2.1 Introducing finer-than-dyadic smooth weights

Here we describe a device that allows us to partition a sum smoothly into finer-than-dyadic intervals. Let $\nu \in (0,1/10)$ be small (we will use $\nu=X^{-{\varepsilon}}$ or $\nu= \log^{-C} X$ ), and fix a non-negative $C^\infty$ -smooth function F supported on $[1-\nu,1+\nu]$ and satisfying

\[|F^{(j)}| \ll_j \nu^{-j},\ j \geq 0 \quad \text{and} \quad \int_{1/2}^2 F(1/t) \frac{dt}{t} = \nu.\]

Suppose that we want to introduce a smooth partition to bound a sum of the form $\sum_{n \leq N} f_n$ . We can write (using a change of variables $t \mapsto t/n$ )

\begin{align*}\sum_{n \leq N} f_n &= \frac{1}{\nu} \sum_{n \leq N} f_n \int_{1/2}^2 F(1/t) \frac{dt}{t} =\frac{1}{\nu} \sum_{n \leq N} \int_{1/2}^{2N} f_n F(n/t) \frac{dt}{t}\\&= \frac{1}{\nu} \int_{1/2}^{2N} \sum_{n \leq N} f_n F(n/t) \frac{dt}{t},\end{align*}

so that

\[\bigg| \sum_{n \leq N} f_n \bigg| \leq \frac{1}{\nu} \int_{1/2}^{2N} \bigg| \sum_{n \leq N} f_n F(n/t) \bigg| \frac{dt}{t}.\]

Hence, at the cost of a factor $\nu^{-1} \log N$ , it suffices to consider sums of the form

\[\sum_{n \leq N} f_n F(n/t)\]

for $t \leq 2N$ . Naturally, if the original sum is $\sum_{n \sim N} f_n$ (dyadic n), then it suffices to consider $\sum_{n \sim N} f_n F(n/t)$ for $t \asymp N$ at a cost $\nu^{-1}$ . The effect is the same as with the usual smooth partition of unity except that we did not need to construct explicitly the partition functions F. We use the notation $F_t (n):=F(n/t)$ .

We also have the following variant on ${\mathbb{R}}/2 \pi {\mathbb{Z}}$ . Fix $\nu_1=X^{-\eta_1}$ for some small $\eta_1 >0$ and let $G: {\mathbb{R}}/2 \pi {\mathbb{Z}} \to {\mathbb{C}}$ be a non-negative $C^\infty$ -smooth function supported on $[-\nu_1,\nu_1]$ , satisfying

\[|G^{(j)}| \ll_j \nu_1^{-j},\ j \geq 0 \quad \text{and} \quad \int G(\theta) \,d \theta = \nu_1.\]

Then, for a sum over Gaussian integers, we may write

\[\sum_{z \in {\mathbb{Z}}[i]} f_z = \frac{1}{\nu_1}\int_{{\mathbb{R}}/2 \pi {\mathbb{Z}}} \sum_{z \in {\mathbb{Z}}[i]} f_z G(\arg z - \theta) \,d\theta\]

to obtain a smooth finer-than-dyadic partition in terms of $\arg z$ .

2.2 Elementary estimates

Lemma 2.1. We have

\[\sum_{1 \leq a \leq A} (a,d) \leq \tau(d) A.\]

Proof. Denoting $c=(a,d)$ and $a=ca'$ , we have

\[\sum_{1 \leq a \leq A} (a,d) \leq \sum_{c|d} c \sum_{1 \leq a' \leq A/c} 1 \leq \tau(d) A.\]

In handling the weights $\mathbf{1}_{(n,P(W))=1}$ we can use the following standard bound for exceptionally smooth numbers, which effectively gives a version of the fundamental lemma of the sieve (see, for instance, [Reference TenenbaumTen15, Chapter III.5, Theorem 1]).

Lemma 2.2. For any $2 \leq Z \leq Y$ , we have

\[\sum_{\substack{n \sim Y \\ P^+(n) < Z}} 1 \, \ll \, Y e^{-u/2},\]

where $u:= \log Y/\log Z.$

We also require the following elementary bound (see, for instance, [Reference Friedlander and IwaniecFI98a, Lemma 1]).

Lemma 2.3. For every square-free integer n and every $k \geq 2$ , there exists some $d|n$ such that $d \leq n^{1/k}$ and

\[\tau(n) \leq 2^k \tau(d)^k.\]

From this we get the more general version.

Lemma 2.4. For every integer n and every $k \geq 2$ , there exists some $d|n$ such that $d \leq n^{1/k}$ and

\[\tau(n) \leq 2^{k^2} \tau(d)^{k^3}.\]

Proof. Write $n=b_1 b_2^2 \cdots b_{k-1}^{k-1} b_k^k$ with $b_1, \dots, b_{k-1}$ square-free, by letting $b_k$ be the largest integer such that $b_k^k | n$ , so that $n/b_k^k$ is k-free and splits uniquely into $b_1 b_2^2 \cdots b_{k-1}^{k-1}$ with $b_j$ square-free. We have

\[\tau(n) \leq \tau(b_1) \tau(b_2)^2 \cdots \tau(b_k)^k.\]

By Lemma 2.3 for all $j \leq k-1$ there is some $d_j|b_j$ with $d_j \leq b_j^{1/k}$ and $\tau(b_j) \leq 2^k\tau(d_j)^k$ . Hence, for $d=d_1\cdots d_{k-1} b_k$ , we have

\[ d \leq (b_1\cdots b_{k-1})^{1/k} b_k \leq (b_1\cdots b_{k-1}b_k^k)^{1/k} \leq n^{1/k}\]

and

\[\tau(n) \leq (2\tau(d_1)\cdots \tau(d_{k-1}) \tau(b_k))^{k^2} \leq 2^{k^2}\tau(d)^{k^3}.\]

2.3 Sieve bounds

The following version of the fundamental lemma of the sieve follows from applying [Reference Friedlander and IwaniecFI10, Theorem 6.9] to the real and the imaginary parts of $a_n$ .

Lemma 2.5 (Fundamental lemma of the sieve). Let ${\mathcal{A}}=(a_n)$ be a sequence of complex coefficients and let $\kappa \geq 0$ , $Z \geq 2,$ and $D \geq Z^{9\kappa +2}$ . Suppose that for some ${\mathbb{X}}$ and for some real-valued multiplicative function g(d), we have, for all square-free d,

\[\sum_{n \equiv 0 \, (d)} a_n = g(d) {\mathbb{X}} + r_d\]

and suppose that, for all p, we have $0 \leq g(p) < 1$ . Suppose that for some $K > 1$ we have for all $W < Z$

\[\prod_{W \leq p < Z} (1-g(p))^{-1} \leq K \bigg( \frac{\log Z}{\log W}\bigg)^\kappa.\]

Use the notation

\[V(Z) := \prod_p (1-g(p)).\]

Then, for some bounded coefficients $\lambda_d$ depending only on $\kappa$ , we have

\[\sum_{(n,P(Z))=1} a_n = {\mathbb{X}} V(Z)(1+O_{\kappa,K}(e^{ -s})) + O\left( \bigg|\sum_{\substack{ d < D \ {\mathfrak{d}}|P(Z)}} \lambda_d r_d\bigg|\right).\]

Remark 2.6. The fact that $\lambda_d$ depend only on $\kappa$ and not the sequence $a_n$ will be important for us since we apply the same sieve to several sequences $a_n^{(b)}$ indexed by $b \in B$ and then use the summation over $b \in B$ while bounding the remainder $\sum_{b \in B}\big|\sum_{d < D} \lambda_d r^{(b)}_d\big|$ (cf. Proposition 4.1). In our set-up, the functions $g^{(b)}(d)$ and $V^{(b)}(Z)$ also depend on b with $g^{(b)}(d)= \mathbf{1}_{(d,2b)=1} \rho(d)/d$ . Thus, we have

\[\sum_{b \in B} V^{(b)}(Z) = V(z) \sum_{b \in B} \prod_{\substack{p| b \\ 2 < p < Z}} (1-\rho(p)/p)^{-1},\]

where the product over $p|b,2 < p< Z$ may be completed to all prime factors $p|b, p\neq 2$ with a negligible error term if Z is not too small.

2.4 Poisson summation

The following lemma gives a truncated version of the Poisson summation formula.

Lemma 2.7 (Poisson summation). Let F be as in § 2.1 for some $\nu \in (0,1/10)$ and denote $F_N(n):=F(n/N)$ . Let $x \gg 1$ and let $q \sim Q$ be an integer. Let ${\varepsilon} >0$ and denote

\[H:= \nu^{-1} (QN)^{{\varepsilon}} Q/N.\]

Then, for any $A >0$ ,

\[\sum_{n \equiv a \, (q)} F_N(n) = \frac{N}{q} \hat{F}(0) + \frac{N}{q} \sum_{1 \leq |h| \leq H} \widehat{F} \bigg( \frac{h N}{q}\bigg) e_q (-ah) + O_{A,\epsilon,F}((QN)^{-A}),\]

where $\hat{f}(h):= \int f(u)e(hu) \,du$ is the Fourier transform.

Proof. By the usual Poisson summation formula we have

\[\sum_{n \equiv a \, (q)} F_N(n)=\frac{N}{q} \sum_{h} \widehat{F} \bigg(\frac{h N}{q}\bigg) e_q (-ah).\]

For $|h| > H$ , we have by integration by parts $j\geq 2$ times

\[\widehat{F} \bigg( \frac{h N}{q}\bigg) = \int F(u)e(u hN/q) \,du \ll_j \nu^{-j+1} (hN/q)^{-j} \ll_j \nu (QN)^{-j{\varepsilon}}(h/H)^{-2},\]

which gives the result.

We will also need the two-dimensional Poisson summation formula.

Lemma 2.8. Let $F: {\mathbb{R}}^2 \to {\mathbb{C}}$ be a $C^\infty$ -smooth compactly supported function such that

\[|F^{(j,k)}| \ll \nu_1^{-j} \nu_2^{-k}, \quad \int_{{\mathbb{R}}^2} |F| \ll \nu_1\nu_2\]

and let $ F_{\boldsymbol{N}}({\mathfrak{n}}) := F(n_1/N_1,n_2/N_2)$ . Then

\[\sum_{\boldsymbol{n} \in {\mathbb{Z}}^2} F_{\boldsymbol{N}}({\mathfrak{n}}) = \sum_{\boldsymbol{h} \in {\mathbb{Z}}^2} \widehat{F_{\boldsymbol{N}}}(\boldsymbol{h})\]

and

\[\widehat{F_{\boldsymbol{N}}}(\boldsymbol{h}) \ll_C \nu_1\nu_2 N_1 N_2 (1+\nu_1|h_1|/N_1)^{-C} (1+\nu_2|h_2|/N_2)^{-C}.\]

2.5 Fourier expansions

We have the following lemma on Mellin transforms, where the construction of the non-negative majorant $\tilde{F}$ is from the proof of [Reference Iwaniec and KowalskiIK04, Lemma 7.1] (cf. the function g(y), with $x_m=\log m$ ).

Lemma 2.9. Fix a non-negative $C^\infty$ -smooth function F supported on $[1-\nu,1+\nu]$ and satisfying

\[|F^{(j)}| \ll_j \nu^{-j},\ j \geq 0 \quad \hbox{and} \quad \int_{1/2}^2 F(1/t) \frac{dt}{t} = \nu.\]

Then, for any $c \in {\mathbb{R}}$ ,

\[F(x) = \frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty} \dot{F}(s) x^{-s} \,ds,\]

where the Mellin transform is

\[\dot{F}(s) := \int_0^\infty F(x) x^{s} \frac{{d}x}{x} \ll_{C,\sigma} \nu (1+\nu |s|)^{-C}.\]

Furthermore, $|\dot{F}(it)|$ has a smooth majorant $\tilde{F}(t)$ such that, for $m,n\sim M$ ,

\[\bigg|\int \tilde{F}(t) (m/n)^{it} \, dt \bigg| \ll \, \mathbf{1}_{|m-n| \leq \nu M}.\]

Similarly, we have the following lemma on Fourier series, which will be applied with the characters

\[ e^{ik\arg z} = \bigg( \frac{z}{|z|}\bigg)^k\]

to expand smooth weights $G(\arg z)$ .

Lemma 2.10. Let $\theta\in {\mathbb{R}}/2\pi {\mathbb{Z}}$ and let G be a bounded smooth function supported on $[\theta-\nu,\theta+\nu]$ and satisfying

\[|G^{(j)}| \ll_j \nu^{-j},\ j \geq 0 \quad \hbox{and} \quad \int_{{\mathbb{R}}/2\pi {\mathbb{Z}}} G(\alpha) \,d \alpha = \nu.\]

Then

\[G(\alpha) = \sum_{k} \check{G}(k) e^{ik\alpha}\]

with

\[\check{G}(k) := \int_{{\mathbb{R}}/2 \pi {\mathbb{Z}}} F(\alpha) e^{-ik \alpha} \,d\alpha \ll_C \nu (1+ \nu |k|)^{-C}.\]

Furthermore, there is a majorant $\widetilde{G}(k) $ of $|\check{G}(k)|$ such that

\[\bigg|\sum_{k} \widetilde{G}(k) e^{ikx} \bigg|\ll \mathbf{1}_{|x| \leq \nu}.\]

2.6 Large sieve bounds

For Type II sums we need the multiplicative large sieve inequality of Bombieri and Davenport (cf., for instance, [Reference Friedlander and IwaniecFI10, (9.52)]).

Lemma 2.11. For any complex numbers $\gamma_n$ , we have

\[\sum_{d\leq D} \frac{d}{\varphi(d)} \sideset{}{^\ast} \sum_{\chi\, (d)} \bigg| \sum_{n \leq N} \gamma_n \chi(n) \bigg|^2 \ll (D^2+N) \sum_{n \leq N} |\gamma_n|^2.\]

For Type I sums we need the large sieve inequality for roots of quadratic congruences (cf. [Reference Fouvry and IwaniecFI97] and, in particular, [Reference Friedlander and IwaniecFI05, Lemma 14.4] for this variant with the twist by $\overline{q}$ ).

Lemma 2.12. Let $q \geq 1$ . For any complex numbers $\gamma_n$ we have

\[\sum_{\substack{d\sim D \\ (d,q)=1}} \sum_{\nu^2+1 \equiv 0 \, (d)} \bigg| \sum_{n \leq N} \gamma_n e_d (\nu n \overline{q}) \bigg|^2 \ll (qD+N) \sum_{n \leq N} |\gamma_n|^2.\]

2.7 Zeros of Hecke L-functions

We say that a Gaussian integer z is primary if $z \equiv 1 \ (2(1+i))$ , so that every odd ideal of ${\mathbb{Z}}[i]$ has a unique primary generator. Note that this definition is multiplicative. We extend any function $\psi:{\mathbb{Z}}[i] \to {\mathbb{C}}$ to a function on odd ideals by defining $\psi({\mathfrak{a}}):= \psi(z)$ if z is the primary generator of ${\mathfrak{a}}$ . For $k \in {\mathbb{Z}}$ , we let $\xi_k$ denote the character

\[\xi_k(z) := \bigg(\frac{z}{|z|}\bigg)^k = e^{i k \arg z},\]

which controls the angular distribution of z. For a Dirichlet character $\chi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}$ with a modulus $u \in {\mathbb{Z}}[i]\setminus\{0\}$ we define the Hecke L-function by

\[L(s,\xi_k \chi) := \sum_{\substack{{\mathfrak{a}} \subseteq {\mathbb{Z}}[i] \\ ({\mathfrak{a}},2)=1}} \frac{ \xi_k({\mathfrak{a}})\chi({\mathfrak{a}})}{({\mathcal{N}} {\mathfrak{a}} )^{s}} = \sum_{\substack{z \in {\mathbb{Z}}[i] \setminus\{0\} \\ z \, \text{primary}}} \frac{\xi_k(z)\chi(z)}{|z|^{2s}}.\]

For a modulus u we define

\[L_u(s,\xi_k) := \prod_{\chi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}}L(s,\xi_k \chi).\]

We have the following lemmas, where all the constants are effectively computable. We will not need the Deuring–Heilbronn zero repulsion as we deal with the case of a Siegel zero via a different method. The first lemma is classical (cf., for instance, [Reference Iwaniec and KowalskiIK04, Chapter 5]).

Lemma 2.13 (Zero-free region, Landau–Page). There is a constant $c_1>0$ such that the function $L_u(s,\xi_k)$ has at most one zero in the region

\[\sigma > 1- \frac{c_1}{\log (|u|(2+|t|)(2+|k|))}.\]

If such a zero exists, then it is real and simple, $k=0$ , and it is a zero of some $L(s,\chi_1)$ with a quadratic character $\chi_1$ .

We let $N^\ast(\alpha,T,K,Q)$ denote the number of zeros of $L(s,\xi_k\chi)$ with primitive characters $\chi$ , $|k| \leq K$ , and modulus $|u|^2 \leq Q$ with $\sigma \geq \alpha,|t| \leq T$ . The following lemma is a generalization of [Reference GallagherGal70, Theorem 6] to Gaussian integers.

Lemma 2.14 (Log-free zero-density estimate). There is some constants $c_2,c_3>0$ such that

\[N^\ast(\alpha,T,K,Q) \leq c_3 (Q^2KT)^{c_2(1-\alpha)}.\]

As a corollary to the zero-free region (Lemma 2.13) we get the following lemma, by taking $\delta>0$ small enough in terms of $c_1$ and for two different moduli $u_1,u_2$ applying Lemma 2.13 with $u=u_1u_2$ .

The following lemma is proved by the same argument as in [Reference Montgomery and VaughanMV07, (11.7)].

Lemma 2.16. Let $\chi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}$ for some $|u|^2 \leq Q$ and let $|k| \leq Q$ . If $L(s,\xi_k\chi)$ has no zeros counted by $N^\ast(\alpha,T,K,Q) $ , then for all $\sigma > (1+\alpha)/2, |t| \leq T, |k| \leq K$ we have

\[|L(s,\xi_k\chi)|^{-1} \, \ll \log Q.\]

2.8 Character sums

We need the following lemma for computing sums over primitive characters.

Lemma 2.17. For any a we have

\[\bigg|\sideset{}{^\ast}\sum_{\chi \, (d)} \chi(a) \bigg| \leq (a-1,d),\]

where the sum extends over primitive Dirichlet characters of $({\mathbb{Z}}/d{\mathbb{Z}})^\times$ .

Proof. By the Chinese remainder theorem we have

\[S(a;d):=\sideset{}{^\ast}\sum_{\chi \, (d)} \chi(a) = \prod_{p^k|| d}S(a; p^k).\]

Thus, it suffices to show that

\[| S(a; p^k) | \leq (a-1,p^k).\]

This follows from

\begin{align*}S(a; p^k) &= \sum_{\chi \, (p^k)} \chi(a) - \sum_{\chi \, (p^{k-1})} \chi(a) \\&= \varphi(p^{k})\mathbf{1}_{a \equiv 1 \, (p^k)} - \varphi(p^{k-1}) \mathbf{1}_{a \equiv 1 \, (p^{k-1})}.\end{align*}

We will need the smoothed Polyá–Vinogradov bound on Gaussian integers, which is a consequence of Poisson summation (Lemma 2.8) and the Gauss sum bound on ${\mathbb{Z}}[i]$ .

Lemma 2.18. Let $F:{\mathbb{R}}^2 \to {\mathbb{C}}$ be as in Lemma 2.8 and for $z=x+iy \in {\mathbb{C}}$ define $F(z) := F(x,y)$ . Let $\chi$ be a character of modulus $u \in {\mathbb{Z}}[i]$ . Then

\[\sum_{z} F(z)\chi(z) \ll |u|.\]

The following lemma is required for the proof of Theorem 1.6.

Lemma 2.19. Let $k\geq 1$ and let p be a prime number. Let $\chi \in \widehat{({\mathbb{Z}}[i]/p^k{\mathbb{Z}}[i])^\times}$ with conductor $p^k$ or $\pi^{k_1}\bar{\pi}^{k_2}$ with $k=\max\{k_1,k_2\}$ . Then

\[\sum_{r \, (p^k)}\chi(r+ip^\ell) \leq 2 p^{k/2+\ell/2}.\]

Proof. Let $q:=p^k$ . We have by $r \mapsto r/s$

\[\sum_{r \, (q)} \chi(r+i p^\ell) = \frac{1}{\varphi(q)} \sum_{(s,q)=1} \sum_{r \, (q)} \chi(r+ip^\ell) = \frac{1}{\varphi(q)} \sum_{r,s \, (q)} \overline{\chi}(s)\chi(r+isp^\ell).\]

The function $ s \mapsto \overline{\chi}(s)$ defines a multiplicative character over the integers and we let $q_1|q$ denote its conductor. By expansion of $\overline{\chi}(s)$ into additive characters we get

\[\bigg|\sum_{r\, (p^k)} \chi(r+ip^\ell) \bigg| \leq \frac{1}{q_1 \varphi(q)} \sum_{a \, (q_1) } \bigg|\sum_{t \, (q_1)} \overline{\chi}(t)e_{q_1}(-at) \bigg| \bigg|\sum_{r,s \, (q)} e_{q_1}(as)\chi(r+isp^\ell)\bigg|.\]

We have (writing $s \mapsto s /p^{\ell}+tp^{k-\ell}, p^\ell|s$ )

\[\sum_{r,s \, (q)} e_{q_1}(as)\chi(r+isp^\ell) = \sum_{\substack{z \, (q) \\ s \equiv 0\, (p^\ell)}} e_{q_1}(as/p^\ell) \chi(z)\sum_{t\, (p^\ell)} e_{q_1}(at p^{k-\ell}).\]

Here

\[\sum_{t\, (p^\ell)} e_{q_1}(at p^{k-\ell}) = p^\ell \mathbf{1}_{a \equiv 0 \, (q_1/(q_1,p^{k-\ell}))}.\]

Hence, denoting $q_2=(q_1,p^{k-\ell})$ and making the change of variables $a\mapsto a q_1/q_2$ , we get

\[\bigg|\sum_{r\, (p^k)} \chi(r+ip^\ell)\bigg| \leq \frac{p^{\ell}}{q_1 \varphi(q)} \sum_{a \, (q_2) } \bigg|\sum_{t \, (q_1)} \overline{\chi}(t)e_{q_2}(-at) \bigg| \bigg|\sum_{\substack{z \, (q) \\ s \equiv 0\, (p^\ell)}} e_{p^\ell q_2}(as) \chi(z) \bigg|.\]

Since $\chi(t)$ is of conductor $q_1$ we have the Gauss sum bound [Reference Montgomery and VaughanMV07, Theorem 9.12]

(2.1) \begin{equation} \bigg|\sum_{t \, (q_1)} \overline{\chi}(t)e_{q_2}(-at)\bigg| \leq q_1^{1/2} .\end{equation}

For the second sum we expand the condition $s \equiv 0 \, (p^\ell)$ with additive characters to get

\[\sum_{\substack{z \, (q) \\ s \equiv 0\, (p^\ell)}} e_{p^\ell q_2}(as) \chi(z) = \frac{1}{p^\ell}\sum_{b \, (p^\ell)} \sum_{\substack{z \, (q)}} e_{p^\ell q_2}(as) e_{p^\ell}(bs) \chi(z).\]

The function

\[z \mapsto e_{p^\ell q_2}(as) e_{p^\ell}(bs) = e \bigg( \frac{(a+q_2 b)s}{q_2p^{\ell}} \bigg) =: e_{q}(cs)\]

is an additive character on ${\mathbb{Z}}[i]/q{\mathbb{Z}}[i]$ since $q_2|p^{k-\ell}$ . If $\chi$ is of conductor $\pi^{k_1}\bar{\pi}^{k_2}$ , then we get by the Gauss sum bound (a direct generalization of [Reference Montgomery and VaughanMV07, Theorem 9.12] to ${\mathbb{Z}}[i]$ , denoting $r:=q/(c,q)$ )

(2.2) \begin{equation} \bigg|\sum_{\substack{z \, (q)}} e_{q}(cs) \chi(z) \bigg| \leq \mathbf{1}_{\pi^{k_1}\bar{\pi}^{k_2} | r} \frac{\varphi_{{\mathbb{Z}}[i]}(q)}{\varphi_{{\mathbb{Z}}[i]}(r)} q \leq q ,\end{equation}

since $\pi^{k_1}\bar{\pi}^{k_2} | r$ and $k=\max\{k_1,k_2\}$ implies that $r=p^k=q$ . We also get the same bound if the conductor of $\chi$ is $p^k$ , as then $\chi$ is primitive. Putting the bounds (2.1) and (2.2) together and using $q_2=(q_1,p^{k-\ell})$ we get (noting that $q/\varphi(q) \leq 2$ for $q=p^k$ )

\begin{align*}\qquad\qquad\qquad\qquad\qquad\bigg|\sum_{r \, (p^k)}\chi(r+ip^\ell) \bigg|&\leq \frac{p^{\ell}}{q_1 \varphi(q)} q_2 q_1^{1/2} q \leq 2 p^{\ell} q_2 q_1^{-1/2} \\&\leq 2 p^\ell q_2^{1/2} \leq 2 p^{k/2+ \ell/2}.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\end{align*}

The previous lemma implies the following.

Lemma 2.20. Let $b \in {\mathbb{Z}}$ and $u\in {\mathbb{Z}}[i]$ and let $Y > |u|^{4}$ . Let $\chi$ be a primitive character modulo u and let $v=(u,b)$ . Then, for any integer $a_0$ ,

\[\sum_{\substack{a \in (Y,Y+Y^{1-\eta}] \\ a \equiv a_0 \, (4)\\ (a,b)=1}} \chi(b+ia) \ll_{\varepsilon} \frac{Y^{1-\eta}}{|u/v|^{1/2-{\varepsilon}}}.\]

Proof. We have

\[\sum_{\substack{a \in (Y,Y+Y^{1-\eta}] \ {\mathfrak{a}} \equiv a_0 \, (4) \\ (a,b)=1}} \chi(b+ia) = \sum_{\substack{c | b \\(c,4 u)=1}} \mu(c) \sum_{\substack{ac \in (Y,Y+Y^{1-\eta}] \\ ac \equiv a_0 \, (4)}} \chi(b+i ac).\]

Let $u=m w$ , where m consists of all prime factors $p\equiv 3 \, (4)$ . Let n denote the smallest integer such that $w| n$ . The contribution from $c > Y/mn$ is trivially bounded by

\[\ll \sum_{\substack{c |b \ {\mathfrak{c}} > Y/mn }} \frac{Y}{c} \ll mn \tau(b).\]

For $c \leq Y/mn$ , we have

\[\sum_{\substack{ac \in (Y,Y+Y^{1-\eta}] \\ ac \equiv a_0 \, (4)}}\chi(b+iac) = \frac{Y^{1-\eta}}{16 cm n} \sum_{\substack{r \, ( 4mn) \\ rc \equiv a_0 \, (4)} }\chi(b+irc) + O(mn) \ll \frac{Y^{1-\eta} (b,mn)^{1/2}}{(mn)^{1/2}},\]

once we show that

\[\sum_{\substack{r \, ( 4mn) \\ rc \equiv a_0 \, (4)} }\chi(b+irc)= \sum_{\substack{r \, ( 4mn) \\ r \equiv a_0 \, (4)} }\chi(b+ir)\leq (mn)^{1/2}(b,mn)^{1/2}.\]

To prove this, write

\[\chi= \prod_{p^k||m} \chi_{p^{k}} \prod_{\pi^k|| w} \chi_{\pi^{k}}.\]

By the Chinese remainder theorem, we get (denoting $k=\max\{k_1,k_2\}$ )

\[\sum_{\substack{r \, ( 4mn) \\ r \equiv a_0 \, (4)} }\chi(b+ir) = \left(\prod_{p^k|| m} \sum_{r \, (p^k)} \chi_{p^k}(b+ir) \right) \left(\prod_{\pi^{k_1} \bar{\pi}^{k_2} || w} \sum_{r \, (p^k)} \chi_{\pi^{k_1}} \chi_{\bar{\pi}^{k_2}}(b+ir)\right)\]

and the claim follows by Lemma 2.19.

3.1 Reduction to bounded $\lambda_b$

We can reduce from coefficients satisfying $|\lambda_b| \leq X^{o(1)}$ to $|\lambda_b| \leq 1$ as follows. Let ${\varepsilon}> 0$ be small enough so that $\delta+2{\varepsilon} < 1/10$ and denote

\[B_0 := \{b: 0<|\lambda_b| \leq 1\} \quad \text{and} \quad B_j:= \{b: 2^{j-1} <|\lambda_b| \leq 2^{j}\}, \quad 1 \leq j \leq \log X.\]

The contribution from $b \in B_j$ with $|B_j| \leq X^{1/2-\delta-{\varepsilon}}$ is negligible, since for such j by $|\lambda_b| \leq X^{o(1)}$

\[\sum_{b \in B_j} |\lambda_b| \,\ll X^{o(1)} |B_j|\, \ll X^{1/2-\delta-{\varepsilon}+o(1)}.\]

For each j such that $|B_j|\, > X^{1/2-\delta-{\varepsilon}}$ , we can renormalize $\lambda_b$ by $2^{-j}$ to get bounded coefficients which satisfy

\[\sum_{b \in B_j} \, |2^{-j}\lambda_b | \geq \frac{1}{2} |B_j| \geq X^{1/2-\delta-2{\varepsilon}}.\]

Then the general case follows by applying the bounded case for each such j separately with the weights $\lambda_{b}^{(j)} := 2^{-j}\lambda_b \mathbf{1}_{b \in B_j} $ . Indeed, if we denote the claim in Theorem 3.3 by $S(\lambda)= \mathrm{M}(\lambda) + O(\mathrm{E}(\lambda))$ , then assuming that the theorem holds for the bounded $\lambda^{(j)}_b$ , we get by linearity

\[S(\lambda) = \sum_j 2^j S(\lambda^{(j)}) = \sum_j 2^j (\mathrm{M}(\lambda^{(j)}) + O(\mathrm{E}(\lambda^{(j)}))) = \mathrm{M}(\lambda) + O(\mathrm{E}(\lambda)).\]

Thus, it suffices to prove Theorem 3.3 for $|\lambda_b| \leq 1$ .

3.2 Reduction from $\lambda_b$ to $\mathbf{1}_B$

We can reduce the proof from general bounded weights $\lambda_b$ to the weights of the type $\mathbf{1}_B(b)$ by a finer-than-dyadic decomposition in terms of the values of $\lambda_b$ . That is, for $\nu=(\log X)^{-C}$ we write

\[I_j := \begin{cases} ((1-\nu)^{j+1},(1-\nu')^{j}] & 0 \leq j\leq (\log X)/\nu ,\\ [0,(1-\nu)^{j}], & j= \lfloor (\log X)/\nu\rfloor,\end{cases}\]

\[\mathbf{1}_{|z|_\infty \leq 1} = \sum_{j_1,j_2} \mathbf{1}_{z \in I_{j_2}\times I_{j_2}}.\]

Since $|\lambda_b| \leq 1$ , we obtain a partition

\[\lambda_b = \sum_{j_1,j_2} \big(\mathbf{1}_{\lambda_b \in I_{j_2}\times I_{j_2}} ((1-\nu)^{j_1} +i (1-\nu)^{j_2}) + O(\nu |\lambda_b| \mathbf{1}_{\lambda_b \in I_{j_2}\times I_{j_2}})\big),\]

where the contribution from the error term is negligible by crude bounds. We consider $\mathbf{1}_B$ for

\[B:= B(j_1,j_2)= \{b \in [\eta X^{1/2},(1-\eta)X^{1/2}]: \lambda_b \in I_{j_2}\times I_{j_2} \}\]

and note that for $j_1,j_2$ with $|B(j_1,j_2)| < X^{1/2-\delta-\eta}$ we can bound the contribution trivially. Hence, it suffices to show Theorem 3.3 for $\lambda_b= \mathbf{1}_B(b)$ .

3.3 Reduction to B in a short interval

Let $B \subseteq [\eta X^{1/2},(2-\eta)X^{1/2}]$ and split

\[B= \bigcup_j B_j, \quad B_j := B \cap [j X^{1/2-\eta},(j+1)X^{1/2-\eta}].\]

The contribution from $B_j$ with $|B_j| \leq X^{-2\eta} |B|$ is negligible by a crude bound. Thus, we only need to deal with $|B_j| \geq |B|X^{-2\eta} = X^{1/2-\delta-2\eta}$ . Therefore, it suffices to prove Theorem 3.3 for $\lambda_b= \mathbf{1}_B(b)$ with

(3.2) \begin{equation} B \subseteq [Y,Y + X^{1/2-\eta}] \quad \text{for some} \ Y\in [\eta X^{1/2},(2-\eta)X^{1/2}].\end{equation}

From now on we always assume that $\lambda_b= \mathbf{1}_B(b)$ for a set B satisfying (3.2).

5.1 Proof of Proposition 5.1

We apply a sieve argument to the sequence ${\mathcal{A}}''= (a''_n)$ over integers defined by

\[ a''_n := \sum_{\substack{\mathcal{N}} {\mathfrak{n}} = n} a'_{\mathfrak{n}}.\]

Note that then

\[S({\mathcal{A}}',2X^{1/2}) = S({\mathcal{A}}'',2X^{1/2}).\]

The proof is essentially the same as in [Reference Friedlander and IwaniecFI10, Proof of Proposition 24.1], but we give it for completeness as it is short. Let $D:=X^{1-2\delta-4\eta}/q^4$ and denote $s= \log D / \log Z$ . By Buchstab’s identity we have

\[S({\mathcal{A}}'',2X^{1/2}) = S({\mathcal{A}}'',Z) - \sum_{Z \leq p< 2X^{1/2}} S({\mathcal{A}}''_p,p).\]

By the fundamental lemma of the sieve (Lemma 2.5, see also Remark 2.6) and Lemma 5.3 we have

\[S({\mathcal{A}}'',Z) = V'(Z) {\mathbb{Y}} (1+ O( e^{-s})) + O(X^{1-\delta-\eta/100}),\]

where

\[ e^{-s} \ll s^{-1} \ll \frac{\log Z}{\log X}.\]

By an upper bound sieve (e.g. using Lemma 2.5) with level $D/p> X^{1/4}$ and Lemma 5.3 we have

\[\sum_{Z \leq p < 2X^{1/2}} S({\mathcal{A}}''_p,p) \leq \sum_{Z \leq {\mathcal{N}} p < 2X^{1/2}} S({\mathcal{A}}''_p,Z) \ll \delta(Z,X) V'(Z) {\mathbb{Y}} +O(X^{1-\delta-\eta/100}).\]

Combining the two estimates we get Proposition 5.1, noting that the bound (5.2) can be proved by a similar argument as in [Reference Friedlander and IwaniecFI10, (24.20)].

Remark 5.4. Instead of taking a small Z and using the fundamental lemma of the sieve, we can take a larger Z and use the linear sieve lower and upper bounds [Reference Friedlander and IwaniecFI10, Theorem 11.12]. Taking $Z=X^{1/4}$ we get that the lower bound for $S({\mathcal{A}}'',2X^{1/2})$ is proportional to (denoting the linear sieve functions by f, F)

\[4 f(4(1-O(\delta))) - {\varepsilon}_14 \int_{1/4}^{1/2} \frac{F(4(1-\alpha - O(\delta)))}{\alpha} d\alpha + O(\delta) = (1-{\varepsilon}_1) 2 \log (3) + O(\delta).\]

To make the right-hand side positive we can certainly take any ${\varepsilon}_1 < 1$ and $\delta = (1- {\varepsilon}_1)/100$ , for instance.

6.1 Sketch of the argument

As the proof of Proposition 6.2 is quite technical, we include here a simplified non-rigorous sketch. In § 6.2 we construct an approximation $\beta^\#_{\mathfrak{n}}$ for $\beta$ such that the difference

\[\beta_{\mathfrak{n}}^{\flat} := \beta_{\mathfrak{n}} - \beta^\#_{\mathfrak{n}}\]

is balanced along arithmetic progressions. Write

\[S(\alpha,\beta) = S(\alpha,\beta^\#)+S(\alpha,\beta^\flat).\]

The approximation will be simple enough that $S(\alpha,\beta^\#)$ can be evaluated by Type I information, so let us consider

\[S(\alpha,\beta^\flat) = \sum_{w} \sum_z \alpha_w \beta^\flat_z \mathbf{1}_{B}({\text{Re}}(\overline{w}z)),\]

where the aim is to capture the oscillations from $\beta^\flat_z$ . By applying Cauchy–Schwarz we get

\[S(\alpha,\beta^\flat) \ll M^{1/2} U^{1/2},\]

with

\[U:= \sum_{z_1,z_2} \beta^\flat_{z_1} \overline{\beta^\flat_{z_2}} \sum_{w} \mathbf{1}_{B}({\text{Re}}(\overline{w}z_1)) \mathbf{1}_{B}({\text{Re}}(\overline{w}z_2)) F_M(|w|^2),\]

for some smooth majorant $F_M$ of the interval [M, 2M]. The goal is to evaluate the sum over w with a main term $M(z_1,z_2)$ and then show that

\[\sum_{z_1,z_2} \beta^\flat_{z_1} \overline{\beta^\flat_{z_2}} M(z_1,z_2) \ll \frac{N|B|^2}{(\log X)^C}\]

is small due to cancellations from the coefficients $\beta^\flat_z$ .

Similarly as in [Reference Friedlander and IwaniecFI98b], we note that by denoting $b_j = {\text{Re}}(\overline{w}z_j)$ and

\[\Delta = {\text{Im}} (\overline{z_1} z_2), \quad a \equiv z_2/z_1 \, (\Delta),\]

we have

(6.1) \begin{equation} i \Delta w =z_2 b_1- z_1b_2.\end{equation}

Note that typically $|\Delta| \approx N$ . The parts where $\Delta=0$ or $|\Delta| < N/ (\log X)^C$ correspond to diagonal contributions and may be bounded by crude estimates. Thus, we assume for simplicity that $|\Delta| \asymp N$ .

Let $b_0:=(b_1,b_2)$ and write $b_j=b_0b_j'$ . Note that in the situation that $B \subseteq q_1 {\mathbb{Z}}$ we have $q_1 | b_0$ , so that $b_0$ can be quite large for a large subset of $B\times B$ . As usual, in most places dealing with greatest common divisors does not cause serious problems but the dependency on $b_0$ will be crucial for our argument.

We have $b_0 | \Delta w$ by (6.1) and, for simplicity, let us assume that $b_0| \Delta$ . Then $w = (z_2 b_1- z_1b_2)/(i\Delta)$ is fixed once we fix $z_j,b_j$ , so that (ignoring the smooth weight $F_M$ ) we have to bound

\[V= \sum_{b_0 }\sum_{\substack{z_1,z_2 \\ |\Delta| \asymp N \\ b_0| \Delta}} \beta^\flat_{z_1} \overline{\beta^\flat_{z_2}} \sum_{b'_2 \equiv a b'_1 \, (\Delta/b_0)} \mathbf{1}_B(b_0b_1') \mathbf{1}_B(b_0b_2').\]

Note that

\[ a \equiv z_2/z_1 \equiv \frac{{\text{Re}} (\overline{z_1}z_2)}{|z_1|^2} \, (\Delta)\]

is congruent to an integer, so that the congruence $b_2' \equiv a b_1' \, (\Delta/b_0)$ lives in ${\mathbb{Z}}/(\Delta/b_0) {\mathbb{Z}}$ .

By expansion with Dirichlet characters and sorting into primitive characters we get (ignoring issues with greatest common divisors)

\[V = \sum_{b_0 }\sum_{\substack{z_1,z_2 \\ |\Delta| \asymp N \\ b_0| \Delta}} \beta^\flat_{z_1} \overline{\beta^\flat_{z_2}} \frac{1}{\varphi(\Delta/b_0)}\sum_{d|\Delta/b_0} \sideset{}{^\ast}\sum_{\chi \, (d)}\overline{\chi}(a)\bigg| \sum_{b_0 b \in B} \chi(b) \bigg|^2,\]

where, morally,

\[\frac{1}{\varphi(\Delta/b_0)} \approx \frac{b_0}{N}.\]

We split this into two parts, $d b_0 > X^{\delta+\eta}$ and $db_0 \leq X^{ \delta+\eta}$ . The contribution from the small d is our main term $M(z_1,z_2)$ referred to in the above.

For the large $db_0$ , we get

\begin{align*}&\sum_{b_0 } b_0 \sum_{d b_0 > X^{\delta+\eta}} \sum_{\substack{D \asymp N \\ D \equiv 0\, (db_0) }} \left(\frac{1}{N}\sum_{\substack{z_1,z_2 \\ |\Delta| =D}} 1 \right) \sideset{}{^\ast}\sum_{\chi \, (d)} \bigg| \sum_{b_0 b \in B} \chi(b) \bigg|^2 \\&\quad \ll N \sum_{b_0 } \sum_{ X^{\delta+\eta} < db_0 \ll N} \frac{1}{d} \sideset{}{^\ast}\sum_{\chi \, (d)} \bigg| \sum_{b_0 b \in B} \chi(b) \bigg|^2,\end{align*}

and applying the large sieve for multiplicative characters (Lemma 2.11) we get

\[\ll N (N + X^{1/2-\delta-\eta}) |B| \ll X^{-\eta} N |B|^2\]

by using $N \ll X^{-\eta} |B|$ . Note that we are applying the large sieve to a very sparse set $B/b_0 \cap {\mathbb{Z}}$ , which causes a loss in the diagonal terms and we are forced to take $d b_0$ at least a bit bigger than $X^\delta$ .

For the small $db_0$ , we can rewrite the conditions $b_0 d| \Delta$ , $a \equiv z_2/z_1 \, (\Delta)$ as $z_2 \equiv a z_1 \, (b_0 d)$ to get

(6.2) \begin{equation} \frac{1}{N} \sum_{b_0 } b_0 \sum_{db_0 \leq X^{\delta+\eta}} \sum_{a \, (b_0 d)} \sideset{}{^\ast}\sum_{\chi \, (d)}\overline{\chi}(a)\bigg| \sum_{b_0b \in B} \chi(b) \bigg|^2 \sum_{z_2 \equiv a z_1 \, (b_0 d)} \beta^\flat_{z_1} \overline{\beta^\flat_{z_2}}.\end{equation}

We now see what precisely is required of the balanced function $\beta_z^\flat$ , we need

\[\sum_{z_2 \equiv a z_1 \, (b_0d)} \beta^\flat_{z_1} \overline{\beta^\flat_{z_2}} \ll \frac{N^2}{\varphi_{{\mathbb{Z}}[i]}(b_0 d) (\log X)^{C}},\]

where the modulus may be as large as $X^{\delta+\eta}$ . This would follow if there were no zeros of $L(s,\chi)$ with a real part $>1- C'\log \log X / \log X$ . Since this is not known, we need to construct the approximation $\beta^\#_{z}$ in a way that takes into account these possible bad characters. By the zero-density estimate (Lemma 2.14) this means that the approximation needs to see $\ll (\log X)^{O(1)}$ of the characters. For technical reasons (due to the smooth weight $F_M$ ) the approximation also needs to see the distribution of $\beta_z$ with respect to $\arg z$ and $|z|^2$ . Note that, in the case, that $B \subseteq q_1 {\mathbb{Z}}$ we have always $q_1|b_0$ , where $q_1$ can be as large as $X^\delta$ .

6.2 An approximation for $\beta_{\mathfrak{n}}$

For the approximation, it is convenient to use a rough finer-than-dyadic partition of unity instead of § 2.1, so that the different parts do not overlap. Let $\nu_2= X^{-\eta_2}$ , and let

\[H_{N'}({\mathfrak{n}}) := \mathbf{1}_{(N',N'(1+\nu_2)]} ({\mathcal{N}} {\mathfrak{n}}),\]

so that

\[\mathbf{1}_{(N,2N]} ({\mathcal{N}} {\mathfrak{n}}) = \sum_{N'= N(1+\nu_2)^{n} \in [N,2N) } H_{N'}({\mathfrak{n}}).\]

We can, of course, choose $\nu_2$ so that $2=(1+\nu_2)^k$ for some $k \asymp \nu_2^{-2}$ .

Let $\beta_{\mathfrak{n}}$ be Q-regular and let $\Psi= \{\xi_{k_j}\chi_j\}$ denote the set of $J \leq (\log X)^{C_2}$ characters, and denote the moduli of the characters by $u_1,\dots,u_J$ and the primitive characters by $\chi_1,\dots,\chi_J$ . For any two coefficients $\alpha,\beta$ , we define their normalized W-rough correlation as

\[{\mathcal{C}}_W(\beta,\alpha) := \left(\sum_{\substack{ ({\mathfrak{n}},P(W))=1 \\ ({\mathfrak{n}},\overline{{\mathfrak{n}}})=1} } \beta_{\mathfrak{n}}\overline{\alpha_{\mathfrak{n}}} \right)\left(\sum_{\substack{({\mathfrak{n}},P(W))=1 \\ ({\mathfrak{n}},\overline{{\mathfrak{n}}})=1} } |\alpha_{\mathfrak{n}}| \right)^{-1}\]

if the denominator is non-zero. We then define the approximation $\beta_{\mathfrak{n}}^\#$ for $\beta_{\mathfrak{n}}$ ,

\[\beta_{\mathfrak{n}}^\#= \beta_{\mathfrak{n}}^\#(\Psi):= \sum_{N'= N(1+\nu_2)^{j} \in [N,2N]}H_{N'}({\mathfrak{n}})\mathbf{1}_{({\mathfrak{n}},\, \overline{{\mathfrak{n}}} P(W))=1} \bigg(\sum_{j \leq J} \overline{\xi_{k_j}\chi_j}({\mathfrak{n}}){\mathcal{C}}_W(\beta, \overline{\xi_{k_j}\chi_j} H_{N'}) \bigg),\]

and the balanced function,

\[\beta_{\mathfrak{n}}^\flat:= \beta_{\mathfrak{n}}-\beta_{\mathfrak{n}}^\#,\]

so that we have a decomposition

\[\beta_{\mathfrak{n}} = \beta_{\mathfrak{n}}^\#+\beta_{\mathfrak{n}}^\flat.\]

Morally, the approximation $\beta^\#$ can be viewed as a kind of ‘expansion’ with respect to a ‘basis’, which is justified since the functions $\xi \chi H_{N'}$ are approximately orthogonal over W-rough number, as the following lemma shows. For the lemma, recall that all functions of odd Gaussian integers z are extended to ${\mathfrak{n}}$ by considering the primary generator, for instance, we write $G(\arg {\mathfrak{n}}) = G(\arg z)$ if z is the primary generator of ${\mathfrak{n}}$ .

Lemma 6.4. Let $\psi,\chi$ be a characters to coprime moduli $u,u_1$ and let $\xi=\xi_k$ with $|k| \ll (\nu_1)^{-2}$ . Let $N > X^\eta |u|^2|u_1|$ . Then, for any $C > 0$ , we have

\[\sum_{\substack{{\mathfrak{n}} }} H_{N'}({\mathfrak{n}})\mathbf{1}_{({\mathfrak{n}},\overline{{\mathfrak{n}}}P(W))=1} (\xi\psi\chi ({\mathfrak{n}}) - \mathbf{1}_{\xi\psi\chi=1}) \ll_C \frac{\nu_2 N}{(\log N)^C}\]

and, for any $({\mathfrak{m}},u)=1$ ,

\[\sum_{\substack{ ({\mathfrak{n}},u)=1}}H_{N'}({\mathfrak{n}})\mathbf{1}_{({\mathfrak{n}},\overline{{\mathfrak{n}}}P(W))=1}\bigg( \mathbf{1}_{{\mathfrak{n}} \equiv {\mathfrak{m}} \, (u)} G(\arg {\mathfrak{n}}) \xi\chi({\mathfrak{n}}) - \frac{\mathbf{1}_{\chi=1}\check{G}(-k)}{\varphi_{{\mathbb{Z}}[i]}(u)} \bigg) \ll_C \frac{\nu_1\nu_2 N}{\varphi_{{\mathbb{Z}}[i]}(u)(\log N)^C}.\]

Proof. We prove the second claim, the first is similar but easier. By applying § 2.1 with a smooth function F with the parameter $\nu_1$ we split ${\mathcal{N}}{\mathfrak{n}}$ smoothly into finer-than-dyadic intervals. The contribution from the edges of the support of $H_{N'}$ gives a negligible contribution by trivial bounds. It then suffices to show that for any $N_1 \sim N$

\[\sum_{\substack{ ({\mathfrak{n}},u)=1}}F_{N_1}({\mathcal{N}}{\mathfrak{n}})\mathbf{1}_{({\mathfrak{n}},\overline{{\mathfrak{n}}}P(W))=1}\left( \mathbf{1}_{{\mathfrak{n}} \equiv {\mathfrak{m}} \, (u)} G(\arg {\mathfrak{n}}) \xi\chi({\mathfrak{n}}) - \frac{\mathbf{1}_{\chi=1}\check{G}(-k)}{\varphi_{{\mathbb{Z}}[i]}(u)} \right) \ll_C \frac{\nu_1^2 N}{\varphi_{{\mathbb{Z}}[i]}(u)(\log N)^C}.\]

We let z denote a primary generator of ${\mathfrak{n}}$ . The condition $(z,\overline{z})=1$ may be dropped with a negligible error term since z is supported on $(z,P(W))=1$ . We write

\[\mathbf{1}_{(z,P(W))=1} = \sum_{\substack{v | (z,P(W)) \\ |v|^2 \leq N^{\eta_1} }}\mu(v) + \sum_{\substack{v | (z,P(W)) \\ |v|^2 > N^{\eta_1} }}\mu(v)\]

and write

\begin{align*}&\sum_{\substack{ ({\mathfrak{n}},u)=1}} F_{N_1}({\mathcal{N}} {\mathfrak{n}})\mathbf{1}_{({\mathfrak{n}},\overline{{\mathfrak{n}}}P(W))=1}\bigg( \mathbf{1}_{{\mathfrak{n}} \equiv {\mathfrak{m}} \, (u)} G(\arg {\mathfrak{n}}) \xi \chi({\mathfrak{n}}) - \frac{\mathbf{1}_{\chi=1}\check{G}(-k)}{\varphi_{{\mathbb{Z}}[i]}(u)} \bigg) ,\\&\qquad= \sum_{z \equiv 1\, (2(1+i))} F_{N_1}(|z|^2)\mathbf{1}_{(z,P(W))=1}\bigg( \mathbf{1}_{z \equiv w\, (u)} G(\arg z) \xi\chi (z) - \frac{\mathbf{1}_{\chi=1}\check{G}(-k)}{\varphi_{{\mathbb{Z}}[i]}(u)} \bigg) .\\&\qquad = S_{\leq} + S_{>}.\end{align*}

For the large v we note that by $v|P(W)$ there is some factor $v_0|v$ such that $|v_0|^2 \in (X^{\eta_1}, X^{\eta_1} W]$ . Thus, by $\check{G}(-k) \ll \nu_1$

\begin{align*}S_{>} \ll &\sum_{\substack{|v_0|^2 \in (X^{\eta_1}, X^{\eta_1} W] \\ v_0| P(W) \\ (v_0,u)=1}} \sum_{z \equiv 0\,(v_0)} \tau(z)^{O(1)} F_{N_1}(|z|^2) \mathbf{1}_{z \equiv w\, (u)} G(\arg z) \\&+ \frac{\nu_1}{\varphi_{{\mathbb{Z}}[i]}(u)}\sum_{\substack{|v_0|^2 \in (X^{\eta_1}, X^{\eta_1} W] \\ v_0| P(W) \\ (v_0,u)=1}} \sum_{z \equiv 0\,(v_0)} \tau(z)^{O(1)} F_{N_1}(|z|^2).\end{align*}

Recall that $\eta$ is large compared with $\eta_1$ . Hence, by counting the sum over $z \equiv 0\, (v_0)$ (using Lemma 2.4 to handle $\tau(z)^{O(1)}$ ) and applying Lemma 2.2, we get

\[S_{>} \ll \frac{\nu_1^2 N}{\varphi_{{\mathbb{Z}}[i]}(u) (\log N)^C}.\]

For small v we split into two cases depending on $\chi\neq 1$ and $\chi=1$ . For $\chi \neq 1$ , by writing $z=uvz'+\alpha$ we have

\begin{align*}S_{\leq} &= \sum_{\substack{v | P(W) \\ |v|^2 \leq N^{\eta_1} \\ (v,uu_1)=1}}\mu(v) \sum_{z \equiv 0\,(v)} F_{N_1}(|z|^2) \mathbf{1}_{z \equiv w\, (u)} G(\arg z) \xi \chi(z) \\&=\sum_{\substack{v | P(W) \\ |v|^2 \leq N^{\eta_1} }}\mu(v) \sum_{z'} F_{N_1}(|uvz'+\alpha|^2) G(\arg uvz + \alpha) \xi(uv z' + \alpha) \chi (uvz'+\alpha) \\& = \sum_{\substack{v | P(W) \\ |v|^2 \leq N^{\eta_1} }}\mu(v) \chi(uv) \sum_{z'} F_{N_1}(|uvz'+\alpha|^2) G(\arg uvz + \alpha) \xi(uv z' + \alpha) \chi (z'+\alpha(uv)^{-1}),\end{align*}

since $(uv,u_1)=1$ . Treating the weight

\[ z' \mapsto F_{N_1}(|uvz'+\alpha|^2) G(\arg uvz + \alpha) \xi(uv z' + \alpha)\]

as a smooth weight, we get by the Polyá–Vinogradov bound (Lemma 2.18)

\[S_{\leq} \ll N^{O(\eta_1)} |u_1| \ll N^{-\eta/2} \frac{\nu_1^2 N}{\varphi_{{\mathbb{Z}}[i]}(u)}\]

by $N^{1-\eta} > |u|^2|u_1|$ since $\eta_1$ is small compared with $\eta$ .

For $\chi=1$ , we have by Lemma 2.10 and $\check{(G\xi_k)}(\ell) =\check{G}(\ell-k) $

\begin{align*}S_{\leq} &= \sum_{\substack{v | P(W) \\ |v|^2 \leq N^{\eta_1} \\ (v,uu_1)=1}}\mu(v) \sum_{z \equiv 0\,(v)} F_{N_1}(|z|^2) \bigg( \mathbf{1}_{z \equiv w\, (u)} G(\arg z) \xi(z) - \frac{\check{G}(-k)}{\varphi_{{\mathbb{Z}}[i]}(u)} \bigg) \\&= \sum_{\substack{v | P(W) \\ |v|^2 \leq N^{\eta_1} \\ (v,uu_1)=1}}\mu(v) \sum_{\ell\neq 0} \check{G}(\ell-k) \ \sum_{z \equiv 0\,(v)} F_{N_1}(|z|^2)\mathbf{1}_{z \equiv w\, (u)} \xi_\ell (z) \\&\quad + \sum_{\substack{v | P(W) \\ |v|^2 \leq N^{\eta_1} \\ (v,uu_1)=1}}\mu(v) \check{G}(-k) \ \sum_{z \equiv 0\,(v)} F_{N_1}(|z|^2) \bigg(\mathbf{1}_{z \equiv w\, (u)} - \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \bigg).\end{align*}

Estimating the contribution from $|\ell| > (\nu_1)^{-3}$ trivially (by Lemma 2.10) and for $|\ell| \leq (\nu_1)^{-3}$ applying the Poisson summation formula on ${\mathbb{Z}}[i]$ we get

\[S_{\leq} \ll N^{-\eta/2} \frac{\nu_1^2 N}{\varphi_{{\mathbb{Z}}[i]}(u)}.\]

Our main lemma about the approximation is the following, which says that $\beta^\flat_{\mathfrak{n}}$ is balanced over arithmetic progressions restricting to small polar boxes.

Lemma 6.5. Let $N> X^\eta Q^3$ . For any $C_1> 0$ , there is some $C_2 \ll_{C_1} 1$ such that the following holds. Let $\xi=\xi_k$ with $k=(\log N)^{O(1)}$ . Let $\beta_{\mathfrak{n}}$ be Q-regular, and let $\beta_{\mathfrak{n}}^\flat$ be as above with $|\Psi|=J \leq (\log X)^{C_2}$ . Suppose that $\beta_{\mathfrak{n}}$ is supported on $({\mathfrak{n}},P(W))=1$ . Let $N'= N(1+\nu_2)^{j} \in [N,2N]$ and $\theta \in {\mathbb{R}}/2\pi {\mathbb{Z}}$ . Then, for any $u \in {\mathbb{Z}}[i]$ with ${\mathcal{M}}(u) \leq Q$ , we have

\[S= \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}} \bigg| \sum_{\substack{{\mathfrak{n}}}} \beta_{\mathfrak{n}}^\flat \psi({\mathfrak{n}})H_{N'}({\mathfrak{n}})G(\arg {\mathfrak{n}} -\theta) \bigg|^2 \ll \frac{\nu_1^2 \nu_2^2 N^2}{\varphi_{{\mathbb{Z}}[i]}(u) (\log X)^{C_1}}.\]

Proof. Let us first show that in the approximation $\beta_{\mathfrak{n}}^\#$ we can replace the characters with conductor dividing u by characters with modulus u. Suppose that $\psi$ is induced by a primitive character $\psi'$ of modulus $u' < u$ . Then

\[\psi({\mathfrak{n}}) = \psi'({\mathfrak{n}}) \mathbf{1}_{(z,u/u')} = \psi'({\mathfrak{n}}) - \psi'({\mathfrak{n}}) \mathbf{1}_{({\mathfrak{n}},u/u') > 1}.\]

Since we have defined the correlation by sums over $({\mathfrak{n}},P(W))=1$ , the characters agree unless ${\mathcal{N}} ({\mathfrak{n}},(u/u')) > W$ , so that we have

\[{\mathcal{C}}_W(\beta, \xi_{k_j}\psi' H_{N'}) = {\mathcal{C}}_W(\beta, \xi_{k_j}\psi H_{N'})+ O(W^{-1} (\log X)^{O(1)}).\]

Thus, if the approximation includes a character whose conductor is a proper divisor of u, we may replace it by the character with $({\mathfrak{n}},u)=1$ at a negligible cost (by using orthogonality of characters). Let us assume that this has been done, so that the moduli of the characters $\chi_j$ satisfy either $u_j=u$ or $(u_j,u)=1$ . Let $\Psi_u$ denote the characters $\xi\psi$ modulo u that are equal to $\xi_{k_j}\chi_{j}$ for some j. By expanding G with Lemma 2.10 we get

\[S= \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times} }} \bigg|\sum_{\substack{k}} \check{G}(k) \bigg( \sum_{\substack{{\mathfrak{n}} }} \beta^\flat_{\mathfrak{n}} \xi_k \psi({\mathfrak{n}}) H_{N'}({\mathfrak{n}}) \bigg) \bigg|^2.\]

Denote

\begin{align*}S(\Psi_u) := & \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times} }} \left|\sum_{\substack{k \\ \xi_k \psi \in \Psi_u}} \check{G}(k) \bigg( \sum_{\substack{ {\mathfrak{n}} }} \beta^\flat_{\mathfrak{n}} \xi_k \psi({\mathfrak{n}}) H_{N'}({\mathfrak{n}})\bigg) \right|^2, \\S(\Psi_u^{\complement}) := & \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times} }} \left|\sum_{\substack{k \\\xi_k \psi \not \in \Psi_u}} \check{G}(k) \bigg( \sum_{\substack{ {\mathfrak{n}} }} \beta^\flat_{\mathfrak{n}} \xi_k \psi({\mathfrak{n}}) H_{N'}({\mathfrak{n}})\bigg) \right|^2.\end{align*}

Then by Cauchy–Schwarz ( $(A+B)^2 \leq 2(A^2+B^2)$ ) we have

\[S \ll S(\Psi_u)+ S(\Psi_u^{\complement}).\]

6.2.1 Bounding $S(\Psi_u)$ .

By the definition of $\beta^\flat_{\mathfrak{n}}$ , we see that for $\xi_k \psi = \xi_{k_{j_0}} \chi_{j_0}$ we have

\[\sum_{\substack{{\mathfrak{n}} }} \beta^\flat_{\mathfrak{n}} \xi_k \psi({\mathfrak{n}})H_{N'}({\mathfrak{n}}) = \sum_{j \neq j_0} {\mathcal{C}}_W(\beta, \overline{\xi_{k_j}\chi_j} H_{N'}) \sum_{\substack{{\mathfrak{n}} }} H_{N'}({\mathfrak{n}})\mathbf{1}_{({\mathfrak{n}},\overline{{\mathfrak{n}}}P(W))=1} \xi_k\psi \overline{\xi_{k_j}\chi_j}({\mathfrak{n}}).\]

Thus, by Cauchy–Schwarz on j and k (recall that for $\psi \xi_k \in \Psi$ we have $|k| \leq (\nu_1)^{-2}$ by Q-regularity) and using $|\check{G}(k)| \ll \nu_1$ we have

By Lemma 6.4 we get

\[S(\Psi_u) \ll \frac{\nu_1^2 \nu_2^2 N^2 }{\varphi_{{\mathbb{Z}}[i]}(u) (\log X)^C} .\]

6.2.2 Bounding $S(\Psi_u^{\complement})$ .

By Cauchy–Schwarz ( $(A+B)^2 \leq 2(A^2+B^2)$ ) we have

\begin{align*}S(\Psi_u^{\complement}) &= \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times} }} \left|\sum_{\substack{k \\\xi_k \psi \not \in \Psi_u}} \check{G}(k) \left( \sum_{\substack{ {\mathfrak{n}} }} \beta^\flat_{\mathfrak{n}} \xi_k \psi({\mathfrak{n}}) H_{N'}({\mathfrak{n}})\right) \right|^2 \\&\ll S_1(\Psi_u^{\complement}) + S_2(\Psi_u^{\complement}),\end{align*}

with

\begin{align*}S_1(\Psi_u^{\complement}) &:= \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times} }} \bigg|\sum_{\substack{k \\\xi_k \psi \not \in \Psi_u}} \check{G}(k) \bigg( \sum_{\substack{ {\mathfrak{n}} }} \beta_{\mathfrak{n}} \xi_k \psi({\mathfrak{n}}) H_{N'}({\mathfrak{n}})\bigg) \bigg|^2, \\S_2(\Psi_u^{\complement}) &:= \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times} }} \bigg|\sum_{j \leq J} {\mathcal{C}}_W(\beta, \overline{\xi_{k_j}\chi_j} H_{N'}) \\& \quad\times \sum_{\substack{k \\\xi_k \psi \not \in \Psi_u}} \check{G}(k) \bigg( \sum_{\substack{ {\mathfrak{n}} }} H_{N'}({\mathfrak{n}})\mathbf{1}_{({\mathfrak{n}},\, \overline{{\mathfrak{n}}} P(W))=1} \xi_k\psi\overline{\xi_{k_j}\chi_j}({\mathfrak{n}}) \bigg)\bigg|^2.\end{align*}

By the assumption that $\beta_{\mathfrak{n}}$ is Q-regular we have

\[S_1(\Psi_u^{\complement}) \ll \frac{\nu_1^2 \nu_2^2 N^2 }{\varphi_{{\mathbb{Z}}[i]}(u) (\log X)^{C_1}}.\]

For $S_2(\Psi_u^{\complement})$ we have by Cauchy–Schwarz on j

\[S_2(\Psi_u^{\complement}) := \sup_{j \leq J}\frac{(\log X)^{C'}}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times} }} \left| \sum_{\substack{k \\\xi_k \psi \not \in \Psi_u }} \check{G}(k) \bigg( \sum_{\substack{ {\mathfrak{n}} }} H_{N'}({\mathfrak{n}})\mathbf{1}_{({\mathfrak{n}},\, \overline{{\mathfrak{n}}} P(W))=1} \xi_k\psi\overline{\xi_{k_j}\chi_j}({\mathfrak{n}})\bigg)\right|^2,\]

we write

\[\sum_{\substack{k \\\xi_k \psi \not \in \Psi_u}} \check{G}(k) = \sum_{\substack{k }} \check{G}(k) - \check{G}(k_j)\mathbf{1}_{ \psi = \chi_j} - \sum_{\substack{k \\ \xi_k \psi \in \Psi_u \\ \xi_k \psi \neq \xi_{k_j}\chi_j}} \check{G}(k).\]

The contribution from the third sum may be extracted from $ S_2(\Psi_u^{\complement})$ by Cauchy–Schwarz and bounded by the same argument as with $ S(\Psi_u)$ . Thus, we are left with bounding

\begin{align*}S_3(\Psi_u^{\complement}) &:= \sup_{j \leq J}\frac{(\log X)^{C'}}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times} }} \bigg| \sum_{\substack{k }} \check{G}(k) \bigg( \sum_{\substack{ {\mathfrak{n}} }} H_{N'}({\mathfrak{n}})\mathbf{1}_{({\mathfrak{n}},\, \overline{{\mathfrak{n}}} P(W))=1} \xi_k\psi\overline{\xi_{k_j}\chi_j}({\mathfrak{n}})\bigg) \\&\quad -\check{G}(k_j)\mathbf{1}_{ \psi = \chi_j} \sum_{\substack{ {\mathfrak{n}} }} H_{N'}({\mathfrak{n}})\mathbf{1}_{({\mathfrak{n}},\, \overline{{\mathfrak{n}}} P(W))=1}\bigg|^2 \\&= \sup_{j \leq J}\frac{(\log X)^{C'}}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times} }} \bigg| \sum_{\substack{ {\mathfrak{n}} }} H_{N'}({\mathfrak{n}}) G(\arg {\mathfrak{n}}) \mathbf{1}_{({\mathfrak{n}},\, \overline{{\mathfrak{n}}} P(W))=1} \psi\overline{\xi_{k_j}\chi_j}({\mathfrak{n}}) \\&\quad- \check{G}(k_j)\mathbf{1}_{ \psi = \chi_j} \sum_{\substack{ {\mathfrak{n}} }} H_{N'}({\mathfrak{n}})\mathbf{1}_{({\mathfrak{n}},\, \overline{{\mathfrak{n}}} P(W))=1}\bigg|^2.\end{align*}

The claim now follows by orthogonality of characters and Lemma 6.4 since $N^{1-\eta} > Q^3$ and $|u|,|u_j| \leq Q$ .

Remark 6.6. When constructing the approximation $\beta^\#_{\mathfrak{n}}$ we have a choice of using either the physical space or the Fourier space. To approximate $\beta_{{\mathfrak{n}}}$ with respect to arithmetic progressions and sectors of ${\mathbb{Z}}[i]$ we use the Fourier space (i.e. characters $\chi,\xi$ ), whereas to approximate $\beta_{{\mathfrak{n}}}$ with respect to the size of ${\mathcal{N}} {\mathfrak{n}}$ we use the physical space (i.e. smooth partition $H_{N'}$ ). These are the most convenient choice for using existing zero-density estimates and information about exceptional characters.

6.3 Partitioning the Type II sum

With the approximation for $\beta_{\mathfrak{n}}$ defined as in § 6.2, we can extract the main term from the Type II sum by writing

\[S(\alpha,\beta) = S(\alpha,\beta^\#)+S(\alpha,\beta^\flat).\]

By $(a,b)=1$ , we may restrict to $(w,\overline{w})=1$ . Since we are working with rough numbers the condition $(a,b)=1$ may be dropped with a negligible error term. By definition, $a_{\mathfrak{n}} = \sum_{u \in \{\pm 1,\pm i\}} a_{uz}$ so that we have

\[S(\alpha,\beta^\flat) = \sum_{\substack{|w|^2 \sim M \\ (w,\overline{w})=1}}\sum_{|z|^2 \sim N} \alpha_w \beta^\flat_z \mathbf{1}_{B}({\text{Re}}(\overline{w}z)) + O_C\bigg(\frac{X^{1/2} |B|}{(\log X)^C}\bigg).\]

The two contributions are bounded by the following two propositions, which together imply Proposition 6.2.

Proposition 6.7. Suppose that the assumptions of Proposition 6.2 hold and let $\beta_{\mathfrak{n}}^\flat$ be as in § 6.2. Then, for every $C_1>0$ , there is some $C_2>0$

\[\sum_{\substack{|w|^2 \sim M \\ (w,\overline{w})=1}} \sum_{|z|^2 \sim N} \alpha_w \beta^\flat_z \mathbf{1}_B ({{\text{Re}}} (\overline{w} z ) ) \ll \frac{X^{1/2}|B|}{ (\log X)^{C_1}}.\]

Proposition 6.8. Suppose that the assumptions of Proposition 6.2 hold and let $\beta^\#_{\mathfrak{n}}$ be as in § 6.2. Then, for any $C>0$ ,

\[\sum_{\substack{\mathfrak{m}},{\mathfrak{n}} } \alpha_{\mathfrak{m}} \beta^{\#}_{\mathfrak{n}} a_{{\mathfrak{m}}{\mathfrak{n}}} = \sum_{j \leq J } \sum_{\substack{\mathfrak{m}},{\mathfrak{n}} } \frac{\alpha_{\mathfrak{m}} \beta_{\mathfrak{n}} \xi_{k_j}\chi_j({\mathfrak{m}}{\mathfrak{n}})}{{\mathcal{N}} {\mathfrak{m}}{\mathfrak{n}}} \sum_{\substack{\mathfrak{a}}} \frac{F({\mathcal{N}}{\mathfrak{m}}{\mathfrak{n}}/{\mathcal{N}} {\mathfrak{a}})}{\widehat{F}(0) } a^\omega_{{\mathfrak{a}}} \overline{\xi_{k_j}\chi_j}({\mathfrak{a}}) + O \bigg( \frac{X^{1/2}|B|}{(\log X)^C}\bigg).\]

Remark 6.9. We want to carry the condition $(w,\overline{w})=1$ through the application of Cauchy–Schwarz. To see why, consider a situation where $B \subseteq d_0 {\mathbb{Z}}$ with $d_0 > X^\eta$ being very smooth so that $\tau(d_0)$ is larger than any fixed power of $\log X$ . Let us split the Type II sum according to the g.c.d. of w and $d_0$ , which gives us

\[\sum_{e|d_0} \sum_{w \equiv 0\, (e)} \sum_{z} \mathbf{1}_B ({\text{Re}} (\overline{w} z ) ).\]

Now in the inner sum $e| w$ means that $e|b$ is automatic, so that the density on the inside is bumped up, that is, we morally have

\[\sum_{w \equiv 0\, (e)} 1 \approx \frac{M}{e^2}, \quad \sum_{z} \mathbf{1}_B ({\text{Re}} (\overline{w} z ) ) \approx e \frac{X^{1/2}|B|}{M}.\]

Prior to Cauchy–Schwarz this is not an issue since we still get converging sum $\sum_{e|d_0} e^{-1}$ . However, after applying Cauchy–Schwarz to w, we get

\[\sum_{e|d_0} \sum_{w \equiv 0\, (e)} \bigg| \sum_{z}\mathbf{1}_B ({\text{Re}} (\overline{w} z ) ) \bigg|^2 \approx N |B|^2 \sum_{e|d_0} 1,\]

which means that we have picked up a large divisor function $\tau(d_0)$ . This would be problematic since we can only save a fixed power of $\log X$ from Lemma 6.5. We resolve this issue by keeping the condition $(w,\overline{w})=1$ so that w has no non-trivial integer divisors, but there are also other ways to deal with this.

6.4 Proof of Lemma 6.1

Note that ${\mathcal{M}}(u) \leq Q$ implies that $|u| \leq Q$ , so that by $N > X^\eta Q^{3}$ we have $N > X^{\eta} |u|^{3}$ . By Lemma 2.14 we can take for $\sigma_Q := 1- \frac{C_2' \log \log Q}{ \log Q}$ with some large $C_2'> 0$ to get

\[J\leq N^\ast(\sigma_Q,X^\eta,X^{\eta},Q^2) \leq (\log X)^{C_2}\]

and let $\Psi$ be the set of primitive characters $\xi\psi$ such that $L(s,\xi\psi)$ has a zero counted in the above with ${\mathcal{M}}(u) \leq Q$ . Recall that we now specify

\[\beta_{\mathfrak{n}} := \mathbf{1}_{{\mathcal{N}} {\mathfrak{n}} \sim N} \mu({\mathcal{N}}{\mathfrak{n}}) \mathbf{1}_{({\mathfrak{n}},P(W))=1}.\]

Then, for Lemma 6.1, we need to show that if $\Psi_u$ denotes the set of characters modulo u which are induced by $\Psi$ , then for any $N' \in [N,2N]$ for any $C_1>0$ there is some $C_2>0$ such that

(6.3) \begin{equation} \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}} \left| \sum_{\substack{k \\ \xi_k \psi \not \in \Psi_u}} \check{G}(k) \left(\sum_{\substack{\mathcal{N}} {\mathfrak{n}} \in (N',N'(1+\nu_2)]} \beta_{ {\mathfrak{n}}} \xi_k\psi({\mathfrak{n}})\right) \right|^2 \ll \frac{\nu_1^2\nu_2^2 N^2}{\varphi_{{\mathbb{Z}}[i]}(d) (\log X)^{C_1}}.\end{equation}

The contribution from the trivial character $\psi=\psi_0$ is bounded by a similar but easier argument as below, using Heath-Brown’s identity and the Vinogradov strength zero-free region of Coleman [Reference ColemanCol90]. We then restrict to $\psi \neq \psi_0$ . We apply § 2.1 to ${\mathcal{N}}{\mathfrak{n}}$ with $\nu_1=X^{-\eta_1}$ , using the assumption that $\eta_2$ is small compared with $\eta_1$ to replace ${\mathcal{N}} {\mathfrak{n}} \in ( N',N'(1+\nu_2)]$ with a smooth weight. It then suffices to show that for any $N_1 \sim N$ , we have

\[\frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}\\ \psi\neq \psi_0}} \left| \sum_{\substack{k \\ \xi_k \psi \not \in \Psi_u}} \check{G}(k) \left(\sum_{\substack{\mathfrak{n}} } \beta_{ {\mathfrak{n}}} \xi_k\psi({\mathfrak{n}}) F_{N_1}({\mathcal{N}}{\mathfrak{n}}) \right) \right|^2 \ll \frac{\nu_1^4 N^2}{\varphi_{{\mathbb{Z}}[i]}(d) (\log X)^{C_1}}.\]

The proof strategy is classical so we will be brief. We use the Heath-Brown identity [Reference Iwaniec and KowalskiIK04, (13.58)] with $K=3$

\[\mu(n) =\sum_{k=1}^3 (-1)^{k+1}\binom{3}{k}\sum_{\substack{n=m_1m_2m_2n_1n_2n_3 \\ m_j \leq 2 N^{1/3}}} \mu(m_1) \mu(m_2) \mu(m_3).\]

Let $F_1$ be as in § 2.1 with $\nu=1/2$ . Using the Heath-Brown identity and a dyadic decomposition (smooth for the free variable ${\mathfrak{a}}$ ), we get sums of Type I

\[S_I := \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}\\ \psi\neq \psi_0}} \left| \sum_{\substack{k \\ \xi_k \psi \not \in \Psi_u}} \check{G}(k) \left( \sum_{\substack{{\mathfrak{a}} \\ {\mathcal{N}} {\mathfrak{m}} \sim M \\ \\ ({\mathfrak{m}}{\mathfrak{a}},P(W))=1}}F_{N_1}({\mathcal{N}}{\mathfrak{m}} {\mathfrak{a}})\alpha({\mathfrak{m}}) F_{1,A}({\mathcal{N}}{\mathfrak{a}}) \xi_k\psi({\mathfrak{m}} {\mathfrak{a}})\right)\right|^2,\]

with $M \ll 2N^{2/3}$ and $AM \sim N$ and sums of Type II

\begin{align*}S_{II} &:= \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}\\ \psi\neq \psi_0}} \left| \sum_{\substack{k \\ \xi_k \psi \not \in \Psi_u}} \check{G}(k) \left(\sum_{\substack{ {\mathcal{N}} {\mathfrak{m}}_j \sim M_j \\ ({\mathfrak{m}}_j,P(W))=1}} F_{N_1}({\mathcal{N}}{\mathfrak{m}}_1{\mathfrak{m}}_2{\mathfrak{m}}_3)\right.\right. \\ &\!\!\!\qquad\qquad\qquad\quad\qquad\qquad\left.\left.\vphantom{\sum_{\substack{ {\mathcal{N}} {\mathfrak{m}}_j \sim M_j \\ ({\mathfrak{m}}_j,P(W))=1}} F_{N_1}({\mathcal{N}}{\mathfrak{m}}_1{\mathfrak{m}}_2{\mathfrak{m}}_3)}\times \alpha_1({\mathfrak{m}}_1) \alpha_2({\mathfrak{m}}_2) \alpha_3({\mathfrak{m}}_3)\xi_k\psi({\mathfrak{m}}_1 {\mathfrak{m}}_2 {\mathfrak{m}}_3)\right) \right|^2,\end{align*}

with $\alpha_1({\mathfrak{m}}) = 1$ or $\alpha_1({\mathfrak{m}}) = \mu({\mathcal{N}} {\mathfrak{m}})$ , $M_1M_2M_3 \sim M$ and

\[N^{1/6} \ll M_1 \ll N^{1/3}.\]

To see this note that we are in the Type I case unless all of the variables $n_j$ are $\ll N^{1/3}$ , and in that case we can take the $M_1$ for the Type II sum to be the range of largest variable $m_j,n_j$ , which must be $\gg N^{1/6}$ .

To show (6.3) it then suffices to show that

\[S_I, \, S_{II} \,\ll \frac{\nu_1^4 N^2}{\varphi_{{\mathbb{Z}}[i]}(d) (\log X)^{C_1}}.\]

For $S_I$ , we let $D= X^{\eta_1}$ and write

\[\mathbf{1}_{({\mathfrak{a}},P(W))=1} = \sum_{\substack{\mathfrak{d}} | ({\mathfrak{a}},P(W))} \mu({\mathfrak{d}}) = \sum_{\substack{{\mathfrak{d}} | ({\mathfrak{a}},P(W)) \\ {\mathcal{N}}{\mathfrak{d}} \leq D}} \mu({\mathcal{N}}{\mathfrak{d}}) +\sum_{\substack{{\mathfrak{d}} | ({\mathfrak{a}},P(W)) \\ {\mathcal{N}}{\mathfrak{d}} > D}} \mu({\mathcal{N}}{\mathfrak{d}}).\]

The contribution from ${\mathcal{N}}{\mathfrak{d}} > D$ is bounded by using Lemma 2.2, after using orthogonality of characters and Lemma 2.10. For $S_I$ , we then need to bound

\[S_I' := \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}\\ \psi\neq \psi_0}} \left| \sum_{\substack{k \\ \xi_k \psi \not \in \Psi_u}} \check{G}(k) \left( \sum_{\substack{ {\mathfrak{a}} \\ {\mathcal{N}} {\mathfrak{m}} \sim M \\ ({\mathfrak{m}},P(W))=1}} \sum_{\substack{{\mathfrak{d}} | ({\mathfrak{n}},P(W)) \\ {\mathcal{N}}{\mathfrak{d}} \sim D}} \mu({\mathfrak{d}}) F_{1,A}({\mathcal{N}} {\mathfrak{d}} {\mathfrak{a}})\alpha({\mathfrak{m}}) \xi_k\psi({\mathfrak{d}}{\mathfrak{m}} {\mathfrak{a}})\right)\right|^2,\]

with $D \ll X^{\eta_1}$ and $M \ll N^{2/3}$ .

Denote

\begin{align*}M(t,\xi\psi) &:= \sum_{\substack{\mathcal{N}} {\mathfrak{m}} \sim M} \mathbf{1}_{({\mathfrak{m}},P(W))=1} \alpha({\mathfrak{m}}) ({\mathcal{N}} {\mathfrak{m}})^{-it}\xi\psi({\mathfrak{m}}), \\M_j(t,\xi\psi) &:= \sum_{\substack{\mathcal{N}} {\mathfrak{m}} \sim M_j} \mathbf{1}_{({\mathfrak{m}},P(W))=1} \alpha_j({\mathfrak{m}}) ({\mathcal{N}} {\mathfrak{m}})^{-it}\xi\psi({\mathfrak{m}}), \\A(t,\xi\psi) &:= \sum_{\substack{\mathfrak{a}}} \sum_{\substack{{\mathfrak{d}}|({\mathfrak{a}},P(W)) \\ {\mathcal{N}}{\mathfrak{d}} \leq D}}\mu({\mathfrak{d}}) F_{1,A}({\mathcal{N}}{\mathfrak{d}}{\mathfrak{a}}) ({\mathcal{N}} {\mathfrak{d}} {\mathfrak{a}})^{-it}\xi\psi({\mathfrak{a}}).\end{align*}

Then, for $\psi \neq \psi_0$ , we have the standard point-wise bounds

(6.4) \begin{equation} A(t,\xi_k\psi) \ll D(1+|t|) (1+|k|)|u|\end{equation}

and for $\alpha_1=1$ or $\alpha_1=\mu$ with $M_1 \gg X^{1/6}$ once $C_2$ is large compared with $C_1$ , with $\xi\psi \not \in \Psi_u$ ,

(6.5) \begin{equation} M_1(t,\xi\psi) \ll \frac{M_1}{(\log X)^{C_1}}.\end{equation}

The bound (6.4) follows by the Polyá–Vinogradov bound (i.e. the convexity bound for $L(s,\xi\psi)$ in the u aspect, Lemma 2.18). The bound (6.5) follows by the truncated Perron’s formula and shifting the contour to $(1+\sigma_Q)/2$ (justified by $\xi \psi \not \in \Psi_u$ ), using the bound Lemma 2.16 for $1/L(s,\xi \psi)$ , and taking $C_2'>0$ in the definition of $\sigma_Q$ sufficiently large.

By Mellin inversion (Lemma 2.9), we get

\[F_{N_1}(x) = \frac{1}{2\pi i}\int \dot{F}(s) N_{1}^{s} x^{-s}\,ds,\]

with

\[|\dot{F}(s)| \ll_C \nu_1 (1+\nu_1|t|)^{-C}.\]

Hence, we have

\[S_I' \ll J_I \quad \text{and} \quad S_{II} \ll J_{II},\]

with

\begin{align*}J_I &:= \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}\\ \psi\neq \psi_0}} \left| \sum_{\substack{k \\ \xi_k \psi \not \in \Psi_u}} |\check{G}(k)|\int |\dot{F}(it)||MA(t,\xi_k \psi)| \,dt \right|^2,\\J_{II}& := \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)}\sum_{\substack{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}\\ \psi\neq \psi_0}} \left| \sum_{\substack{k \\ \xi_k \psi \not \in \Psi_u}} |\check{G}(k)|\int |\dot{F}(it)||M_1M_2 M_3(t,\xi_k \psi) |\,dt \right|^2.\end{align*}

To bound $J_I$ we apply Cauchy–Schwarz on t, k, (6.4), and orthogonality of characters to get, for some coefficients $\gamma_{\mathfrak{n}}$ and for some $t,\xi$ ,

\begin{align*}J_I& \ll \nu_1^{O(1)} \frac{D^2|u|^2}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}} \bigg| M (t,\xi \psi)\bigg|^2 \\& \ll \nu_1^{O(1)} D^2 |u|^2 \sum_{\substack{{\mathfrak{n}}_1 \equiv {\mathfrak{n}}_2 \, (u) \\ {\mathcal{N}} {\mathfrak{n}}_1, {\mathcal{N}} {\mathfrak{n}}2 \ll M }} |\gamma_{{\mathfrak{n}}_1} \overline{\gamma_{{\mathfrak{n}}_2}}| \\& \ll \nu_1^{O(1)} D^2 M^2 + \nu_1^{O(1)} |u| D^2 M.\end{align*}

By $N > N^{\eta} |u|^{3} $ , $M \ll N^{2/3}$ , and $D =X^{\eta_1}$ we get

\[J_I \ll N^{4/3 + O(\eta_1)} + |u| N^{2/3 + O(\eta_1)} \ll N^{-\eta}\frac{N^2}{\varphi_{{\mathbb{Z}}[i]}(u)}\]

since $\eta_1$ is small compared with $\eta$ , which is sufficient for bounding $J_I$ .

For $J_{II}$ we apply the bound (6.5) for $M_1$ and Cauchy–Schwarz in the t, k variables to get

\begin{align*}J_{II} &\ll \frac{M_1^{2}}{(\log X)^{C_1}} \iint |\dot{F}(it_1)\dot{F}(it_2)| \sum_{k_1,k_2} |\check{G}(k_1)\check{G}(k_2) | \\&\quad \times \frac{1}{\varphi_{{\mathbb{Z}}[i]}(u)} \sum_{\psi \in \widehat{({\mathbb{Z}}[i]/u{\mathbb{Z}}[i])^\times}} |M_2(t_1,\xi_{k_1} \psi)M_3(t_2,\xi_{k_2} \psi)|^2 \,dt_1 \,d t_2.\end{align*}

By orthogonality of characters and Lemmas 2.10 and 2.9 we get

\[J_{II} \ll \frac{M_1^{2}}{(\log X)^{C_1}} T^2 \sum_{\substack{{\mathfrak{m}}_{21},{\mathfrak{m}}_{22},{\mathfrak{m}}_{31} {\mathfrak{m}}_{32} \\ {\mathcal{N}} {\mathfrak{m}}_{jk} \sim M_j \\ {\mathfrak{m}}_{21} {\mathfrak{m}}_{31} \equiv {\mathfrak{m}}_{22} {\mathfrak{m}}_{32} \, (u) \\ |{\mathcal{N}} {\mathfrak{m}}_{21} - {\mathcal{N}} {\mathfrak{m}}_{22} |\ll \nu_1 M_2 \\|\arg {\mathfrak{m}}_{21} - \arg {\mathfrak{m}}_{22} |\ll \nu_1 \\ | {\mathcal{N}}{\mathfrak{m}}_{31} - {\mathcal{N}}{\mathfrak{m}}_{32} | \ll \nu_1 M_3 \\ | \arg{\mathfrak{m}}_{31} - \arg {\mathfrak{m}}_{32} | \ll \nu_1}} 1.\]

Let us denote ${\mathfrak{m}}_{2j} = (w_j)$ and ${\mathfrak{m}}_{3j}=(z_{j})$ so that

\[J_{II} \ll \frac{M_1^{2}}{(\log X)^{C_1}} T^2 \sum_{\substack{w_1,w_2,z_1,z_2 \\ |w_j|^2 \sim M_2 \\ |z_j|^2 \sim M_3 \\ w_1z_1 \equiv w_2z_2 \, (u) \\ | w_1 - w_2 |^2 \ll \nu_1 M_2 \\ |z_1 - z_2 |^2 \ll \nu_1 M_3 }} 1.\]

Writing $w_2=w_1+u$ , $z_1=z_2+v$ we get (using Lemma 2.4 to handle $\tau(w) \tau(z) $ )

\begin{align*}J_{II} &\ll \frac{M_1^{2}}{(\log X)^{C_1}} T^2 \sum_{\substack{w_1,u,v,z_2 \\ |w_1|^2 \sim M_2 \\ |u|^2 \ll \nu_1 M_2 \\ |z_2|^2 \sim M_3 \\ v \ll \nu_1 M_3 \\ w_1v \equiv uz_2 \, (u) \\ }} 1 \\& \ll \frac{M_1^{2}}{(\log X)^{C_1}} T^2 \sum_{\substack{ |w|^2,|z|^2 \ll \nu_1 M_2 M_3 \\ w \equiv z \, (u) \\ }} \tau(w) \tau(z)\\&\ll \frac{ \nu^4 N^2}{\varphi_{{\mathbb{Z}}[i]}(u) (\log N)^{C_1-O(1)}},\end{align*}

using $M_1 \ll N^{1/3}$ , $M_1M_2 M_3 \asymp N$ to get $M_2M_3 \gg N^{2/3} \gg X^{\eta} |u|^2$ .

7.1 Cauchy–Schwarz

Let $F_M(m)=F(m/M)$ with a fixed smooth majorant F for the interval [1,2], supported on $[1/2,3]$ . By applying Cauchy–Schwarz we get

\[\sum_{\substack{|w|^2 \sim M \\ (w,\overline{w})=1}} \sum_{|z|^2 \sim N} \alpha_w \beta^\flat_z \mathbf{1}_B ({\text{Re}} (\overline{w} z )) \ll M^{1/2} U(\beta)^{1/2},\]

where

\[U(\beta) := \sum_{|z_1|^2, |z_2|^2 \sim N} \beta^\flat_{z_1} \overline{\beta^\flat_{z_2}} \sum_{(w,\overline{w})=1} F_M(|w|^2)\mathbf{1}_B ({\text{Re}} (\overline{w} z_1 ) ) .\]

It then suffices to show that

(7.1) \begin{equation} U(\beta) \ll \frac{N |B|^2}{(\log X)^{C_1}}.\end{equation}

Define

\[\Delta=\Delta(z_1,z_2) := {\text{Im}} (\overline{z_1} z_2) = |z_1z_2| \sin (\arg z_2 - \arg z_1).\]

Note that typically $|\Delta| \asymp N$ . We partition the sum into a main term and diagonal terms by writing

\[U(\beta) = V(\beta) + O(U_0(\beta) + U_1(\beta) ),\]

where

\begin{align*}V(\beta)& := \sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ \Delta \neq 0 \\ (z_1,z_2) = 1}} \beta^\flat_{z_1} \overline{\beta^\flat_{z_2}} \sum_{(w,\overline{w})=1} F_M(|w|^2)\mathbf{1}_B ({\text{Re}} (\overline{w} z_1 ) ) \mathbf{1}_B ({\text{Re}} (\overline{w} z_2 ) ) ,\\U_0(\beta) &:= \sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ \Delta = 0}} |\beta^\flat_{z_1} \beta^\flat_{z_2} |\sum_{(w,\overline{w})=1} F_M(|w|^2)\mathbf{1}_B ({\text{Re}} (\overline{w} z_1 ) ) \mathbf{1}_B ({\text{Re}} (\overline{w} z_2 ) ),\\U_1(\beta) &:= \sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ \Delta \neq 0 \\ (z_1,z_2) > 1}} |\beta^\flat_{z_1} \beta^\flat_{z_2}|\sum_{(w,\overline{w})=1} F_M(|w|^2)\mathbf{1}_B ({\text{Re}} (\overline{w} z_1 ) ) \mathbf{1}_B ({\text{Re}} (\overline{w} z_2 ) .\end{align*}

For $V(\beta)$ we apply § 2.1 with $G:{\mathbb{R}}/2\pi{\mathbb{Z}} \to {\mathbb{R}}$ being a non-negative smooth function with the parameter $\nu_1=X^{-\eta_1}$ to the variables $\arg z_1$ and $\arg z_2$ to get

\[V(\beta) = \nu_1^{-2} \int_{({\mathbb{R}}/2\pi{\mathbb{Z}})^2} V(\beta,\boldsymbol{\theta}) \,d \theta_1 \,d \theta_2,\]

with

\[V(\beta,\boldsymbol{\theta}) = \sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ \Delta \neq 0 \\(z_1,z_2) = 1}} \beta^\flat_{z_1,\theta_1} \overline{\beta^\flat_{z_2,\theta_2}} \sum_{(w,\overline{w})=1} F_M(|w|^2)\mathbf{1}_B ({\text{Re}} (\overline{w} z_1 ) ) \mathbf{1}_B ({\text{Re}} (\overline{w} z_2 ) )\]

and

\[\beta^\flat_{z,\theta} = \beta^\flat_{z} G( \arg z -\theta ).\]

We now further partition $V(\beta)$ according to the size of $|\Delta|$ . Let $\nu_3=X^{-\eta_3}$ and write

\[V(\beta) = V_{> \nu_3}(\beta) + V_{\leq \nu_3}(\beta),\]

where $V_{> \nu_3}(\beta)$ is the part where $|\sin (\theta_1-\theta_2) | > \nu_3$ , which implies (since $\eta_3$ is small compared with $\eta_1$ )

\[|\Delta| = |z_1z_2|\, |\sin(\arg z_2-\arg z_1)| \gg \nu_3 N.\]

We then have the following lemmas, which together imply Proposition 6.7.

Lemma 7.1 (Off-diagonal contribution). For $|\sin (\theta_1-\theta_2) | > \nu_3$ ,

\[V(\beta,\boldsymbol{\theta}) \ll \frac{\nu_1^2 N |B|^2}{|\sin (\theta_1-\theta_2) | (\log X)^{C_1}}.\]

Lemma 7.2 (Diagonal contribution). We have

\[U_0(\beta) \ll_{\varepsilon} X^{1/2+{\varepsilon}} |B|\]

Lemma 7.3 (Pseudo-diagonal contribution I). We have

\[U_1(\beta) \ll \frac{N |B|^2 }{W^{1-\eta}}\]

Lemma 7.4 (Pseudo-diagonal contribution II). We have

\[V_{\leq \nu_3}(\beta) \ll_{\varepsilon} X^{\varepsilon} \nu_3 N |B|^2\]

7.2 Proof of Lemma 7.1

It suffices to show that for $|\theta_1 - \theta_2\pmod{\pi}| > \nu_3$ we have

(7.2) \begin{align}V(\beta,\boldsymbol{\theta}) &= \sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ (z_1,z_2) = 1}} \beta^\flat_{z_1,\theta_1} \overline{\beta^\flat_{z_2,\theta_2}} \nonumber\\&\quad \times \sum_{(w,\overline{w})=1} F_M(|w|^2)\mathbf{1}_B ({\text{Re}} (\overline{w} z_1 ) ) \mathbf{1}_B ({\text{Re}} (\overline{w} z_2 ) ) \nonumber\\& \ll \frac{\nu_1^2 N |B|^2}{|\sin (\theta_1-\theta_2) |(\log X)^C}.\end{align}

We note that $(z_1,z_2)=(z_1,\overline{z_1}) = (z_2,\overline{z_2}) = 1$ implies $(\Delta,|z_1|^2|z_2|^2)=1$ . By symmetry, this can be seen from

\[\overline{(\Delta,\overline{z_1})} = (\Delta,z_1) = ({\text{Im}}(z_2 \overline{z_1}),z_1) = (z_2 \overline{z_1},z_1) = 1.\]

Denoting ${\text{Re}} (\overline{w} z_1 ) = b_j$ , we have

\[ i \Delta w = z_2 b_1 -z_1b_2.\]

Let $b_0:= (b_1,b_2)$ . Since $(w,\overline{w})=1$ , we know that $b_0|\Delta$ . Thus,

(7.3) \begin{equation} i (\Delta/b_0) w = z_2 b_1/b_0 -z_1b_2/b_0.\end{equation}

Denoting $b_j= b_0b_j'$ , we have

\[V(\beta,\boldsymbol{\theta}) = \sum_{\substack{b_0\geq 1 }} \sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ (z_1,z_2) = 1 \\ \Delta \equiv 0 \, (b_0)}} \beta^\flat_{z_1,\theta_1} \overline{\beta^\flat_{z_2,\theta_2}} \sum_{\substack{b'_2 \equiv a b'_1 \, ( \Delta/b_0) \\ (b'_1b'_2, \Delta/b_0)=1 \\ (b'_1,b'_2)=1 \\ (w,\overline{w})=1}} \mathbf{1}_B(b_0b'_1)\mathbf{1}_B(b_0b'_2)F_M\bigg(\bigg|b_0 \frac{z_2b'_1-z_1b'_2}{\Delta}\bigg|^2\bigg).\]

We now wish to remove the smooth weight $F_M$ . Recall that already $b_0b'_j \in [Y,Y + X^{1/2-\eta}]$ by (3.2). We introduce a rough finer-than-dyadic partition for $|z_1|^2,|z_2|^2$ by using $H_{N'}$ as in § 6.2 with $\nu_2=X^{-\eta_2}$ . Let $N_{1},N_{1}\sim N$ , and denote

\[\beta^\flat_{z,i} := \beta^\flat_{z}H_{N_{i}}(z)G(\arg z-\theta_i).\]

To prove (7.2) it then suffices to prove that for $N_{1},N_{1}\sim N$ and for $|\sin (\theta_1-\theta_2) | \geq \nu_3$ we have

(7.4) \begin{align}V'(\beta,\boldsymbol{N},\boldsymbol{\theta}) &:= \sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ (z_1,z_2) = 1}} \beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2}}\nonumber \\&\quad \times \sum_{(w,\overline{w})=1} F_M(|w|^2)\mathbf{1}_B ({\text{Re}} (\overline{w} z_1 ) ) \mathbf{1}_B ({\text{Re}} (\overline{w} z_2 ) ) \nonumber \\& \ll \frac{\nu_1^2 \nu_2^2 N |B|^2}{|\sin (\theta_1-\theta_2) |(\log X)^C}.\end{align}

Using $|\sin (\theta_1-\theta_2) | > \nu_3$ we have $\Delta = |z_1z_2| \sin (\arg_2-\arg z_1) \gg \nu_3 N$ , so that for some constant $F_M(\boldsymbol{N},\boldsymbol{\theta})$

(7.5) \begin{equation} F_{M} \bigg( \bigg| b_0 \frac{z_2b'_1 -z_1b'_2}{\Delta} \bigg|^2 \bigg) = F_M(\boldsymbol{N},\boldsymbol{\theta}) + O(\nu_2\nu_3^{-2} ).\end{equation}

Therefore, we have

\[V'(\beta,\boldsymbol{N},\boldsymbol{\theta}) = F_M(\boldsymbol{N},\boldsymbol{\theta})V(\beta,\boldsymbol{N},\boldsymbol{\theta}) + \nu_2 \nu_3^{-2} O(W(\beta,\boldsymbol{N},\boldsymbol{\theta}) ),\]

where

\begin{align*}V(\beta,\boldsymbol{N},\boldsymbol{\theta}) &:= \sum_{b_0}\sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ (z_1,z_2) = 1 \\ \Delta \equiv 0 \, (b_0)}} \beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2}} \sum_{\substack{b'_2 \equiv a b'_1 \, (\Delta/b_0) \\ (b'_1b'_2, \Delta/b_0)=1\\ (b'_1,b'_2)=1 \\ (w,\overline{w})=1}} \mathbf{1}_B(b_0b'_1)\mathbf{1}_B(b_0b'_2) \\W(\beta,\boldsymbol{N},\boldsymbol{\theta}) &:= \sum_{b_0}\sum_{\substack{(|z_1|^2, |z_2|^2 \sim N \\ (z_1,z_2) = 1 \\ \Delta \equiv 0 \, (b_0)}} |\beta^\flat_{z_1,1}| |\beta^\flat_{z_2,2} |\sum_{\substack{b'_2 \equiv a b'_1 \, (\Delta/b_0) \\ (b'_1b'_2, \Delta/b_0)=1\\ (b'_1,b'_2)=1 \\ (w,\overline{w})=1}} \mathbf{1}_B(b_0b'_1)\mathbf{1}_B(b_0b'_2).\end{align*}

Then (7.4) follows from the following two lemmas once $\eta_3$ is small compared with $\eta_2$ so that $\nu_2 \nu_3^{-3}=X^{-\eta_2+3\eta_3} < X^{-\eta_2/2}$ , say.

Lemma 7.5. For any $C_1 >0$ , there is some $C_2>0$ for $\beta^\flat_{\mathfrak{n}}$ as in § 6.2 such that for $|\sin (\theta_1-\theta_2)|> \nu_3$ we have

\[V(\beta,\boldsymbol{N},\boldsymbol{\theta}) \ll \nu_1^2\nu_2^2\frac{N |B|^2}{|\sin (\theta_1-\theta_2)| (\log X)^{C_1}}.\]

Lemma 7.6. For $|\sin (\theta_1-\theta_2)| > \nu_3$ , we have

\[W(\beta,\boldsymbol{N},\boldsymbol{\theta}) \ll \frac{\nu_1^2 \nu_2^2 N |B|^2 (\log X)^{O(1)}}{|\sin (\theta_1-\theta_2)|}.\]

7.2.1 Proof of Lemma 7.5.

The condition $(w,\overline{w})=1$ has served its purpose so we remove it by expanding

\[\mathbf{1}_{(w,\overline{w})=1} = \sum_{\ell|w} \mu (\ell),\]

where $\ell$ runs over integers. Writing $w=\ell w'$ , we get from (7.3)

(7.6) \begin{equation} i (\Delta/b_0) \ell w' = z_2 b_1' -z_1b_2'.\end{equation}

We have

(7.7) \begin{equation} \bigg(\frac{\ell \Delta}{b_0},b_j' \bigg) = 1.\end{equation}

To see this, by using $(b_1',b_2')=1$ and (7.6), we get that for $j \in \{1,2\}$ ,

\[\bigg(\frac{\ell \Delta}{b_0},b_j' \bigg)\, \bigg|\,\bigg(\frac{\ell \Delta}{b_0},z_j\bigg),\]

and (7.7) follows since the left-hand side is an integer and by $(z_j,\overline{z_j})=1$ the only integer dividing $z_j$ is 1. Similarly, we also get $(z_1z_2,\ell) =1$ by (7.6). Thus,

\[ a:= z_2/z_1 \equiv b_2'/b_1' \, (\ell)\]

is congruent to a rational integer, which is equivalent to saying that $\Delta \equiv 0 \, (\ell)$ . Thus, we get

\[V(\beta,\boldsymbol{N},\boldsymbol{\theta}) = \sum_{b_0,\ell }\mu(\ell)\sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ (z_1,z_2) = 1 \\ \Delta \equiv 0 \, ([b_0,\ell])}} \beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2}} \sum_{\substack{b'_2 \equiv a b'_1 \, (\ell\Delta/b_0) \\ (b'_1b'_2,\ell \Delta/b_0)=1\\ (b'_1,b'_2)=1}} \mathbf{1}_B(b_0b'_1)\mathbf{1}_B(b_0b'_2).\]

By expanding the congruence $b'_2 \equiv a b'_1 \, (\ell\Delta/b_0) $ into Dirichlet characters and splitting into primitive characters we get

\begin{align*}V(\beta,\boldsymbol{N},\boldsymbol{\theta}) &= \sum_{b_0, \ell }\mu(\ell)\sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ (z_1,z_2) = 1 \\ \Delta \equiv 0 \, ([b_0,\ell])}} \beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2}} \\&\quad\times\frac{1}{\varphi (\ell \Delta/b_0)} \sum_{d| \ell \Delta/b_0} \sideset{}{^\ast}\sum_{\chi\, (d)} \chi(a) \left( \sum_{\substack{b_0b'_1,b_0b'_2 \in B \\ (b'_1,b'_2)=1 \\ (b_1'b_2',\ell \Delta/(db_0))=1}} \chi(b'_1) \overline{\chi}(b'_2) \right).\end{align*}

We separate $db_0 > X^{\delta+\eta/2}$ and $ db_0 \leq X^{ \delta +\eta/2}$ , that is, write

\[V(\beta,\boldsymbol{N},\boldsymbol{\theta}) = V_{ \leq}(\beta,\boldsymbol{N},\boldsymbol{\theta})+ V_{>}(\beta,\boldsymbol{N},\boldsymbol{\theta}).\]

For the large $db_0$ , we use the expansion

\[\mathbf{1}_{(b_1',b_2')=1} = \sum_{c|(b_1',b_2')} \mu(c)\]

to get

\begin{align*}V_{>}(\beta,\boldsymbol{N},\boldsymbol{\theta}) &= \sum_{b_0,\ell } \mu (\ell)\sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ (z_1,z_2) = 1 \\ \Delta \equiv 0 \, ([b_0,\ell])}} \beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2} } \\&\quad \times \frac{1}{\varphi (\ell \Delta/b_0)} \sum_{\substack{d| \ell\Delta/b_0 \\ db_0 > X^{\delta+\eta/2}}}\sum_{(c,d)=1} \mu(c) \sideset{}{^\ast}\sum_{\chi\, (d)} \chi(a) \left| \sum_{\substack{cb_0b \in B \\ (cb,\ell \Delta/(db_0))=1}} \chi(b) \right|^2.\end{align*}

We split the sum according to $a_0=(b_0,\ell)$ which gives us

\begin{align*}V_{>}(\beta,\boldsymbol{N},\boldsymbol{\theta}) &= \sum_{\substack{b_0 \\ a_0 | b_0}} \sum_{(\ell,b_0)=a_0} \mu (\ell)\sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ (z_1,z_2) = 1 \\ \Delta \equiv 0 \, (b_0\ell/a)}} \beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2} } \\&\quad \times \frac{1}{\varphi (\ell \Delta/b_0)} \sum_{\substack{d| \ell\Delta/b_0 \\ db_0 > X^{\delta+\eta/2}}}\sum_{(c,d)=1} \mu(c) \sideset{}{^\ast}\sum_{\chi\, (d)} \chi(a) \left| \sum_{\substack{cb_0b \in B \\ (cb,\ell \Delta/(db_0))=1}} \chi(b) \right|^2.\end{align*}

Recall that by $|\sin (\theta_1-\theta_2)| \gg \nu_3$ we have $|\Delta| \gg \nu_3 N$ . For any fixed D with $\nu_3 N \ll D \ll N$ we have (combining the variables $z_2\overline{z_1}=z$ , using a divisor bound, and recalling that $\Delta= {\text{Im}} (z_2\overline{z_1})$ )

\[\sum_{\substack{|z_1|^2,|z_2|^2 \sim N \\ |\Delta|=D \gg \nu_3 N }} \frac{1}{\varphi (\ell \Delta/b_0)} \ll X^{\eta/200}\sum_{\substack{|z|^2 \sim N^2\\ |{\text{Im}} (z)| = D \gg \nu_3 N }} \frac{b_0 }{\ell D} \ll \frac{ X^{\eta/200} b_0 }{\nu_3 \ell },\]

which gives us (since $\eta_3$ is small compared with $\eta$ )

\begin{align*}V_{>}(\beta,\boldsymbol{N},\boldsymbol{\theta}) &\ll \sup_{L \ll X} \frac{X^{\eta/100}}{L} \sum_{\substack{b_0 \\ a_0| b_0 }} \sum_{\substack{\ell \sim L \\ \ell \equiv 0 \, (a_0)}} b_0 \sum_{\substack{D \ll N \\ D \equiv 0 \, (b_0 \ell/a_0)}}\sum_{\substack{d| \ell D/b_0 \\ db_0 > X^{\delta+\eta/2}}}\sum_{(c,d)=1} \\&\quad\times \sideset{}{^\ast}\sum_{\chi\, (d)} \left| \sum_{\substack{cb_0b \in B \\ (cb,\ell D/(db_0))=1}} \chi(b) \right|^2.\end{align*}

By a dyadic partition, we get (denoting $D'=\ell D/(db_0) $

\begin{align*}V_{>}(\beta,\boldsymbol{N},\boldsymbol{\theta}) &\ll \sup_{\substack{B_0,D_0,L \\ X^{\delta+\eta/2} \ll D_0 B_0 \ll LN}} \frac{X^{\eta/50} B_0}{L} \sum_{\substack{b_0 \sim B_0 \\ a_0| b_0 }} \sum_{\substack{\ell \sim L \\ \ell \equiv 0 \, (a_0)}} \sum_{\substack{D' \sim L N / B_0 D_0 \\ D' \equiv 0 \, (\ell^2/a_0)}}\\& \quad \times\sum_{d \sim D_0} \sum_{(c,d)=1} \sideset{}{^\ast}\sum_{\chi\, (d)} \left| \sum_{\substack{cb_0b \in B \\ (cb,D')=1}} \chi(b) \right|^2 \\&\ll \sup_{\substack{B_0,D_0,L \\ X^{\delta+\eta/2} \ll D_0 B_0 \ll LN}} \frac{X^{\eta/50} B_0}{L} \sum_{\substack{b_0 \sim B_0 \\ a_0| b_0 }} \sum_{\substack{\ell \sim L \\ \ell \equiv 0 \, (a_0)}} \sum_{\substack{D' \sim L N / B_0 D_0 \\ D' \equiv 0 \, (\ell^2/a_0)}} \\& \quad \times \sup_{D''}\sum_{d \sim D_0} \sum_{(c,d)=1} \sideset{}{^\ast}\sum_{\chi\, (d)} \left| \sum_{\substack{cb_0b \in B \\ (cb,D'')=1}} \chi(b) \right|^2 \\&\ll X^{\eta/40} N \sup_{\substack{ B_0,D_0,L \\ X^{\delta+\eta/2} \ll D_0 B_0 \ll LN}} \frac{1}{D_0 L} \sum_{b_0 \sim B_0 } \sum_c \sup_{D''} \sum_{d \sim D_0} \sideset{}{^\ast}\sum_{\chi\, (d)} \left| \sum_{\substack{cb_0b \in B \\ (cb,D'')=1}} \chi(b) \right|^2.\end{align*}

By the large sieve for multiplicative characters (Lemma 2.11) we obtain

\[V_{>}(\beta,\boldsymbol{N},\boldsymbol{\theta}) \ll X^{\eta/30} N (N + X^{1/2-\delta-\eta/2} ) |B| \ll X^{-\eta/4} N |B|^2,\]

since $N \leq X^{-\eta} |B|$ . This is sufficient for Lemma 7.5 since $\eta_1,\eta_2$ are small compared with $\eta$ .

To bound the contribution from the small $d b_0$ recall that

\begin{align*}V_{\leq}(\beta,\boldsymbol{N},\boldsymbol{\theta}) &= \sum_{b_0,\ell } \mu(\ell) \sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ (z_1,z_2) = 1 \\ \Delta \equiv 0 \, ([b_0,\ell])}} \beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2} } \\&\quad \times \frac{1}{\varphi (\ell \Delta/b_0)} \sum_{\substack{ d| \ell \Delta/b_0 \\ db_0 \leq X^{\delta+\eta/2}}} \sideset{}{^\ast}\sum_{\chi\, (d)} \chi(a) \left( \sum_{\substack{b_0b'_1,b_0b'_2 \in B \\ (b'_1,b'_2)=1\\ (b_1'b_2', \ell \Delta/(db_0))=1}} \chi(b'_1) \overline{\chi}(b'_2) \right).\end{align*}

We expand the conditions $(b_1'b_2',\ell \Delta/(db_0))=1$ by using the Möbius function to get

\begin{align*}V_{\leq}(\beta,\boldsymbol{N},\boldsymbol{\theta}) &= \sum_{(e_1,e_2)=1} \mu (e_1) \mu(e_2)\sum_{b_0,\ell } \mu(\ell)\sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ (z_1,z_2) = 1 \\ \Delta \equiv 0 \, ([b_0,\ell])}} \beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2} } \\& \quad\times \frac{1}{\varphi (\ell \Delta/b_0)} \sum_{\substack{de_1e_2|\ell \Delta/b_0 \\ d b_0\leq X^{\delta+\eta/2}}} \sideset{}{^\ast}\sum_{\chi\, (d)} \chi(a) \left( \sum_{\substack{b_0e_1 b'_1,b_0e_2 b'_2 \in B \\ (b'_1,b'_2)=1 \\ e_j|b_j'}} \chi(e_1b''_1) \overline{\chi}(e_2b''_2) \right).\end{align*}

We have

\[\frac{1}{\varphi (\ell \Delta/b_0)} = \frac{b_0}{\ell |\Delta|} \prod_{p| \ell \Delta/b_0} \bigg(1+\frac{1}{p-1}\bigg) = \frac{b_0}{\ell |\Delta|} \sum_{\substack{ d_1 | \ell\Delta/b_0}} \frac{|\mu(d_1)|}{\varphi(d_1)}.\]

Since $z_1,z_2$ are restricted to polar boxes, we have

(7.8) \begin{equation} |\Delta| = N_1^{1/2} N_2^{1/2} |\sin (\theta_2-\theta_2)| (1+O(\nu_2)) := D(1+O(\nu_2)).\end{equation}

Plugging this in, we get

(7.9) \begin{equation} V_{\leq}(\beta,\boldsymbol{N},\boldsymbol{\theta}) = V'_{\leq}(\beta,\boldsymbol{N},\boldsymbol{\theta})+O(W_{\leq }(\beta,\boldsymbol{N},\boldsymbol{\theta})),\end{equation}

with

Let us first consider $V'_{\leq}(\beta,\boldsymbol{N},\boldsymbol{\theta})$ . We note that the sums over $d_1,e_2,e_2,\ell$ converge quickly, so that we expect to be able to bound the contribution from large ranges of $d_1,e_2,e_2$ by crude estimates. We have

\[\frac{\ell}{(b_0,\ell)}\! \left|\Delta/b_0 \quad \text{and} \quad \frac{[d_1,de_1e_2]}{([d_1,de_1e_2],\ell)} \right|\! \Delta/b_0.\]

Denoting

\[ f:=f(\ell,d_1,d,e_1,e_2) := b_0\cdot \bigg[\frac{[d_1,de_1e_2]}{([d_1,de_1e_2],\ell)} , \frac{\ell}{(b_0,\ell)}\bigg],\]

we get $f | \Delta$ , which is equivalent to saying that

\[ z_2 \equiv a z_1 \, (f)\]

for some $a \, (f)$ . Note that $|f| \ll N$ . Thus,

\begin{align*}V'_{\leq}(\beta,\boldsymbol{N},\boldsymbol{\theta}) &= \frac{1}{D}\sum_{\substack{ d_1 }} \frac{|\mu(d_1)|}{\varphi(d_1)} \sum_{\substack{(e_1,e_2)=1 }} \mu (e_1) \mu(e_2)\sum_{b_0,\ell} \frac{b_0\mu(\ell)}{\ell} \\& \quad\times \sum_{d b_0 \leq X^{\delta+\eta/2}} \sum_{a \, (f)} \sideset{}{^\ast}\sum_{\chi\, (d)} \chi(a) \left( \sum_{\substack{b_0b'_1,b_0b'_2 \in B \\ (b'_1,b'_2)=1 \\ e_j|b_j'}} \chi(b'_1) \overline{\chi}(b'_2) \right) \sum_{\substack{z_2 \equiv a z_1 \, (f) \\ (z_1,z_2)=1}} \beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2} }.\end{align*}

We write

(7.10) \begin{equation} V'_{ \leq}(\beta,\boldsymbol{N}, \boldsymbol{\theta}) = V''_{\leq}(\beta,\boldsymbol{N}, \boldsymbol{\theta}) + E_{ \leq}(\beta,\boldsymbol{N}, \boldsymbol{\theta}),\end{equation}

where $E_{\leq}(\beta,\boldsymbol{N}, \boldsymbol{\theta})$ is the part where $f > X^{\delta +\eta}$ .

For $f \leq X^{\delta + \eta}$ , we note by $(z_1,z_2)=1$ we have $(z_1z_2,f)=1$ . By dropping the condition $(z_1,z_2)=1$ , we get

(7.11) \begin{equation} V''_{ \leq}(\beta,\boldsymbol{N}, \boldsymbol{\theta}) = V'''_{ \leq}(\beta,\boldsymbol{N}, \boldsymbol{\theta}) + O(Z_{ \leq}(\beta,\boldsymbol{N}, \boldsymbol{\theta}))\end{equation}

with

\begin{align*}V'''_{\leq}(\beta,\boldsymbol{N},\boldsymbol{\theta}) &= \frac{1}{D}\sum_{\substack{ d_1 }} \frac{|\mu(d_1)|}{\varphi(d_1)} \sum_{\substack{(e_1,e_2)=1 }} \mu (e_1) \mu(e_2)\sum_{b_0,\ell} \frac{b_0\mu(\ell)}{\ell} \\&\quad \times \sum_{\substack{ d b_0 \leq X^{\delta+\eta/2} \\ f \leq X^{\delta+\eta}}} \sum_{a \, (f)} \sideset{}{^\ast}\sum_{\chi\, (d)} \chi(a) \left( \sum_{\substack{b_0b'_1,b_0b'_2 \in B \\ (b'_1,b'_2)=1 \\ e_j|b_j'}} \chi(b'_1) \overline{\chi}(b'_2) \right) \sum_{\substack{z_2 \equiv a z_1 \, (f) \\ (z_1z_2,f)=1}} \beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2} }\end{align*}

and

\begin{align*}Z_{ \leq}(\beta,\boldsymbol{N}, \boldsymbol{\theta}) &:= \frac{1}{D}\sum_{\substack{ d_1 }} \frac{1}{\varphi(d_1)} \sum_{\substack{(e_1,e_2)=1 }} \sum_{b_0,\ell} \frac{b_0}{\ell} \\&\quad \times \sum_{\substack{ d b_0 \leq X^{\delta+\eta/2} \\ f \leq X^{\delta+\eta}}} \sum_{a \, (f)} \left|\sideset{}{^\ast} \sum_{\chi\, (d)} \chi(a) \left( \sum_{\substack{b_0b'_1,b_0b'_2 \in B \\ (b'_1,b'_2)=1 \\ e_j|b_j'}} \chi(b'_1) \overline{\chi}(b'_2) \right)\right| \sum_{\substack{z_2 \equiv a z_1 \, (f) \\ (z_1z_2,f)=1\\ |(z_1,z_2)|^2 \geq W }} |\beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2} } |.\end{align*}

By expanding $z_2 \equiv a z_1 \, (f)$ with Dirichlet characters we have

\begin{align*}V'''_{\leq}(\beta,\boldsymbol{N}) &\leq \frac{1}{D} \sum_{\substack{ d_1 }} \frac{1}{d_1} \sum_{\substack{(e_1,e_2)=1}} \sum_{b_0, \ell } \frac{b_0}{\ell}\sum_{d b_0\leq X^{\delta+\eta/2}} \sum_{\substack{a \, (f)}} \left| \sideset{}{^\ast}\sum_{\chi\, (d)} \chi(a) \left( \sum_{\substack{b_0b_1',b_0b_2' \in B \\ (b_1',b_2')=1 \\ e_j| b_j'}} \chi(b_1') \overline{\chi}(b_2') \right) \right|\\&\quad\times\frac{\mathbf{1}_{f \leq X^{\delta+\eta}}}{\varphi_{{\mathbb{Z}}[i]}(f)}\sum_{\psi \in \widehat{({\mathbb{Z}}[i]/f {\mathbb{Z}}[i])^\times}} \bigg|\sum_{z_1,z_2} \beta^\flat_{z_1,1} \psi (z_1) \overline{\beta^\flat_{z_2,2} \psi (z_2)} \bigg|.\end{align*}

By applying Cauchy–Schwarz on $\psi$ and using Lemma 6.5 we get

\begin{align*}|V'''_{\leq}(\beta,\boldsymbol{N}) | &\ll \frac{\nu_1^2\nu_2^2 N^2}{D (\log X)^{C_1}}\sum_{\substack{d_1,e_1,e_2 }} \sum_{b_0 ,\ell }\sum_{db_0 \leq X^{\delta+\eta/2}} \frac{b_0}{d_1 \ell f^2} \\&\quad\times\sum_{\substack{a \, (f) }} \left|\sideset{}{^\ast}\sum_{\chi\, (d)} \chi(a) \left( \sum_{\substack{b_0b_1',b_0b_2' \in B \\ (b'_1,b'_2)=1 \\ e_j | b_j'}} \chi(b'_1) \overline{\chi}(b'_2) \right) \right|.\end{align*}

By Lemma 2.17 (with the argument $ab'_1/b'_2$ ) and Lemma 2.1 this is bounded by

\begin{align*}&\ll \frac{\nu_1^2\nu_2^2 N^2}{D (\log X)^{C_1}} \sum_{d_1,b_0,\ell } \sum_{\substack{b_0 b'_1,b_0 b'_2 \in B \\(b'_1,b'_2)=1}}\sum_{\substack{e_1|b'_1 \\ e_2|b'_2}} \sum_{\substack{ d b_0\leq X^{\delta+\eta/2} \\ (b'_1b'_2,d)=1 }}\frac{b_0}{d_1\ell f^2} \sum_{\substack{ a \, (f)\\(a,d)=1}} (d,b'_2-ab'_1) \\& \ll \frac{ \nu_1^2 \nu_2^2 N^2}{D(\log X)^{C_1}} \sum_{d_1,b_0,\ell } \sum_{\substack{b_0 b'_1,b_0 b'_2 \in B \\(b'_1,b'_2)=1}}\sum_{\substack{e_1|b'_1 \\ e_2|b'_2}} \sum_{\substack{ d b_0\leq X^{\delta+\eta/2} \\ (b_1b_2,d)=1 }}\frac{\tau(d)b_0}{d_1\ell f}.\end{align*}

Using

\[ f = b_0\cdot \bigg[\frac{[d_1,de_1e_2]}{([d_1,de_1e_2],\ell)} , \frac{\ell}{(b_0,\ell)}\bigg] \geq \frac{b_0 d}{(d,\ell)},\]

by Lemma 2.1 and (7.8), we get

\begin{align*}|V'''_{\leq}(\beta,\boldsymbol{N}) | & \ll \frac{\nu_1^2\nu_2^2 N^2}{D(\log X)^{C_1}}\sum_{\substack{b_1,b_2 \in B }} \sum_{\substack{ d,\ell \leq X }}\frac{\tau(d) (\ell,d)}{d \ell} \\& \ll \frac{\nu_1^2\nu_2^2 N|B|^2}{|\sin(\theta_2-\theta_1)|(\log X)^{C_1}},\end{align*}

which is sufficient.

The argument for bounding $ Z_{ \leq}(\beta,\boldsymbol{N}, \boldsymbol{\theta}) $ from (7.11) is the same except that instead of expanding into Hecke characters and Lemma 6.5 we use the trivial bound

\begin{align*}\sum_{\substack{z_2 \equiv a z_1 \, (f) \\ (z_1z_2,f)=1 \\ |(z_1,z_2)|^2 \geq W }} |\beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2} } | &= \sum_{|z_0|^2 \geq W} \sum_{\substack{z'_2 \equiv a z'_1 \, (f) \\ (z_1z_2,f)=1 }} |\beta^\flat_{z_0z_1,1} \overline{\beta^\flat_{z_0z_2,2} } | \\&\ll (\log X)^{O(1)} \sum_{ W < |z_0|^2 \ll N} \bigg( \frac{\nu_1 \nu_2 N}{|z_0|^2} + 1\bigg)\bigg( \frac{\nu_1 \nu_2 N}{|z_0|^2 f^2} + 1\bigg) \\&\ll (\log X)^{O(1)} \bigg(\frac{\nu_1^2 \nu_2^2 N^2}{W f^2}+ \nu_1 \nu_2 N (\log X) + N \bigg)\\&\ll (\log X)^{O(1)} \frac{\nu_1^2 \nu_2^2 N^2}{W f^2},\end{align*}

where the last bound holds since $N > X^{3\delta+3\eta}$ and $f^2 \leq X^{\delta+2\eta}$ . This gives us

\[Z_{ \leq}(\beta,\boldsymbol{N}, \boldsymbol{\theta}) \ll (\log X)^{O(1)} \frac{\nu_1^2\nu_2^2 N |B|^2}{|\sin \theta_2-\theta_1| W },\]

which suffices for Lemma 7.5.

For $E_{\leq}(\beta,\boldsymbol{N},\boldsymbol{\theta})$ from (7.10) with large f we will need to use a slightly different argument since the modulus f can be as large as N which would make $f^2$ much bigger than N. We write

\begin{align*}E_{\leq}(\beta,\boldsymbol{N},\boldsymbol{\theta}) &= \frac{1}{D}\sum_{\substack{ d_1 }} \frac{|\mu(d_1)|}{\varphi(d_1)} \sum_{\substack{(e_1,e_2)=1 }} \mu (e_1) \mu(e_2)\sum_{b_0,\ell} \frac{b_0\mu(\ell)}{\ell} \\&\quad \times \sum_{\substack{d b_0 \leq X^{\delta+\eta/2} \\ f > X^{\delta + \eta}}} \sum_{a \, (f)} \sideset{}{^\ast}\sum_{\chi\, (d)} \chi(a) \left( \sum_{\substack{b_0b'_1,b_0b'_2 \in B \\ (b'_1,b'_2)=1 \\ e_j|b_j'}} \chi(b'_1) \overline{\chi}(b'_2) \right) \sum_{z_2 \equiv a z_1 \, (f)} \beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2} } \\&\ll \frac{1}{D}\sum_{\substack{ d_1 }} \frac{1}{\varphi(d_1)} \sum_{\substack{e_1,e_2 }} \sum_{b_0,\ell} \frac{b_0}{\ell} \\&\quad \times \sum_{\substack{d b_0 \leq X^{\delta+\eta/2} \\ f > X^{\delta + \eta}}} \sideset{}{^\ast}\sum_{\chi\, (d)}\bigg| \sum_{\substack{b_0b'_1,b_0b'_2 \in B \\ (b'_1,b'_2)=1 \\ e_j|b_j'}} \chi(b'_1) \overline{\chi}(b'_2) \bigg| \sum_{\substack{z_1,z_2 \\ f| \Delta }} |\beta^\flat_{z_1,1} \overline{\beta^\flat_{z_2,2} } | \\&\ll \frac{N^2}{D} \sum_{\substack{ d_1 }} \frac{1}{\varphi(d_1)} \sum_{\substack{e_1,e_2}} \sum_{b_0,\ell} \frac{b_0}{\ell} \\&\quad \times \sum_{\substack{d b_0 \leq X^{\delta+\eta/2} \\ f > X^{\delta + \eta}}} \frac{1}{|f|} \sideset{}{^\ast}\sum_{\chi\, (d)}\bigg| \sum_{\substack{b_0b'_1,b_0b'_2 \in B \\ (b'_1,b'_2)=1 \\ e_j|b_j'}} \chi(b'_1) \overline{\chi}(b'_2) \bigg|.\end{align*}

By using $f > X^{\delta + \eta}$ and $db_0 \leq X^{\delta + \eta/2}$ we get $f^{-1} \leq X^{-\eta/2} (b_0 d)^{-1}$ , which gives us

By using the expansion

\[\mathbf{1}_{(b_1',b_2')=1} = \sum_{c|(b_1',b_2')} \mu(c),\]

Cauchy–Schwarz, orthogonality of characters, and a divisor bound, we have

\begin{align*}&\sideset{}{^\ast}\sum_{\chi\, (d)} \sum_{e_1, e_2} \bigg| \sum_{\substack{b_0b'_1,b_0b'_2 \in B \\ (b'_1,b'_2)=1 \\ e_j|b_j'}} \chi(b'_1) \overline{\chi}(b'_2) \bigg| \leq \sum_{c} \sum_{\chi\, (d)} \sum_{e_1 ,e_2} \bigg| \sum_{\substack{b_0b'_1,b_0b'_2 \in B \\ e_j|b_j' \\ c|b_j'}} \chi(b'_1) \overline{\chi}(b'_2) \bigg| \\&\quad \ll_{\varepsilon} X^{{\varepsilon}} \left( \sum_{e_1 ,e_1'} \sum_{\substack{b_0 b_1,b_0b_1' \in B \\ e_1| b_1 \\ e_1'|b_1' }} d\mathbf{1}_{b_1 \equiv b_1' \, (d)} \right)^{1/2} \left( \sum_{e_2 ,e_2'} \sum_{\substack{b_0 b_2,b_0 b_2' \in B \\ e_2| b_2 \\ e_2'|b_2' }}d \mathbf{1}_{b_2 \equiv b_2' \, (d)} \right)^{1/2} \\&\quad\ll X^{\eta/4} \sum_{e_1 ,e_1'} \sum_{\substack{b_0 b_1,b_0b_1' \in B \\ e_1| b_1 \\ e_1'|b_1' }} d\mathbf{1}_{b_1 \equiv b_1' \, (d)} ,\end{align*}

where the last bound follows by symmetry. Hence, we have

\[E_{\leq}(\beta,\boldsymbol{N},\boldsymbol{\theta}) \ll X^{-\eta/4} \frac{N^2}{D}\sum_{\substack{b_0,d,e_1,e_1' \\ db_0 \leq X^{\delta+\eta/2} }} \sum_{\substack{ b_0b_1,b_0b_1' \in B \\ e_1| b_1 \\ e_1' |b_1' }} \mathbf{1}_{b_1 \equiv b_1' \, (d)}.\]

The contribution from $b_1'=b_1$ is bounded by

\[\ll X^{-\eta/8} \frac{N^2}{D}|B| X^{\delta+\eta/2} \ll X^{-\eta/8} \frac{N^2}{D}|B|^2.\]

The contribution from $b_1'\neq b_1$ is bounded by

\[\ll X^{-\eta/4} \frac{N^2}{D}\sum_{\substack{b_0,e_1,e_1' }} \sum_{\substack{ b_0b_1,b_0b_1' \in B \\ e_1| b_1 \\ e_1' |b_1' }} \mathbf{1}_{b_1 \neq b_1'} \tau(b_1'-b_1) \ll X^{-\eta/8} \frac{N^2}{D}|B|^2\]

by applying the divisor bound $\tau(n) \ll_{\varepsilon} n^{\varepsilon}$ . Therefore, by (7.8) we get (recall that $\nu_3$ is small compared with $\nu$ )

\[E_{\leq}(\beta,\boldsymbol{N},\boldsymbol{\theta}) \ll X^{-\eta/10} N|B|^2.\]

Finally, the error term $W_{\leq }(\beta,\boldsymbol{N},\boldsymbol{\theta})$ from (7.9) is bounded by exactly the same argument as above except that in the part $f \leq X^{\delta+\eta}$ we use the trivial estimate

\[\sum_{\substack{|z_1|^2, |z_2|^2 \sim N \\ z_2 \equiv a z_1 \, (f)}} |\beta^\flat_{z_1,1}| |\beta^\flat_{z_2,2} | \ll (\log X)^{O(1)}\frac{\nu_1^4 N^2}{|f|^2}\]

instead of expanding into Hecke characters and Lemma 6.5.