mathop{\prod }_{i=1}^{n}w\bigg({\displaystyle \frac{x_{i}}{B}}-{\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}\bigg).\end{eqnarray}$$

4.2 A level of distribution result

Let us now take the opportunity to record a level of distribution result that will be the main input in the forthcoming sieving arguments. For $\mathbf{k}\in \mathbb{N}^{n}$ with $\text{gcd}(\widetilde{\mathbf{k}},W)=1$ and each $k_{i}$ being square-free let $w:\mathbb{R}\rightarrow \mathbb{R}_{{\geqslant}0}$ be a smooth weight as above. We let

(4.6) $$\begin{eqnarray}N_{w}(B;\mathbf{k}):=\mathop{\sum }_{\substack{ \mathbf{x}\in \mathscr{A} \\ k_{i}|x_{i}}}\mathop{\prod }_{i=1}^{n}w\bigg({\displaystyle \frac{x_{i}}{B}}-{\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}\bigg).\end{eqnarray}$$

Recall the definition of the matrix $\unicode[STIX]{x1D750}$ in (1.7). Our result will involve an error term related to the following function, defined for $\mathbf{m}\in \mathbb{N}^{n}$ and $B\geqslant 1$ :

$$\begin{eqnarray}E(B;\mathbf{m}):=\mathop{\sum }_{i=1}^{3}B^{-\unicode[STIX]{x1D716}_{i,1}}|\mathbf{m}|^{\unicode[STIX]{x1D716}_{i,2}}\min \{m_{\!j}\}^{\unicode[STIX]{x1D716}_{i,3}}.\end{eqnarray}$$

Furthermore, extend the function $\unicode[STIX]{x1D71B}$ defined in (3.5) to $\mathbb{N}^{n}$ by letting for $\mathbf{k}\in \mathbb{N}^{n}$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D71B}(\mathbf{k}):=\mathop{\prod }_{p|k_{1}\cdots k_{n}}\unicode[STIX]{x1D71B}(p^{\unicode[STIX]{x1D708}_{p}(k_{1})},\ldots ,p^{\unicode[STIX]{x1D708}_{p}(k_{n})})\end{eqnarray}$$

and if $\text{gcd}(k_{1}\cdots k_{n},W)=1$ we define $\unicode[STIX]{x1D749}\in (\mathbb{Z}\,\cap \, [0,W))^{n}$ via $\langle \unicode[STIX]{x1D749}\mathbf{k}\rangle \equiv \mathbf{y}~(\text{mod}~W)$ . Finally, we let

$$\begin{eqnarray}\mathfrak{S}(\mathbf{f},W):=\mathop{\prod }_{p|W}\unicode[STIX]{x1D70E}_{p}(\mathbf{f}(\unicode[STIX]{x1D749}+W\mathbf{s}))\mathop{\prod }_{p\nmid W}\unicode[STIX]{x1D70E}_{p}(\mathbf{f})\end{eqnarray}$$

and

$$\begin{eqnarray}\mathscr{J}_{w}(\mathbf{f},W):={\displaystyle \frac{1}{W^{n}}}\int _{\mathbb{R}^{R}}\int _{\mathbb{R}^{n}}e(\unicode[STIX]{x1D738}\cdot \mathbf{f}(\mathbf{u}))\mathop{\prod }_{i=1}^{n}w\bigg(u_{i}-{\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}\bigg)\,d\mathbf{u}\,d\unicode[STIX]{x1D738}.\end{eqnarray}$$

Lemma 4.1. Assume $\mathfrak{B}(\mathbf{f})>2^{d-1}R(R+1)(d-1)$ and that $\mathbf{k}\in \mathbb{N}^{n}$ satisfies

$$\begin{eqnarray}\text{gcd}(k_{1}\cdots k_{n},W)=1\quad \text{and}\quad |\mathbf{k}|\leqslant B^{1/\unicode[STIX]{x1D70C}}(\log B)^{-1},\end{eqnarray}$$

where $B\in \mathbb{R}_{{\geqslant}1}$ and the constant $\unicode[STIX]{x1D70C}$ was defined in (1.6). Then for each $\unicode[STIX]{x1D716}>0$ we have

$$\begin{eqnarray}N_{w}(B;\mathbf{k})=\mathscr{J}_{w}(\mathbf{f},W)\mathfrak{S}(\mathbf{f},W){\displaystyle \frac{\unicode[STIX]{x1D71B}(\mathbf{k})}{\widetilde{\mathbf{k}}}}B^{n-Rd}+O\bigg({\displaystyle \frac{B^{n+\unicode[STIX]{x1D716}}}{\widetilde{\mathbf{k}}}}E(B;\mathbf{k})\bigg).\end{eqnarray}$$

Proof. Defining $\mathbf{g}(\mathbf{s}):=\mathbf{f}(\langle \mathbf{k}(\unicode[STIX]{x1D749}+W\mathbf{s})\rangle )$ gives

$$\begin{eqnarray}N_{w}(B;\mathbf{k})=\mathop{\sum }_{\substack{ \mathbf{s}\in \mathbb{Z}^{n} \\ \mathbf{g}(\mathbf{s})=\mathbf{0}}}\mathop{\prod }_{i=1}^{n}w\bigg({\displaystyle \frac{s_{i}}{B/k_{i}W}}-\bigg({\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}-{\displaystyle \frac{\unicode[STIX]{x1D70F}_{i}}{B/k_{i}}}\bigg)\bigg).\end{eqnarray}$$

We shall apply Theorem 2.1 at this point; before doing so we need to verify that

$$\begin{eqnarray}\bigg|{\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}-{\displaystyle \frac{\unicode[STIX]{x1D70F}_{i}}{B/k_{i}}}\bigg|\leqslant 1\end{eqnarray}$$

and that condition (2.2) is met. The former is easy to verify due to $|\unicode[STIX]{x1D749}|\leqslant W\ll 1$ and $\unicode[STIX]{x1D70C}>1$ , which implies that $B/k_{i}\geqslant B^{1-1/\unicode[STIX]{x1D70C}}(\log B)\rightarrow +\infty$ . Regarding (2.2), the obvious equality $\mathbf{g}^{\natural }(\mathbf{s})=W^{d}\mathbf{f}(\langle \mathbf{k}\mathbf{s}\rangle )$ presents us with $\max \{\Vert \mathbf{g}^{\natural }\Vert ,\Vert \mathbf{g}\Vert \}\ll |\mathbf{k}|^{d}$ , thus the growth condition on $|\mathbf{k}|$ in our lemma is sufficient. The last issue to be commented regards the real densities. The real density provided by the application of Theorem 2.1 is

$$\begin{eqnarray}\int _{\mathbb{R}^{R}}\int _{\mathbb{R}^{n}}e(W^{d}\boldsymbol{\unicode[STIX]{x1D6FD}}\cdot \mathbf{f}(\langle \mathbf{k}\mathbf{s}\rangle ))\mathop{\prod }_{i=1}^{n}w\bigg({\displaystyle \frac{s_{i}}{B/Wk_{i}}}-\bigg({\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}-{\displaystyle \frac{\unicode[STIX]{x1D70F}_{i}}{B/k_{i}}}\bigg)\bigg)\,d\mathbf{s}\,d\boldsymbol{\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$

Note that the proof of Theorem 2.1 in fact shows that the real density can also be replaced by its inhomogeneous version,

$$\begin{eqnarray}\int _{\mathbb{R}^{R}}\int _{\mathbb{R}^{n}}e(\boldsymbol{\unicode[STIX]{x1D6FD}}\cdot \mathbf{f}(\langle \mathbf{k}\unicode[STIX]{x1D749}\rangle +W\langle \mathbf{k}\mathbf{s}\rangle ))\mathop{\prod }_{i=1}^{n}w\bigg({\displaystyle \frac{s_{i}}{B/Wk_{i}}}-\bigg({\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}-{\displaystyle \frac{\unicode[STIX]{x1D70F}_{i}}{B/k_{i}}}\bigg)\bigg)\,d\mathbf{s}\,d\boldsymbol{\unicode[STIX]{x1D6FD}}.\end{eqnarray}$$

For this we note that the major arc analysis initially came in its inhomogeneous form, namely having $\mathbf{f}(\langle \mathbf{k}\unicode[STIX]{x1D749}\rangle +W\langle \mathbf{k}\mathbf{s}\rangle )$ in the exponential. Moreover, by shifting the centre of the weight functions, one sees that Lemma 2.15 still applies to the inhomogeneous form and then everything stays exactly the same with regard to the error terms.

To continue the proof of our lemma we perform the linear change of variables $s_{i}\mapsto u_{i}$ and $\unicode[STIX]{x1D6FD}_{i}\mapsto \unicode[STIX]{x1D6FE}_{i}$ given by $k_{i}(\unicode[STIX]{x1D70F}_{i}+Ws_{i})=Bu_{i}$ , $B^{d}\unicode[STIX]{x1D6FD}_{i}=\unicode[STIX]{x1D6FE}_{i}$ . This leads to the following expression for the real density in our lemma:

$$\begin{eqnarray}{\displaystyle \frac{B^{n-Rd}}{W^{n}\widetilde{\mathbf{k}}}}\int _{\mathbb{R}^{R}}\int _{\mathbb{R}^{n}}e(\unicode[STIX]{x1D738}\cdot \mathbf{f}(\mathbf{u}))\mathop{\prod }_{i=1}^{n}w\bigg(u_{i}-{\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}\bigg)\,d\mathbf{u}\,d\unicode[STIX]{x1D738},\end{eqnarray}$$

which equals $\mathscr{J}_{w}(\mathbf{f},W)\widetilde{\mathbf{k}}^{-1}B^{n-Rd}$ .◻

The most noteworthy property of Lemma 4.1 is related to the presence of $\widetilde{\mathbf{k}}^{-1}$ in the error term; this allows us to drastically improve the level of distribution in the forthcoming applications.

4.3 Using the Rosser–Iwaniec sieve

By (4.5) we have the following whenever $z$ satisfies $z_{0}<z<B$ :

$$\begin{eqnarray}\mathop{\sum }_{\substack{ \mathbf{x}\in (\mathbb{N}\cap [-B,B])^{n} \\ \mathbf{f}(\mathbf{x})=\mathbf{0},P^{-}(\widetilde{\mathbf{x}})>z}}\hspace{-2.84544pt}1\geqslant w_{0}^{-n}\mathop{\sum }_{\substack{ \mathbf{x}\in \mathscr{A} \\ P^{-}(\widetilde{\mathbf{x}})>z}}\mathop{\prod }_{i=1}^{n}w\bigg({\displaystyle \frac{x_{i}}{B}}-{\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}\bigg).\end{eqnarray}$$

Let us now bring into play a lower bound sieve sequence $\unicode[STIX]{x1D706}_{k}^{-}$ of dimension $n$ . Recall the definition of $\unicode[STIX]{x1D703}^{\prime }$ in (1.5). We shall make use of the terminology in [Reference Friedlander and Iwaniec16, §11.8]; in doing so we shall call the support of $\unicode[STIX]{x1D706}^{-}$ by $D:=B^{\unicode[STIX]{x1D6FF}}$ , for some constant $\unicode[STIX]{x1D6FF}\in (0,\unicode[STIX]{x1D703}^{\prime }).$ Using $(1\ast \unicode[STIX]{x1D707})(l)\geqslant (1\ast \unicode[STIX]{x1D706}^{-})(l)$ for $l=\text{gcd}(P(z_{0},z),\widetilde{\mathbf{x}})$ yields

$$\begin{eqnarray}\mathop{\sum }_{\substack{ \mathbf{x}\in (\mathbb{N}\cap [-B,B])^{n} \\ \mathbf{f}(\mathbf{x})=\mathbf{0},P^{-}(\widetilde{\mathbf{x}})>z}}\hspace{-2.84544pt}1\geqslant w_{0}^{-n}\mathop{\sum }_{\substack{ k|P(z_{0},z) \\ k\leqslant B^{\unicode[STIX]{x1D6FF}}}}\unicode[STIX]{x1D706}_{k}^{-}\mathop{\sum }_{\substack{ \mathbf{x}\in \mathscr{A} \\ k|\widetilde{\mathbf{x}}}}\mathop{\prod }_{i=1}^{n}w\bigg({\displaystyle \frac{x_{i}}{B}}-{\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}\bigg).\end{eqnarray}$$

The proof of [Reference Brüdern and Fouvry7, Lemma 8] can be directly adapted in the setting of arbitrary dimension, thus providing the equality of the inner sum over $\mathbf{x}$ to

$$\begin{eqnarray}\unicode[STIX]{x1D707}(k)\mathop{\sum }_{\substack{ \mathbf{k}\in \mathbb{N}^{n} \\ p|\widetilde{\mathbf{k}}\Leftrightarrow p|k}}\unicode[STIX]{x1D707}(\mathbf{k})N_{w}(B;\mathbf{k}),\end{eqnarray}$$

where here and throughout the rest of the paper we will use the notation

$$\begin{eqnarray}\unicode[STIX]{x1D707}(\mathbf{k}):=\unicode[STIX]{x1D707}(k_{1})\cdots \unicode[STIX]{x1D707}(k_{n}).\end{eqnarray}$$

A moment’s thought reveals that the succeeding function is multiplicative,

(4.7) $$\begin{eqnarray}g(k):=\mathbf{1}_{(k,W)=1}(k)\unicode[STIX]{x1D707}(k)\mathop{\sum }_{\substack{ \mathbf{k}\in \mathbb{N}^{n} \\ p|\widetilde{\mathbf{k}}\Leftrightarrow p|k}}\unicode[STIX]{x1D707}(\mathbf{k})\unicode[STIX]{x1D71B}(\mathbf{k})\widetilde{\mathbf{k}}^{-1},\end{eqnarray}$$

a notation which allows us to assort our conclusions so far in the following form:

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ \mathbf{x}\in (\mathbb{N}\cap [-B,B])^{n} \\ \mathbf{f}(\mathbf{x})=\mathbf{0},P^{-}(\widetilde{\mathbf{x}})>z}}1 & \gg & \displaystyle B^{n-Rd}\mathop{\sum }_{\substack{ k|P(z_{0},z) \\ k\leqslant B^{\unicode[STIX]{x1D6FF}}}}\unicode[STIX]{x1D706}_{k}^{-}g(k)\nonumber\\ \displaystyle & & \displaystyle +\,O\bigg(B^{n+\unicode[STIX]{x1D716}}\mathop{\sum }_{k\leqslant B^{\unicode[STIX]{x1D6FF}}}|\unicode[STIX]{x1D707}(k)|\mathop{\sum }_{\substack{ \mathbf{k}\in \mathbb{N}^{n} \\ p|\widetilde{\mathbf{k}}\Leftrightarrow p|k}}{\displaystyle \frac{|\unicode[STIX]{x1D707}(\mathbf{k})|}{\widetilde{\mathbf{k}}}}E(B;\mathbf{k})\bigg).\nonumber\end{eqnarray}$$

In bounding the error term we will be confronted with sums of the form

$$\begin{eqnarray}b_{k}:=|\unicode[STIX]{x1D707}(k)|\mathop{\sum }_{\substack{ \mathbf{k}\in \mathbb{N}^{n} \\ p|\widetilde{\mathbf{k}}\Leftrightarrow p|k}}{\displaystyle \frac{|\unicode[STIX]{x1D707}(\mathbf{k})|}{\widetilde{\mathbf{k}}}}|\mathbf{k}|^{\unicode[STIX]{x1D6FC}_{1}}\min \{k_{i}\}^{\unicode[STIX]{x1D6FC}_{2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{i}\geqslant 0$ . Each $\mathbf{k}$ making a contribution to $b_{k}$ satisfies $|\mathbf{k}|\leqslant k\leqslant \widetilde{\mathbf{k}}$ , therefore

$$\begin{eqnarray}b_{k}\ll |\unicode[STIX]{x1D707}(k)|k^{\unicode[STIX]{x1D6FC}_{1}+\unicode[STIX]{x1D6FC}_{2}-1+\unicode[STIX]{x1D716}}.\end{eqnarray}$$

We deduce that for each $1\leqslant j\leqslant 3$ , the quantity

$$\begin{eqnarray}B^{-\unicode[STIX]{x1D716}_{j,1}}\mathop{\sum }_{k\leqslant B^{\unicode[STIX]{x1D6FF}}}k^{\unicode[STIX]{x1D716}_{j,2}+\unicode[STIX]{x1D716}_{j,3}-1}\ll B^{-\unicode[STIX]{x1D716}_{j,1}+\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D716}_{j,2}+\unicode[STIX]{x1D716}_{j,3})}\end{eqnarray}$$

becomes $\ll B^{-Rd-\unicode[STIX]{x1D716}^{\prime }}$ for some $\unicode[STIX]{x1D716}^{\prime }>0$ due to $\unicode[STIX]{x1D6FF}<\unicode[STIX]{x1D703}^{\prime }$ . Therefore, we can see that for each $\unicode[STIX]{x1D6FF}\in (0,\unicode[STIX]{x1D703}^{\prime })$ and $\unicode[STIX]{x1D716}>0$ there exists $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}(\unicode[STIX]{x1D716},\unicode[STIX]{x1D6FF})>0$ such that

$$\begin{eqnarray}B^{n+\unicode[STIX]{x1D716}}\mathop{\sum }_{k\leqslant B^{\unicode[STIX]{x1D6FF}}}|\unicode[STIX]{x1D707}(k)|\mathop{\sum }_{\substack{ \mathbf{k}\in \mathbb{N}^{n} \\ p|\widetilde{\mathbf{k}}\Leftrightarrow p|k}}{\displaystyle \frac{|\unicode[STIX]{x1D707}(\mathbf{k})|}{\widetilde{\mathbf{k}}}}E(B;\mathbf{k})\ll B^{n-Rd-\unicode[STIX]{x1D702}}.\end{eqnarray}$$

This leads to the conclusion that subject to the assertion

(4.8) $$\begin{eqnarray}\mathop{\sum }_{\substack{ k|P(z_{0},z) \\ k\leqslant B^{\unicode[STIX]{x1D6FF}}}}\unicode[STIX]{x1D706}_{k}^{-}g(k)\gg (\log B)^{-n}\end{eqnarray}$$

we can establish Theorem 1.5 due to

$$\begin{eqnarray}\mathop{\sum }_{\substack{ \mathbf{x}\in (\mathbb{N}\cap [-B,B])^{n} \\ \mathbf{f}(\mathbf{x})=\mathbf{0},P^{-}(\widetilde{\mathbf{x}})>z}}1\gg {\displaystyle \frac{B^{n-Rd}}{(\log B)^{n}}}.\end{eqnarray}$$

To prove (4.8) we shall use [Reference Friedlander and Iwaniec16, Theorem 11.12]. To this end, for any polynomials $h_{i}\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ we abbreviate

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{p}(p|\mathbf{h}(\mathbf{x})):=\lim _{l\rightarrow +\infty }p^{-(n-R)l}\sharp \{1\leqslant \mathbf{x}\leqslant p^{l}:p^{l}|\mathbf{f}(\mathbf{x}),p|\mathbf{h}(\mathbf{x})\}.\end{eqnarray}$$

Proof. The definition (4.7) furnishes

$$\begin{eqnarray}g(p)\unicode[STIX]{x1D70E}_{p}=\mathop{\sum }_{m=1}^{n}{\displaystyle \frac{(-1)^{m-1}}{p^{m}}}\mathop{\sum }_{\substack{ \mathbf{j}\in \{0,1\}^{n} \\ |\mathbf{j}|_{1}=m}}\unicode[STIX]{x1D6FF}(\mathbf{j}),\end{eqnarray}$$

thus, letting $N_{\mathbf{j}}(p^{l}):=\sharp \{1\leqslant x_{1},\ldots ,x_{l}\leqslant p^{l}:\mathbf{f}((p^{j_{i}}x_{i}))\equiv \mathbf{0}~(\text{mod}~p^{l})\}$ , we conclude that

(4.9) $$\begin{eqnarray}g(p)\lim _{l\rightarrow +\infty }{\displaystyle \frac{N_{\mathbf{\mathbf{0}}}(p^{l})}{p^{(n-R)l}}}=\lim _{l\rightarrow +\infty }\mathop{\sum }_{m=1}^{n}{\displaystyle \frac{(-1)^{m-1}}{p^{m}}}\mathop{\sum }_{\substack{ \mathbf{j}\in \{0,1\}^{n} \\ |\mathbf{j}|_{1}=m}}{\displaystyle \frac{N_{\mathbf{j}}(p^{l})}{p^{(n-R)l}}}.\end{eqnarray}$$

Obviously, if $j_{i}=1$ and $y_{i}\equiv x_{i}~(\text{mod}~p^{l-1})$ then $p^{j_{i}}y_{i}\equiv p^{j_{i}}x_{i}~(\text{mod}~p^{l})$ . Therefore we may split the interval $[1,p^{l}]$ into $p$ subintervals of length $p^{l-1}$ to obtain

$$\begin{eqnarray}\displaystyle N_{\mathbf{j}}(p^{l}) & = & \displaystyle p^{|\mathbf{j}|_{1}}\sharp \{\!j_{i}=1\Rightarrow 1\leqslant x_{i}\leqslant p^{l-1},\nonumber\\ \displaystyle & & \displaystyle \qquad \,\,\,j_{i}=0\Rightarrow 1\leqslant x_{i}\leqslant p^{l}:\mathbf{f}((p^{j_{i}}x_{i}))\equiv \mathbf{0}~(\text{mod}~p^{l})\!\}.\nonumber\end{eqnarray}$$

One can see that this entity equals $\sharp \{\mathbf{x}\leqslant p^{l}:\mathbf{f}(\mathbf{x})\equiv \mathbf{0}~(\text{mod}~p^{l}),j_{i}=1\Rightarrow p|x_{i}\}$ , hence, combining this with (4.9) yields the desired result.◻

Lemma 4.3. There exists $\unicode[STIX]{x1D716}_{0}\in (0,1)$ such that one has

$$\begin{eqnarray}g(p)={\displaystyle \frac{n}{p}}+O(p^{-1-\unicode[STIX]{x1D716}_{0}}).\end{eqnarray}$$

Proof. For a prime $p$ and $t\in \mathbb{N}$ let $M(p^{t}):=\sharp \{1\leqslant \mathbf{x}\leqslant p^{t}:p^{t}|\mathbf{f}(\mathbf{x}),~p\nmid x_{1}\cdot \ldots \cdot x_{n}\}$ . Then [Reference Cook and Magyar11, Lemmas 11–12] imply that there exists a positive $\unicode[STIX]{x1D716}_{0}>0$ such that

$$\begin{eqnarray}\bigg(1-{\displaystyle \frac{1}{p}}\bigg)^{-n}\lim _{t\rightarrow \infty }p^{-t(n-R)}M(p^{t})=1+O(p^{-1-\unicode[STIX]{x1D716}_{0}}).\end{eqnarray}$$

We observe that $\lim _{t\rightarrow \infty }p^{-t(n-R)}M(p^{t})=\unicode[STIX]{x1D70E}_{p}-\unicode[STIX]{x1D70E}_{p}(p|x_{1}\ldots x_{n})$ , thus Lemma 4.2 reveals that

$$\begin{eqnarray}\displaystyle g(p) & = & \displaystyle {\displaystyle \frac{\unicode[STIX]{x1D70E}_{p}(p|x_{1}\ldots x_{n})}{\unicode[STIX]{x1D70E}_{p}}}\nonumber\\ \displaystyle & = & \displaystyle 1-\unicode[STIX]{x1D70E}_{p}^{-1}\lim _{t\rightarrow \infty }p^{-t(n-R)}M(p^{t})\nonumber\\ \displaystyle & = & \displaystyle 1-\bigg(1-{\displaystyle \frac{1}{p}}\bigg)^{n}{\displaystyle \frac{1}{\unicode[STIX]{x1D70E}_{p}}}+O(\unicode[STIX]{x1D70E}_{p}^{-1}p^{-1-\unicode[STIX]{x1D716}_{0}}).\nonumber\end{eqnarray}$$

The work of Birch [Reference Birch2] establishes the existence of a positive $\unicode[STIX]{x1D716}_{1}$ such that $\unicode[STIX]{x1D70E}_{p}=1+O(p^{-1-\unicode[STIX]{x1D716}_{1}}).$ This is sufficient for our lemma.◻

Enlarging $z_{0}$ if necessary, ensures that for all primes $p$ we have

$$\begin{eqnarray}0\leqslant g(p)<1\quad \text{and}\quad g(p)\leqslant {\displaystyle \frac{n}{p}}+O(p^{-1-\unicode[STIX]{x1D716}_{0}}).\end{eqnarray}$$

This means that one can take $\unicode[STIX]{x1D705}=n$ in [Reference Friedlander and Iwaniec16, equation (11.129)], hence our sieve problem has dimension $n$ . By [Reference Diamond and Halberstam15, Theorem 17.2, Proposition 17.3], the sieving limit $\unicode[STIX]{x1D6FD}$ fulfils $\unicode[STIX]{x1D6FD}\leqslant 3.75n$ , thus [Reference Friedlander and Iwaniec16, Theorem 11.12], in combination with Lemma 4.3, guarantees the veracity of (4.8) under the condition

$$\begin{eqnarray}{\displaystyle \frac{\log D}{\log z}}>3.75n.\end{eqnarray}$$

This concludes the proof of Theorem 1.5.

4.4 Using the weighted sieve

In the last section we saw that sieving out small prime divisors of $x_{1}\cdots x_{n}$ for integer zeros of $\mathbf{f}(\mathbf{x})=\mathbf{0}$ gives rise to a sieve of dimension $n$ . When the dimension of the sieve increases then the weighted sieve gives better results for the number of prime divisors in our sequence. We would like to use the weighted sieve in the form given in the Cambridge Tract of Diamond and Halberstam [Reference Diamond and Halberstam15, Theorem 11.1], however we shall need a more flexible version of their work; one that allows the use of smooth weights. This will follow from a generalization of the weighted sieve that will be given in §4.4.1. This generalization permits the use of any suitable non-negative function rather than just a smooth weight as well as sieving in multisets.

4.4.1 The weighted sieve with smooth weights

We assume that $\mathscr{M}$ is any set equipped with two functions $\unicode[STIX]{x1D70B}:\mathscr{M}\rightarrow \mathbb{N},h:\mathscr{M}\rightarrow \mathbb{R}$ such that

(4.10) $$\begin{eqnarray}h(\mathscr{M})\subset [0,1],\quad h\neq 0,\sharp \mathscr{M}<\infty .\end{eqnarray}$$

For convenience of presentation we shall prefer the notation $\overline{m}=\unicode[STIX]{x1D70B}(m)$ . We also assume that there exists a set of primes $\mathscr{P}$ , a constant $X\in \mathbb{R}_{{>}0}$ and a multiplicative function $\unicode[STIX]{x1D714}:\mathbb{N}\rightarrow \mathbb{R}_{{\geqslant}0}$ such that, when letting

$$\begin{eqnarray}r_{\!\mathscr{M},h}(k):=\mathop{\sum }_{\substack{ m\in \mathscr{M} \\ b|\overline{m}}}h(m)-{\displaystyle \frac{\unicode[STIX]{x1D714}(b)}{b}}X\quad (b\in \mathbb{N}),\end{eqnarray}$$

there exist constants $\unicode[STIX]{x1D70F}\in (0,1]$ , $\unicode[STIX]{x1D705}\in \mathbb{N}$ , $A_{1}\geqslant 1$ and $A_{2}\geqslant 1$ such that

(4.11) $$\begin{eqnarray}\mathop{\sum }_{1\leqslant b\leqslant X^{\unicode[STIX]{x1D70F}}(\log X)^{-A_{1}}}\unicode[STIX]{x1D707}(b)^{2}4^{\unicode[STIX]{x1D708}(b)}|r_{\!\mathscr{ M},h}(b)|\leqslant A_{2}{\displaystyle \frac{X}{(\log X)^{\unicode[STIX]{x1D705}+1}}},\end{eqnarray}$$

where the function $\unicode[STIX]{x1D714}$ enjoys the following properties for some constants $\unicode[STIX]{x1D705}\geqslant 1,A>1$ :

(4.12) $$\begin{eqnarray}0\leqslant \unicode[STIX]{x1D714}(p)<p\,\,(p\in \mathscr{P}),\quad \unicode[STIX]{x1D714}(p)=0\,\,(p\notin \mathscr{P})\end{eqnarray}$$

(4.13) $$\begin{eqnarray}\mathop{\prod }_{w_{1}\leqslant p<w}\bigg(1-{\displaystyle \frac{\unicode[STIX]{x1D714}(p)}{p}}\bigg)^{-1}\leqslant \bigg({\displaystyle \frac{\log w}{\log w_{1}}}\bigg)^{\unicode[STIX]{x1D705}}\bigg(1+{\displaystyle \frac{A}{\log w_{1}}}\bigg),\quad 2\leqslant w_{1}<w.\end{eqnarray}$$

We furthermore demand that

(4.14) $$\begin{eqnarray}m\in \mathscr{M},\quad p|\overline{m}\Rightarrow p\in \mathscr{P},\end{eqnarray}$$

and that there exists a constant $\unicode[STIX]{x1D707}_{0}>0$ such that

(4.15) $$\begin{eqnarray}\max \{|\overline{m}|:m\in \mathscr{M}\}\leqslant X^{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D707}_{0}}.\end{eqnarray}$$

Lastly, we shall say that the property $\mathbf{Q}(u,v)$ holds for two real positive numbers $u<v$ if

(4.16) $$\begin{eqnarray}\mathbf{Q}(u,v):\mathop{\sum }_{\substack{ X^{1/v}\leqslant p\leqslant X^{1/u} \\ p\in \mathscr{P}}}\mathop{\sum }_{\substack{ m\in \mathscr{M} \\ p^{2}|\overline{m}}}h(m)\ll {\displaystyle \frac{X}{\log X}}\mathop{\prod }_{\substack{ p\in \mathscr{P} \\ p<X^{1/v}}}\bigg(1-{\displaystyle \frac{\unicode[STIX]{x1D714}(p)}{p}}\bigg).\end{eqnarray}$$

Before stating the main theorem in this section recall the definition of $f=f_{\unicode[STIX]{x1D705}},F=F_{\unicode[STIX]{x1D705}}$ and $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}$ in [Reference Diamond and Halberstam15, Theorem 6.1] through certain differential equations. The inequality $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}<\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D705}}$ is proved for $\unicode[STIX]{x1D705}\geqslant 200$ in [Reference Diamond and Halberstam14, Theorem 2]; here $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D705}}$ is the Ankeni–Onishi sieving limit [Reference Ankeny and Onishi1] that satisfies $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D705}}\sim c\unicode[STIX]{x1D705}$ as $\unicode[STIX]{x1D705}\rightarrow +\infty$ , where

$$\begin{eqnarray}c={\displaystyle \frac{2}{e\log 2}}\exp \bigg(\int _{0}^{2}{\displaystyle \frac{\text{e}^{u}-1}{u}}\,du\bigg)=2.445\ldots .\end{eqnarray}$$

In particular there exists an absolute positive constant $c_{0}$ such that $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}\leqslant c_{0}\unicode[STIX]{x1D705}$ for all $\unicode[STIX]{x1D705}\geqslant 1$ .

Theorem 4.4 (Diamond–Halberstam–Richert).

Assume that $\unicode[STIX]{x1D705}\geqslant 1,\mathscr{M},X,\unicode[STIX]{x1D714},\unicode[STIX]{x1D707}_{0}$ are as above, that each one of the conditions (4.10)–(4.15) holds, that $r$ is a natural number satisfying $r>N(u,v;\unicode[STIX]{x1D705},\unicode[STIX]{x1D707}_{0},\unicode[STIX]{x1D70F})$ , where

$$\begin{eqnarray}N(u,v;\unicode[STIX]{x1D705},\unicode[STIX]{x1D707}_{0},\unicode[STIX]{x1D70F}):=\unicode[STIX]{x1D70F}\unicode[STIX]{x1D707}_{0}u-1+{\displaystyle \frac{\unicode[STIX]{x1D705}}{f_{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D70F}v)}}\int _{u}^{v}F_{\unicode[STIX]{x1D705}}\bigg(v\bigg(\unicode[STIX]{x1D70F}-{\displaystyle \frac{1}{s}}\bigg)\bigg)\bigg(1-{\displaystyle \frac{u}{s}}\bigg){\displaystyle \frac{ds}{s}}\end{eqnarray}$$

and $u,v$ satisfy $\mathbf{Q(u,v)},\unicode[STIX]{x1D70F}v>\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}$ as well as $1/\unicode[STIX]{x1D70F}<u<v$ . Then we have

$$\begin{eqnarray}\sharp \{m\in \mathscr{M},P^{-}(\overline{m})\geqslant X^{1/v},\unicode[STIX]{x1D6FA}(\overline{m})\leqslant r\}\gg X\mathop{\prod }_{\substack{ p\in \mathscr{P} \\ p<X^{1/v}}}\bigg(1-{\displaystyle \frac{\unicode[STIX]{x1D714}(p)}{p}}\bigg).\end{eqnarray}$$

Proof. The proof is merely a careful recast of the proof of Theorem $11.1$ in [Reference Diamond and Halberstam15, §11]. In place of the function defined in [Reference Diamond and Halberstam15, equation (11.6)] we shall use the following function that combines the classical weights related to the weighted sieve in addition to the new weight $h$ :

$$\begin{eqnarray}W_{h}(\mathscr{M},\mathscr{ P},z,y,\unicode[STIX]{x1D706}):=\mathop{\sum }_{\substack{ m\in \mathscr{M} \\ \text{gcd}(\overline{m},P(z))=1}}h(m)\bigg\{\unicode[STIX]{x1D706}-\mathop{\sum }_{\substack{ p\in \mathscr{P},p|\overline{m} \\ z\leqslant p<y}}\bigg(1-{\displaystyle \frac{\log p}{\log y}}\bigg)\bigg\},\end{eqnarray}$$

where $P(z):=\prod \{p:p\in \mathscr{P},p<z\}$ . A statement analogous to [Reference Diamond and Halberstam15, equation (11.9)] can be verified once the entities $S(\mathscr{A},\mathscr{P},X^{1/v})$ and $S(\mathscr{A}_{p},\mathscr{P},X^{1/v})$ are replaced by

$$\begin{eqnarray}\mathop{\sum }_{\substack{ m\in \mathscr{M} \\ \text{gcd}(\overline{m},P(X^{1/v}))=1}}h(m)\quad \text{and}\quad \mathop{\sum }_{\substack{ m\in \mathscr{M},p|\overline{m} \\ \text{gcd}(\overline{m},P(X^{1/v}))=1}}h(m)\end{eqnarray}$$

respectively. The rest of the arguments in [Reference Diamond and Halberstam15, §11.2] are carried automatically to our setting since, once the level of distribution result (4.11) is applied, all information regarding $\mathscr{M}$ and $h$ is absorbed into $X$ . The only point of departure is the use of various sieve estimates from previous chapters of the book. These sieve estimates boil down to the use of the fundamental lemma of sieve theory and the Selberg sieve, both of which can be adapted to our setting. This is due to the non-negativity of the function $h$ , which allows various combinatorial inequalities to be adapted once multiplied by $h$ . One example of this is in the case of an upper bound sieve, say $\unicode[STIX]{x1D706}^{+}$ : opening up the convolution in the right side of $(1\ast \unicode[STIX]{x1D707})\leqslant (1\ast \unicode[STIX]{x1D706}^{+})$ gives

$$\begin{eqnarray}\mathop{\sum }_{\substack{ m\in \mathscr{M} \\ \text{gcd}(\overline{m},P(z))=1}}h(m)\leqslant \mathop{\sum }_{k|P(z)}\unicode[STIX]{x1D706}_{k}^{+}\mathop{\sum }_{\substack{ m\in \mathscr{M} \\ k|\overline{m}}}h(m),\end{eqnarray}$$

and one can now use (4.11) to absorb $\mathscr{M}$ and $h$ in $X$ for the rest of the argument.

For the proof of the present theorem it remains to adapt the arguments in [Reference Diamond and Halberstam15, §11.3]. First, the contribution towards $\sum _{m}h(m)$ of those $m\in \mathscr{M}$ such that $\overline{m}$ is divisible by the square of a prime $p\in \mathscr{P}$ in the range $X^{1/v}\leqslant p\leqslant X^{1/u}$ can be safely ignored due to condition (4.16). An inspection of [Reference Diamond and Halberstam15, §11] reveals that condition $\mathbf{Q_{0}}$ in [Reference Diamond and Halberstam15, equation (11.2)] is used in the proof of [Reference Diamond and Halberstam15, Theorem 11.1] only to deal with this particular sum over primes in $\mathscr{P}\cap [X^{1/v},X^{1/u}]$ . We are thus free to focus our attention exclusively on the contribution of the elements of the set

$$\begin{eqnarray}\mathscr{M}^{\prime }:=\{m^{\prime }\in \mathscr{M}:\text{ there is no prime }p\in \mathscr{P}\cap [X^{1/v},X^{1/u}]\text{ such that }p^{2}\mid \overline{m}^{\prime }\}.\end{eqnarray}$$

The last inequality in [Reference Diamond and Halberstam15, p. 140] becomes

$$\begin{eqnarray}\mathop{\sum }_{\substack{ X^{1/v}\leqslant p<X^{1/u} \\ p\in \mathscr{P},p|\overline{m}^{\prime }}}\bigg(1-{\displaystyle \frac{u\log p}{\log X}}\bigg)\geqslant \unicode[STIX]{x1D6FA}(\overline{m}^{\prime })-{\displaystyle \frac{u\log |\overline{m}^{\prime }|}{\log X}},\end{eqnarray}$$

which, when multiplied by $h(m^{\prime })$ , gives, as in [Reference Diamond and Halberstam15, p. 141],

$$\begin{eqnarray}W_{h}(\mathscr{M}^{\prime },\mathscr{P},z,y,\unicode[STIX]{x1D706})\leqslant (r+1)\mathop{\sum }_{\substack{ m^{\prime }\in \mathscr{M}^{\prime },\unicode[STIX]{x1D6FA}(\overline{m}^{\prime })\leqslant r \\ \text{gcd}(\overline{m}^{\prime },P(X^{1/v}))=1}}h(m^{\prime })\end{eqnarray}$$

for the choice of $\unicode[STIX]{x1D706}$ and $r$ made in [Reference Diamond and Halberstam15, p. 141]. The property $h(\mathscr{M})\subset [0,1]$ shows that

$$\begin{eqnarray}\displaystyle & & \displaystyle \sharp \{m\in \mathscr{M}:P^{-}(\overline{m})\geqslant X^{1/v},\unicode[STIX]{x1D6FA}(\overline{m})\leqslant r\}\nonumber\\ \displaystyle & & \displaystyle \qquad \geqslant \sharp \{m^{\prime }\in \mathscr{M}^{\prime }:P^{-}(\overline{m}^{\prime })\geqslant X^{1/v},\unicode[STIX]{x1D6FA}(\overline{m}^{\prime })\leqslant r\}\nonumber\\ \displaystyle & & \displaystyle \qquad \geqslant \mathop{\sum }_{\substack{ m^{\prime }\in \mathscr{M},\unicode[STIX]{x1D6FA}(\overline{m}^{\prime })\leqslant r \\ \text{gcd}(\overline{m}^{\prime },P(X^{1/v}))=1}}h(m^{\prime })\geqslant {\displaystyle \frac{1}{r+1}}W_{h}(\mathscr{M}^{\prime },\mathscr{P},z,y,\unicode[STIX]{x1D706}),\nonumber\end{eqnarray}$$

which allows the rest of the proof of [Reference Diamond and Halberstam15, Theorem 11.1] to be adapted to our case. Finally, the choice of the constants $v$ and $r$ given in our theorem is borrowed from the inequalities succeeding [Reference Diamond and Halberstam15, equation (11.22)].◻

Remark 4.5. The setting of Theorem 4.4 includes that of [Reference Diamond and Halberstam15, Theorem 1.1]; indeed, one can choose $(\mathscr{M},\unicode[STIX]{x1D70B},h)=(\mathscr{A},\text{id},1)$ .

Remark 4.6. In most cases it is easy to verify $\mathbf{Q}(u,v)$ for all $u,v>0$ , however this is not the case for the problem of prime factors of $x_{1}\cdots x_{n}$ for integer solutions $\mathbf{x}=(x_{1},\ldots ,x_{n})$ of general Diophantine equations, since, as explained in §3, quite often a prime could divide two coordinates of $\mathbf{x}$ .

Remark 4.7. A table of estimates for $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}$ for $1\leqslant \unicode[STIX]{x1D705}\leqslant 10$ is given in [Reference Diamond and Halberstam15, p. 227]. Furthermore, [Reference Diamond and Halberstam15, equation (11.21)] contains estimates for $r$ that are slightly weaker but simpler than that of [Reference Diamond and Halberstam15, Theorem 11.1]. For example, the choice $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}$ in [Reference Diamond and Halberstam15, equation (11.21)] shows that, as long as $\mathbf{Q}((2\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}-1)/\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}},(2\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}-1)/\unicode[STIX]{x1D70F})$ holds, then the conclusion of Theorem 4.4 remains valid with $v=(2\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}-1)/\unicode[STIX]{x1D70F}$ and for all natural numbers $r$ satisfying

(4.17) $$\begin{eqnarray}r\geqslant \unicode[STIX]{x1D707}_{0}-1+(\unicode[STIX]{x1D707}_{0}-\unicode[STIX]{x1D705})(1-1/\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}})+(\unicode[STIX]{x1D705}+1)\log \unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}.\end{eqnarray}$$

In fact [Reference Diamond and Halberstam15, equation (11.21)] with $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}$ shows that if $\mathbf{Q}(u,v)$ holds for some $u>1/\unicode[STIX]{x1D70F}$ and any $v>u$ , then letting

$$\begin{eqnarray}v^{\prime }:={\displaystyle \frac{\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}-1}{\unicode[STIX]{x1D70F}-1/u}}\end{eqnarray}$$

we deduce that the conclusion of Theorem 4.4 still holds with any $r$ satisfying

(4.18) $$\begin{eqnarray}r\geqslant \unicode[STIX]{x1D70F}\unicode[STIX]{x1D707}_{0}u-1+\bigg(\unicode[STIX]{x1D705}+{\displaystyle \frac{u}{v^{\prime }}}\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D705}}\bigg)\log {\displaystyle \frac{v^{\prime }}{u}}-\unicode[STIX]{x1D705}\bigg(1-{\displaystyle \frac{u}{v^{\prime }}}\bigg).\end{eqnarray}$$

To prove Theorem 1.6 we take

$$\begin{eqnarray}\mathscr{M}:=\{\mathbf{x}\in \mathbb{N}^{n}:\mathbf{f}(\mathbf{x})=\mathbf{0},\mathbf{x}\equiv \mathbf{y}~(\text{mod}~W),|\mathbf{x}|\leqslant B\},\unicode[STIX]{x1D70B}(\mathbf{x}):=\widetilde{\mathbf{x}},\end{eqnarray}$$

and we let

$$\begin{eqnarray}h(\mathbf{x}):=\mathop{\prod }_{i=1}^{n}w\bigg({\displaystyle \frac{x_{i}}{B}}-{\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}\bigg).\end{eqnarray}$$

Then for $\mathscr{P}$ being the set of all primes $p>z_{0}$ , $g$ as in (4.7), $\unicode[STIX]{x1D703}^{\prime }$ as in (1.5) and any $0<\unicode[STIX]{x1D716}<\unicode[STIX]{x1D703}^{\prime }$ we can verify all conditions (4.11)–(4.15) with

$$\begin{eqnarray}\displaystyle X & := & \displaystyle \mathscr{J}_{w}(\mathbf{f},W)\mathfrak{S}(\mathbf{f},W)B^{n-Rd},\qquad \unicode[STIX]{x1D714}(b):=bg(b),\nonumber\\ \displaystyle \unicode[STIX]{x1D705} & := & \displaystyle n,\qquad \unicode[STIX]{x1D70F}:=\unicode[STIX]{x1D703}^{\prime }-\unicode[STIX]{x1D716},\qquad \unicode[STIX]{x1D707}_{0}={\displaystyle \frac{n}{n-Rd}}{\displaystyle \frac{1+\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D703}^{\prime }-\unicode[STIX]{x1D716}}}\nonumber\end{eqnarray}$$

with an argument that is identical to that in §4.3. It remains to check condition $\mathbf{Q}(u,v)$ and for this we note that in our setting, the sum in (4.16) is at most

$$\begin{eqnarray}\mathop{\sum }_{X^{1/v}<p\leqslant X^{1/u}}\mathop{\sum }_{\substack{ \mathbf{k}\in \mathbb{N}^{n} \\ \widetilde{\mathbf{k}}=p^{2}}}\mathop{\sum }_{\substack{ \mathbf{x}\in \mathscr{A} \\ k_{i}|x_{i}}}\mathop{\prod }_{i=1}^{n}w\bigg({\displaystyle \frac{x_{i}}{B}}-{\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}\bigg).\end{eqnarray}$$

Invoking Lemma 4.1 we see that, if $u>2(n-Rd)\unicode[STIX]{x1D70C}$ , where $\unicode[STIX]{x1D70C}$ is defined in (1.6), this is

$$\begin{eqnarray}\displaystyle & & \displaystyle \ll B^{n-Rd}\bigg(\mathop{\sum }_{X^{1/v}\leqslant p\leqslant X^{1/u}}p^{-2}\mathop{\sum }_{\substack{ \mathbf{k}\in \mathbb{N}^{n} \\ \widetilde{\mathbf{k}}=p^{2}}}\unicode[STIX]{x1D71B}(\mathbf{k})\bigg)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,B^{n+\unicode[STIX]{x1D716}}\bigg(\mathop{\sum }_{X^{1/v}<p\leqslant X^{1/u}}p^{-2}\mathop{\sum }_{\substack{ \mathbf{k}\in \mathbb{N}^{n} \\ \widetilde{\mathbf{k}}=p^{2}}}E(B;\mathbf{k})\bigg).\nonumber\end{eqnarray}$$

Assuming $\max \{(d^{2}-1)R^{2}2^{d-1},(d-1)R^{2}2^{d-1}+2(R+1)\}<\mathfrak{B}(\mathbf{f})$ , we obtain via Corollary 3.6 that the first sum over $\mathbf{k}$ above is $\ll 1$ , thus, when $v>0$ , the first term contributes

$$\begin{eqnarray}\ll B^{n-Rd-(n-Rd)/v}\ll {\displaystyle \frac{B^{n-Rd}}{(\log B)^{n}}}.\end{eqnarray}$$

It remains to verify that there exists $\unicode[STIX]{x1D716}^{\prime }>0$ such that

(4.19) $$\begin{eqnarray}\mathop{\sum }_{X^{1/v}<p\leqslant X^{1/u}}p^{-2}\mathop{\sum }_{\substack{ \mathbf{k}\in \mathbb{N}^{n} \\ \widetilde{\mathbf{k}}=p^{2}}}E(B;\mathbf{k})\ll B^{-\unicode[STIX]{x1D716}^{\prime }-Rd}.\end{eqnarray}$$

For this we note that each $\unicode[STIX]{x1D716}_{i,2}$ is at least $1/2$ owing to $K\geqslant \max \{Rd,R^{2}(d-1)\}$ and $K\geqslant 1$ . Thus the error term above becomes

$$\begin{eqnarray}\ll \mathop{\sum }_{i=1}^{3}B^{-\unicode[STIX]{x1D716}_{i,1}}\mathop{\sum }_{X^{1/v}<p\leqslant X^{1/u}}p^{-2+2\unicode[STIX]{x1D716}_{i,2}}\ll \mathop{\sum }_{i=1}^{3}B^{-\unicode[STIX]{x1D716}_{i,1}+((n-Rd)/u)(2\unicode[STIX]{x1D716}_{i,2}-1)}.\end{eqnarray}$$

Therefore, if

$$\begin{eqnarray}u>\max \bigg\{{\displaystyle \frac{(n-Rd)(2\unicode[STIX]{x1D716}_{i,2}-1)}{\unicode[STIX]{x1D716}_{i,1}-Rd}}:1\leqslant i\leqslant 3\bigg\}\end{eqnarray}$$

then (4.19) holds. Now define $u_{0}:=(1+\unicode[STIX]{x1D716})\max \{u_{1},1/(\unicode[STIX]{x1D703}^{\prime }-\unicode[STIX]{x1D716}),2(n-Rd)\unicode[STIX]{x1D70C}\}$ , where

$$\begin{eqnarray}u_{1}:=\max \bigg\{{\displaystyle \frac{(n-Rd)(2\unicode[STIX]{x1D716}_{i,2}-1)}{\unicode[STIX]{x1D716}_{i,1}-Rd}}:1\leqslant i\leqslant 3\bigg\}.\end{eqnarray}$$

Then applying (4.18) with $u=u_{0}$ and $v^{\prime }:=(nc_{n}-1)/(\unicode[STIX]{x1D703}^{\prime }-\unicode[STIX]{x1D716}-1/u_{0})$ , allows us to take

$$\begin{eqnarray}r\geqslant {\displaystyle \frac{n}{n-Rd}}(1+\unicode[STIX]{x1D716})u_{0}-1+n\bigg(1+{\displaystyle \frac{u_{0}}{v^{\prime }}}c_{n}\bigg)\log {\displaystyle \frac{v^{\prime }}{u_{0}}}-n\bigg(1-{\displaystyle \frac{u_{0}}{v^{\prime }}}\bigg),\end{eqnarray}$$

where $c_{n}:=\unicode[STIX]{x1D6FD}_{n}/n$ satisfies $\lim _{n\rightarrow +\infty }c_{n}=2.44\ldots .$ Letting $\unicode[STIX]{x1D716}>0$ be arbitrarily close to zero concludes the proof of the lower bound claimed in Theorem 1.6. This is because the quantities $u^{\prime \prime },\widehat{u},\widehat{v}$ introduced in (1.8) and (1.9) are such that for fixed $\mathbf{f},n,d,R$ we have

$$\begin{eqnarray}\lim _{\unicode[STIX]{x1D716}\rightarrow 0}(u_{1},u_{0},v)=(u^{\prime \prime },\widehat{u},\widehat{v}).\end{eqnarray}$$

To complete the proof of Theorem 1.6 it remains to verify the estimates regarding $\widehat{v}$ and $r_{0}$ , where $r_{0}$ is defined in (1.10). It is easy to see that $u^{\prime \prime }/(n-Rd)$ is a function of $K$ that is bounded away from $0$ and $+\infty$ , while a similar remark applies to $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D703}^{\prime }$ . This implies that $\widehat{u}\ll _{d,R}n$ and noting that $\widehat{u}<\widehat{v}$ , one has

$$\begin{eqnarray}{r_{0}\ll }_{d,R}\widehat{u}\,+n\log {\displaystyle \frac{\widehat{v}}{\widehat{u}}}\ll _{d,R}n\bigg(1+\log {\displaystyle \frac{\widehat{v}}{\widehat{u}}}\bigg),\end{eqnarray}$$

where the implied constant is independent of $K$ and $n$ . The identity $nc_{n}-1=\unicode[STIX]{x1D70F}\widehat{v}-\widehat{v}/\widehat{u}$ shows that

$$\begin{eqnarray}\widehat{v}/\widehat{u}\ll n+\widehat{v}\ll n+{\displaystyle \frac{n}{\unicode[STIX]{x1D703}^{\prime }-1/\widehat{u}}}\ll n,\end{eqnarray}$$

therefore $r_{0}=O_{d,R}(n\log n)$ , with an implied constant depending at most on $d$ and $R$ .

5.1 Bounds for $\unicode[STIX]{x1D71B}$

One has to be careful when adapting the approach [Reference Brüdern and Fouvry7] to homogeneous equations. The reason is that in the case of Lagrange’s equation there exists a multiplicative function $\widetilde{\unicode[STIX]{x1D71B}}$ satisfying

$$\begin{eqnarray}\unicode[STIX]{x1D71B}(\mathbf{m})\leqslant \mathop{\prod }_{i=1}^{n}\widetilde{\unicode[STIX]{x1D71B}}(m_{i})\end{eqnarray}$$

and such that for all large primes $p$ one has $\widetilde{\unicode[STIX]{x1D71B}}(p)\leqslant 2$ , see [Reference Brüdern and Fouvry7, Lemma 12, part (iii)]. It is easy to see that bounds of this quality fail to hold rather spectacularly for systems of forms $\mathbf{f}=\mathbf{0}$ as in Theorem 1.1. Indeed,

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D71B}(p,\ldots ,p)\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D70E}_{p}^{-1}\lim _{l\rightarrow \infty }p^{-l(n-R)}\sharp \{\mathbf{x}~(\text{mod}~p^{l}):\mathbf{f}(p\mathbf{x})\equiv 0~(\text{mod}~p^{l})\}\nonumber\\ \displaystyle & & \displaystyle \quad =p^{Rd}\unicode[STIX]{x1D70E}_{p}^{-1}\lim _{l\rightarrow \infty }p^{-(l-d)(n-R)}\sharp \{\mathbf{x}~(\text{mod}~p^{l-d}):\mathbf{f}(\mathbf{x})\equiv 0~(\text{mod}~p^{l-d})\}\nonumber\\ \displaystyle & & \displaystyle \quad =p^{Rd}.\nonumber\end{eqnarray}$$

To confront this issue our first task is to control the contribution towards $\unicode[STIX]{x1D6F4}(D_{1},z_{1})$ of integer vectors $\mathbf{k}$ such that there exists $i<j$ with $k_{ij}:=\text{gcd}(k_{i},k_{\!j})$ attaining a large value. Define

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{\ast }(D_{1},z_{1})=\mathop{\sum }_{\substack{ \mathbf{k}\in \mathbb{N}^{n} \\ k_{i}|P(z_{0},z_{1}) \\ \max k_{ij}\leqslant \unicode[STIX]{x1D6E5}}}{\displaystyle \frac{\unicode[STIX]{x1D71B}(\mathbf{k})}{\widetilde{\mathbf{k}}}}\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D706}_{k_{i}}^{+}\end{eqnarray}$$

and recall the definition of $\unicode[STIX]{x1D6F6}$ in (1.4).

Lemma 5.2. Assuming $\max \{(d-1)R^{2}2^{d-1}+(R+1)(\unicode[STIX]{x1D6F6}+1),(d^{2}-1)R^{2}2^{d-1}\}<\mathfrak{B}(\mathbf{f})$ , one has

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}(D_{1},z_{1})-\unicode[STIX]{x1D6F4}^{\ast }(D_{1},z_{1})\ll \unicode[STIX]{x1D6E5}^{-1+\unicode[STIX]{x1D716}}(\log z_{1})^{n}.\end{eqnarray}$$

Proof. The quantity under investigation is $\ll \sum _{1\leqslant l_{1}<l_{2}\leqslant n}\mathscr{E}(l_{1},l_{2})$ , where

$$\begin{eqnarray}\mathscr{E}(l_{1},l_{2}):=\mathop{\sum }_{\unicode[STIX]{x1D6FF}>\unicode[STIX]{x1D6E5}}\unicode[STIX]{x1D707}(\unicode[STIX]{x1D6FF})^{2}\mathop{\sum }_{\substack{ k_{i}|P(z_{0},z_{1}) \\ \unicode[STIX]{x1D6FF}|k_{l_{1}},\unicode[STIX]{x1D6FF}|k_{l_{2}}}}{\displaystyle \frac{\unicode[STIX]{x1D71B}(\mathbf{k})}{\widetilde{\mathbf{k}}}}.\end{eqnarray}$$

We may now use the multiplicative properties of $\unicode[STIX]{x1D71B}$ to deduce that

$$\begin{eqnarray}\mathscr{E}(l_{1},l_{2})\ll \mathop{\sum }_{\unicode[STIX]{x1D6FF}>\unicode[STIX]{x1D6E5}}\bigg(\mathop{\prod }_{\substack{ z_{0}<p\leqslant z_{1} \\ p|\unicode[STIX]{x1D6FF}}}\mathop{\sum }_{\substack{ \mathbf{j}\in \{0,1\}^{n} \\ j_{l_{1}}=j_{l_{2}}=1}}{\displaystyle \frac{\unicode[STIX]{x1D71B}(p^{\mathbf{j}})}{p^{|\mathbf{j}|_{1}}}}\bigg)\bigg(\mathop{\prod }_{\substack{ z_{0}<p\leqslant z_{1} \\ p\nmid \unicode[STIX]{x1D6FF}}}\mathop{\sum }_{\mathbf{j}\in \{0,1\}^{n}}{\displaystyle \frac{\unicode[STIX]{x1D71B}(p^{\mathbf{j}})}{p^{|\mathbf{j}|_{1}}}}\bigg).\end{eqnarray}$$

Fix $\unicode[STIX]{x1D702}\in (0,1/4)$ and let us denote $s_{0}:=\unicode[STIX]{x1D6F6}+1+\unicode[STIX]{x1D702}$ . By Corollary 3.8 we obtain

$$\begin{eqnarray}\mathop{\sum }_{|\mathbf{j}|_{1}\geqslant s_{0}}{\displaystyle \frac{\unicode[STIX]{x1D71B}(p^{\mathbf{j}})}{p^{|\mathbf{j}|_{1}}}}\ll p^{\unicode[STIX]{x1D6F6}}\mathop{\sum }_{s_{0}\leqslant s\leqslant n}\binom{n}{s}p^{-s}\ll p^{-1-\unicode[STIX]{x1D702}}.\end{eqnarray}$$

The assumptions of our lemma allow us to apply Corollary 3.6 whenever $|\mathbf{j}|_{1}\leqslant s_{0}$ . Thus it supplies us with some $\unicode[STIX]{x1D706}>0$ such that $\unicode[STIX]{x1D71B}(p^{\mathbf{j}})=1+O(p^{-\unicode[STIX]{x1D706}})$ , which yields

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\mathbf{j}\in \{0,1\}^{n}}{\displaystyle \frac{\unicode[STIX]{x1D71B}(p^{\mathbf{j}})}{p^{|\mathbf{j}|_{1}}}}=1+{\displaystyle \frac{n}{p}}+O(p^{-1-\unicode[STIX]{x1D716}})\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\substack{ \mathbf{j}\in \{0,1\}^{n} \\ j_{l_{1}}=j_{l_{2}}=1}}{\displaystyle \frac{\unicode[STIX]{x1D71B}(p^{\mathbf{j}})}{p^{|\mathbf{j}|_{1}}}}=p^{-2}+O(p^{-2-\unicode[STIX]{x1D716}}),\nonumber\end{eqnarray}$$

for some $\unicode[STIX]{x1D716}>0$ . Assorting all related estimates we obtain for square-free $\unicode[STIX]{x1D6FF}$ that

$$\begin{eqnarray}\mathop{\prod }_{\substack{ z_{0}<p\leqslant z_{1} \\ p|\unicode[STIX]{x1D6FF}}}\mathop{\sum }_{\substack{ \mathbf{j}\in \{0,1\}^{n} \\ j_{l_{1}}=j_{l_{2}}=1}}{\displaystyle \frac{\unicode[STIX]{x1D71B}(p^{\mathbf{j}})}{p^{|\mathbf{j}|_{1}}}}\ll \unicode[STIX]{x1D6FF}^{-2+\unicode[STIX]{x1D716}},\end{eqnarray}$$

and

$$\begin{eqnarray}\mathop{\prod }_{\substack{ z_{0}<p\leqslant z_{1} \\ p\nmid \unicode[STIX]{x1D6FF}}}\mathop{\sum }_{\mathbf{j}\in \{0,1\}^{n}}{\displaystyle \frac{\unicode[STIX]{x1D71B}(p^{\mathbf{j}})}{p^{|\mathbf{j}|_{1}}}}\ll \mathop{\prod }_{z_{0}<p\leqslant z_{1}}\bigg(1+{\displaystyle \frac{n}{p}}+O(p^{-1-\unicode[STIX]{x1D716}})\bigg)\ll (\log z_{1})^{n}.\end{eqnarray}$$

These estimates prove that $\mathscr{E}(l_{1},l_{2})\ll (\log z_{1})^{n}\sum _{\unicode[STIX]{x1D6FF}>\unicode[STIX]{x1D6E5}}\unicode[STIX]{x1D6FF}^{-2+\unicode[STIX]{x1D716}}$ , which is sufficient.◻

For any square-free integer $m$ and index $1\leqslant i\leqslant n$ define

$$\begin{eqnarray}\unicode[STIX]{x1D71B}_{i}(m):=\unicode[STIX]{x1D71B}(1,\ldots ,1,m,1,\ldots ,1),\end{eqnarray}$$

where $m$ appears in the $i$ th position. For $\unicode[STIX]{x1D716}>0$ define the multiplicative function

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(m):=\mathop{\prod }_{\substack{ p|m \\ p>z_{0}}}(1+p^{-\unicode[STIX]{x1D716}}).\end{eqnarray}$$

Note that if assumptions of Corollary 3.6 hold for $|\mathbf{j}|_{1}=1$ then there exists $\unicode[STIX]{x1D716}=\unicode[STIX]{x1D716}(\mathbf{f})>0$ such that $\unicode[STIX]{x1D70E}_{p}(p^{\mathbf{e}_{i}}|\mathbf{x})=1/p+O(p^{-1-\unicode[STIX]{x1D716}})$ . Enlarging $z_{0}$ and replacing $\unicode[STIX]{x1D716}$ by a smaller positive constant if needed yields the following result.

Lemma 5.3. Assume that $\mathfrak{B}(\mathbf{f})>\max \{R^{2}2^{d-1}(d^{2}-1),R^{2}2^{d-1}(d-1)+(R+1)\}$ . Then there exists $\unicode[STIX]{x1D716}=\unicode[STIX]{x1D716}(\mathbf{f})>0$ such that for all square-free integers $m$ ,

$$\begin{eqnarray}\max _{1\leqslant i\leqslant n}\unicode[STIX]{x1D71B}_{i}(m)\leqslant \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(m).\end{eqnarray}$$

Observe that for all $\mathbf{d}\in \mathbb{N}^{n}$ with $\unicode[STIX]{x1D707}(\mathbf{d})^{2}=1$ the expression

$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x1D71B}(\mathbf{d})}{\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D71B}_{i}(d_{i})}}\end{eqnarray}$$

is a function of the vector $(\text{gcd}(d_{i},d_{\!j}))_{1\leqslant i<j\leqslant n}$ . To see this, it is enough to consider the case when $\widetilde{\mathbf{d}}$ is divisible by a single prime, say $p$ . We need to show that if $\mathbf{h},\mathbf{k}\in \{0,1\}^{n}$ and

(5.6) $$\begin{eqnarray}i\neq j\Rightarrow \min (k_{i},k_{\!j})=\min (h_{i},h_{\!j})\end{eqnarray}$$

then

(5.7) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x1D71B}(p^{\mathbf{k}})}{\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D71B}_{i}(p^{k_{i}})}}={\displaystyle \frac{\unicode[STIX]{x1D71B}(p^{\mathbf{h}})}{\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D71B}_{i}(p^{h_{i}})}}.\end{eqnarray}$$

Obviously this holds in the case that $\mathbf{k}=\mathbf{h}$ and we can therefore assume that $\mathbf{k}\neq \mathbf{h}$ . A little thought reveals that in this case (5.6) guarantees that there exist $l,m,i\neq j$ such that $(\mathbf{k},\mathbf{h})$ equals one of the following:

$$\begin{eqnarray}(\mathbf{e}_{l},\mathbf{0}),(\mathbf{0},\mathbf{e}_{m}),(\mathbf{e}_{i},\mathbf{e}_{\!j}).\end{eqnarray}$$

For any such instance we can verify that both sides of (5.7) equal $1$ , hence our claim holds. We have proved that there exists a function $\widehat{g}:\mathbb{N}^{\binom{n}{2}}\rightarrow \mathbb{R}_{{\geqslant}0}$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D707}(\mathbf{d})^{2}=1\Rightarrow \unicode[STIX]{x1D71B}(\mathbf{d})=\widehat{g}((d_{i,j}))\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D71B}_{i}(d_{i}).\end{eqnarray}$$

The function $\unicode[STIX]{x1D71B}_{i}(d_{i})$ keeps track of the probability that $d_{i}|x_{i}$ and the function $\widehat{g}((d_{i,j}))$ takes values close to $1$ when the events $d_{i}|x_{i}$ are independent (in a suitable sense) but can obtain larger values in general.

Defining

$$\begin{eqnarray}S((u_{i,j})):=\mathop{\sum }_{\substack{ \mathbf{k}\in \mathbb{N}^{n} \\ k_{i}|P(z_{0},z_{1}) \\ (k_{i},k_{\!j})=u_{i,j}}}\mathop{\prod }_{i=1}^{n}{\displaystyle \frac{\unicode[STIX]{x1D706}_{k_{i}}^{+}\unicode[STIX]{x1D71B}_{i}(k_{i})}{k_{i}}}\end{eqnarray}$$

enables us to write

(5.8) $$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{\ast }(D_{1},z_{1})=\mathop{\sum }_{\substack{ u_{i,j}\leqslant \unicode[STIX]{x1D6E5} \\ 1\leqslant i<j\leqslant n}}\widehat{g}((u_{i,j}))S((u_{i,j})).\end{eqnarray}$$

We may now use the expression $(\unicode[STIX]{x1D707}\ast 1)((k_{i}/u_{i,j},k_{\!j}/u_{i,j}))$ to detect the condition $(k_{i},k_{\!j})=u_{i,j}$ , thus inferring

(5.9) $$\begin{eqnarray}S((u_{i,j}))=\mathop{\sum }_{\substack{ (l_{i,j})\in \mathbb{N}^{\binom{n}{2}} \\ 1\leqslant i\neq j\leqslant n \\ u_{i,j}l_{i,j}|P(z_{0},z_{1})}}\unicode[STIX]{x1D707}(\mathbf{l})\mathop{\prod }_{i=1}^{n}\Bigg(\mathop{\sum }_{\substack{ k\in \mathbb{N} \\ k|P(z_{0},z_{1}) \\ \unicode[STIX]{x1D709}_{i}|k}}{\displaystyle \frac{\unicode[STIX]{x1D706}_{k}^{+}\unicode[STIX]{x1D71B}_{i}(k)}{k}}\Bigg),\end{eqnarray}$$

where

$$\begin{eqnarray}\unicode[STIX]{x1D709}_{i}:=\text{rad}\Bigg(\mathop{\prod }_{\substack{ 1\leqslant j\leqslant n \\ j\neq i}}u_{i,j}l_{i,j}\Bigg)\end{eqnarray}$$

and $\text{rad}$ stands for the radical of a positive integer. Under the assumptions of Lemma 5.3 we thus obtain the following estimate for all square-free integers $\unicode[STIX]{x1D6FF}$ :

$$\begin{eqnarray}\bigg|\mathop{\sum }_{\substack{ k\in \mathbb{N} \\ k|P(z_{0},z_{1}) \\ \unicode[STIX]{x1D6FF}|k}}{\displaystyle \frac{\unicode[STIX]{x1D706}_{k}^{+}\unicode[STIX]{x1D71B}_{i}(k)}{k}}\bigg|\leqslant {\displaystyle \frac{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D6FF})}{\unicode[STIX]{x1D6FF}}}\mathop{\prod }_{z_{0}<p\leqslant z_{1}}(1+p^{-1}+p^{-1-\unicode[STIX]{x1D716}})\ll {\displaystyle \frac{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D6FF})}{\unicode[STIX]{x1D6FF}}}\log z_{1}.\end{eqnarray}$$

Note that the succeeding inequality holds for all divisors $m^{\prime }$ of $m$ ,

$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(m)}{m}}\leqslant {\displaystyle \frac{\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(m^{\prime })}{m^{\prime }}}.\end{eqnarray}$$

Letting $\unicode[STIX]{x1D709}_{i}^{\ast }$ be the radical of $\prod _{j\neq i}l_{i,j}$ and using the last inequalities with $\unicode[STIX]{x1D6FF}=m=\unicode[STIX]{x1D709}_{i}$ and $m^{\prime }=\unicode[STIX]{x1D709}_{i}^{\ast }$ allows us to truncate the sum in (5.9) to the range $l_{i,j}\leqslant \unicode[STIX]{x1D6E5}^{B_{1}}$ , where $B_{1}>0$ is a constant that will be chosen in due course. The contribution of $l_{1,2}>\unicode[STIX]{x1D6E5}^{B_{1}}$ is

$$\begin{eqnarray}\ll (\log z_{1})^{n}\mathop{\sum }_{\substack{ l_{i,j}\leqslant D_{1},l_{i,j}|P(z_{0},z_{1}) \\ l_{1,2}>\unicode[STIX]{x1D6E5}^{B_{1}}}}{\displaystyle \frac{\unicode[STIX]{x1D707}(\mathbf{l})^{2}}{\widetilde{\boldsymbol{\unicode[STIX]{x1D709}}^{\ast }}}}\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D709}_{i}^{\ast }),\end{eqnarray}$$

where $D_{1}$ is the support of $\unicode[STIX]{x1D706}^{+}$ .

We may now use the inequality

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D709}_{i}^{\ast })\leqslant \mathop{\prod }_{j\neq i}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(l_{i,j})\end{eqnarray}$$

to obtain

$$\begin{eqnarray}\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D709}_{i}^{\ast })\leqslant \mathop{\prod }_{1\leqslant i\neq j\leqslant n}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(l_{i,j})^{2}.\end{eqnarray}$$

Hence the last sum is

$$\begin{eqnarray}{\leqslant}\mathop{\sum }_{\substack{ l_{1,2}>\unicode[STIX]{x1D6E5}^{B_{1}}}}{\displaystyle \frac{\unicode[STIX]{x1D707}(\mathbf{l})^{2}}{\unicode[STIX]{x1D709}_{1}^{\ast }\cdots \unicode[STIX]{x1D709}_{n}^{\ast }}}\mathop{\prod }_{1\leqslant i\neq j\leqslant n}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(l_{i,j})^{2}.\end{eqnarray}$$

This is really a summation over the variables $l_{1,2},\ldots ,l_{n-1,n}$ because each expression $\unicode[STIX]{x1D709}_{i}^{\ast }$ is a function of some of these variables. We first perform a summation over $l_{n-1,n}$ . Recalling that

$$\begin{eqnarray}\unicode[STIX]{x1D709}_{i}^{\ast }=\text{rad}\bigg(\mathop{\prod }_{j\neq i}l_{i,j}\bigg)\end{eqnarray}$$

we see that only $\unicode[STIX]{x1D709}_{n-1}^{\ast }$ and $\unicode[STIX]{x1D709}_{n}^{\ast }$ depend on $l_{n-1,n}$ , since they satisfy

$$\begin{eqnarray}\unicode[STIX]{x1D709}_{n}^{\ast }=[l_{n-1,n},\unicode[STIX]{x1D709}_{n}^{\ast \ast }],\qquad \unicode[STIX]{x1D709}_{n-1}^{\ast }=[l_{n-1,n},\unicode[STIX]{x1D709}_{n-1}^{\ast \ast }],\end{eqnarray}$$

where both $\unicode[STIX]{x1D709}_{n-1}^{\ast \ast }$ and $\unicode[STIX]{x1D709}_{n}^{\ast \ast }$ are defined as $\unicode[STIX]{x1D709}_{n-1}^{\ast }$ and $\unicode[STIX]{x1D709}_{n}^{\ast }$ but with the variable $l_{n-1,n}$ missing, i.e.

$$\begin{eqnarray}\unicode[STIX]{x1D709}_{n-1}^{\ast \ast }:=\text{rad}\bigg(\mathop{\prod }_{j\neq n-1,n}l_{n-1,j}\bigg),\qquad \unicode[STIX]{x1D709}_{n}^{\ast \ast }:=\text{rad}\bigg(\mathop{\prod }_{j\neq n-1,n}l_{n,j}\bigg).\end{eqnarray}$$

Hence the sum over $l_{n-1,n}$ is

$$\begin{eqnarray}\mathop{\sum }_{l_{n-1,n}}{\displaystyle \frac{\unicode[STIX]{x1D707}(l_{n-1,n})^{2}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(l_{n-1,n})^{2}}{[l_{n-1,n},\unicode[STIX]{x1D709}_{n-1}^{\ast \ast }][l_{n-1,n},\unicode[STIX]{x1D709}_{n}^{\ast \ast }]}},\end{eqnarray}$$

which equals

$$\begin{eqnarray}{\displaystyle \frac{1}{\unicode[STIX]{x1D709}_{n-1}^{\ast \ast }\unicode[STIX]{x1D709}_{n}^{\ast \ast }}}\mathop{\sum }_{l_{n-1,n}}{\displaystyle \frac{\unicode[STIX]{x1D707}(l_{n-1,n})^{2}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D716}}(l_{n-1,n})^{2}}{l_{n-1,n}^{2}}}\text{gcd}(\unicode[STIX]{x1D709}_{n-1}^{\ast \ast },l_{n-1,n})\text{gcd}(\unicode[STIX]{x1D709}_{n}^{\ast \ast },l_{n-1,n}).\end{eqnarray}$$

The last sum is

$$\begin{eqnarray}{\leqslant}\mathop{\prod }_{p|\unicode[STIX]{x1D709}_{n-1}^{\ast \ast }\unicode[STIX]{x1D709}_{n}^{\ast \ast }}(2+2p^{-\unicode[STIX]{x1D716}}+p^{-2\unicode[STIX]{x1D716}})\mathop{\prod }_{p}(1+p^{-2}(1+p^{-\unicode[STIX]{x1D716}})^{2})\ll \unicode[STIX]{x1D70F}(\unicode[STIX]{x1D709}_{n-1}^{\ast \ast })^{A}\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D709}_{n}^{\ast \ast })^{A},\end{eqnarray}$$

where $A=3$ . Of course we can bound any $\unicode[STIX]{x1D709}_{k}^{\ast \ast }$ by the product of all available variables except $l_{n-1,n}$ , i.e. $\prod _{\{i,j\}\neq \{n-1,n\}}l_{i,j}$ , thus we obtain

$$\begin{eqnarray}\ll {\displaystyle \frac{1}{\unicode[STIX]{x1D709}_{n-1}^{\ast \ast }\unicode[STIX]{x1D709}_{n}^{\ast \ast }}}\mathop{\prod }_{\{i,j\}\neq \{n-1,n\}}\unicode[STIX]{x1D70F}(l_{i,j})^{2A}.\end{eqnarray}$$

The process above is the first step of a finite induction that eliminates all variables $l_{i,j}$ , beginning from $l_{n-1,n}$ and terminating with $l_{1,2}$ . At each step expressions of the form

$$\begin{eqnarray}\mathop{\sum }_{\substack{ l_{1,2}>\unicode[STIX]{x1D6E5}^{B_{1}}}}{\displaystyle \frac{\unicode[STIX]{x1D707}(\mathbf{l})^{2}}{\unicode[STIX]{x1D709}_{1}^{\prime }\cdots \unicode[STIX]{x1D709}_{n}^{\prime }}}\mathop{\prod }_{1\leqslant i\neq j\leqslant n}^{\flat }\unicode[STIX]{x1D70F}(l_{i,j})^{A}\end{eqnarray}$$

are bounded by

$$\begin{eqnarray}\ll \mathop{\sum }_{\substack{ l_{1,2}>\unicode[STIX]{x1D6E5}^{B_{1}}}}{\displaystyle \frac{\unicode[STIX]{x1D707}(\mathbf{l})^{2}}{\unicode[STIX]{x1D709}_{1}^{^{\prime \prime }}\cdots \unicode[STIX]{x1D709}_{n}^{^{\prime \prime }}}}\mathop{\prod }_{1\leqslant i\neq j\leqslant n}^{\flat \flat }\unicode[STIX]{x1D70F}(l_{i,j})^{100A},\end{eqnarray}$$

where the notation $\unicode[STIX]{x1D709}^{\prime },\prod ^{\flat }$ means that some of the variables $l_{i,j}$ have been eliminated, the notation $\unicode[STIX]{x1D709}^{^{\prime \prime }},\prod ^{\flat \flat }$ that one further variable has been eliminated and the constant $A^{\prime }$ depends at most on $A$ and $\mathbf{f}$ . At the last step of the induction we will arrive at the expression

$$\begin{eqnarray}\mathop{\sum }_{l_{1,2}>\unicode[STIX]{x1D6E5}^{B_{1}}}{\displaystyle \frac{\unicode[STIX]{x1D707}(l_{1,2})^{2}}{l_{1,2}^{2}}}\unicode[STIX]{x1D70F}(l_{1,2})^{C},\end{eqnarray}$$

where $C=C(\mathbf{f})$ . Obviously this is $\ll \unicode[STIX]{x1D6E5}^{-B_{1}/2}$ . The arguments above show that

(5.10) $$\begin{eqnarray}S((u_{i,j}))=\!\mathop{\sum }_{\substack{ (l_{i,j})\in \mathbb{N}^{\binom{n}{2}} \\ l_{i,j}\leqslant \unicode[STIX]{x1D6E5}^{B_{1}} \\ u_{i,j}l_{i,j}|P(z_{0},z_{1})}}\unicode[STIX]{x1D707}(\mathbf{l})\mathop{\prod }_{i=1}^{n}\Bigg(\!\mathop{\sum }_{\substack{ k\in \mathbb{N} \\ k|P(z_{0},z_{1}) \\ \unicode[STIX]{x1D709}_{i}|k}}\!{\displaystyle \frac{\unicode[STIX]{x1D706}_{k}^{+}\unicode[STIX]{x1D71B}_{i}(k)}{k}}\Bigg)+O((\log z_{1})^{n}\unicode[STIX]{x1D6E5}^{-B_{1}/2}),\end{eqnarray}$$

where the implied constant is independent of the $u_{i,j}$ .

We now aim to use a consequence of the linear case of the Rosser–Iwaniec sieve (in fact the linear case was settled first by Jurkat and Richert [Reference Jurkat and Richert25]) that is given in [Reference Brüdern and Fouvry7, Lemma 11]. We shall find it convenient to use the error term appearing in [Reference Iwaniec24, Theorem 1], this will lead us to replace the term $\text{e}^{\sqrt{L-s}}(\log D)^{-1/3}$ in [Reference Brüdern and Fouvry7, Lemma 10] and [Reference Brüdern and Fouvry7, Lemma 11] by

$$\begin{eqnarray}\text{e}^{\sqrt{L}}Q(s)(\log D)^{-1/3}\end{eqnarray}$$

where, as stated in [Reference Iwaniec24, equation (1.6)], the function $Q(s)$ satisfies

$$\begin{eqnarray}Q(s)<\exp \{-s\log s+s\log \log 3s+O(s)\},\quad s\geqslant 3.\end{eqnarray}$$

The constant $L$ in our case will depend at most on the coefficients of $\mathbf{f}$ , which is considered constant throughout our paper, thus we can assume that the terms above are $\ll _{\mathbf{f}}\,s^{-s}(\log D)^{-1/3}$ , with an implied constant depending at most on $\mathbf{f}$ . Let us choose the set of primes

$$\begin{eqnarray}\mathscr{P}:=\{p\text{ prime: }p>z_{0}\}.\end{eqnarray}$$

Moreover, we observe that $\unicode[STIX]{x1D71B}_{i}(k)$ is a multiplicative function for all $1\leqslant i\leqslant n$ . We define the modified multiplicative function $\widetilde{\unicode[STIX]{x1D71B}_{i}}(k)$ by

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D71B}}_{i}(p):=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D71B}_{i}(p) & \text{if }p>z_{0},\\ 0 & \text{if }p\leqslant z_{0}.\end{array}\right.\end{eqnarray}$$

So far we can only assume that $\unicode[STIX]{x1D71B}_{i}(p)\leqslant 1+p^{-\unicode[STIX]{x1D716}},$ whereas in [Reference Brüdern and Fouvry7] they work with the stronger statement that $\unicode[STIX]{x1D71B}(p)\leqslant 1+1/(p-1).$ However, we still get the bound present in [Reference Brüdern and Fouvry7, equation (3.10)] for a uniform $L$ . For this we observe that

$$\begin{eqnarray}\displaystyle \log \mathop{\prod }_{w_{1}<p\leqslant w_{2}}\bigg(1-{\displaystyle \frac{\widetilde{\unicode[STIX]{x1D71B}}_{i}(p)}{p}}\bigg)^{-1} & {\leqslant} & \displaystyle \mathop{\sum }_{w_{1}<p\leqslant w_{2}}\log \bigg(1-{\displaystyle \frac{1}{p}}-{\displaystyle \frac{C}{p^{1+\unicode[STIX]{x1D716}}}}\bigg)^{-1}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{w_{1}<p\leqslant w_{2}}\bigg({\displaystyle \frac{1}{p}}+{\displaystyle \frac{C}{p^{1+\unicode[STIX]{x1D716}}}}\bigg)+O(w_{1}^{-1})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \log \log w_{2}-\log \log w_{1}+O\bigg({\displaystyle \frac{1}{\log w_{1}}}\bigg),\nonumber\end{eqnarray}$$

by Mertens’ theorem. This leads to the bound

$$\begin{eqnarray}\mathop{\prod }_{w_{1}<p\leqslant w_{2}}\bigg(1-{\displaystyle \frac{\widetilde{\unicode[STIX]{x1D71B}}_{i}(p)}{p}}\bigg)^{-1}\leqslant \bigg({\displaystyle \frac{\log w_{2}}{\log w_{1}}}\bigg)\bigg(1+{\displaystyle \frac{L}{\log w_{1}}}\bigg),\end{eqnarray}$$

for a uniform constant $L=L(\mathbf{f})$ . We can now directly apply [Reference Brüdern and Fouvry7, Lemma 11] to the inner sums appearing in (5.10). Introduce the constant $s_{0}$ through $s_{0}:=(\log D_{1})/(\log z_{1})$ , which we demand fulfils $s_{0}\geqslant 3$ , and set

$$\begin{eqnarray}U_{i}(z_{1},\unicode[STIX]{x1D709}_{i}):=\unicode[STIX]{x1D707}(\unicode[STIX]{x1D709}_{i})\mathop{\prod }_{\substack{ p|\unicode[STIX]{x1D709}_{i} \\ p>z_{0}}}{\displaystyle \frac{\unicode[STIX]{x1D71B}_{i}(p)}{p-\unicode[STIX]{x1D71B}_{i}(p)}}\mathop{\prod }_{\substack{ z_{0}<p\leqslant z_{1}}}\bigg(1-{\displaystyle \frac{\unicode[STIX]{x1D71B}(p)}{p}}\bigg).\end{eqnarray}$$

This provides us with

$$\begin{eqnarray}\mathop{\sum }_{\substack{ k\in \mathbb{N} \\ k|P(z_{0},z_{1}) \\ \unicode[STIX]{x1D709}_{i}|k}}{\displaystyle \frac{\unicode[STIX]{x1D706}_{k}^{+}\unicode[STIX]{x1D71B}_{i}(k)}{k}}=U_{i}(z_{1},\unicode[STIX]{x1D709}_{i})+O(\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D709}_{i})s_{0}^{-s_{0}}).\end{eqnarray}$$

Owing to the apparent bounds $0\leqslant \unicode[STIX]{x1D71B}_{i}(p)<p/2$ , valid for $p>z_{0}$ (as long as $z_{0}$ is enlarged) we deduce that $|U_{i}(z_{1},\unicode[STIX]{x1D709}_{i})|\leqslant 1$ for all $1\leqslant i\leqslant n$ and divisors $\unicode[STIX]{x1D709}_{i}|P(z_{0},z_{1})$ . We use this approximation in (5.10), to obtain

$$\begin{eqnarray}\displaystyle & & \displaystyle S((u_{i,j}))-\mathop{\sum }_{\substack{ l_{i,j}\leqslant \unicode[STIX]{x1D6E5}^{B_{1}} \\ u_{i,j}l_{i,j}|P(z_{0},z_{1})}}\unicode[STIX]{x1D707}(\mathbf{l})\mathop{\prod }_{i=1}^{n}U_{i}(z_{1},\unicode[STIX]{x1D709}_{i})\nonumber\\ \displaystyle & & \displaystyle \quad \ll (\log z_{1})^{n}\unicode[STIX]{x1D6E5}^{-B_{1}/2}+\unicode[STIX]{x1D6E5}^{B_{1}\binom{n}{2}+1/100}(s_{0}^{-s_{0}}+s_{0}^{-s_{0}}(\log D_{1})^{-1/3}).\nonumber\end{eqnarray}$$

Assume that the assumptions in Lemma 5.2 are satisfied. Together with equation (5.8) we now obtain

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}(D_{1},z_{1})=\unicode[STIX]{x1D6F4}^{MT}(D_{1},z_{1})+\unicode[STIX]{x1D6F4}^{ET}(D_{1},z_{1}),\end{eqnarray}$$

with a main term given by

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{MT}(D_{1},z_{1})=\mathop{\sum }_{\substack{ u_{i,j}\leqslant \unicode[STIX]{x1D6E5} \\ 1\leqslant i<j\leqslant n}}\widehat{g}((u_{i,j}))\mathop{\sum }_{\substack{ (l_{i,j})\in \mathbb{N}^{\binom{n}{2}} \\ l_{i,j}\leqslant \unicode[STIX]{x1D6E5}^{B_{1}} \\ u_{i,j}l_{i,j}|P(z_{0},z_{1})}}\unicode[STIX]{x1D707}(\mathbf{l})\mathop{\prod }_{i=1}^{n}U_{i}(z_{1},\unicode[STIX]{x1D709}_{i}),\end{eqnarray}$$

and an error term satisfying

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{ET}(D_{1},z_{1})\ll {\displaystyle \frac{(\log z_{1})^{n}}{\unicode[STIX]{x1D6E5}^{1-\unicode[STIX]{x1D716}}}}+{\displaystyle \frac{\unicode[STIX]{x1D6E5}^{C+\binom{n}{2}-B_{1}/2}}{(\log z_{1})^{-n}}}+\unicode[STIX]{x1D6E5}^{C+(B_{1}+1)\binom{n}{2}+1/100}s_{0}^{-s_{0}},\end{eqnarray}$$

where $C=C(\mathbf{f})>0$ is such that

$$\begin{eqnarray}|\widehat{g}((u_{i,j}))|\ll \max \{u_{i,j}\}^{C}.\end{eqnarray}$$

We will assume that such a $C$ exists for the moment, this will be proved later in Lemma 5.4. Therefore we may choose $B_{1}>0$ large enough so that $C+\binom{n}{2}-B_{1}/2<-1$ . We can then obtain

(5.11) $$\begin{eqnarray}\unicode[STIX]{x1D6F4}(D_{1},z_{1})-\unicode[STIX]{x1D6F4}^{MT}(D_{1},z_{1})\ll {\displaystyle \frac{(\log z_{1})^{n}}{\unicode[STIX]{x1D6E5}^{1-\unicode[STIX]{x1D716}}}}+\unicode[STIX]{x1D6E5}^{c}s_{0}^{-s_{0}},\end{eqnarray}$$

where $c=c(\mathbf{f})>0$ . Note that here we implicitly assume that $s_{0}\geqslant 3$ , thus $\log D_{1}/\!\log z_{1}\geqslant 3$ .

Lemma 5.4. Assume that $\mathfrak{B}(\mathbf{f})>\max \{R^{2}2^{d-1}(d^{2}-1),R^{2}2^{d-1}(d-1)+(R+1)\}$ . Let $\mathbf{u}\in \mathbb{N}^{\binom{n}{2}}$ be such that $\unicode[STIX]{x1D707}^{2}(\mathbf{u})=1$ and such that $p|\widetilde{\mathbf{u}}$ implies that $p>z_{0}$ . Then, for $z_{0}$ sufficiently large one has

$$\begin{eqnarray}\widehat{g}((u_{i,j}))\ll \bigg(\mathop{\prod }_{i\neq j}u_{i,j}\bigg)^{(d\mathfrak{B}(\mathbf{f})/(d-1)2^{d-1})(d-1/R)+R+\unicode[STIX]{x1D716}}.\end{eqnarray}$$

Proof. First we recall that

(5.12) $$\begin{eqnarray}\widehat{g}((u_{i,j}))\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D71B}_{i}(u_{i})=\unicode[STIX]{x1D71B}(\mathbf{u}),\end{eqnarray}$$

where we have $u_{i,j}=\text{gcd}(u_{i},u_{\!j})$ . For bounding $\widehat{g}((u_{i,j}))$ we may make the following assumption: if $p$ is a prime with $p|u_{i}$ for some $1\leqslant i\leqslant n$ , then there is a $1\leqslant j\leqslant n$ , $j\neq i$ such that $p|u_{i,j}$ . Otherwise we could replace in (5.12) the vector $\mathbf{u}$ with a vector $\widetilde{\mathbf{u}}$ where $\widetilde{u_{k}}=u_{k}$ for $k\neq i$ and $\widetilde{u_{k}}=u_{k}/p$ for $k=i$ . In particular, we may assume that

$$\begin{eqnarray}u_{i}\leqslant \mathop{\prod }_{j\neq i}u_{ij},\end{eqnarray}$$

for every $1\leqslant i\leqslant n$ .

Next we observe that

$$\begin{eqnarray}\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D71B}_{i}(u_{i})=\mathop{\prod }_{i=1}^{n}\mathop{\prod }_{p|u_{i}}\unicode[STIX]{x1D71B}_{i}(p).\end{eqnarray}$$

We recall the identity $\unicode[STIX]{x1D71B}_{i}(p)=p\unicode[STIX]{x1D70E}(p^{\mathbf{e}_{i}}|\mathbf{x})\unicode[STIX]{x1D70E}_{p}^{-1}$ . By Corollary 3.6 we have

$$\begin{eqnarray}\unicode[STIX]{x1D70E}(p^{\mathbf{e}_{i}}|\mathbf{x})={\displaystyle \frac{1}{p}}+O(p^{-1-\unicode[STIX]{x1D716}}).\end{eqnarray}$$

Therefore we obtain

$$\begin{eqnarray}\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D71B}_{i}(u_{i})=\mathop{\prod }_{i=1}^{n}\mathop{\prod }_{p|u_{i}}(1+O(p^{-1-\unicode[STIX]{x1D716}}))^{-1}(1+O(p^{-\unicode[STIX]{x1D716}})),\end{eqnarray}$$

and $\prod _{i=1}^{n}\unicode[STIX]{x1D71B}_{i}(u_{i})^{-1}\ll _{\unicode[STIX]{x1D707}}\,(u_{1}\cdots u_{n})^{\unicode[STIX]{x1D707}},$ for any $\unicode[STIX]{x1D707}>0$ . By Corollary 3.8 we have

$$\begin{eqnarray}\unicode[STIX]{x1D71B}(p^{\mathbf{j}})\ll p^{(d\mathfrak{B}(\mathbf{f})/(d-1)2^{d-1})(d-1/R)+R}.\end{eqnarray}$$

Injecting these bounds into (5.12) yields

$$\begin{eqnarray}\displaystyle \widehat{g}((u_{i,j})) & \ll & \displaystyle \bigg(\mathop{\prod }_{p|\widetilde{\mathbf{u}}}p\bigg)^{(d\mathfrak{B}(\mathbf{f})/(d-1)2^{d-1})(d-1/R)+R+\unicode[STIX]{x1D716}}\nonumber\\ \displaystyle & \ll & \displaystyle \bigg(\mathop{\prod }_{i\neq j}u_{i,j}\bigg)^{(d\mathfrak{B}(\mathbf{f})/(d-1)2^{d-1})(d-1/R)+R+\unicode[STIX]{x1D716}},\nonumber\end{eqnarray}$$

thus concluding the proof. ◻

As in [Reference Brüdern and Fouvry7, p. 90], we now observe that $\unicode[STIX]{x1D6F4}^{MT}(D_{1},z_{1})$ is independent of $D_{1}$ . We set

$$\begin{eqnarray}D_{2}:=\max (D_{1},3^{z_{1}})\end{eqnarray}$$

and with equation (5.11) applied to $D_{2}$ instead of $D_{1}$ , we obtain that

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}(D_{1},z_{1})-\unicode[STIX]{x1D6F4}(D_{2},z_{1})\ll {\displaystyle \frac{(\log z_{1})^{n}}{\unicode[STIX]{x1D6E5}^{1-\unicode[STIX]{x1D716}}}}+\unicode[STIX]{x1D6E5}^{c}s_{0}^{-s_{0}}.\end{eqnarray}$$

For this choice of $D_{2}$ we have $\unicode[STIX]{x1D706}_{d}^{+}=\unicode[STIX]{x1D707}(d)$ for $d|P(z_{0},z_{1})$ (note that with the change of $D_{1}$ to $D_{2}$ also the sieve weights $\unicode[STIX]{x1D706}$ change). Hence we can compute $\unicode[STIX]{x1D6F4}(D_{2},z_{1})$ as

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}(D_{2},z_{1})=\mathop{\sum }_{\substack{ \mathbf{d}\in \mathbb{N}^{n} \\ d_{i}|P(z_{0},z_{1})}}{\displaystyle \frac{\unicode[STIX]{x1D71B}(\mathbf{d})}{\widetilde{\mathbf{d}}}}\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D707}(d_{i})=\mathop{\prod }_{z_{0}<p\leqslant z_{1}}\bigg(1-{\displaystyle \frac{g(p)}{p}}\bigg),\end{eqnarray}$$

with $g(p)$ defined as in (4.7). Injecting our estimates for $\unicode[STIX]{x1D6F4}(D_{1},z_{1})$ into (5.5) yields the upper bound in the next result.

Proposition 5.5. Assuming $l_{i}|P(z_{1},z),|\mathbf{l}|D_{1}\log B\leqslant B^{1/\unicode[STIX]{x1D70C}}$ and that $\mathfrak{B}(\mathbf{f})$ exceeds

$$\begin{eqnarray}\max \{2^{d-1}(d-1)R(R+1),2^{d-1}(d-1)R^{2}+(R+1)(\unicode[STIX]{x1D6F6}+1),2^{d-1}(d^{2}-1)R^{2}\}\end{eqnarray}$$

we have

$$\begin{eqnarray}\displaystyle G(B,z_{1};\mathbf{l}) & = & \displaystyle X_{\mathbf{l}}\mathop{\prod }_{z_{0}<p\leqslant z_{1}}\bigg(1-{\displaystyle \frac{g(p)}{p}}\bigg)\nonumber\\ \displaystyle & & \displaystyle +\,O\bigg(\unicode[STIX]{x1D71B}(\mathbf{l}){\displaystyle \frac{B^{n-Rd}}{\widetilde{\mathbf{l}}}}\bigg({\displaystyle \frac{(\log z_{1})^{n}}{\unicode[STIX]{x1D6E5}^{1-\unicode[STIX]{x1D716}}}}+\unicode[STIX]{x1D6E5}^{c}s_{0}^{-s_{0}}\bigg)\bigg)\nonumber\\ \displaystyle & & \displaystyle +\,O\bigg({\displaystyle \frac{B^{n+\unicode[STIX]{x1D716}}}{\widetilde{\mathbf{l}}}}\mathop{\sum }_{\substack{ |\mathbf{k}|\leqslant D_{1} \\ p|\widetilde{\mathbf{k}}\Rightarrow z_{0}<p\leqslant z_{1}}}\unicode[STIX]{x1D707}(\mathbf{k})^{2}{\displaystyle \frac{E(B;(k_{1}l_{1},\ldots ,k_{n}l_{n}))}{\widetilde{\mathbf{k}}}}\bigg).\nonumber\end{eqnarray}$$

The lower bound can be procured upon writing

$$\begin{eqnarray}G(B,z_{1};\mathbf{l})=\mathop{\sum }_{\substack{ \mathbf{x}\in \mathscr{A} \\ \mathbf{l}|\mathbf{x}}}\bigg(\mathop{\prod }_{i=1}^{n}w\bigg({\displaystyle \frac{x_{i}}{B}}-{\displaystyle \frac{\unicode[STIX]{x1D701}_{i}}{2|\unicode[STIX]{x1D73B}|}}\bigg)\bigg)\bigg(\mathop{\prod }_{i=1}^{n}(1\ast \unicode[STIX]{x1D707})(\text{gcd}(P(z_{0},z_{1}),x_{i}))\bigg)\end{eqnarray}$$

and using Lemma 5.1 to obtain

$$\begin{eqnarray}G(B,z_{1};\mathbf{l})\geqslant \mathop{\sum }_{i=0}^{n}c_{i}M_{i},\end{eqnarray}$$

where for $1\leqslant i\leqslant n$ we define $c_{i}:=1$ and

$$\begin{eqnarray}M_{i}:=\mathop{\sum }_{\substack{ k_{i}|P(z_{0},z_{1})}}\bigg(\unicode[STIX]{x1D706}_{k_{i}}^{-}\mathop{\prod }_{j\neq i}\unicode[STIX]{x1D706}_{k_{\!j}}^{+}\bigg)N_{w}(B;(k_{1}l_{1},\ldots ,k_{n}l_{n})),\end{eqnarray}$$

in addition to $c_{0}:=-(n-1)$ and

$$\begin{eqnarray}M_{0}:=\mathop{\sum }_{\substack{ k_{i}|P(z_{0},z_{1})}}\bigg(\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D706}_{k_{i}}^{+}\bigg)N_{w}(B;(k_{1}l_{1},\ldots ,k_{n}l_{n})).\end{eqnarray}$$

The treatment of each individual $M_{i},(i\neq 0)$ , is identical to the treatment of $M_{0}$ earlier in this section. The only difference arises at the last step (the calculation of the Euler products in the main term). Here the coefficients $c_{i}$ satisfy $\sum _{0\leqslant i\leqslant n}c_{i}=1$ , thus completing the proof of Proposition 5.5.