2.1. Preparatory results

Lemma 2.2. Let $q = p^e$ be a prime power and fix $a,b,c \in {\mathbb{N}}$ . We have

(2.2) \begin{align} \# \big\{ (v_1, v_2) \in ({\mathbb{N}} \cap [1,x])^2\,:\, q | v_1^{a} v_2^{b} \big(v_1^c - v_2^c\big) \big\} \ll \frac {x^2}{q^{1 / (2\, {\max}(a,b,c))}}. \end{align}

Proof. Let $d = \gcd (v_1, v_2)$ and $v_1 = d u_1, v_2 = d u_2$ . We have

\begin{align*}v_1^a v_2^b \big(v_1^c - v_2^c\big) = d^{a+b+c} u_1^a u_2^b \big(u_1^c - u_2^c\big).\end{align*}

The left hand side of (2.2) is $\leqslant \sum _{k \geqslant 0} S_k$ , where

\begin{align*} S_k = \sum _{\substack {d \leqslant x \\ v_p(d) = k}} \#\big\{ (u_1, u_2) \in ({\mathbb{N}} \cap [1,x/d])^2\,:\, \gcd (u_1, u_2) = 1, p^{e} | p^{k(a+b+c)} u_1^{a} u_2^{b} \big(u_1^c - u_2^c\big) \big\}. \end{align*}

When $k(a+b+c) \leqslant e$ , we denote $e' = e - k(a+b+c)$ . Then we know that

\begin{align*} S_k & = \sum _{\substack {d \leqslant x \\ v_p(d) = k}} \#\big\{ (u_1, u_2) \in ({\mathbb{N}} \cap [1,x/d])^2\,:\, \gcd (u_1, u_2) = 1, p^{e'} | u_1^{a} u_2^{b} \big(u_1^c - u_2^c\big) \big\}. \end{align*}

Since $\gcd (u_1,u_2) = 1$ , the three integers $u_1^a, u_2^b$ , and $u_1^c - u_2^c$ are pairwise coprime, and so $p^{e'}$ divides exactly one of them. For fixed $d$ , there are at most $(x/d)^2 \cdot (p^{-e'/a} + p^{-e'/b})$ pairs of $(u_1, u_2)$ such that $p^{e'}$ divides $u_1^a$ or $u_2^b$ . According to [Reference Browning, Sofos and Teräväinen1, Lemma 2.3], we have

\begin{align*} \#\big\{ (u_1, u_2) \in ({\mathbb{N}} \cap [1,x/d])^2\,:\, \gcd (u_1, u_2) = 1, p^{e} | \big(u_1^c - u_2^c\big) \big\} \ll \frac {(x/d)^2}{p^{e' / 2c}}. \end{align*}

This yields

\begin{align*} S_k \ll \sum _{\substack {d \leqslant x \\ v_p(d) = k}} \frac {(x/d)^2}{p^{e' / (2\, {\max}(a,b,c))}} \ll \frac {(x / p^k)^2}{p^{ (e - k(a+b+c) ) / (2\, {\max}(a,b,c))}}. \end{align*}

When $k(a+b+c) \gt e$ , we consider the trivial bound $S_k \leqslant (x / p^k)^2$ . By combining these two ranges of $k$ , the left-hand side of (2.2) is

\begin{align*} & \ll \sum _{\substack {k \geqslant 0 \\ k(a+b+c) \leqslant e}} \frac {(x/p^k)^2}{p^{(e-k(a+b+c))/(2\, {\max} (a,b,c))}} + \sum _{\substack {k \geqslant 0 \\ k(a+b+c) \gt e}} (x/p^k)^2 \\ & \leqslant \sum _{\substack {k \geqslant 0 \\ k(a+b+c) \leqslant e}} \frac {x^2}{p^{e/(2\, {\max}(a,b,c))}} + \sum _{\substack {k \geqslant 0}} \frac {1}{p^{2k}} \cdot \frac {x^2}{p^{2e/(a+b+c)}} \\ & \ll \frac {x^2}{q^{1/(2\, {\max}(a,b,c))}} + \frac {x^2}{q^{2/(a+b+c)}} \\ & \ll \frac {x^2}{q^{1/ (2\max (a,b,c))}}. \end{align*}

Lemma 2.3. Let $0 \leqslant k \lt l \leqslant d$ . Let $g$ be a binary form in the form ( 2.1 ) of degree $d$ with $(c_0,\ldots ,c_d)\in S(H)$ . Then for any $m, n \in {\mathbb{Z}}$ , we have

\begin{eqnarray*} \gcd (g(m, n), m^k n^{d-l}) \leqslant \gcd (m,n)^d \gcd (c_0,m)^k \gcd (c_d,n)^{d-l}. \end{eqnarray*}

Proof. Let $t = \gcd (m, n)$ and $m = m_0t, n = n_0t$ . Then

\begin{align*} \gcd \big(g(m,n), m^k n^{d-l}\big) & = t^{k+d-l} \gcd \big(t^{l-k}g(m_0,n_0), m_0^kn_0^{d-l}\big) \\ & \leqslant t^{d} \gcd \big(g(m_0,n_0), m_0^k n_0^{d-l}\big) \\ & = t^{d} \gcd (g(m_0,n_0), m_0^k) \gcd \big(g(m_0,n_0), n_0^{d-l}\big) \\ & \leqslant t^{d} \gcd (g(m_0,n_0), m_0)^k \gcd (g(m_0,n_0), n_0)^{d-l} \\ & \leqslant t^{d} \gcd (c_0, m_0)^k \gcd (c_d, n_0)^{d-l} \\ & \leqslant t^d \gcd (c_0, m)^k \gcd (c_d, n)^{d-l}. \end{align*}

Lemma 2.4. For all $q \in {\mathbb{N}}$ , $z \geqslant 1$ , and $\alpha \geqslant 1/2$ , we have

\begin{align*}\sum _{\substack {d | q \\ d \geqslant z}} d^{-\alpha } \ll z^{-\alpha +1/2} \exp \left( \frac {4 ({\log}(3q))^{1/2}}{({\log} \,{\log} (3q))^{3/2}} \right).\end{align*}

Proof. Writing $d^{-\alpha } = d^{-\alpha + 1/2} \cdot d^{-1/2} \leqslant z^{-\alpha + 1/2} d^{-1/2}$ for $d \geqslant z$ , and bounding the divisor sum by the Euler product $\sum _{d \mid q} d^{-1/2} \leqslant \prod _{p \mid q}(1 - p^{-1/2})^{-1}$ via multiplicativity and the geometric series, we obtain the upper bound

\begin{align*}\sum _{\substack {d|q \\ d \geqslant z}} d^{-\alpha } \leqslant z^{-\alpha +1/2} \prod _{p | q} ( 1 - p^{-1/2} )^{-1}.\end{align*}

The remaining proof is the same as in the proof of [Reference Browning, Sofos and Teräväinen1, Lemma2.5], where this Euler-product estimate is used.

We will prove a generalization of [Reference Browning, Sofos and Teräväinen1, Lemma 2.7].

Lemma 2.5. Let $A \gt 1, d \in {\mathbb{N}}$ and $0 \leqslant k \lt l \leqslant d$ be fixed, and let $x \gg _A 1$ . Let $g$ be a binary form in the form ( 2.1 ) of degree $d$ with $(c_0,\ldots ,c_d)\in S(x^A)$ . Define

(2.3) \begin{align} \mathscr{M}_A=\left \{\mathbf (\mathbf{m}, \mathbf{n}) \in ({\mathbb{N}} \cap [1,x])^4\,:\, \begin{array}{l} m_i, n_i \gt x / ({\log} \,x)^{A}, i = 1,2 \\[2pt] \gcd (m_i, n_j) \lt ({\log} \,x)^{A}, 1 \leqslant i,j \leqslant 2 \\[2pt] |n_1m_2 - n_2m_1| \gt x^2/ ({\log} \,x)^{A+1} \\[2pt] \gcd (g(m_1,n_1), m_1^k n_1^{d-l}) \lt x_2 \end{array} \right \}, \end{align}

where $x_2 = \sqrt {x_1} = \exp (\sqrt {\log x}/ (2\log \log x ))$ . Then

\begin{eqnarray*} \#(({\mathbb{N}} \cap [1,x])^4 \setminus \mathscr{M}_A) \ll x^4 / ({\log} \,x)^A. \end{eqnarray*}

Proof. It is sufficient to show that the set of $(n_1, n_2)$ that fail any individual property in (2.3) has size less than the claimed bound. The proof for the first two properties is identical to that presented in [Reference Browning, Sofos and Teräväinen1, Lemma 2.7].

For the third property, if we arbitrarily choose $n_1$ and $m_2$ , the number of choices of $n_2$ and $m_1$ is at most

\begin{align*} \sum _{|t - n_1m_2| \leqslant x^2/({\log} \,x)^{A+1}} \tau (t) & \ll x^2 / ({\log} \,x)^A. \end{align*}

Hence, the total possible choices are $\ll x^4 / ({\log} \,x)^A.$

Finally, it suffices to estimate the size of pairs satisfying the second property that fail the last property. If $\gcd (g(m_1,n_1), m_1^k n_1^{d-l}) \gt x_2$ , then it follows from Lemma 2.3 that

\begin{align*}\gcd (m_1,n_1)^d \gcd (c_0, m_1)^k \gcd (c_d, n_1)^{d-l} \gt x_2.\end{align*}

Since we assume that $\gcd (m_1, n_1) \lt ({\log} \,x)^A$ , we have

\begin{align*}\gcd (c_0, m_1)^k \gcd (c_d, n_1)^{d-l} \gt x_2 / ({\log} \,x)^{Ad}.\end{align*}

This implies that

\begin{align*}\gcd (c_0, m_1) \cdot \gcd (c_d, n_1) \gt x_2^{1/d} / ({\log} \,x)^{A}.\end{align*}

Therefore, by Lemma 2.4 and the assumption $0 \lt |c_0|, |c_d| \leqslant x^A$ , the number of tuples satisfying the second property but failing the last property is

\begin{align*} & \leqslant x^2 \sum _{\substack {a | c_0, b | c_d\\ {x_2}^{1/d} / ({\log} \,x)^{A} \lt a b \leqslant x}} \ \sum _{\substack {m_1, n_1 \leqslant x \\ a | m_1, b | n_1}} 1 \\ & \leqslant x^4 \, \sum _{a | c_0} \frac {1}{a} \sum _{\substack {b | c_d \\ a b \gt {x_2}^{1/d}({\log} \,x)^{-A}}} \frac {1}{b} \\ & \ll x^4 \, \sum _{a | c_0} \frac {1}{a} \left( \frac {{x_2}^{1/d}}{a ({\log} \,x)^A} \right)^{-1/2} \exp \left( \frac {4({\log} \,(3 x^A))^{1/2}}{({\log} \,{\log}(3 x^A))^{3/2}} \right) \\ & \ll x^4 \left( \frac {{x_2}^{1/d}}{({\log} \,x)^A} \right)^{-1/2} \exp \left( \frac {8({\log} \,(3 x^A))^{1/2}}{({\log} \,{\log}(3 x^A))^{3/2}} \right) \\ & \ll x^4 ({\log} \,x)^{A/2} \exp \left( \frac {8 \sqrt {A}({\log} \,x)^{1/2}}{({\log} \,\log x)^{3/2}} - \frac {1}{4d} \frac {({\log} \,x)^{1/2}}{\log \log x}\right).\\ & \ll x^4 ({\log} \,x)^{-A}. \end{align*}

We would like to recall the fact

(2.4) \begin{align} \tau _B(mn) \leqslant \tau _B(m) \tau _B(n), \end{align}

and the standard upper bound

(2.5) \begin{align} \sum _{r \leqslant R} \tau _B(r)^k \ll R({\log} \,R)^{B^k - 1}, \end{align}

for any $B \geqslant 1$ and $k \geqslant 1$ . We will also apply [Reference Browning, Sofos and Teräväinen1, Lemma 2.8] quite often, which has shown that

(2.6) \begin{align} \sum _{y \leqslant n \leqslant x + y} \tau _B(rn + a)^k \ll \tau _B({\gcd} (a, r))^k x ({\log} \,x)^{B^k - 1} \log \log x, \end{align}

for fixed $k \geqslant 1$ , $A, B \geqslant 1, x \geqslant 3, y \leqslant x^A$ and $1 \leqslant a, r \leqslant x^A$ .

2.2. Proof of Theorem 2.1

Let $x' \in [x/2, x]$ and $g \in {\mathbb{Z}}[x,y]$ be chosen so that

\begin{eqnarray*} \sum _{|a|,|b| \leqslant H} \Big | \sum _{\substack {1 \leqslant m, n \leqslant x'}} \alpha _{m,n} F(am^kn^{d-k} + bm^ln^{d-l} + g(m, n)) \Big |^2 \end{eqnarray*}

is maximized. Opening up the square and recalling that $|\alpha _{m, n}| \leqslant 1$ , it is sufficient to show that

(2.7) \begin{align} \sum _{\substack {1 \leqslant m_i, n_i \leqslant x' \\ i = 1, 2}} \alpha _{m_1, n_1} \overline {\alpha }_{m_2, n_2} \sum _{\substack {|a|, |b| \leqslant H}} F(h_{a,b}(m_1, n_1)) \overline {F}(h_{a,b}(m_2,n_2)) \ll H^2 x^4 ({\log} \,x)^{-A}. \end{align}

Here

\begin{eqnarray*} h_{a,b}(m, n) = am^kn^{d-k} + bm^ln^{d-l} + g(m, n). \end{eqnarray*}

Let $\mathscr{M} = \mathscr{M}_{3A}$ , where $\mathscr{M}_{3A}$ is defined in (2.3). We denote

(2.8) \begin{align} \Delta = (n_1m_2)^{l-k} - (n_2m_1)^{l-k}. \end{align}

We consider separately the following two sets:

\begin{align*} \mathscr{N} & = \{ (\mathbf{m},\mathbf{n}) \in ([1, x'] \cap {\mathbb{Z}})^4\,:\, \exists \, q \in \mathscr{Q} \text{ s.t. } q | \Delta m_1^k n_1^{d-l} \} \cap \mathscr{M}, \\ \mathscr{N}^{\kern3pt c} & = \{(\mathbf{m}, \mathbf{n}) \in ([1, x'] \cap {\mathbb{Z}})^4\} \setminus \mathscr{N}. \end{align*}

Let $v_1 = n_1m_2$ and $v_2 = n_2m_1$ . By Lemma 2.2, (2.5), Cauchy–Schwarz and the assumption $\sum _{q \in \mathscr{Q}} q^{-1/(4d)} \ll ({\log} \,x)^{-3(A+1)}$ , it follows that

\begin{align*} & \ \# \big\{ (\mathbf{m},\mathbf{n}) \in ({\mathbb{N}} \cap [1, x'])^4\,:\, \exists \, q \in \mathscr{Q} \text{ s.t. } q | \Delta m_1^k n_1^{d-l} \big\} \\ \leqslant & \ \# \big\{ (\mathbf{m},\mathbf{n}) \in ({\mathbb{N}} \cap [1, x'])^4\,:\, \exists \, q \in \mathscr{Q} \text{ s.t. } q | \Delta v_2^k v_1^{d-l} \big\} \\ \leqslant & \ \sum _{q \in \mathscr{Q}} \sum _{\substack {v_1, v_2 \leqslant x^2 \\ q | (v_1^{l-k} - v_2^{l-k}) v_2^k v_1^{d-l} }} \tau (v_1) \tau (v_2) \\ \leqslant & \ \sum _{q \in \mathscr{Q}} \Bigg( \sum _{\substack {v_1, v_2 \leqslant x^2}} \tau (v_1)^2 \tau (v_2)^2 \Bigg)^{1/2} \Bigg(\sum _{\substack {v_1, v_2 \leqslant x^2 \\ q | (v_1^{l-k} - v_2^{l-k})v_2^k v_1^{d-l}}} 1 \Bigg)^{1/2} \\ \ll & \ \sum _{q \in \mathscr{Q}} \frac {x^4 \log ^3 x}{q^{1/{(4d)}}} \ll x^4({\log} \,x)^{-3A}. \end{align*}

Combining with Lemma 2.5, we may conclude that

(2.9) \begin{align} \# \mathscr{N}^{\kern3pt c} \ll x^4 ({\log} \,x)^{-3A}. \end{align}

Contribution of $(\mathbf{m}, \mathbf{n}) \in \mathscr{N}^{\kern3pt c}$ . It follows from assumption (1) that the contribution to the left-hand side of (2.7) from $(\mathbf{m},\mathbf{n})\in \mathscr{N}^{\kern3pt c}$ is

\begin{align*} \ll \sum _{(\mathbf{m}, \mathbf{n}) \in \mathscr{N}^{\kern3pt c}} \sum _{|a|, |b| \leqslant H} \tau _B(h_{a, b}(m_1, n_1)) \tau _B(h_{a, b}(m_2, n_2)). \end{align*}

By Cauchy–Schwarz, this is

(2.10) \begin{align} \ll (\# \mathscr{N}^{\kern3pt c})^{1/2} \, \Bigg ( \sum _{\substack { |a|, |b| \leqslant H \\ 1 \leqslant m_1, m_2, n_1, n_2 \leqslant x}} \tau _B(h_{a, b}(m_1, n_1))^2 \tau _B(h_{a, b}(m_2, n_2))^2 \Bigg)^{1/2}. \end{align}

From (2.4), (2.5), (2.6), and Cauchy–Schwarz, we know

\begin{align*} & \sum _{\substack { |a|, |b| \leqslant H \\ 1 \leqslant m_1, m_2, n_1, n_2 \leqslant x}} \tau _B(h_{a, b}(m_1, n_1))^2 \tau _B(h_{a, b}(m_2, n_2))^2 \\ & = \sum _{\substack { |a| \leqslant H \\ 1 \leqslant m_1, m_2, n_1, n_2 \leqslant x}} \sum _{|b| \leqslant H} \tau _B(h_{a, b}(m_1, n_1))^2 \tau _B(h_{a, b}(m_2, n_2))^2 \\ & \leqslant \sum _{\substack { |a| \leqslant H \\ 1 \leqslant m_1, m_2, n_1, n_2 \leqslant x}} \bigg( \sum _{|b| \leqslant H} \tau _B(h_{a, b}(m_1, n_1))^4 \bigg)^{1/2} \bigg ( \sum _{|b| \leqslant H} \tau _B(h_{a, b}(m_2, n_2))^4 \bigg )^{1/2} \\ & \ll \sum _{\substack { |a| \leqslant H \\ 1 \leqslant m_1, m_2, n_1, n_2 \leqslant x}} \big ( H ({\log} \,H)^{B^4} \tau _B\big(m_1^l n_1^{d-l}\big)^4 \big )^{1/2} \big( H ({\log} \,H)^{B^4} \tau _B\big(m_2^l n_2^{d-l}\big)^4 \big)^{1/2} \end{align*}

\begin{align*} \ll & \ H^2 ({\log} \,H)^{B^4} \sum _{\substack { 1 \leqslant m_1, m_2, n_1, n_2 \leqslant x}} \tau _B\big(m_1^l n_1^{d-l}\big)^2 \tau _B\big(m_2^l n_2^{d-l}\big)^2\qquad \qquad\qquad\quad\\ \ll & \ H^2 ({\log} \,H)^{B^4} \bigg (\sum _{1 \leqslant m \leqslant x} \tau _B(m)^{2d} \bigg )^4 \\ \ll & \ H^2 ({\log} \,H)^{B^4} x^4 ({\log} \,x)^{4 B^{2d}} \\ \ll & \ H^2 x^4 ({\log} \,x)^{100B^4 + 4B^{2d}}. \end{align*}

Therefore, by (2.9) we know that (2.10) is $\ll H x^4 ({\log} \,x)^{-A}$ , since $50 B^4 + 2B^{2d} \leqslant 100B^{2d+4} \leqslant A/2$ by assumption.

Contribution of $(\mathbf{m},\mathbf{n}) \in \mathscr{N}$ . By the triangle inequality and assumption (1), the left-hand side of (2.7) is

(2.11) \begin{align} \ll \sum _{(\mathbf{m}, \mathbf{n}) \in \mathscr{N}} \sum _{u_2 \in {\mathbb{Z}}} \tau _B(u_2) \bigg | \sum _{u_1 \in {\mathbb{Z}}} F(u_1) \gamma (\mathbf{u}) \bigg |, \end{align}

where

\begin{align*} \gamma (\mathbf{u}) = \#\{ (a,b) \in ({\mathbb{Z}} \cap [{-}H,H])^2\,:\, u_i = h_{a,b}(m_i, n_i), i = 1,2 \}. \end{align*}

We make the change of variables $u_i' = (u_i - g(m_i, n_i))/(m_i^{k}n_i^{d-l})$ to write this as

\begin{align*} \sum _{(\mathbf{m}, \mathbf{n}) \in \mathscr{N}} \sum _{u_2' \in {\mathbb{Z}}} \tau _B\big(u_2' m_2^k n_2^{d-l} + g(m_2,n_2)\big) \bigg | \sum _{u_1' \in {\mathbb{Z}}} F\big(u_1'm_1^kn_1^{d-l} + g(m_1,n_1)\big) \gamma '(\mathbf{u}') \bigg |, \end{align*}

where

(2.12) \begin{align} \gamma '(\mathbf{u}') = \#\big\{ (a,b) \in ({\mathbb{Z}} \cap [{-}H,H])^2\,:\, u_i' = a n_i^{l-k} + b m_i^{l-k}, i = 1,2 \big\}. \end{align}

From (2.8), we have

\begin{align*}|\Delta | \geqslant |n_1m_2 - n_2m_1| \cdot \min (n_1m_2, n_2m_1)^{l-k-1}.\end{align*}

When $(\mathbf{m},\mathbf{n}) \in \mathscr{N}$ , it follows (2.3) that

(2.13) \begin{align} \frac {x^{2(l-k)}}{({\log} \,x)^{6A(l-k)}} \ll |\Delta | \ll x^{2(l-k)} . \end{align}

The solution of (2.12) is $a = \Delta ^{-1}(m_2^{l-k}u_1' - m_1^{l-k}u_2')$ and $b = \Delta ^{-1}({-}n_2^{l-k}u_1' + n_1^{l-k}u_2').$ Hence, we require

(2.14) \begin{align} m_2^{l-k}u_1' \equiv m_1^{l-k}u_2' \ (\mathrm{mod}\, \Delta ), \quad n_2^{l-k}u_1' \equiv n_1^{l-k}u_2' \ (\mathrm{mod}\, \Delta ). \end{align}

This system has an integer solution if and only if

(2.15) \begin{align} \gcd \big(m_2^{l-k}, \Delta \big) \, \big | \, m_1^{l-k}u_2', \quad \gcd \big(n_2^{l-k}, \Delta \big) \, \big | \, n_1^{l-k}u_2'. \end{align}

In this case, the equation (2.14) has a unique integer solution $u_1' \equiv u_1''$ modulo $\Delta _0$ , where

\begin{align*} \Delta _0 = {\mathrm{lcm}} \left( \frac {\Delta }{\gcd (m_2^{l-k}, \Delta )}, \frac {\Delta }{\gcd (n_2^{l-k}, \Delta )} \right). \end{align*}

Note that

\begin{align*} \gcd \big(n_2^{l-k}, \Delta \big) = \gcd \big(n_2^{l-k}, (m_2n_1)^{l-k}\big) \leqslant ( \gcd (n_2, m_2) \gcd (n_2,n_1) )^{l-k} \leqslant ({\log} \,x)^{6A(l-k)}, \end{align*}

by (2.3). Hence, by (2.13), we have

(2.16) \begin{align} \frac {x^{2(l-k)}}{({\log} \,x)^{12A(l-k)}} \ll \Delta _0 \ll x^{2(l-k)} . \end{align}

The condition $|a|, |b| \leqslant H$ is equivalent to

\begin{align*} \big|m_2^{l-k}u_1' - m_1^{l-k}u_2'\big| \leqslant |\Delta | H, \quad \big|-n_2^{l-k}u_1' + n_1^{l-k}u_2'\big| \leqslant |\Delta | H. \end{align*}

For fixed $(\mathbf{m}, \mathbf{n}, u_2')$ , we require that $u_1'$ lies in some interval $J(\mathbf{m},\mathbf{n}, u_2')$ with

(2.17) \begin{align} |J(\mathbf{m},\mathbf{n}, u_2')| \leqslant \frac {2 |\Delta | H}{\max (m_2, n_2)^{l-k}} \ll H x^{l-k}({\log} \,x)^{3A(l-k)} . \end{align}

By (2.12), we also need $|u_2'| \leqslant (m_2^{l-k} + n_2^{l-k})H$ .

We may now conclude that (2.11) is

(2.18) \begin{align} \leqslant \sum _{(\mathbf{m},\mathbf{n}) \in \mathscr{N}} \sum _{|u_2'| \leqslant (m_2^{l-k} + n_2^{l-k}) H} \tau _B\big( u_2' m_2^k n_2^{d-l} + g(m_2,n_2)\big) | S_1 |, \end{align}

where

\begin{align*} S_1 & = \sum _{\substack {u_1' \in J(\mathbf{m},\mathbf{n},u_2') \\ u_1' \equiv u_1'' \, \mathrm{mod}\, \Delta _0}} F\big (u_1' m_1^k n_1^{d-l} + g(m_1, n_1)\big ), \end{align*}

if (2.15) holds, and $S_1 = 0$ if it does not. Here, $u_1' \equiv u_1''$ is the unique (if possible) solution for (2.14) modulo $\Delta _0$ . We may rewrite

\begin{align*} S_1 = \sum _{\substack {u \in I \\ u \equiv v \, \mathrm{mod}\, q}} F(u), \end{align*}

with modulus $q = {\mathrm{lcm}}(\Delta _0, m_1^k n_1^{d-l})$ and interval length $|I| = m_1^k n_1^{d-l} J(\mathbf{m},\mathbf{n}, u_2')$ . It follows that

(2.19) \begin{align} \gcd (v, q) & \leqslant \gcd \big(g(m_1,n_1), m_1^k n_1^{d-l}\big) \gcd \big(u_1'' m_1^k n_1^{d-l} + g(m_1,n_1), \Delta _0\big). \end{align}

By definition (2.3), we see that

(2.20) \begin{align} \gcd (g(m_1,n_1), m_1^k n_1^{d-l}) \lt x_2 = \exp \big(\sqrt {\log x}/2\log \log x\big) \end{align}

We will now estimate (2.18) in three different subcases.

Subcase I. $\gcd (u_1'' m_1^k n_1^{d-l} + g(m_1,n_1), \Delta _0) \leqslant x_2$ and $|I| \gt H^{1 - \varepsilon }x^{l-k}$ .

In this subcase, the assumption (2) of Theorem2.1 is now satisfied, according to (2.19) and (2.20). We have

\begin{align*}|S_1| \ll \frac {|I|}{q ({\log} \,H)^{40 Ad^2}} = \frac {\gcd \big(\Delta _0, m_1^k n_1^{d-l}\big) J(\mathbf{m}, \mathbf{n}, u_2')}{\Delta _0 \, ({\log} \,H)^{40Ad^2}}.\end{align*}

It follows from (2.3) that

\begin{align*} \gcd \big(\Delta _0, m_1^k n_1^{d-l}\big) & \leqslant \gcd \big(\Delta , (m_1n_1)^d\big) \\ & \leqslant \gcd \big((m_2n_2)^{d(l-k)}, (m_1n_1)^d\big) \\ & \leqslant \gcd (m_2n_2, m_1n_1)^{d^2} \\ & \ll ({\log} \,x)^{12Ad^2}. \end{align*}

Hence, (2.16) and (2.17) yield that

\begin{align*} |S_1| \ll \frac {H x^{l-k} ({\log} \,x)^{12Ad^2 + 15A(l-k)}}{x^{2(l-k)} ({\log} \,H)^{40Ad^2}} \leqslant \frac {H}{x^{l-k} ({\log} \,H)^{12Ad^2}}. \end{align*}

By (2.4), (2.5), and (2.6), we know (2.18) is

\begin{align*} & \ll \sum _{m_1,m_2,n_1,n_2 \leqslant x} \ \sum _{|u_2'| \leqslant (m_2^{l-k} + n_2^{l-k}) H} \tau _B\big( u_2' m_2^k n_2^{d-l} + g(m_2,n_2)\big) \frac {H}{x^{l-k} ({\log} \,H)^{12Ad^2}} \\ & \ll \sum _{m_1,m_2,n_1,n_2 \leqslant x} \tau _B\big(m_2^k n_2^{d-l}\big) H^2 ({\log} \,H)^{B - 12Ad^2} \\ & \ll x^2 H^2 ({\log} \,H)^{B - 12 Ad^2} \sum _{m_2 \leqslant x} \tau _B(m_2)^k \sum _{n_2 \leqslant x} \tau _B(n_2)^{d-l} \\ & \ll x^4 H^2 ({\log} \,H)^{B + B^k + B^{d-l} - 12 Ad^2}. \end{align*}

As $B+B^k+B^{d-l} \leqslant 3B^d \leqslant 2A$ and $({\log} \,H)^{-1} \leqslant (2d \log x)^{-1}$ by assumption, this shows that (2.18) is $\ll x^4H^2 ({\log} \,x)^{-A}$ .

Subcase II. $\gcd (u_1'' m_1^k n_1^{d-l} + g(m_1,n_1), \Delta _0) \leqslant x_2$ and $|I| \leqslant H^{1 - \varepsilon } x^{l-k}$ .

We apply the crude bound $|F(n)| \leqslant \tau _B(|n|) \ll (Hx^d)^{\varepsilon /10}$ . This leads to the estimate

\begin{align*} |S_1| \ll (H x^d)^{\varepsilon /10} \left( \frac {|I|}{q} + 1 \right). \end{align*}

We use the assumption $|I| \leqslant H^{1 - \varepsilon } x^{l-k}$ , the fact $q \geqslant \Delta _0$ and (2.16) to deduce that

\begin{align*} \frac {(H x^d)^{\varepsilon /10} \, |I|}{q} \ll \frac {H^{1 - 0.7 \varepsilon }x^{l-k}}{\Delta _0} \ll \frac {H}{x^{l-k}({\log} \,H)^{12Ad^2}}. \end{align*}

According to the assumption $x \leqslant H^{1/(2d)}$ in Theorem2.1, we get

\begin{align*} |S_1| \ll \frac {H}{x^{l-k}({\log} \,H)^{12Ad^2}} + (H x^d)^{\varepsilon /10} \ll \frac {H}{x^{l-k}({\log} \,H)^{12Ad^2}}. \end{align*}

The remaining part is similar to subcase I.

Subcase III. $\gcd (u_1'' m_1^k n_1^{d-l} + g(m_1,n_1), \Delta _0) \gt x_2$ .

Let $u_1' = u' \Delta _0 + u_1''$ . We rewrite

\begin{align*}S_1 = \sum _{u' \in J'} F\big(u' \Delta _0 m_1^k n_1^{d-l} + u_1'' m_1^k n_1^{d-l} + g(m_1,n_1)\big),\end{align*}

where

\begin{align*}|J'| \ll |J| / \Delta _0 \ll H x^{k-l} ({\log} \,x)^{15A(l-k)},\end{align*}

by (2.16) and (2.17). Note that (2.6) along with the assumptions $|F(n)| \leqslant \tau _B(|n|)$ and $\log H \leqslant ({\log} \,x)^{100}$ allow us to bound

\begin{align*} S_1 & \ll \tau _B\big(\Delta _0 m_1^k n_1^{d-l}\big) |J'| ({\log} \,|J'|)^B \\[4pt]& \ll H x^{k-l} ({\log} \,x)^{15A(l-k) + 101B} \tau _B\big(\Delta _0 m_1^k n_1^{d-l}\big). \end{align*}

For brevity, we denote

\begin{align*}\sum\nolimits^{*}{}_{\kern-16pt\raise-10pt\hbox{${}_{u_2'}$}}\,\ \ \ = \sum _{\substack {|u_2'| \leqslant (m_2^{l-k} + n_2^{l-k}) H \\ \gcd (u_1'' m_1^k n_1^{d-l} + g(m_1,n_1), \Delta _0) \gt x_2}}.\end{align*}

We may bound (2.18) by

\begin{align*} \ll H x^{k-l} ({\log} \,x)^{15A(l-k)+101B} \sum _{(\mathbf{m},\mathbf{n}) \in \mathscr{N}} \sum\nolimits^{*}{}_{\kern-16pt\raise-10pt\hbox{${}_{u_2'}$}}\,\ \ \ \tau _B \big(u_2' m_2^k n_2^{d-l} + g(m_2,n_2)\big) \tau _B\big(\Delta _0 m_1^kn_1^{d-l}\big). \end{align*}

By applying Cauchy–Schwarz, this is

(2.21) \begin{align} \leqslant H x^{k-l} ({\log} \,x)^{15A(l-k)+101B} S_2^{1/2} S_3^{1/2}, \end{align}

where

\begin{align*} S_2 = \sum _{(\mathbf{m},\mathbf{n})\in \mathscr{N}} \sum _{\substack {|u_2'| \leqslant (m_2^{l-k} + n_2^{l-k}) H}} \tau _B \big(u_2' m_2^k n_2^{d-l} + g(m_2,n_2)\big)^2 \tau _B\big(\Delta _0 m_1^kn_1^{d-l}\big)^2, \end{align*}

and

\begin{align*} S_3 = \sum _{(\mathbf{m},\mathbf{n}) \in \mathscr{N}} \sum\nolimits^{*}{}_{\kern-16pt\raise-10pt\hbox{${}_{u_2'}$}}\,\ \ \ 1. \end{align*}

It follows from (2.14) that

\begin{align*} \gcd \big(u_1'' m_1^k n_1^{d-l} + g(m_1,n_1), \Delta _0\big) & \leqslant \, \gcd \big(n_2^{l-k} u_1'' m_1^k n_1^{d-l} + n_2^{l-k} g(m_1,n_1), \Delta \big) \\[2pt] & = \, \gcd \big(u_2' m_1^kn_1^{d-k} + n_2^{l-k} g(m_1,n_1), \Delta \big). \end{align*}

Hence,

\begin{align*}S_3 \leqslant \sum _{(\mathbf{m},\mathbf{n}) \in \mathscr{N}} \sum _{\substack {|u_2'| \leqslant (m_2^{l-k} + n_2^{l-k}) H \\ \gcd (u_2' m_1^kn_1^{d-k} + n_2^{l-k} g(m_1,n_1), \Delta ) \gt x_2}} 1.\end{align*}

By (2.3), we know

\begin{align*} \gcd \big(m_1^kn_1^{d-k}, \Delta \big) & \leqslant \ \gcd \big(m_1^k, (m_2n_1)^{l-k}\big) \gcd \big(n_1^{d-k}, (m_1n_2)^{l-k}\big) \\ & \ll ({\gcd} (m_1, m_2) \gcd (m_1, n_1) \gcd (n_1, m_1) \gcd (n_1, n_2) ) ^{d^2} \\ & \ll \, ({\log} \,x)^{12Ad^2}. \end{align*}

As a consequence of [Reference Browning, Sofos and Teräväinen1, Lemma 2.6] and the fact $\tau (\Delta ) \ll |\Delta |^{0.1/(l-k)} \ll x^{0.2}$ , it follows that

(2.22) \begin{align} S_3 \ll & \ x^4 \left ( \frac {x^{l-k} H ({\log} \,x)^{12Ad^2}}{x_2^{1/2}} \exp \left( \frac {4({\log} \,(3|\Delta |))^{1/2}}{({\log} \,{\log}(3|\Delta |))^{3/2}} \right ) + \tau (\Delta ) \right ) \nonumber \\ \ll & \ \frac {x^{4 +l-k} H ({\log} \,x)^{12Ad^2}}{x_2^{1/2}} \exp \left( \frac {8 d^{1/2} ({\log} \,(3x))^{1/2}}{({\log} \,\log x)^{3/2}} \right) + x^{4.2} \nonumber \\ \ll & \ \frac {x^{4+l-k} H}{\exp (({\log} \,x)^{0.49})}. \end{align}

Now (2.4), (2.6), and Cauchy–Schwarz imply that

\begin{align*} S_2 & \ll x^{l-k} H ({\log} \,H)^B \sum _{(\mathbf{m},\mathbf{n}) \in \mathscr{N}} \tau _B(\Delta )^2 \tau _B\big(m_1^k\big)^2 \tau _B\big(m_2^k\big)^2 \tau _B\big(n_1^{d-l}\big)^2 \tau _B\big(n_2^{d-l}\big)^2 \\ & \ll x^{l-k} H ({\log} \,x)^{100B} \Bigg( \sum _{\substack {m_1,m_2,n_1,n_2 \leqslant x \\ m_1n_2 \neq m_2n_1}} \tau _B(\Delta )^4 \Bigg)^{1/2} \bigg ( \sum _{m \leqslant x} \tau _B(m)^{4d} \bigg )^{2}. \end{align*}

Following from (2.5), [Reference Browning, Sofos and Teräväinen1, Lemma 2.9], and Cauchy–Schwarz, we obtain that

\begin{align*} \sum _{\substack {m_1,m_2,n_1,n_2 \leqslant x \\ m_1n_2 \neq m_2n_1}} \tau _B(\Delta )^4 & \ll \sum _{1 \leqslant t_1 \lt t_2 \leqslant x^2} \tau _B\big(t_1^{l-k} - t_2^{l-k}\big)^4 \tau (t_1) \tau (t_2) \\ & \ll \bigg ( \sum _{1 \leqslant t_1 \lt t_2 \leqslant x^2} \tau _B(t_1^{l-k} - t_2^{l-k})^8 \bigg )^{1/2} \sum _{t \leqslant x^2} \tau (t)^2 \\ & \ll x^4 ({\log} \,x)^{dB^8/2 + 3} \end{align*}

Combine with (2.5), we know

\begin{align*} S_2 \ll x^{4+l-k} H ({\log} \,x)^{100B + dB^8/4 + 3/2 + 2 B^{4d}}. \end{align*}

Therefore, along with (2.21) and (2.22), the last subcase is crudely bounded by $\ll x^4 H^2 ({\log} \,x)^{-A}$ .

3.1. Proof of Theorem 1.2

We will apply Theorem2.1 and Proposition 3.1 to establish Theorem1.2. We use Chebyshev’s inequality in the form that a second moment bound controls the number of coefficient vectors for which the corresponding summand exceeds a prescribed threshold. Our goal is to prove that

(3.4) \begin{eqnarray} \sum _{g \in \mathscr{C}} \bigg | \sum _{u,v \leqslant x} \lambda (g(u,v)) \bigg |^2 \ll x^4 H^2 ({\log} \,x)^{-A}, \end{eqnarray}

uniformly for $x \in [H^c, 2H^c]$ , where $0\lt c\lt 5/(31d)$ is as in Theorem1.2. From this, Theorem1.2 will then follow directly from Chebyshev’s inequality.

Suppose that $\mathscr{C}$ has the $k$ th and $l$ th coordinates as two of its free coordinates, with $k\lt l$ . In order to prove (3.4), it suffices to show that

(3.5) \begin{eqnarray} \sum _{|a|,|b| \leqslant H} \Bigg | \sum _{m,n \leqslant x} \lambda (am^kn^{d-k} + bm^ln^{d-l} + g(m,n)) \Bigg |^2 \ll H^2 x^4 ({\log} \,x)^{-A}, \end{eqnarray}

uniformly for binary forms $g \in {\mathbb{Z}}[s,t]$ of degree $\leqslant d$ , which have coefficients in $[{-}H,H]$ and satisfying zero $s$ -coefficients at degree $k$ and $l$ . We claim it suffices to prove that, for any coefficients $\alpha _{m,n} \in \mathbb{C}$ such that $|\alpha _{m,n}| \leqslant 1$ , we have

(3.6) \begin{eqnarray} \sum _{|a|,|b| \leqslant H} \Bigg | \sum _{m,n \leqslant x} \alpha _{m, n}\lambda (am^kn^{d-k} + bm^ln^{d-l} + g(m,n)) \Bigg |^2 \ll H^2 x^4 ({\log} \,x)^{-A}, \end{eqnarray}

uniformly for binary forms $g \in {\mathbb{Z}}[s,t]$ of degree $\leqslant d$ which have coefficients in $[{-}H,H]$ and satisfying the conditions $c_{d-k} = c_{d-l} = 0$ and $c_0 \cdot c_d \neq 0$ , where

\begin{align*}g(m, n) = c_0 m^d + c_1 m^{d-1}n + \cdots + c_{d-1}m n^{d-1} + c_d n^d.\end{align*}

Let $i \geqslant 0$ be the minimal integer such that $c_{d-i} \neq 0$ , taking $i = d + 1$ if $g$ vanishes identically. Similarly let $j \leqslant d$ be the maximal integer such that $c_j \neq 0$ , taking $j = -1$ if $g$ vanishes identically. Note that we have $d-i \geqslant j$ .

If $i \leqslant k$ and $d-j \geqslant l$ , we see that

\begin{align*} \Sigma = \sum _{|a|,|b| \leqslant H} \Bigg | \sum _{m,n \leqslant x} \alpha _{m,n} \lambda (am^{k-i}n^{d-k-j} + bm^{l-i}n^{d-l-j} + g^{*}(m,n)) \Bigg |^2, \end{align*}

where $\alpha _{m,n} = \lambda (m)^i \lambda (n)^j$ and $g^{*}(m,n) = c_{j}m^{d-i-j} + \cdots + c_{d-i}n^{d-i-j}$ with $c_j \cdot c_{d-i} \neq 0$ . This is exactly of the form (3.6).

If $i \gt k$ and $d-j \geqslant l$ , we have

\begin{align*} \Sigma & = \sum _{|a|,|b| \leqslant H} \Bigg | \sum _{m,n \leqslant x} \alpha _{m,n} \lambda (an^{d-i-j} + bm^{l-i}n^{d-l-j} + g^{\dagger }(m,n)) \Bigg |^2 \\ & = \sum _{|a|,|b| \leqslant H} \Bigg | \sum _{m,n \leqslant x} \alpha _{m,n} \lambda \big ((a+1)n^{d-i-j} + bm^{l-i}n^{d-l-j} + g^{\dagger }(m,n)\big ) \Bigg |^2 + O(Hx^4), \end{align*}

on shifting $a$ by $1$ , where $\alpha _{m, n} = \lambda (m)^k \lambda (n)^j$ and

\begin{align*}g^{\dagger }(m, n) = c_j m^{d-i-k} + \ldots + c_{d-i}m^{i-k}n^{d-i-j}.\end{align*}

The first term is exactly of the form (3.6), with $g(m, n) = g^{\dagger }(m, n) + n^{d-i-j}$ , and the error term of $O(Hx^4)$ is acceptable. If $i \leqslant k$ and $d-j \lt l$ , the argument follows similarly.

Finally, if $i \gt k$ and $d-j \lt l$ , we have

\begin{align*} \Sigma & = \sum _{|a|,|b| \leqslant H} \Bigg | \sum _{m,n \leqslant x} \alpha _{m,n} \lambda (an^{d-i-j} + bm^{d-i-j} + g^{\ddagger }(m,n)) \Bigg |^2 \\ & = \sum _{|a|,|b| \leqslant H} \Bigg | \sum _{m,n \leqslant x} \alpha _{m,n} \lambda ( (a+1)n^{d-i-j} + (b+1)m^{d-i-j} + g^{\ddagger }(m,n)) \Bigg |^2+ O(Hx^4), \end{align*}

on shifting both $a$ and $b$ by $1$ , where $\alpha _{m, n} = \lambda (m)^k \lambda (n)^{d-l}$ , and

\begin{align*}g^{\ddagger }(m, n) = c_j m^{d-j-k} n^{j-d+l} + \ldots + c_{d-i}m^{i-k}n^{l-i}.\end{align*}

The first term is exactly of the form (3.6), with $g(m, n) = g^{\ddagger }(m,n) + n^{d-i-j} + m^{d-i-j}$ , and the error term of $O(Hx^4)$ is acceptable.

3.2. Proof of Theorem 1.3

Let

(3.7) \begin{align} w = \exp \big(\sqrt {\log x}\big), \quad W = \prod _{p \leqslant w} p. \end{align}

Define a modified Cramér model (or a $W$ -tricked model) for the von Mangoldt function by

\begin{align*}\Lambda _w(n) = \frac {W}{\varphi (W)} \mathbf{1}_{\gcd (n, W) = 1}.\end{align*}

Note that $\Lambda _w(n) \leqslant W / \varphi (W) \leqslant \log x$ , for any $n \in {\mathbb{N}}$ . Recall the definition of $\mathfrak{S}_{g_1, \ldots , g_r}(x)$ in Theorem1.3, for an $r$ -tuple $g_1, \ldots , g_r \in {\mathbb{Z}}[s, t]$ .

Lemma 3.3. Let $A \geqslant 1$ and $d, r \in {\mathbb{N}}$ be fixed, let $x \geqslant 2$ , and let $w$ be as in ( 3.7 ). Let $g_1, \ldots , g_r \in {\mathbb{Z}}[s,t]$ be binary forms of degree $\leqslant d$ . Then

(3.8) \begin{align} \frac {1}{x^2}\sum _{m, n \leqslant x} \Lambda _w(g_1(m,n)) \cdots \Lambda _w(g_r(m,n)) = \mathfrak{S}_{g_1, \ldots , g_r}(x) + O({\log} \,(x)^{-A}). \end{align}

Proof. Let

\begin{align*}h(p) = \mathbf{1}_{p \leqslant w} \frac {\#\big\{(u, v) \in {\mathbb{F}}_p^2\,:\, g_1(u, v) \cdots g_r(u, v) = 0 \big\}}{p^2}.\end{align*}

When $h(p) = 1$ for some prime $p$ , Lemma 3.3 holds trivially. This is because in that case both $\Lambda _w(g_1(m,n)) \cdots \Lambda _w(g_r(m,n))$ and $\mathfrak{S}_{g_1, \ldots , g_r}(x)$ would be zero. When $h(p) = 0$ for every prime $p \leqslant w$ , $g_1 \cdots g_r$ does not have a fixed prime divisor $p \leqslant w$ . In particular, for each $g_i$ , $1 \leqslant i \leqslant r$ , the content of $g_i$ (the greatest common divisor of its coefficients) is not divisible by $p$ . By Lagrange’s theorem on roots of a polynomial over a field, for any $1 \leqslant i \leqslant r$ , we have

\begin{align*} \#\big\{(u, v) \in {\mathbb{F}}_p^2\,:\, g_i(u, v) = 0 \big\} & \leqslant p + \#\big\{(u, v) \in {\mathbb{F}}_p^2\,:\, u \neq 0, g_i(u, v) = 0 \big\} \\ & \leqslant p + (p-1)d \lt p (d+1). \end{align*}

We have

\begin{align*}h(p) \leqslant (d+1)rp/p^2 = (d+1)r/p,\end{align*}

whenever $p \leqslant w$ . It follows from Mertens’ theorem that

\begin{align*}\prod _{a \leqslant p \lt b}(1 - h(p))^{-1} \leqslant \prod _{\substack {a \leqslant p \lt b \\ p \leqslant w}} \left ( 1 - \frac {(d+1)r}{p} \right )^{-1} \leqslant \left ( \frac {\log b}{\log a} \right )^{(d+1)r} \left ( 1 + \frac {K_{d, r}}{\log a} \right ), \end{align*}

where $K_{d, r}$ is a constant depending on $d$ and $r$ . Following the proof of [Reference Browning, Sofos and Teräväinen1, Lemma5.1], we use the fundamental lemma of sieve theory [Reference Iwaniec and Kowalski29, Fundamental Lemma 6.3] with $\kappa = (d+1)r$ to obtain the desired estimate.

By Lemma 3.3, the triangle inequality, Chebyshev’s inequality, and the induction argument in [Reference Browning, Sofos and Teräväinen1, Proof of Theorem 1.4], it suffices to prove for any coefficients $|\alpha _n| \leqslant 1$ , we have

(3.9) \begin{align} \sum _{g \in \mathscr{C}} \left | \sum _{m, n \leqslant x} \alpha _n(\Lambda - \Lambda _w)(g(m,n)) \right |^2 \ll \frac {x^2H^2}{({\log} \,x)^A}, \end{align}

uniformly for $x \in [H^c, 2H^c]$ . Suppose $\mathscr{C}$ has the $k$ th and $l$ th coordinates as two of its free coordinates, with $k \lt l$ . It then suffices to prove that

(3.10) \begin{align} \sum _{|a|, |b| \leqslant H} \left | \alpha _n(\Lambda - \Lambda _w) (a m^{k}n^{d-k} + bm^ln^{d-l} + g(m,n)) \right |^2 \ll \frac {x^2H^2}{({\log} \,x)^A}, \end{align}

uniformly for sequences $|\alpha _n| \leqslant 1$ , and for binary forms $g \in {\mathbb{Z}}[s, t]$ of degree $\leqslant d$ having coefficients in $[{-}H, H]$ , and with $g(0,1) \cdot g(1, 0) \neq 0$ . The condition $g(0,1) \cdot g(1, 0) \neq 0$ can indeed be imposed, since if $k = 0$ and $g(0,1) = 0$ , or $l = d$ and $g(1, 0) = 0$ , we can change the value of $g(0, 1)$ or $g(1, 0)$ by $O(1)$ without changing the validity of (3.10). Alternatively, if $k \gt 0$ or $l \lt d$ , the $\mathscr{C}$ consists of binary forms that are not irreducible, a case that was excluded.

Proof of Theorem 1.3. As in the last paragraph of [Reference Browning, Sofos and Teräväinen1, Proof of Theorem 1.4], we can use [Reference Iwaniec and Kowalski29, Fundamental Lemma6.3] to conclude from Proposition 3.2 that

\begin{align*}\Bigg| \sum _{\substack {x \leqslant n \leqslant x + h \\ n \equiv a \, (\mathrm{mod}\, q)}} (\Lambda (n) - \Lambda _w(n)) \Bigg| \ll \frac {h}{q({\log} \,x)^A},\end{align*}

provided that $x \geqslant h \geqslant x^{1-5/36+\varepsilon }, 1 \leqslant a \leqslant q \leqslant x^{5/36}, \gcd (a, q) \leqslant \exp (\sqrt {\log x})$ , and $q$ is not a multiple of any element of the set $\mathscr{Q}$ present in Proposition 3.2. Note that the function $F(n) = (\Lambda (n) - \Lambda _w(n))/({\log} \,x)$ is $1$ -bounded. Hence, Assumptions (1) and (2) of Theorem2.1 both hold, provided that $(2Hx^d)^{1-5/36} \leqslant H^{1-\varepsilon }$ for some small $\varepsilon \gt 0$ , which indeed holds by the assumption that $x \leqslant 2H^c$ for some fixed $c \lt 5/(31d)$ . Now (3.9) follows by applying Theorem2.1 to $F(n)$ .

4.1. A localized counting function

Let

(4.8) \begin{align} k(x) = \lfloor 1000dA \log \log x \rfloor \end{align}

and, for the remainder of the norm-form argument, redefine

(4.9) \begin{align} W = \prod _{p \leqslant w} p^{k(x)}\ \text{ for }\ w =\exp (\sqrt {\log x}). \end{align}

This differs from the quantity denoted by $W$ in (3.7): the present $W$ is the previous one raised to the power $k(x)$ , and the notation follows [Reference Browning, Sofos and Teräväinen1, §7]. In particular, by the prime number theorem, we have $\log W = (1 + o(1)) w k(x)$ , and thus, $W$ exceeds any power of $x$ .

For any $a \in {\mathbb{Z}}$ and $q \in {\mathbb{N}}$ , we define

\begin{align*} \rho _K(q, a) = \frac {1}{q^{e-1}} \# \{ \mathbf{s} \in ({\mathbb{Z}} / q{\mathbb{Z}})^e\,:\, \mathbf{N}_K(\mathbf{s}) \equiv a \, (\mathrm{mod}\, \, q) \}. \end{align*}

This function is multiplicative in $q$ . With this notation, we put

(4.10) \begin{align} \hat {R}_K(n;\, B) = \rho _K(W, n) \cdot \omega (n;\, B), \end{align}

where $\omega (n;\, B)$ is an archimedean factor that we proceed to define.

By (4.4), suppose now that $y = \mathbf{N}_K(\mathbf{x})$ with $\mathbf{x} \in B{\mathscr{B}}$ . Then, by the implicit function theorem, we can express $x_j$ in terms of $\mathbf{x}' = (x_1, \ldots , \widehat {x_j}, \ldots , x_e)$ and $y$ as $x_j = x_j(\mathbf{x}', y)$ . We now define

(4.11) \begin{align} \omega (y) = \omega (y;\, B) = \int _{{\mathscr{B}}(y)} \Big | \frac {\partial \mathbf{N}_K}{\partial x_j}(x_j(\mathbf{x}', y), \mathbf{x}') \Big |^{-1} d \mathbf{x}', \end{align}

where ${\mathscr{B}}(y) = \{ \mathbf{x}' \in {\mathbb{R}}^{e-1}\,:\, (x_j(\mathbf{x}', y), \mathbf{x}') \in B {\mathscr{B}} \}$ . This expression represents the real density of solutions to the equation $y = \mathbf{N}_K(\mathbf{x})$ . A similar real density was originally introduced in [Reference Browning and Heath-Brown24, Lemma 9].

We recall [Reference Browning, Sofos and Teräväinen1, Lemma 7.1], which collects some basic facts about $\omega (y)$ .

We now define the localized counting function to be

(4.12) \begin{align} \hat {N}_{\mathbf{c}}(x) = \sum _{\substack {-x \leqslant m, n \leqslant x \\ n \neq 0}} \hat {R}_K(g_{\mathbf{c}}(m, n);\, B), \end{align}

for ${\mathbf{c}} \in {\mathbb{Z}}^{d+1}$ . The proof of Proposition 4.1 proceeds in two steps. First, we shall show that $\hat {N}_{\mathbf{c}}(x)$ is usually a good approximation to $N_{\mathbf{c}}(x)$ , for suitable ranges of $x$ and $H$ . The remaining task will then be to show that $\hat {N}_{\mathbf{c}}(x)$ is rarely smaller than it should be.

6.1. The archimedean densities are rarely small

First, we turn our attention to the archimedean aspect.

For $\mathbf{u} = (u, v) \in {\mathbb{R}}^2 \setminus \{0\}$ , let $(r, \theta ) = (r(\mathbf{u}), \theta (\mathbf{u}))$ be its polar coordinates so that

\begin{align*}u = r \cos \theta , \quad v = r \sin \theta , \quad \text{ with } r \gt 0 \text{ and } 0 \leqslant \theta \lt 2\pi .\end{align*}

For $0 \leqslant \theta \lt 2\pi$ , we also define

\begin{align*}g_{\mathbf{c}}(\theta ) = g_{\mathbf{c}}(\cos \theta , \sin \theta ).\end{align*}

Since $g_{\mathbf{c}}(\mathbf{u})$ is a binary form of degree $d$ , we have

\begin{align*}g_{\mathbf{c}}(\mathbf{u}) = r^d g_{\mathbf{c}}(\theta ).\end{align*}

For $\epsilon \in \{\pm \}$ and ${\mathbf{c}} \in S_\epsilon (H, \tilde {H})$ , we choose $\mathbf{u}_{\mathbf{c}} = (u_{\mathbf{c}}, v_{\mathbf{c}})$ with $|\mathbf{u}_{\mathbf{c}}| = 1$ satisfying $g_{\mathbf{c}}(\mathbf{u}_{\mathbf{c}}) = b^\epsilon _{\mathbf{c}}$ . Thus, by the definitions (5.5) and (5.6), we have

\begin{align*}\frac {1}{2}\tilde {H} \lt |g_{\mathbf{c}}(\mathbf{u}_{\mathbf{c}})| \leqslant \tilde {H}.\end{align*}

Let $\theta _{\mathbf{c}} = \theta (\mathbf{u}_{\mathbf{c}})$ , then $g_{\mathbf{c}}(\theta _{\mathbf{c}}) = r(\mathbf{u}_{\mathbf{c}})^{-d} g_{\mathbf{c}}(\mathbf{u}_{\mathbf{c}})$ . Since $1 \leqslant r(\mathbf{u}_{\mathbf{c}}) \leqslant \sqrt {2}$ , we know

\begin{align*}(\sqrt {2})^{-d} |g_{\mathbf{c}}(\mathbf{u}_{\mathbf{c}})| \leqslant |g_{\mathbf{c}}(\theta _{\mathbf{c}})| \leqslant |g_{\mathbf{c}}(\mathbf{u}_{\mathbf{c}})|.\end{align*}

This implies that

(6.1) \begin{align} (\sqrt {2})^{-d-2} \tilde {H} \leqslant |g_{\mathbf{c}}(\theta _{\mathbf{c}})| \leqslant \tilde {H}. \end{align}

Lemma 6.2. Let $\epsilon \in \{\pm \}$ and ${\mathbf{c}} \in S_\epsilon (H, \tilde {H})$ . Then there exists a small constant $0 \lt \eta \ll 1$ such that if

\begin{align*}\Big |r/x - \left(\frac {1}{2}\tilde {H}/|g_{\mathbf{c}}(\theta _{\mathbf{c}})|\right)^{1/d} \Big | \lt \eta ^2 \quad and\quad \left \lVert \theta - \theta _{\mathbf{c}}\right \rVert \leqslant ({\log} \,H)^{-2A},\end{align*}

then

\begin{align*}\omega (r^d g_{\mathbf{c}}(\theta );\, B) \gg 1.\end{align*}

Here $\left \lVert \theta - \theta _{\mathbf{c}}\right \rVert = \min \{|\theta - \theta _{\mathbf{c}} + 2\pi n|\,:\, n \in {\mathbb{Z}} \}.$

Proof. It follows from Lemma 4.2 (iv) that there exists $0 \lt \eta \ll 1$ such that $\omega (y;\, B) \gg 1$ whenever

\begin{align*}|\epsilon y/B^e - 1/2| \lt \eta .\end{align*}

If $\big |r/x - \big (\frac {1}{2}\tilde {H}/|g_{\mathbf{c}}(\theta _{\mathbf{c}})|\big )^{1/d} \big | \lt \eta ^2$ , then

(6.2) \begin{align} (r/x)^d = \frac {\tilde {H}}{2|g_{\mathbf{c}}(\theta _{\mathbf{c}})|} + O(\eta ^2), \end{align}

noting from (6.1) that we have $g_{\mathbf{c}}(\theta _{\mathbf{c}}) \gg _d \tilde {H}$ . Moreover, Taylor’s theorem yields

\begin{align*}\Big ||g_{\mathbf{c}}(\theta )| - |g_{\mathbf{c}}(\theta _{\mathbf{c}})|\Big | \leqslant |g_{\mathbf{c}}(\theta ) - g_{\mathbf{c}}(\theta _{\mathbf{c}})| \ll \sum _{k \geqslant 1} \left \lVert \theta - \theta _{\mathbf{c}}\right \rVert ^k \cdot \Big | \frac {\partial ^k g_{\mathbf{c}}}{\partial \theta ^k} (\theta _{\mathbf{c}}) \Big | \ll H({\log} \,H)^{-2A}. \end{align*}

Since $r \leqslant \sqrt {2}x$ , it follows that

(6.3) \begin{align} r^d g_{\mathbf{c}}(\theta ) = r^d g_{\mathbf{c}}(\theta _{\mathbf{c}}) + O(x^dH({\log} \,H)^{-2A}). \end{align}

Recall from (5.7) that $B^e = \tilde {H}x^d$ . It follows from (6.1) that

\begin{align*}\frac {|g_{\mathbf{c}}(\theta _{\mathbf{c}})|}{\tilde {H}} \leqslant 1.\end{align*}

By (6.2) and (6.3), we obtain

\begin{align*} \epsilon r^d g_{\mathbf{c}}(\theta ) / B^e - 1/2 & = r^d |g_{\mathbf{c}}(\theta )| / B^e - 1/2 \\ & = \frac {2r^d |g_{\mathbf{c}}(\theta _{\mathbf{c}})| - \tilde {H}x^d + O(x^dH({\log} \,H)^{-2A})}{2\tilde {H}x^d} \\ & = O \left ( \frac {\eta ^2|g_{\mathbf{c}}(\theta _{\mathbf{c}})|}{\tilde {H}} + \frac {H}{\tilde {H}({\log} \,H)^{2A}} \right ) \\ & = O\left (\eta ^2 + \frac {H}{\tilde {H}({\log} \,H)^{2A}} \right ), \end{align*}

The right-hand side is $\ll \eta$ , since $\tilde {H} \geqslant 2H({\log} \,H)^{-A}$ . The statement of the lemma follows by choosing $\eta \gt 0$ small enough.

For $\epsilon \in \{\pm \}$ and ${\mathbf{c}} \in S_\epsilon (H, \tilde {H})$ , define

\begin{align*}r_0 = \left (\frac {1}{2}\tilde {H}/|g_{\mathbf{c}}(\theta _{\mathbf{c}})|\right )^{1/d} x.\end{align*}

From (6.1), we know $2^{-1/d} x \leqslant r_0 \leqslant \sqrt {2} x$ . We now introduce the region

(6.4) \begin{align} D_{\mathbf{c}} = \left \{(u, v) \in [0, x]^2 \,:\, \begin{array}{l} |r - r_0| \leqslant \eta ^2 x, \\[2pt] \|\theta - \theta _{\mathbf{c}}\| \leqslant 3({\log} \,H)^{-2A}, \\[2pt] \|\theta \pm \frac {\pi }{2}\| \geqslant ({\log} \,H)^{-2A} \end{array} \right \}. \end{align}

The area of $D_{\mathbf{c}}$ satisfies $|D_{\mathbf{c}}| \leqslant x^2$ . Moreover, by (6.1), we know

(6.5) \begin{align} |D_{\mathbf{c}}| & \gg \pi \big(\big(r_0 + \eta ^2x\big)^2 - \big(r_0 - \eta ^2x\big)^2 \big) \cdot ({\log} \,H)^{-2A} \nonumber \\ & = 4\pi \eta ^2 \left (\frac {1}{2}\tilde {H}/|g_{\mathbf{c}}(\theta _{\mathbf{c}})|\right )^{1/d} x^2({\log} \,H)^{-2A} \nonumber \\ & \gg _\eta x^2({\log} \,H)^{-2A}. \end{align}

Recall from (4.12) the definition of the localized counting function. By Lemma 6.2, we have

(6.6) \begin{align} \#\big\{ {\mathbf{c}} \in S^{\mathrm{loc}}(H, \tilde {H})\,:\, \hat {N}_{\mathbf{c}}(x) \lt \delta x^2 \big\} \leqslant \#\big\{ {\mathbf{c}} \in S^{\mathrm{loc}}(H, \tilde {H})\,:\, \tilde {N}_{\mathbf{c}}(D_{\mathbf{c}}) \ll \delta x^2 \big\}, \end{align}

where we define

\begin{align*}\tilde {N}_{\mathbf{c}}(D_{\mathbf{c}}) = \sum _{(m, n) \in {D_{\mathbf{c}} \cap {\mathbb{Z}}^2}} \rho _K(W, g_{\mathbf{c}}(m,n)).\end{align*}

6.2. Results on binary forms

In this section, we collect several results concerning binary forms that will be needed in the subsequent analysis. For a vector ${\mathbf{c}} \in (c_0, \ldots , c_d) \in {\mathbb{Z}}^{d+1}$ , we define its content by

\begin{align*}h_{\mathbf{c}} = \gcd (c_0, \ldots , c_d).\end{align*}

We define the content of a binary form $g_{\mathbf{c}}$ in terms of the content of its coefficient vector.

We recall the definition (4.9) of the modulus $W$ . As in [Reference Browning, Sofos and Teräväinen1, Section 9], the very small prime factors of $W$ or the factors of $W$ that share a common factor with $h_{\mathbf{c}}$ need to be treated separately. Let $M_{d, K}$ be a sufficiently large positive constant that only depends on $d$ and the number field $K$ . For any ${\mathbf{c}} \in S^{\mathrm{loc}}(H, \tilde {H})$ , we define

(6.7) \begin{align} W_0 = \prod _{\substack {p \leqslant M_{d,K} \text{ or } p \mid h_{\mathbf{c}}}} p^{k(x)} \quad \text{and} \quad W_1 = \prod _{\substack {M_{d,K} \lt p \leqslant w \\ p \nmid h_{\mathbf{c}}}} p^{k(x)}. \end{align}

We also introduce

(6.8) \begin{align} \sigma _{\mathbf{c}}(W_0) = \frac {1}{W_0^{e+1}} \#\{(\mathbf{x}, s, t) \in ({\mathbb{Z}} / W_0 {\mathbb{Z}})^{e+2}\,:\, \mathbf{N}_K(\mathbf{x}) \equiv g_{\mathbf{c}}(s,t) \, (\mathrm{mod}\, W_0)\}. \end{align}

Moreover, we set

(6.9) \begin{align} P_{\mathbf{c}}(w) = \prod _{\substack {M_{d, K} \lt p \leqslant w \\ p \nmid h_{\mathbf{c}}}} p. \end{align}

Let $g\in {\mathbb{Z}}[s,t]$ be a binary form of degree $d$ . We say that $g$ is separable, if the discriminant of $g$ is non-zero. Let

(6.10) \begin{align} \lambda _g(q) = \# \{(u, v) \in ({\mathbb{Z}} / q{\mathbb{Z}})^2\,:\, g(u, v) \equiv 0 \, (\mathrm{mod}\, q) \}, \end{align}

for any integer $q \geqslant 1$ . We now prove an upper bound for $\lambda _g(p^k)$ when $g$ is separable.

Proof. The first statement is trivial. Now we assume that $\sigma \lt k$ . For a fixed integer $n$ , we denote

(6.11) \begin{align} g_n(m) = g(m, n) = c_0m^d + (c_1n)m^{d-1} + \ldots + (c_{d-1}n^{d-1})m + c_d n^d, \end{align}

as a polynomial in $m$ with coefficients depending on $n$ . Note that when $n$ is non-zero, the polynomial $g_n$ is separable if and only if the binary form $g$ is separable. We know

(6.12) \begin{align} h_{g_n} = \gcd (c_0, c_1n, \ldots , c_dn^d) \leqslant h_g \cdot n^d. \end{align}

Let $\sigma _n = v_p(h_{g_n})$ . According to [Reference Browning, Sofos and Teräväinen1, Lemma 6.1], we know $\lambda _{g_n}(p^k) = p^k$ when $\sigma _n \geqslant k$ . If $\sigma _n \lt k$ , then

(6.13) \begin{align} \lambda _{g_n}(p^k) \leqslant d \min \big\{ p^{k(1 - \frac {1}{d} + \frac {\sigma _n}{d})}, p^{k-1} \big\}. \end{align}

By using (6.12) and the first bound of (6.13), we obtain

\begin{align*} \lambda _g(p^k) & = \sum _{1 \leqslant n \leqslant p^k} \lambda _{g_n}(p^k) \\ & \leqslant d\sum _{1 \leqslant n \leqslant p^k} p^{k(1-1/d) + \sigma _n/d} \\ & \leqslant d\sum _{1 \leqslant n \leqslant p^k} p^{k(1-1/d) + \sigma _n/d + v_p(n)} \\ & \leqslant d \sum _{0 \leqslant v_p(n) \leqslant k} p^{k-v_p(n)} \cdot p^{k(1 - 1/d) + \sigma _n/d + v_p(n)} \\ & = d \sum _{0 \leqslant v \leqslant k} p^{k + k(1 - 1/d) + \sigma _n/d} \\ & \leqslant d(k+1) p^{k(2 - \frac {1}{d}) + \frac {\sigma _n}{d}}. \end{align*}

By using the second bound of (6.13), we know

\begin{align*} \lambda _g(p^k) & = \sum _{1 \leqslant n \leqslant p^k} \lambda _{g_n}(p^k) \\ & = \sum _{\substack {1 \leqslant n \leqslant p^k \\ p | n}} \lambda _{g_n}(p^k) + \sum _{\substack {1 \leqslant n \leqslant p^k \\ p \nmid n}} \lambda _{g_n}(p^k) \\ & \leqslant p^{k-1} \cdot p^{k} + p^k \cdot d p^{k-1} = (d+1)p^{2k-1}. \end{align*}

Combining these two estimates completes the proof

Next, let $\tau$ be the divisor function. We shall make frequent use of the following bound on the average size of $\tau$ evaluated at binary form arguments.

Lemma 6.4. Let $\delta \gt 0$ and $C\gt 0$ . Let $g \in {\mathbb{Z}}[s, t]$ be a separable binary form of degree $d\geqslant 1$ . Let $\|g\|$ be the maximum modulus of the coefficients of $g$ and assume that the content $h_g$ of $g$ is at most $({\log} \,x)^C$ . Then there exist a positive constant $K$ , depending only on $\delta$ , $d$ and $C$ , such that for $x \geqslant \|g\|^{\delta }$ we have

\begin{align*} \sum _{\substack {m, n\leqslant x}} \tau (|g(m, n)|)^C \ll x^2 ({\log} \,x)^{K}. \end{align*}

Proof. We use the notation (6.11). Let $h_{g_n}$ be the content of $g_n$ so that $g_n(m) = h_{g_n} \hat {g}_n(m)$ , where $\hat {g}_n$ has content $1$ . By [Reference Browning, Sofos and Teräväinen1, Lemma6.2], we know

\begin{align*}\sum _{m \leqslant x} \tau (|g_n(m)|)^C \ll x({\log} \,x)^{K'},\end{align*}

for some $K' \gt 0$ , depending only on $\delta , d$ and $C$ . By using the assumption $h_g \leqslant ({\log} \,x)^C$ , we know $\tau (h_g) \ll h_g^{1/C} \ll \log x$ . We then apply (2.4) and (6.12) to obtain

\begin{align*} \sum _{m, n \leqslant x} \tau (|g(m, n)|)^C & = \sum _{n \leqslant x} \sum _{m \leqslant x} \tau (|g_n(m)|)^C \\ & \leqslant \sum _{n \leqslant x} \tau (h_{g_n})^C \sum _{m \leqslant x} \tau (|\hat {g}_n(m)|)^C\end{align*}

\begin{align*} \qquad\qquad\qquad\quad & \ll x({\log} \,x)^{K'} \sum _{n \leqslant x} \tau (h_g)^C \tau (n^d)^C \\ & \ll x({\log} \,x)^{K' + 1} \sum _{n \leqslant x} \tau (n)^{dC} \\ & \ll x({\log} \,x)^K, \end{align*}

for some $K \gt 0$ , depending only on $\delta , d$ and $C$ .

6.3. A transition to non-archimedean densities

Let $K$ be a number field with ring of integers $\mathfrak{o}_K$ . For an integral ideal $\mathfrak{a} \subset \mathfrak{o}_K$ we define its norm by

\begin{align*}\mathrm{N}\mathfrak{a}=\# (\mathfrak{o}_K/\mathfrak{a}).\end{align*}

The Dedekind zeta function associated to $K$ is defined for $\mathrm{Re}(s)\gt 1$ by

\begin{align*} \zeta _K(s)=\sum _{\substack {\mathfrak{a} \subset \mathfrak{o}_K\\\mathfrak{a}\neq (0)}} \frac {1}{(\mathrm{N} \mathfrak{a})^s} = \sum _{m=1}^\infty \frac {r_K(m)}{m^{s}}, \end{align*}

where

\begin{align*} r_K(m)=\#\{\mathfrak{a} \subset \mathfrak{o}_K\,:\, \mathrm{N} \mathfrak{a}=m\}. \end{align*}

It is well known that $r_K(m)$ is multiplicative and that $\zeta _K(s)$ admits a meromorphic continuation to the entire complex plane with a simple pole at $s=1$ .

For any prime $p$ and any integer $k \geqslant 1$ , we define

\begin{align*} \alpha _p = \left( 1 - \frac {1}{p} \right)^{-1} \prod _{\mathfrak{p} | p} \left( 1 - \frac {1}{\mathrm{N} \mathfrak{p}} \right) \end{align*}

and

\begin{align*} \beta _{p^k} = p^k \sum _{j \geqslant k} \frac {r_K(p^j)}{p^j} \prod _{\mathfrak{p} | p} \left ( 1 - \frac {1}{\mathrm{N} \mathfrak{p}} \right ). \end{align*}

Next, define the multiplicative function $\tilde {r}_K(n)$ on prime powers, with $k(x)$ as in (4.8), by

\begin{align*} \tilde {r}_K(p^j) = \begin{cases} r_K(p^j), & \text{if } j \lt k(x), \\ \alpha _p^{-1} \beta _{p^{k(x)}}, & \text{if } j \geqslant k(x). \end{cases} \end{align*}

Define also $b = \tilde {r}_K * \mu$ . Then [Reference Browning, Sofos and Teräväinen1, Formula (7.27)] shows that

(6.14) \begin{align} |b(k)| \leqslant \tau (k)^{e+1}. \end{align}

We now state the main result of this section.

Proposition 6.5. Let $A, C \geqslant 1$ be fixed. Take $B = \tilde {H}^{1/e} x^{d/e}$ and assume ( 5.4 ) holds. For $\epsilon \in \{\pm \}$ and $\delta = ({\log} \,x)^{-C}$ , there exists a constant $M = M_{C, d, K} \geqslant 1$ such that

\begin{align*} \#\big\{ {\mathbf{c}}\in S_\epsilon ^{\text{loc}}(\tilde H, H)\,:\, \hat N_{\mathbf{c}}(x) \lt \delta x^2\big\} & \ll \frac {H^{d+1}}{({\log} \,x)^{dC}}\\ & \quad + \# \left \{ {\mathbf{c}}\in S_\epsilon ^{\text{loc}}(\tilde H, H)\,:\, \sigma _{{\mathbf{c}}}(W_0) \leqslant \delta M ({\log} \,x)^{c_{d,e}} ({\log} \,H)^{2A}\right \}, \end{align*}

where $c_{d,e} = e + d^2(d+1)^{e+2}$ .

The following class of binary forms plays an important role in the subsequent analysis.

The following result is the key step in the proof of Proposition 6.5.

Lemma 6.7. Let $C \geqslant 1$ and let ${\mathbf{c}} \in S(H, \tilde {H})$ . Suppose that $g_{\mathbf{c}}$ is $C$ -admissible. Then

\begin{align*} \tilde {N}_{\mathbf{c}}(D_{\mathbf{c}}) = \sigma _{\mathbf{c}}(W_0)|D_{\mathbf{c}}| \sum _{\substack {k \in \mathbb{N} \\ p \mid k \Rightarrow p \mid P_{\mathbf{c}}(w)}} \frac {c_{\mathbf{c}}(w) b(k) \lambda _{g_{\mathbf{c}}}(k)}{k^2} + O_C \left ( x^2 \exp \left ( - ({\log} \,x)^{1/8} \right ) \right ), \end{align*}

where $D_{\mathbf{c}}$ is given by ( 6.4 ), $P_{\mathbf{c}}(w)$ is given by ( 6.9 ) and

\begin{align*}c_{\mathbf{c}}(w) = \prod _{p | P_{\mathbf{c}}(w)} \alpha _p.\end{align*}

Proof. By Definition 6.6, the content of $g_{\mathbf{c}}$ is $\leqslant ({\log} \,x)^C$ . Recall from (6.7) the definition of $W_0$ . It follows from [Reference Browning, Sofos and Teräväinen1, Formula (9.4)] that

(6.15) \begin{align} W_0 \leqslant \exp \big (C'({\log} \,\log x)^3\big ), \end{align}

where $C'$ is a constant only depends on $d, K$ and $C$ . According to [Reference Browning, Sofos and Teräväinen1, Formula (9.5)], we may write

(6.16) \begin{align} \tilde {N}_{\mathbf{c}}(D_{\mathbf{c}}) = \sum _{\nu _1, \nu _2 \, (\mathrm{mod}\, W_0)} \rho _K(W_0, g_{\mathbf{c}}(\nu _1, \nu _2))T(D_{\mathbf{c}};\, \nu _1, \nu _2), \end{align}

where

(6.17) \begin{align} T(D_{\mathbf{c}};\, \nu _1, \nu _2) = \sum _{\substack {(m,n) \in D_{\mathbf{c}} \\ (m, n) \equiv (\nu _1, \nu _2) \, (\mathrm{mod}\, W_0)}} \rho _K(W_1, g_{\mathbf{c}}(m, n)). \end{align}

A further application of [Reference Browning, Sofos and Teräväinen1, Formula (9.6)] gives

(6.18) \begin{align} T(D_{\mathbf{c}};\, \nu _1, \nu _2) = U(D_{\mathbf{c}};\, \nu _1, \nu _2) \prod _{p | P_{\mathbf{c}}(w)} \alpha _p, \end{align}

where

\begin{align*}U(D_{\mathbf{c}};\, \nu _1, \nu _2) = \sum _{\substack {(m,n) \in D_{\mathbf{c}} \\ (m,n) \equiv (\nu _1, \nu _2) \, (\mathrm{mod}\, W_0)}} \tilde {r}_K(g_{\mathbf{c}}(m,n)_W),\end{align*}

with the notation

\begin{align*}n_W = \prod _{\substack {p^\nu \mid \mid n \\ p \mid P(w)}} p^\nu .\end{align*}

We know $g_{\mathbf{c}}(m, n) \neq 0$ for all $(m, n) \in D_{\mathbf{c}} \cap {\mathbb{Z}}^2$ . Write

\begin{align*}\mathscr{C}_w = \{n \in {\mathbb{N}}\,:\, p | n \, \Rightarrow \, p | P_{\mathbf{c}}(w)\}.\end{align*}

Let $d' = g_{\mathbf{c}}(m, n)_W$ . Then there exists an integer $d''$ such that $g_{\mathbf{c}}(m, n) = d'd''$ , with $d' \in \mathscr{C}_w$ and $\gcd (d'', P_{\mathbf{c}}(w)) = 1$ . An application of Möbius inversion then yields

\begin{align*}U(D_{\mathbf{c}};\, \nu _1, \nu _2) = \sum _{k \in \mathscr{C}_w} b(k) \sum _{\substack {(m, n) \in D_{\mathbf{c}} \\ (m, n) \equiv (\nu _1, \nu _2) \, (\mathrm{mod}\, W_0) \\ k | g_{{\mathbf{c}}}(m, n)}} 1,\end{align*}

where $b = \tilde {r}_K * \mu$ . Then we break the outer sum into two ranges, leading to

(6.19) \begin{align} U(D_{\mathbf{c}};\, \nu _1, \nu _2) = \Sigma _1 + \Sigma _2, \end{align}

where

\begin{align*}\Sigma _1 = \sum _{\substack {k \in \mathscr{C}_w \\ k \leqslant x^{1/(2d)}}} b(k) \sum _{\substack {(m, n) \in D_{\mathbf{c}} \\ (m, n) \equiv (\nu _1, \nu _2) \, (\mathrm{mod}\, W_0) \\ k | g_{{\mathbf{c}}}(m, n)}} 1\end{align*}

and

\begin{align*}\Sigma _2 = \sum _{\substack {k \in \mathscr{C}_w \\ k \gt x^{1/(2d)}}} b(k) \sum _{\substack {(m, n) \in D_{\mathbf{c}} \\ (m, n) \equiv (\nu _1, \nu _2) \, (\mathrm{mod}\, W_0) \\ k | g_{{\mathbf{c}}}(m, n)}} 1 .\end{align*}

For the contribution $\Sigma _2$ , by (6.14), we have

\begin{align*}\Sigma _2 \ll \sum _{\substack {k \in \mathscr{C}_w \\ k \gt x^{1/(2d)}}} \tau (k)^{e+1} \sum _{\substack {m, n \leqslant x \\ k|g_{\mathbf{c}}(m,n)}}1.\end{align*}

Suppose that $\omega (k) = r$ . Then any $k$ in the sum satisfies

\begin{align*}x^{1/(2d)} \lt k = p_1^{j_1} \cdots p_r^{j_r} \leqslant w^{k(x)r},\end{align*}

hence $\omega (g_{\mathbf{c}}(m, n)) \geqslant \omega (k) = r \gt K_0$ , with

\begin{align*}K_0 = \frac {\log x}{2k(x) d \log w} = \frac {\sqrt {\log x}}{2k(x)d}.\end{align*}

It then follows that $1 \leqslant 2^{-K_0}2^{\omega (g_{\mathbf{c}}(m, n))} \leqslant 2^{-K_0} \tau (|g_{\mathbf{c}}(m, n)|)$ , so

\begin{align*}\Sigma _2 \ll \frac {1}{2^{K_0}} \sum _{m,n\leqslant x} \tau (|g_{\mathbf{c}}(m, n)|)^{e+2}.\end{align*}

Since $g_{\mathbf{c}}$ has content $\leqslant ({\log} \,x)^C$ , it now follows from Lemma 6.4 that

(6.20) \begin{align} \Sigma _2 \ll _A \frac {x({\log} \,x)^{K_1}}{2^{K_0}} \ll x \exp \left ({-}({\log} \,x)^{1/4} \right ), \end{align}

for a suitable constant $K_1$ only depending on $d, e$ and $A$ .

For the contribution $\Sigma _1$ , recall that the region $D_{\mathbf{c}}$ compact and is the union of at most two compact regions, each formed by the difference of two convex regions.

By [Reference Schmidt31, Lemma 1], we obtain

\begin{align*}\Sigma _1 = \sum _{\substack {k \in \mathscr{C}_w \\ k \leqslant x^{1/(2d)}}} b(k) \lambda _{g_{\mathbf{c}}}(k) \bigg ( \frac {|D_{\mathbf{c}}|}{W_0^2 k^2} + O(x) \bigg ).\end{align*}

From (6.14), we know $|b(k)| = O_\varepsilon (k^\varepsilon )$ . We use the trivial bound $\lambda _{g_{\mathbf{c}}}(k) \leqslant k^2$ to obtain

(6.21) \begin{align} \Sigma _1 = \frac {|D_{\mathbf{c}}|}{W_0^2} \sum _{\substack {k \in \mathscr{C}_w \\ k \leqslant x^{1/(2d)}}} \frac {b(k) \lambda _{g_{\mathbf{c}}}(k)}{k^2} + O_{A, \varepsilon }(x^{1 + \frac {3}{2d} + \varepsilon }). \end{align}

Since $|D_{\mathbf{c}}| \leqslant x^2$ , we see that

\begin{align*} \frac {|D_{\mathbf{c}}|}{W_0^2} \sum _{\substack {k \in \mathscr{C}_w \\ k \leqslant x^{1/(2d)}}} \frac {b(k) \lambda _{g_{\mathbf{c}}}(k)}{k^2} = \frac {|D_{\mathbf{c}}|}{W_0^2} \sum _{\substack {k \in \mathscr{C}_w}} \frac {b(k) \lambda _{g_{\mathbf{c}}}(k)}{k^2} + O(x^2 \Upsilon (x)), \end{align*}

where

(6.22) \begin{align} \Upsilon (x) = \sum _{\substack {k \in \mathscr{C}_w \\ k \gt x^{1/(2d)}}} \frac {|b(k)| \lambda _{g_{\mathbf{c}}}(k)}{k^2}. \end{align}

We now claim that

(6.23) \begin{align} \Upsilon (x) \ll x^{-1/(4d)}. \end{align}

To prove this, we write $k = k_1k_2$ for coprime $k_1, k_2$ , where $k_1$ is square-free and $k_2$ is square-full. Moreover, we may assume that $v_p(k_2) \leqslant k(x)$ for any prime $p | k_2$ , since otherwise $b(k) = 0$ . Furthermore any $p | k_1k_2$ is coprime to the content of $g_{\mathbf{c}}$ , since $k \in \mathscr{C}_w$ . Note that $\lambda _{g_{\mathbf{c}}}(k) = \lambda _{g_{\mathbf{c}}}(k_1) \lambda _{g_{\mathbf{c}}}(k_2)$ and

\begin{align*}\lambda _{g_{\mathbf{c}}}(k_1) \leqslant \prod _{p \| k_1 }(d+1) \leqslant (d+1)^{\omega (k_1)} \leqslant \tau (k_1)^{d+1},\end{align*}

by Lemma 6.3. Then, combining this with (6.14), we deduce that

\begin{align*}\Upsilon (x)\ll \sum _{\substack {k_2\in \mathscr{C}_w\\p^\nu \| k_2\Rightarrow 2\leqslant \nu \leqslant k(x)}} \frac {\tau (k_2)^{e+1}\lambda _{g_{\mathbf{c}}}(k_2)}{k_2^2} \Sigma _{d+e+2}\left (\frac {x^{1/(2d)}}{k_2}\right ),\end{align*}

where

\begin{align*} \Sigma _B(z)=\sum _{\substack {p\mid k \Rightarrow p\leqslant w\\ k\gt z}} \frac {\tau (k)^{B}}{k^2} \ll _B \sum _{k \gt z} \frac {k^\varepsilon }{k^2} \ll z^{-1+\varepsilon }, \end{align*}

for any $\varepsilon \gt 0$ . In particular,

\begin{align*}\Sigma _{d+e+2}\left (\frac {x^{1/(2d)}}{k_2}\right ) \ll \frac {k_2}{x^{1/(2d)-\varepsilon }} ,\end{align*}

since $\log w = \sqrt {\log x}$ . Hence we obtain

\begin{align*} \Upsilon (x) & \ll \ \frac {k_2}{x^{1/(2d) - 2\varepsilon }} \sum _{\substack {k_2\in \mathscr{C}_w\\ p^\nu \| k_2\Rightarrow 2\leqslant \nu \leqslant k(x)}} \frac {\tau (k_2)^{e+1} \lambda _{g_{\mathbf{c}}}(k_2)}{k_2^2} \\ & = \ \frac {1}{x^{1/(2d) - 2\varepsilon }} \sum _{\substack {k_2\in \mathscr{C}_w\\ p^\nu \| k_2\Rightarrow 2\leqslant \nu \leqslant k(x)}} \frac {\tau (k_2)^{e+1} \lambda _{g_{\mathbf{c}}}(k_2)}{k_2}. \end{align*}

It follows from multiplicativity and Lemma 6.3 that

\begin{align*} \sum _{\substack {k_2\in \mathscr{C}_w\\ p^\nu \| k_2\Rightarrow 2\leqslant \nu \leqslant k(x)}} \frac {\tau (k_2)^{e+1} \lambda _{g_{\mathbf{c}}}(k_2)}{k_2} & \leqslant \prod _{p \leqslant w} \left ( 1 + \sum _{2 \leqslant j \leqslant d} \frac {d(j+1)^{e+2}}{p} + \sum _{j \geqslant d+1} \frac {d(j+1)^{e+2}}{p^{j/d}} \right ) \\ & = \prod _{p \leqslant w} \left ( 1 + O \left ( \frac {1}{p} \right ) \right ) \\ & \ll ({\log} \,x)^B. \end{align*}

This proves (6.23).

By (6.23), it follows that

\begin{align*} \Sigma _1 = \frac {|D_{\mathbf{c}}|}{W_0^2} \sum _{\substack {k \in \mathscr{C}_w}} \frac {b(k) \lambda _{g_{\mathbf{c}}}(k)}{k^2} + O_A\big(x^{2 - 1/(4d)}\big), \end{align*}

in (6.21). Combining this with (6.20) in (6.19), we obtain

\begin{align*} U(D_{\mathbf{c}};\, \nu _1, \nu _2) = \frac {|D_{\mathbf{c}}|}{W_0^2} \sum _{\substack {k \in {\mathbb{N}} \\ p | k \Rightarrow p | P_{\mathbf{c}}(w)}} \frac {b(k) \lambda _{g_{\mathbf{c}}}(k)}{k^2} + O_A\left (x^2 \exp ({-}({\log} \,x)^{1/4})\right ). \end{align*}

We now insert this into (6.18) to obtain

(6.24) \begin{align} T(D_{\mathbf{c}};\, \nu _1, \nu _2) = \frac {|D_{\mathbf{c}}|}{W_0^2} \sum _{\substack {k \in {\mathbb{N}} \\ p | k \Rightarrow p | P_{\mathbf{c}}(w)}} \frac {c_{\mathbf{c}}(w)b(k) \lambda _{g_{\mathbf{c}}}(k)}{k^2} + O_A\left (x^2 \exp ({-}({\log} \,x)^{1/4})\right ). \end{align}

We further substitute this into (6.16), noting that $c_{\mathbf{c}}(w) \leqslant W/\varphi (W) \ll \sqrt {\log x}$ and

\begin{align*} \sum _{\nu _1, \nu _2 \, (\mathrm{mod}\, W_0)} \frac {\rho _K(W_0, g_{\mathbf{c}}(\nu _1, \nu _2))}{W_0^2} = \sigma _{\mathbf{c}}(W_0), \end{align*}

in the notation of (6.8). The main term therefore agrees with the statement of the lemma. By (6.15), we see

\begin{align*}c_{\mathbf{c}}(w) x^2 \exp \left ({-}({\log} \,x)^{1/4}\right ) \sum _{\nu _1, \nu _2 \, (\mathrm{mod}\, W_0)} \frac {\rho _K(W_0, g_{\mathbf{c}}(\nu _1, \nu _2))}{W_0^2} \ll x^2 \exp \left ({-}({\log} \,x)^{1/8}\right ).\end{align*}

This completes the proof of the lemma.

Lemma 6.8. Let $g \in {\mathbb{Z}}[s, t]$ be a separable binary form of degree $d$ and content $h$ . Define

\begin{align*}P = \prod _{\substack {M_{d,K} \lt p \leqslant w \\ p \nmid h}} p.\end{align*}

Then we have

\begin{align*}\sum _{\substack {k \in \mathbb{N} \\ p \mid k \Rightarrow p \mid P}} \frac {c_{\mathbf{c}}(w) b(k) \lambda _{g}(k)}{k^2} \gg ({\log} \,x)^{-c_{d, e}}, \end{align*}

where $c_{d, e} = e + d^2(d+1)^{e+2}$ .

Proof. By multiplicativity, we may write

\begin{align*}\sum _{\substack {k \in \mathbb{N} \\ p \mid k \Rightarrow p \mid P}} \frac {c_{\mathbf{c}}(w) b(k) \lambda _{g}(k)}{k^2} = \prod _{p | P} \alpha _p \xi _p,\end{align*}

where

\begin{align*} \xi _p & = 1 + \sum _{j \geqslant 1} \frac {b(p^j)\lambda _{g}(p^j)}{p^{2j}}. \end{align*}

Using the bound (6.14) and Lemma 6.3, we obtain

\begin{align*} \xi _p & \geqslant 1 - \sum _{j \geqslant 1} \frac {d (j+1) \tau (p^j)^{e+1} \min (p^{2j-1}, p^{j(2 - 1/d)})}{p^{2j}} \\ & = 1 - \sum _{j = 1}^d \frac {d(j+1)^{e+2}}{p} - \sum _{j \geqslant d+1} \frac {d(j+1)^{e+2}}{p^{j/d}} \\ & \geqslant 1 - \sum _{j = 1}^d \frac {d(j+1)^{e+2}}{p} - \sum _{j \geqslant d+1} \frac {d(j+1)^{e+2}}{p^{j/d}}, \end{align*}

On the other hand, we also have

\begin{align*}\alpha _p \geqslant 1 - \frac {e-1}{p} + O \left ( \frac {1}{p^2} \right ),\end{align*}

by [Reference Browning, Sofos and Teräväinen1, Formula (6.8)]. Thus,

\begin{align*}\alpha _p \xi _p \geqslant 1 - \frac {e + d^2(d+1)^{e+2}}{p} + O \left ( \frac {1}{p^2} \right ).\end{align*}

On assuming that $M_{d, K}$ is sufficiently large, the asserted lower bound follows from Mertens’ theorem.

Proof of Proposition 6.5. By the argument in the proof of [Reference Browning, Sofos and Teräväinen1, Proposition 9.1], the number of coefficient vectors $\mathbf{c}$ for which $g_{\mathbf{c}}$ either fails to be separable or has content greater than $({\log} \,x)^C$ is $\ll H^{d+1}({\log} \,x)^{-dC}.$ Suppose $g_{\mathbf{c}}(m, n) = 0$ for non-zero $(m, n) \in {\mathbb{Z}}^2$ . Without loss of generality, we assume $n \neq 0$ . Then the polynomial

\begin{align*}f_{\mathbf{c}}(x) = c_0 x^d + c_1 x^{d-1} + \ldots + c_d\end{align*}

has a rational solution $x = m/n$ . According to [Reference Kuba32], there are $O(H^d \log H)$ choices of ${\mathbf{c}} \in S^{\mathrm{loc}}(H, \tilde {H})$ for which $f_{\mathbf{c}}$ is reducible over $\mathbb{Q}$ . It follows that there are $O(H^{d}\log H)$ choices of ${\mathbf{c}} \in S^{\mathrm{loc}}(H, \tilde {H})$ for which $f_{\mathbf{c}}$ has a rational root, since these are the polynomials which admit a linear factor over $\mathbb{Q}$ . Therefore, we know

\begin{align*}\# \left \{ {\mathbf{c}} \in S^{\mathrm{loc}}(H, \tilde {H})\,:\, g_{\mathbf{c}}\ \text{ is not}\ C\hbox{-}\text{admissible} \right \} \ll \frac {H^{d+1}}{({\log} \,x)^{dC}}.\end{align*}

On the other hand, when $g_{\mathbf{c}}$ is $C$ -admissible, we have

\begin{align*}\tilde {N}_{\mathbf{c}}(D_{\mathbf{c}}) \gg \sigma _{\mathbf{c}}(W_0) \frac {x^2}{({\log} \,x)^{c_{d,e}} ({\log} \,H)^{2A}} + O \left ( x^2 \exp \left ( - ({\log} \,x)^{1/8} \right ) \right ),\end{align*}

by (6.5), Lemmas 6.7 and 6.8. The statement of Proposition 6.5 now follows.

6.4. The non-archimedean densities are rarely small

Recall the definition (6.7) of $W_0$ . In this section, we establish an upper bound for

\begin{align*} M(H, \tilde {H};\, \Delta ) = \#\{{\mathbf{c}} \in S^{\mathrm{loc}}(H, \tilde {H})\,:\, \sigma _{\mathbf{c}}(W_0) \lt \Delta \}. \end{align*}

Let $U(W_0) \subset ({\mathbb{Z}} / W_0 {\mathbb{Z}})^{d+1}$ be the image of the set $S^{\mathrm{loc}}(H, \tilde {H})$ under reduction modulo $W_0$ . By (6.15), we know $W_0 \leqslant \tilde {H} \leqslant H$ . By partitioning according to the congruence class modulo $W_0$ , we have

\begin{align*} M(H, \tilde {H};\, \Delta ) & \leqslant \sum _{\substack {\mathbf{u} \in U(W_0) \\ \sigma _{\mathbf{u}}(W_0) \lt \Delta }} \# \{ {\mathbf{c}} \in S^{\mathrm{loc}}(H, \tilde {H})\,:\, |{\mathbf{c}}| \leqslant H, {\mathbf{c}} \equiv \mathbf{u} \, (\mathrm{mod}\, W_0) \} \\ & \ll \sum _{\substack {\mathbf{u} \in U(W_0) \\ \sigma _{\mathbf{u}}(W_0) \lt \Delta }} \frac {\tilde {H}}{W_0} \left( \frac {H}{W_0} \right)^d. \end{align*}

It follows from Rankin’s trick that

\begin{align*} M(H, \tilde {H};\, \Delta ) \ll \frac {\Delta ^\kappa \tilde {H} H^d}{W_0^{d+1}} \sum _{\mathbf{u} \in U(W_0)} \frac {1}{\sigma _{\mathbf{u}}(W_0)^\kappa }, \end{align*}

for any $\kappa \gt 0$ . By multiplicativity, we may write

(6.25) \begin{align} M(H, \tilde {H};\, \Delta ) \ll \frac {\Delta ^\kappa \tilde {H} H^d}{W_0^{d+1}} \prod _{p | W_0} \sum _{\mathbf{u} \in U(p^{k(x)})} \frac {1}{\sigma _{\mathbf{u}}(p^{k(x)})^\kappa }, \end{align}

The next result provides a good lower bound for $\sigma _{\mathbf{u}}(p^k)$ whenever the equation $\mathbf{N}_K(\mathbf{x}) = g_{\mathbf{u}}(s,t)$ has a $p$ -adic solution which is not too singular. It is the binary form analogue of [Reference Browning, Sofos and Teräväinen1, Lemma 9.5], and the proof is similar.

Lemma 6.9. Let $p^k$ be a prime power and let $\mathbf{u} \in ({\mathbb{Z}} / p^k {\mathbb{Z}})^{d+1}$ . Assume that there exists an integer $\alpha \geqslant 0$ and a solution $(\mathbf{x}_0, s_0, t_0) \in {\mathbb{Z}}_p^{e+2}$ satisfying

(6.26) \begin{align} \mathbf{N}_K(\mathbf{x}_0) \equiv g_{\mathbf{u}}(s_0, t_0) \, \mathrm{mod}\, p^k \quad and \quad p^\alpha \| (\nabla \mathbf{N}_K(\mathbf{x}_0), \nabla g_{\mathbf{u}}(s_0, t_0)). \end{align}

Then $\sigma _{\mathbf{u}}(p^k) \geqslant p^{-(\alpha +1)(e+1)}$ .

We now state our final bound for $M(H, \tilde {H};\, \Delta )$ .

Proposition 6.10. We have

\begin{align*}M(H, \tilde {H};\, \Delta ) \ll \Delta ^{\frac {1}{d^2(e+1)}} \tilde {H} H^d.\end{align*}

Proof. We stratify the set $U(p^{k(x)})$ according to the $p$ -adic valuation $\alpha$ of the vector

\begin{align*} (\nabla \mathbf{N}_K(\mathbf{x}_0), \nabla g_{\mathbf{u}}(s_0, t_0)).\end{align*}

For $0\leqslant \alpha \leqslant k(x)$ , let $U_\alpha (p^{k(x)})$ be the set of $\mathbf{u} \in ({\mathbb{Z}}/p^{k(x)}{\mathbb{Z}})^{d+1}$ such that $p\nmid \mathbf{u}$ , and for which there exists $ (\mathbf{x}_0,s_0, t_0)\in {\mathbb{Z}}_p^{e+1}$ such that (6.26) holds with $k=k(x)$ . Note that $U_{k(x)}(p^{k(x)})$ should actually be defined with the condition that

\begin{align*}p^{k(x)} | (\nabla \mathbf{N}_K(\mathbf{x}_0), \nabla g_{\mathbf{u}}(s_0, t_0))\end{align*}

in (6.26). Then we have

\begin{align*} U(p^{k(x)})=U_0(p^{k(x)})\sqcup U_1(p^{k(x)})\sqcup \cdots \sqcup U_{{k(x)}}(p^{k(x)}). \end{align*}

From Lemma 6.9, we know

\begin{align*} \sum _{\mathbf{u}\in U(p^{k(x)})} \frac {1}{\sigma _{\mathbf{u}}(p^{k(x)})^\kappa } \leqslant \sum _{0\leqslant \alpha \leqslant {k(x)}} p^{(\alpha +1)(e+1) \kappa } \cdot \#U_\alpha (p^{k(x)}). \end{align*}

According to the proof of [Reference Browning, Sofos and Teräväinen1, Proposition 9.6], we know

\begin{align*}\#U_\alpha (p^{k(x)}) \ll p^{k(x) (d+1) - \frac {\alpha }{d(d-1)}}.\end{align*}

Thus, summing over $\alpha$ we deduce

\begin{align*}\sum _{\mathbf{u} \in U(p^{k(x)})} \frac {1}{\sigma _{\mathbf{u}}(p^{k(x)})^\kappa } \ll \sum _{0 \leqslant \alpha \leqslant k(x)} p^{(\alpha +1)(e+1)\kappa } \cdot p^{k(x)(d+1) - \frac {\alpha }{d(d-1)}}.\end{align*}

If we choose

\begin{align*}\kappa = \frac {1}{d^2(e+1)},\end{align*}

then $e \kappa \lt \frac {1}{d(d-1)}$ . Note that $p|W_0 = O(1)$ , we obtain

\begin{align*}\sum _{\mathbf{u} \in U(p^{k(x)})} \frac {1}{\sigma _{\mathbf{u}}(p^{k(x)})^\kappa } \ll p^{k(x)(d+1)}.\end{align*}

Making this choice of $\kappa$ , we return to (6.25) and therefore arrive at the statement of the proposition.