2.1 Automorphic forms

Let $f\in \mathcal {B}_0(1)$ be a Hecke–Maass cusp form with the spectral parameter $t_f$ for $\operatorname {SL}(2,\mathbb {Z})$ with the normalized Fourier coefficients $\lambda _f(n)$ . Let $\theta _2$ be the bound toward the Ramanujan conjecture, and we have $\theta _2\leq 7/64$ due to Kim–Sarnak [Reference Kim25]. Rankin–Selberg theory gives (see Iwaniec [Reference Iwaniec21, Lemma 1])

(2.1) $$ \begin{align} \sum_{n\leq N} |\lambda_f(n)|^2 \ll t_f^{\varepsilon} N. \end{align} $$

Let $\phi $ be a Hecke–Maass cusp form of type $(\nu _1,\nu _2)$ for $\operatorname {SL}(3,\mathbb {Z})$ with the normalized Fourier coefficients $A(r,n)$ . Rankin–Selberg theory gives

(2.2) $$ \begin{align} \sum_{r^2n\leq N} |A(r,n)|^2 \ll_{\phi} N. \end{align} $$

We record the Hecke relation

$$\begin{align*}A(r,n) = \sum_{d\mid (r,n)} \mu(d) A\left(\frac{r}{d},1\right) A\left(1,\frac{n}{d}\right) \end{align*}$$

which follows from Möbius inversion and [Reference Goldfeld12, Theorem 6.4.11]. Hence, we have the individual bounds

$$\begin{align*}A(r,n) \ll (rn)^{\theta_3+\varepsilon}, \end{align*}$$

where $\theta _3\leq 5/14$ is the bound toward the Ramanujan conjecture on $\operatorname {GL}(3)$ ([Reference Kim25]). Thus, we have

(2.3) $$ \begin{align} \sum_{n\sim N} |A(r,n)| \ll \sum_{n_1\mid r^{\infty}} \sum_{\substack{n\sim N/n_1 \\ (n,r)=1}} |A(r,nn_1)| \leq \sum_{n_1\mid r^{\infty}}|A(r,n_1)| \sum_{\substack{n\sim N/n_1 \\ (n,r)=1}} |A(1,n)| \ll r^{\theta_3+\varepsilon} N \end{align} $$

and

(2.4) $$ \begin{align} \sum_{n\sim N} |A(r,n)|^2 \ll \sum_{n_1\mid r^{\infty}} \sum_{\substack{n\sim N/n_1 \\ (n,r)=1}} |A(r,nn_1)|^2 \leq \sum_{n_1\mid r^{\infty}}|A(r,n_1)|^2 \sum_{\substack{n\sim N/n_1 \\ (n,r)=1}} |A(1,n)|^2 \ll r^{2\theta_3+\varepsilon} N. \end{align} $$

Those bounds depend on $\phi $ and $\varepsilon $ . Here, we have used (2.2) and the fact $ \sum _{d\mid r^{\infty }} d^{-\sigma } \ll r^{\varepsilon }, \; \textrm {for } \sigma>0. $

2.2 The approximate functional equation

The Rankin–Selberg L-function $L(s,\phi \times f)$ has the following functional equation

$$\begin{align*}\Lambda(s,\phi\times f) = \epsilon_{\phi\times f} \Lambda(1-s,\tilde\phi\times f), \end{align*}$$

where

$$\begin{align*}\Lambda(s,\phi\times f) = \pi^{-3s} \prod_{j=1}^{3} \prod_{\pm} \Gamma\left(\frac{s-\alpha_j\pm i t_f}{2}\right) L(s,\phi\times f) \end{align*}$$

is the completed L-function and $\epsilon _{\phi \times f}$ is the root number, which has absolute value one. Here, $\alpha _j$ are the Langlands parameters of $\phi $ , and $\tilde \phi $ is the dual form of $\phi $ . We have the following approximate functional equation.

Lemma 2.1. Assume $t\geq 0$ . Let $T=t+t_f$ and $T'=t-t_f$ . Then we have

$$\begin{align*}L(1/2+it,\phi\times f) \ll_{\phi,\varepsilon} T^{\varepsilon} \sup_{ 1 \leq N \leq T^{3/2+\varepsilon}(|T'|+1)^{3/2}} \frac{|S(N)|}{N^{1/2}} + T^{-2021}, \end{align*}$$

where $S(N)$ is a sum of the form

$$\begin{align*}S(N) := \sum_{r\geq1}\sum_{n\geq1} A(r,n) \lambda_f(n) (r^2n)^{-it} V\left(\frac{r^2n}{N}\right) \end{align*}$$

for some smooth function V, such that $\int _{\mathbb {R}}V(x)\mathrm {d} x=1$ , $\operatorname {supp} V \subset [1,2]$ , and $V^{(j)}(x) \ll _j 1$ for any integer $j\geq 0$ .

If $|T'| \leq T^{3/5}$ , then Lemma 2.1 gives

$$\begin{align*}L(1/2+it,\phi \times f) \ll_{\phi,\varepsilon} T^{3/4+\varepsilon} (|T'|+1)^{3/4} \ll T^{6/5+\varepsilon}, \end{align*}$$

which is better than (1.2). Hence, to prove Theorem 1.1, we only need to consider the case $|T'| \geq T^{3/5}$ , which we assume from now on. We will always write

$$\begin{align*}T=t+t_f \quad \textrm{and} \quad T'=t-t_f. \end{align*}$$

We first estimate the contribution from large values of r. By (2.1) and (2.4), we have

$$ \begin{align*} \sum_{r\geq R} & \left|\sum_{n\geq1} A(r,n) \lambda_f(n) (r^2n)^{-it} V\left(\frac{r^2n}{N}\right)\right| \\ & \ll \sum_{ R \leq r \ll \sqrt{N}} \left(\sum_{n\asymp N/r^2} |A(r,n)|^2 \right)^{1/2} \left(\sum_{n\asymp N/r^2} |\lambda_f(n)|^2\right)^{1/2} \\ & \ll \sum_{ R \leq r \ll \sqrt{N}} r^{\theta_3+\varepsilon} \frac{N}{r^2} \ll N \sum_{ R \leq r \ll \sqrt{N}} r^{-23/14+\varepsilon} \ll N^{1/2} T^{3/4+\varepsilon} |T'|^{3/4} R^{-9/14}, \end{align*} $$

for $N \ll T^{3/2+\varepsilon }|T'|^{3/2}$ . Take

(2.5) $$ \begin{align} R = \left\{ \begin{array}{ll} |T'|^{77/180}T^{-7/36}, & \textrm{ if } T^{5/6} \leq |T'|\leq T, \\ |T'|^{25/36}T^{-15/36}, & \textrm{ if } T^{3/5}\leq |T'|\leq T^{5/6}. \end{array} \right. \end{align} $$

The contribution from those terms to $L(1/2+it,\pi \times f )$ is bounded by $T^{3/4+\varepsilon } |T'|^{3/4} R^{-9/14}$ , which is good enough for Theorem 1.1. Hence, we get

(2.6) $$ \begin{align} L(1/2+it,\phi\times f) \ll T^{\varepsilon} \sum_{r\leq R} \frac{1}{r} \sup_{ N \leq \frac{T^{3/2+\varepsilon}|T'|^{3/2}}{r^2}} \frac{|S_r(N)|}{N^{1/2}} + T^{7/8+\varepsilon} |T'|^{19/40} + T^{57/56+\varepsilon} |T'|^{17/56} , \end{align} $$

where

$$\begin{align*}S_r(N) := \sum_{n\geq1} A(r,n) \lambda_f(n) n^{-it} V\left(\frac{n}{N}\right). \end{align*}$$

Thus, to prove Theorem 1.1, we only need to prove the following proposition.

Proposition 2.2. Assume $|T'| \geq T^{3/5}$ . For $r\leq R$ and $ N \leq \frac {T^{3/2+\varepsilon }|T'|^{3/2}}{r^2}$ , we have

$$\begin{align*}S_r(N) \ll N^{1/2+\varepsilon} \left(T^{7/8} |T'|^{19/40} + T^{57/56} |T'|^{17/56} \right). \end{align*}$$

2.3 Summation formulas

We first recall the Poisson summation formula over an arithmetic progression.

Lemma 2.3. Let $\beta \in \mathbb {Z}$ and $c\in \mathbb {Z}_{\geq 1}$ . For a Schwartz function $f:\mathbb {R}\rightarrow \mathbb {C}$ , we have

$$\begin{align*}\sum_{\substack{n\in\mathbb{Z}\\ n\equiv \beta \bmod{c}}} f(n) = \frac{1}{c} \sum_{n\in\mathbb{Z}} \hat{f}\left(\frac{n}{c}\right) e\left(\frac{n\beta}{c}\right), \end{align*}$$

where $\hat {f}(y)=\int _{\mathbb {R}} f(x) e(-xy)\mathrm {d} x$ is the Fourier transform of f.

Now, we turn to the Voronoi summation formula for $\operatorname {SL}(2,\mathbb {Z})$ . Let f be a weight zero Hecke–Maass cusp form for $\operatorname {SL}(2,\mathbb {Z})$ with spectral parameter $t_f$ . Let $\epsilon _f=\pm 1$ depending on if f is even or odd. Let $g(x),\ \psi (x)$ be smooth functions with compact support on the positive reals. Let $q\in \mathbb {Z}_{\geq 1}$ and $a\in \mathbb {Z}$ with $(q,a)=1$ . Define $\bar {a}$ as the inverse of a modulo q, that is, $a\bar {a}\equiv 1 \pmod {q}$ .

Lemma 2.4. With the notation as above. Then we have

(2.7) $$ \begin{align} \sum_{n\geq1} \lambda_f(n) e\left(\frac{an}{q}\right) g(n) = q \sum_{\pm} \sum_{n\geq1} \frac{ \lambda_{f}(n) }{n} e\left(\mp\frac{\bar{a} n}{q}\right) G^{\pm} \left(\frac{n}{q^2} \right), \end{align} $$

where

(2.8) $$ \begin{align} G^{\pm}(y) & = \frac{\epsilon_f^{(1\mp 1)/2}}{4\pi^2 i} \int_{(\sigma)} (\pi^2 y)^{-s} \left( \frac{\Gamma(\frac{1+s+it_f}{2})\Gamma(\frac{1+s-it_f}{2})} {\Gamma(\frac{-s+it_f}{2})\Gamma(\frac{-s-it_f}{2})} \mp \frac{\Gamma(\frac{2+s+it_f}{2})\Gamma(\frac{2+s-it_f}{2})} {\Gamma(\frac{1-s+it_f}{2})\Gamma(\frac{1-s-it_f}{2})} \right) \tilde{g}(-s) \mathrm{d} s \nonumber\\ & = \epsilon_f^{(1\mp 1)/2} y \int_{0}^{\infty} g(x) J^{\pm}_{f}\left( 4\pi\sqrt{yx} \right) \mathrm{d} x, \end{align} $$

with $\sigma>\theta _2-1$ and $\tilde {g}(s) = \int _{0}^{\infty } g(x) x^{s-1} \mathrm {d} x$ the Mellin transform of g, and

$$\begin{align*}J^+_{f}\left( x \right) = \frac{-\pi}{\sin(\pi i t_f)} \left( J_{2it_f}(x) - J_{-2it_f}(x) \right), \qquad J^-_f(x) = 4\cosh(\pi t_f) K_{2it_f}(x). \end{align*}$$

We also recall the Voronoi summation formula for $\operatorname {SL}(3,\mathbb {Z})$ . Let $\psi $ be a smooth compactly supported function on $(0,\infty )$ , and let $\tilde {\psi }$ be the Mellin transform of $\psi $ . For $\sigma>5/14$ , we define

(2.9) $$ \begin{align} \Psi^{\pm}(z) := z \frac{1}{2\pi i} \int_{(\sigma)} (\pi^3z)^{-s} \gamma^{\pm}(s) \tilde{\psi}(1-s)\mathrm{d} s , \end{align} $$

with

(2.10) $$ \begin{align} \gamma^{\pm}(s) := \prod_{j=1}^{3} \frac{\Gamma\left(\frac{s+\alpha_j}{2}\right)} {\Gamma\left(\frac{1-s-\alpha_j}{2}\right)} \pm \frac{1}{i} \prod_{j=1}^{3} \frac{\Gamma\left(\frac{1+s+\alpha_j}{2}\right)} {\Gamma\left(\frac{2-s-\alpha_j}{2}\right)}, \end{align} $$

where $\alpha _j$ are the Langlands parameters of $\phi $ as above. Note that changing $\psi (y)$ to $\psi (y/N)$ for a positive real number N has the effect of changing $\Psi ^{\pm }(z)$ to $\Psi ^{\pm }(zN)$ . The Voronoi formula on $\operatorname {GL}(3)$ was first proved by Miller–Schmid [Reference Miller and Schmid37]. The present version is due to Goldfeld–Li [Reference Goldfeld and Li13] with slightly renormalized variables (see Blomer [Reference Blomer4, Lemma 3]).

Lemma 2.5. Let $c,d,\bar {d}\in \mathbb Z$ with $c\neq 0$ , $(c,d)=1$ , and $d\bar {d}\equiv 1\pmod {c}$ . Then we have

$$ \begin{align*} \begin{aligned} \sum_{n=1}^{\infty} A(r,n)e\left(\frac{n\bar{d}}{c}\right)\psi(n) = \frac{c\pi^{3/2}}{2} \sum_{\pm} \sum_{n_1|cr} \sum_{n_2=1}^{\infty} \frac{A(n_2,n_1)}{n_1n_2} S\left(rd,\pm n_2;\frac{rc}{n_1}\right) \Psi^{\pm}\left(\frac{n_1^2n_2}{c^3r}\right), \end{aligned} \end{align*} $$

where $S(a,b;c) := \mathop {{\sum }^*}_{d(c)} e\left (\frac {ad+b\bar {d}}{c}\right )$ is the classical Kloosterman sum.

2.4 The delta method

There are several oscillatory factors contributing to the convolution sums. Our method is based on separating these oscillations using the delta/circle method. In the present situation, we will use a version of the delta method of Duke, Friedlander, and Iwaniec. More specifically, we will use the expansion (20.157) given in [Reference Iwaniec and Kowalski22, Section 20.5]. Let $\delta :\mathbb {Z}\rightarrow \{0,1\}$ be defined by

$$ \begin{align*}\delta(n)=\begin{cases} 1&\text{if}\;\;n=0;\\ 0&\text{otherwise}.\end{cases} \end{align*} $$

We seek a Fourier expansion which matches with $\delta (n)$ .

Lemma 2.6. Let Q be a large positive number. Then we have

(2.11) $$ \begin{align} \delta(n)=\frac{1}{Q}\sum_{1\leq q\leq Q} \;\frac{1}{q}\; \sideset{}{^{\star}}\sum_{a\bmod{q}}e\left(\frac{na}{q}\right) \int_{\mathbb{R}} g(q,x) e\left(\frac{nx}{qQ}\right)\mathrm{d}x, \end{align} $$

where $g(q,x)$ is a weight function that satisfies that

(2.12) $$ \begin{align} g(q,x)=1+O\left(\frac{Q}{q}\left(\frac{q}{Q}+|x|\right)^A\right), \quad g(q,x)\ll |x|^{-A}, \quad \textrm{for any } A>1, \end{align} $$

and

(2.13) $$ \begin{align} \frac{\partial^j}{\partial x^j} g(q,x) \ll |x|^{-j} \min(|x|^{-1},Q/q) \log Q, \quad j\geq1. \end{align} $$

Here, the $\star $ on the sum indicates that the sum over a is restricted by the condition $(a,q)=1$ .

2.5 Weight functions

Let $\mathcal {F}$ be an index set and $X=X_T:\mathcal {F}\rightarrow \mathbb {R}_{\geq 1}$ be a function of $T\in \mathcal {F}$ . A family of $\{w_T\}_{T\in \mathcal {F}}$ of smooth functions supported on a product of dyadic intervals in $\mathbb {R}_{>0}^d$ is called X-inert if for each $j=(j_1,\ldots ,j_d) \in \mathbb {Z}_{\geq 0}^d$ we have

$$\begin{align*}\sup_{T\in\mathcal{F}} \sup_{(x_1,\ldots,x_d) \in \mathbb{R}_{>0}^d} X_T^{-j_1-\cdots -j_d} \left| x_1^{j_1} \cdots x_d^{j_d} w_T^{(j_1,\ldots,j_d)} (x_1,\ldots,x_d) \right| \ll_{j_1,\ldots,j_d} 1. \end{align*}$$

For a $T^{\varepsilon }$ -inert function V, we may separate variables in $V(x_1, \ldots , x_d)$ by first inserting a redundant function $V (x_1) \cdots V (x_d)$ that is 1 on the support of V and then applying the Mellin inversion

$$ \begin{align*} V (x_1, \ldots , x_d) &= V (x_1, \ldots , x_d)V (x_1) \cdots V (x_d) \\& = \frac{1}{(2\pi i)^d} \int_{(0)}\cdots \int_{(0)} \tilde{V}(s_1,\ldots,s_d) (V (x_1) \cdots V (x_d) x_1^{-s_1} \cdots x_n^{-s_d} ) \mathrm{d} s_1 \cdots \mathrm{d} s_d, \end{align*} $$

where $\tilde {V}(s_1,\ldots ,s_d)=\int _{0}^{\infty }\cdots \int _{0}^{\infty } V(x_1, \ldots , x_d) x_1^{s_1-1} \cdots x_d^{s_d-1} \mathrm {d} x_1 \cdots \mathrm {d} x_d$ is the Mellin transform of V. Here, we can truncate the vertical integrals at height $|\operatorname {Im} s_j| \ll T^{2\varepsilon }$ at the cost of a negligible error $O_A(T^{-A})$ . We will often separate variables in this way without explicit mention.

2.6 Oscillatory integrals

We will use the following integration by parts and stationary phase lemmas several times.

Lemma 2.7. Let $Y\geq 1$ . Let $X,\; V,\; R,\; Q>0$ , and suppose that $w=w_T$ is a smooth function with $\operatorname {supp} w \subseteq [\alpha ,\beta ]$ satisfying $w^{(j)}(\xi ) \ll _j X V^{-j}$ for all $j\geq 0$ . Suppose that on the support of w, $h=h_T$ is smooth and satisfies that $h'(\xi )\gg R$ and $ h^{(j)}(\xi ) \ll Y Q^{-j}$ , for all $j\geq 2.$ Then, for arbitrarily large A, we have

$$\begin{align*}I = \int_{\mathbb{R}} w(\xi) e(h(\xi)) \mathrm{d} \xi \ll_A (\beta-\alpha) X \left[ \left(\frac{QR}{\sqrt{Y}}\right)^{-A} + (RV)^{-A} \right]. \end{align*}$$

Lemma 2.8. Suppose $w_T$ is X-inert in $t_1,\ldots ,t_d$ , supported on $t_i\asymp X_i$ for $i=1,2,\ldots ,d$ . Suppose that on the support of $w_T$ , $h=h_T$ satisfies that

$$\begin{align*}\frac{\partial^{a_1+a_2+\cdots +a_d}}{\partial t_1^{a_1}\cdots \partial t_d^{a_d}} h(t_1,t_2,\ldots,t_d) \ll_{a_1,\ldots,a_d} \frac{Y}{X_1^{a_1} X_2^{a_2}\cdots X_d^{a_d}}, \end{align*}$$

for all $a_1,\ldots ,a_d\in \mathbb {Z}_{\geq 0}$ . Let

$$\begin{align*}I = \int_{\mathbb{R}} w_T(t_1,t_2,\ldots,t_d) e^{i h(t_1,t_2,\ldots,t_d)} \mathrm{d} t_1. \end{align*}$$

Suppose $\frac {\partial ^{2}}{\partial t_1^{2}} h(t_1,t_2,\ldots ,t_d) \gg \frac {Y}{X_1^2}$ for all $(t_1,t_2,\ldots ,t_d)\in \operatorname {supp} w_T$ , and there exists $t_0 \in \mathbb {R}$ , such that $ \frac {\partial }{\partial t_1} h(t_0,t_2,\ldots ,t_d)=0$ . Suppose that $Y/X^2 \geq R \geq 1$ . Then

$$\begin{align*}I = \frac{X_1}{\sqrt{Y}} e^{i h(t_0,t_2,\ldots,t_d)} W_T(t_2,\ldots,t_d) + O_A(X_1 R^{-A}), \end{align*}$$

for some X-inert family of functions $W_T$ and any $A>0$ .

In the applications of Lemma 2.8, we will explicitly give estimates of the derivatives for the first variable. For other derivatives, we will also check all those conditions but may not write them down explicitly.

2.7 Stirling’s formula

For fixed $\sigma \in \mathbb {R}$ , real $|t|\geq 10$ , and any $J>0$ , we have Stirling’s formula

$$ \begin{align*} \Gamma(\sigma+it) = e^{-\frac{\pi}{2}|t|} |t|^{\sigma-\frac{1}{2}} \exp\left( it\log\frac{|t|}{e} \right) \left( g_{\sigma,J}(t) + O_{\sigma,J}(|t|^{-J}) \right), \end{align*} $$

where

$$\begin{align*}t^j \frac{\partial^j}{\partial t^j} g_{\sigma,J}(t) \ll_{j,\sigma,J} 1 \end{align*}$$

for all fixed $j\in \mathbb {N}_0$ . Similarly, we have

$$ \begin{align*} \frac{1}{\Gamma(\sigma+it)} = e^{\frac{\pi}{2}|t|} |t|^{-\sigma+\frac{1}{2}} \exp\left( -it\log\frac{|t|}{e} \right) \left( h_{\sigma,J}(t) + O_{\sigma,J}(|t|^{-J}) \right), \end{align*} $$

where

$$\begin{align*}t^j \frac{\partial^j}{\partial t^j} h_{\sigma,J}(t) \ll_{j,\sigma,J} 1 \end{align*}$$

for all fixed $j\in \mathbb {N}_0$ . Hence

(2.14) $$ \begin{align} \frac{\Gamma(\sigma+it)}{\Gamma(\sigma-it)} = \exp\left( 2it\log\frac{|t|}{e} \right) \left( w_{\sigma,J}(t) + O_{\sigma,J}(|t|^{-J}) \right), \end{align} $$

where

$$\begin{align*}t^j \frac{\partial^j}{\partial t^j} w_{\sigma,J}(t) \ll_{j,\sigma,J} 1 \end{align*}$$

for all fixed $j\in \mathbb {N}_0$ .

2.8 Bessel functions

We need the following asymptotic formula for Bessel functions when $x\gg T^{\varepsilon } |\tau |$ . For $\tau \in \mathbb {R}$ , $|\tau |>1$ and $x>0$ , we have [Reference Erdélyi, Magnus, Oberhettinger and Tricomi11, Equation 7.13.2 (17)]

(2.15) $$ \begin{align} \frac{J_{2i\tau}(2x)}{\cosh(\pi \tau)} = \sum_{\pm} e^{\pm 2i \omega(x,\tau)} \frac{g_A^{\pm}(x,\tau)}{x^{1/2}+|\tau|^{1/2}} + O(x^{-A}), \end{align} $$

where $g_A^{\pm }(x,\tau )$ is an $1$ -inert function and

(2.16) $$ \begin{align} \omega(x,\tau) = |\tau| \cdot \operatorname{arcsinh} \frac{|\tau|}{x} - \sqrt{x^2+\tau^2}. \end{align} $$

For $x\geq T^{\varepsilon } |\tau |$ , we have [Reference Erdélyi, Magnus, Oberhettinger and Tricomi11, Equation 7.13.2 (18)]

(2.17) $$ \begin{align} K_{2i\tau}(2x) \cosh(\pi \tau) \ll x^{-1/2} \exp(-2x+ \pi|\tau|) \ll x^{-6} \exp(-x), \end{align} $$

for T large enough.

4.1 The oscillating cases

If $\frac {NX}{PQ}\gg T^{\varepsilon }$ , then we have

$$ \begin{align*} S_{r}^{\pm}(N,X,P) = \frac{1}{Q}&\sum_{\substack{q\sim P}} \int_{\mathbb{R}} V\left(\frac{\pm x}{X}\right) \;\frac{1}{q}\; \sideset{}{^{\star}}\sum_{\substack{a\bmod{q} }} \frac{q\pi^{3/2}}{2} \sum_{n_1|qr} \sum_{n_2=1}^{\infty} \frac{A(n_2,n_1)}{n_1n_2} S\left(-r\bar{a},\pm n_2;\frac{rq}{n_1}\right) \\ & \cdot \left(\pm \frac{Nx}{qQ}\right)^{3/2} e\left(\pm 2 \frac{( n_1^2n_2 Q)^{1/2}}{r^{1/2} q(\pm x)^{1/2}}\right) w\left(\frac{n_1^2n_2 Q^3}{r N^2 (\pm x )^{3}}\right) \\ & \cdot \sum_{\substack{m\geq1}} \lambda_f(m) e\left(\frac{m a}{q}\right) e\left(\frac{m x}{qQ}\right) m^{-it} W\left(\frac{m}{N}\right) \mathrm{d}x + O(T^{-A}). \end{align*} $$

We first deal with the x-integral. Making a change of variable $x = \pm X \xi $ , we get

$$ \begin{align*} &S_{r}^{\pm}(N,X,P) = X \frac{1}{Q}\sum_{\substack{q\sim P}} \;\frac{1}{q}\; \sideset{}{^{\star}}\sum_{\substack{a\bmod{q} }} \frac{q\pi^{3/2}}{2} \sum_{n_1|qr} \sum_{n_2=1}^{\infty} \frac{A(n_2,n_1)}{n_1n_2} S\left(-r\bar{a},\pm n_2;\frac{rq}{n_1}\right) \\&\quad \cdot \left(\frac{NX}{qQ}\right)^{3/2} \sum_{\substack{m\geq1}} \lambda_f(m) e\left(\frac{ma}{q}\right) m^{-it} W\left(\frac{m}{N}\right) \\&\quad \cdot \int_{\mathbb{R}} w\left(\frac{n_1^2n_2 Q^3}{r N^2X^{3}\xi^3}\right) V\left(\xi\right) \xi^{3/2} e\left(\frac{\pm m X\xi}{qQ}\right) e\left(\pm 2 \frac{( n_1^2n_2 Q)^{1/2}}{r^{1/2} qX^{1/2} \xi^{1/2}}\right) \mathrm{d}\xi + O(T^{-A}). \end{align*} $$

We can remove the weight function $w\left (\frac {n_1^2n_2 Q^3}{r N^2X^{3}\xi ^3}\right )$ by the Mellin technique as in Section 2.5. Then we have

$$ \begin{align*} S_{r}^{\pm}(N,X,P) \ll T^{\varepsilon} \bigg| &\frac{X}{Q}\sum_{\substack{q\sim P}} \; \sideset{}{^{\star}}\sum_{\substack{a\bmod{q} }} \sum_{n_1|qr} \sum_{n_2=1}^{\infty} \frac{A(n_2,n_1)}{n_1n_2} S\left(-r\bar{a},\pm n_2;\frac{rq}{n_1}\right) V_1\left(\frac{n_1^2n_2 Q^3}{r N^2X^{3}}\right) \\ & \cdot \left(\frac{NX}{qQ}\right)^{3/2} \sum_{\substack{m\geq1}} \lambda_f(m) e\left(\frac{ma}{q}\right) m^{-it} W\left(\frac{m}{N}\right) \\ & \cdot \int_{\mathbb{R}} V_2\left(\xi\right) e\left(\frac{\pm m X\xi}{qQ} \pm 2 \frac{( n_1^2n_2 Q)^{1/2}}{r^{1/2} qX^{1/2} \xi^{1/2}}\right) \mathrm{d}\xi \bigg| + O(T^{-A}), \end{align*} $$

for some $T^{\varepsilon }$ -inert functions $V_1$ and $V_2$ with support in $[1,2]$ . We now consider the $\xi $ -integral above. Let (temporarily)

$$\begin{align*}h(\xi) = \frac{\pm m X\xi}{qQ} \pm 2 \frac{( n_1^2n_2 Q)^{1/2}}{r^{1/2} qX^{1/2} \xi^{1/2}}. \end{align*}$$

Then

$$\begin{align*}h'(\xi) = \frac{\pm m X }{qQ} \mp \frac{( n_1^2n_2 Q)^{1/2}}{r^{1/2} qX^{1/2} \xi^{3/2}}, \end{align*}$$

and

$$\begin{align*}h"(\xi) = \pm \frac{3}{2} \frac{( n_1^2n_2 Q)^{1/2}}{r^{1/2} qX^{1/2} \xi^{5/2}} , \quad h^{(j)}(\xi) \asymp_j \frac{NX}{PQ}, \quad j\geq2. \end{align*}$$

The solution of $h'(\xi )=0$ is $\xi _0 = \frac {( n_1^2n_2)^{1/3} Q}{r^{1/3} m^{2/3}X}$ . Note that

$$\begin{align*}h(\xi_0) = \pm 3\frac{m^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \quad \textrm{and} \quad h"(\xi_0) = \pm \frac{3}{2} \frac{mX}{q Q \xi_0 }. \end{align*}$$

Now, by Lemma 2.8 with

$$\begin{align*}X= T^{\varepsilon}, t_1=\xi, t_2=n_1^2n_2, t_3=m, t_4=q, X_1=1, X_2=\frac{rN^2X^3}{Q^3}, X_3=N, X_4=P, \textrm{ and } Y=\frac{NX}{PQ}, \end{align*}$$

we get

$$ \begin{align*} &\int_{\mathbb{R}} V_2\left(\xi\right) e \left(\frac{\pm m X\xi}{qQ} \pm 2 \frac{( n_1^2n_2 Q)^{1/2}}{r^{1/2} qX^{1/2} \xi^{1/2}}\right) \mathrm{d}\xi \\&\quad = \left(\frac{NX}{qQ}\right)^{-1/2} V_3\left(\frac{ n_1^2n_2 Q^3}{r N^{2}X^3},\frac{m}{N},\frac{q}{P}\right) e\left(\pm 3\frac{m^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \right) + O_A(T^{-A}), \end{align*} $$

where $V_3$ is a $T^{\varepsilon }$ -inert function with compact support in $\mathbb {R}_{>0}^3$ . Hence, we obtain

(4.2) $$ \begin{align} S_{r}^{\pm}(N,X,P) & \ll T^{\varepsilon} \bigg| \frac{X}{Q}\sum_{\substack{q\sim P}} \;\frac{1}{q}\; \sideset{}{^{\star}}\sum_{\substack{a\bmod{q} }} \!\frac{NX}{Q} \sum_{n_1|qr} \sum_{\substack{n_2=1 \\ n_1^2n_2 \asymp \frac{r N^{2}X^3} {Q^3} }}^{\infty} \!\!\frac{A(n_2,n_1)}{n_1n_2} S\left(-r\bar{a},\pm n_2;\frac{rq}{n_1}\right) V\left(\frac{ n_1^2n_2 Q^3}{r N^{2}X^3}\right) \notag\\&\quad \cdot \sum_{\substack{m\geq1}} \lambda_f(m) e\left(\frac{ma}{q}\right) e\left(\pm 3\frac{m^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \right) m^{-it} W\left(\frac{m}{N}\right) \bigg| + O(T^{-A}). \end{align} $$

Here, we have removed the weight function $V_3$ by the Mellin technique again to separate the variables $n_2$ and m, and modified the weight functions W and V accordingly. Note that W and V are $T^{\varepsilon }$ -inert functions with compact support in $\mathbb {R}_{>0}$ .

We now apply the Voronoi summation formula (see Lemma 2.4) to the sum over m getting

(4.3) $$ \begin{align} \sum_{\substack{m\geq1}} \lambda_f(m) e\left(\frac{ma}{q}\right) e\left(\pm 3\frac{m^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \right) & m^{-it} W\left(\frac{m}{N}\right) \nonumber \\ & = q \sum_{\pm_1} \sum_{m\geq1} \frac{ \lambda_{f}(m) }{m} e\left(\frac{\pm_1\bar{a} m}{q}\right) G^{\pm_1} \left(\frac{m}{q^2} \right), \end{align} $$

where $g(m)= e\left (\pm 3\frac {m^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \right ) m^{-it} W\left (\frac {m}{N}\right )$ and $G^{\pm _1}$ is defined as in (2.8).

Proof. (i) First, we use the second expression in (2.8), getting

$$ \begin{align*} G^{\pm_1}(y) = \epsilon_f^{(1\mp_1 1)/2} y \int_{0}^{\infty} e\left(\pm 3\frac{u^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \right) u^{-it} W\left(\frac{u}{N}\right) V\left(\frac{ n_1^2n_2 Q^3}{r u^{2}X^3}\right) J^{\pm_1}_{f}\left( 4\pi\sqrt{yu} \right) \mathrm{d} u. \end{align*} $$

Making a change of variable $u=N\xi $ , we have $G^{\pm _1}(y)$ is equal to

$$ \begin{align*} \epsilon_f^{(1\mp_1 1)/2} y N^{1-it} \int_{0}^{\infty} e\left(\pm 3\frac{N^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \xi^{1/3} \right) \xi^{-it} W\left(\xi\right) V\left(\frac{ n_1^2n_2 Q^3}{r N^2 \xi^{2}X^3}\right) J^{\pm_1}_{f}\left( 4\pi\sqrt{yN\xi} \right) \mathrm{d} \xi. \end{align*} $$

If $yN\gg t_f^{2}T^{\varepsilon }$ , then by (2.17), we have $G^{-}(y) \ll _A y^{-6} T^{-A}$ for any $A>0$ . If $yN\gg T^{2+\varepsilon } + (\frac {NX}{PQ})^{2+\varepsilon }$ , then by (2.15), we have

$$ \begin{align*} G^+(y) &= y N^{1-it} \int_{0}^{\infty} e\left(\pm 3\frac{N^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \xi^{1/3} \right) \xi^{-it} W\left(\xi\right) V\left(\frac{ n_1^2n_2 Q^3}{r N^2 \xi^{2}X^3}\right) \\&\quad\cdot \sum_{\pm} e^{\pm 2i \omega(2\pi\sqrt{yN\xi},t_f)} \frac{g_A^{\pm}(2\pi\sqrt{yN\xi},t_f)}{(2\pi\sqrt{yN\xi})^{1/2}+t_f^{1/2}} \mathrm{d} \xi + O_A(y^{-6} T^{-A}). \end{align*} $$

Let (temporarily)

$$\begin{align*}h(\xi) = \pm 6\pi\frac{N^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \xi^{1/3} -t\log \xi \pm 2 \omega(2\pi\sqrt{yN\xi},t_f). \end{align*}$$

Then we have

$$\begin{align*}h'(\xi) = \pm 2\pi\frac{N^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \xi^{-2/3} -\frac{t}{\xi} \mp \frac{\sqrt{(2\pi\sqrt{yN\xi})^2+t_f^2}}{\xi} \gg (yN)^{1/2}, \end{align*}$$

$$\begin{align*}h^{(j)}(\xi) \ll (yN)^{1/2}, \quad j\geq2. \end{align*}$$

By Lemma 2.7 with

$$\begin{align*}X=V=1, Y= (yN)^{1/2}, Q=1, \textrm{ and } R= (yN)^{1/2}, \end{align*}$$

we get $G^+(y) \ll _A y^{-6} T^{-A}$ . Hence, we have $G^{\pm _1}(y) \ll _A y^{-6} T^{-A}$ if $yN\gg T^{2+\varepsilon } + (\frac {NX}{PQ})^{2+\varepsilon }$ .

(ii) For $yN \ll T^{2+\varepsilon } + (\frac {NX}{PQ})^{2+\varepsilon }$ , we use the first expression in (2.8). Writing $s=\sigma +i\tau $ with $\sigma =-1/2$ and making a change of variable $\tau \rightsquigarrow \tau -t$ , we get

(4.4) $$ \begin{align} G^{\pm_1}(y) = \frac{\epsilon_f^{(1\mp_1 1)/2}}{4\pi^2} \int_{\mathbb{R}} (\pi^2 y)^{1/2-i\tau+it} \gamma_2^{\pm_1}(-1/2+i\tau-it) \tilde{g}(1/2-i\tau+it) \mathrm{d} \tau, \end{align} $$

where

$$\begin{align*}\gamma_2^{\pm_1}(-1/2+i\tau-it) = \frac{\Gamma(\frac{1/2+i\tau-iT}{2})\Gamma(\frac{1/2+i\tau-iT'}{2})} {\Gamma(\frac{1/2-i\tau+iT}{2})\Gamma(\frac{1/2-i\tau+iT'}{2})} \pm_1 \frac{\Gamma(\frac{3/2+i\tau-iT}{2})\Gamma(\frac{3/2+i\tau-iT'}{2})} {\Gamma(\frac{3/2-i\tau+iT}{2})\Gamma(\frac{3/2-i\tau+iT'}{2})}. \end{align*}$$

If $\frac {NX}{PQ}\gg T^{\varepsilon }$ , then

$$ \begin{align*} \tilde{g}(1/2-i\tau+it) = \int_{\mathbb{R}} e\left(\pm 3\frac{u^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \right) W\left(\frac{u}{N}\right) u^{-1/2-i\tau} \mathrm{d} u. \end{align*} $$

Making a change of variable $u=N\xi ^3$ , we have

$$ \begin{align*} \tilde{g}(1/2-i\tau+it) = N^{1/2-i\tau} \int_{\mathbb{R}} e\left(\pm 3\frac{N^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \xi - \frac{3 \tau}{2\pi} \log \xi \right) W\left(\xi^3\right) 3 \xi^{1/2} \mathrm{d} \xi. \end{align*} $$

Let (temporarily)

$$\begin{align*}h(\xi) = \pm 3\frac{N^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \xi - \frac{3\tau}{2\pi} \log \xi. \end{align*}$$

Then

$$\begin{align*}h'(\xi) = \pm 3 \frac{N^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} - \frac{3 \tau}{2\pi \xi} , \end{align*}$$

and

$$\begin{align*}h"(\xi) = \frac{3\tau}{2\pi \xi^2} , \quad h^{(j)}(\xi) \asymp_j |\tau|, \quad j\geq2. \end{align*}$$

By Lemma 2.7 with

$$\begin{align*}X=1, V=T^{-\varepsilon}, Y=|\tau|, Q=1, \textrm{ and } R= |\tau|+\frac{NX}{PQ}, \end{align*}$$

we have $\tilde {g}(1/2-i\tau +it)$ is negligibly small unless $\operatorname {sgn}(\tau )=\pm $ and $\tau \asymp \frac {N^{1/3} ( n_1^2n_2)^{1/3} }{r^{1/3} q} \asymp \frac {NX}{PQ}$ , in which case, the solution of $h'(\xi )=0$ is

$$\begin{align*}\xi_0 = \frac{\pm \tau}{2\pi } \frac{r^{1/3} q} {N^{1/3} ( n_1^2n_2)^{1/3} }. \end{align*}$$

Note that

$$\begin{align*}h(\xi_0) = -\frac{3\tau}{2\pi} \log \frac{\pm \tau r^{1/3} q} {2\pi e N^{1/3} ( n_1^2n_2)^{1/3} }. \end{align*}$$

Now, by Lemma 2.8 with

$$\begin{align*}X= T^{\varepsilon}, t_1=\xi, t_2=n_1^2n_2, t_3=\tau, t_4=q, X_1=1, X_2=\frac{rN^2X^3}{Q^3}, X_3=\frac{NX}{PQ}, X_4=P, \text{ and } Y=\frac{NX}{PQ}, \end{align*}$$

we get

(4.5) $$ \begin{align} \tilde{g}(1/2-i\tau+it) = N^{1/2-i\tau} \left(\frac{NX}{PQ}\right)^{-1/2} e\left(-\frac{3\tau}{2\pi} \log \frac{\pm \tau } {2\pi e B }\right) W_1\left(\frac{ n_1^2n_2 Q^3}{r N^{2}X^3}, \frac{\pm \tau} { \frac{NX}{PQ} } ,\frac{q}{P} \right) + O_A(T^{-A}), \end{align} $$

where $B=\frac { N^{1/3} ( n_1^2n_2)^{1/3} }{ r^{1/3} q} $ and $W_1$ is a $T^{\varepsilon }$ -inert function with compact support in $\mathbb {R}_{>0}^3$ . Note that we have $B \asymp \frac {NX}{PQ}$ .

Now, we consider the case $ \frac {NX}{PQ}\gg |T'|^{1-\varepsilon }$ . By (4.4) and (4.5), we have

(4.6) $$ \begin{align} G^{\pm_1}(y) &= \frac{\epsilon_f^{(1\mp_1 1)/2}}{4\pi} (\pi^2 y)^{it} (yN)^{1/2} \left(\frac{NX}{PQ}\right)^{-1/2}\!\! \int_{\mathbb{R}} (\pi^2 yN )^{-i\tau} \left(\frac{ N n_1^2n_2}{ r q^3} \right)^{i\tau}\! W_1\!\left(\frac{ n_1^2n_2 Q^3}{r N^{2}X^3}, \frac{\pm \tau} { \frac{NX}{PQ} } ,\frac{q}{P} \right)\notag \\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \ \cdot \gamma_2^{\pm_1}(-1/2+i\tau-it) e\left(-\frac{3\tau}{2\pi} \log \frac{\pm \tau } {2\pi e}\right) \mathrm{d} \tau + O(T^{-A}). \end{align} $$

Taking $w^{\pm _1}(\tau ) = \frac {\epsilon _f^{(1\mp _1 1)/2}}{4\pi } \pi ^{2it} (\pi ^2 r)^{-i\tau } \gamma _2^{\pm _1}(-1/2+i\tau -it) e\left (-\frac {3\tau }{2\pi } \log \frac {\pm \tau } {2\pi e}\right )$ , and noting that $w^{\pm _1}(\tau )\ll 1$ , we complete the proof of Lemma 4.2 (ii).

(iii) We now consider the case $T^{\varepsilon } \ll \frac {NX}{PQ}\ll |T'|^{1-\varepsilon }$ . By Stirling’s formula, for $\tau \ll |T'|^{1-\varepsilon }$ , we have

$$ \begin{align*} \gamma_2^{\pm_1}(-1/2+i\tau-it)& = \exp\left( i(\tau-T)\log\frac{T-\tau}{2e} + i(\tau-T')\log\frac{|T'-\tau|}{2e} \right) \\ &\qquad\qquad\qquad\qquad\qquad\qquad\cdot \left( w_{\epsilon_f,J}^{\pm_1}(\tau-T) w_{\epsilon_f,J}^{\pm_1}(\tau-T') + O_{J}(T^{-J}) \right), \end{align*} $$

where

$$\begin{align*}t^j \frac{\partial^j}{\partial t^j} w_{\epsilon_f,J}^{\pm_1}(t) \ll_{j,J} 1 \end{align*}$$

for all fixed $j\in \mathbb {N}_0$ . Hence, together with (4.4) and (4.5), we have

$$ \begin{align*} &G^{\pm_1}(y) = \frac{\epsilon_f^{(1\mp_1 1)/2}}{4\pi } (\pi^2 y)^{it} (Ny)^{1/2} \left(\frac{NX}{PQ}\right)^{-1/2} \!\!\int_{\mathbb{R}} w_{\epsilon_f,J}^{\pm_1}\left(\tau - T\right) w_{\epsilon_f,J}^{\pm_1}\left(\tau-T'\right) W_1\!\left(\frac{B^3 q^3 Q^3}{N^{3}X^3}, \frac{\pm \tau} { \frac{NX}{PQ} } , \frac{q}{P}\right) \\&\quad\cdot e\left( - \frac{\tau}{2\pi} \log(\pi^2 yN) + \frac{(\tau - T)}{2\pi}\log\frac{T-\tau}{2e} + \frac{(\tau - T')}{2\pi}\log\frac{|T'-\tau|}{2e} -\frac{3\tau}{2\pi} \log \frac{\pm \tau } {2\pi e B } \right) \mathrm{d} \tau + O_A(T^{-A}). \end{align*} $$

Making a change of variable $\tau = \pm B \xi $ , we get

$$ \begin{align*} G^{\pm_1}(y) &= (\pi^2 y)^{it} (Ny)^{1/2} \left(\frac{NX}{PQ}\right)^{1/2} \int_{\mathbb{R}} W_2^{\pm_1}\left(\xi, \frac{B} { \frac{NX}{PQ} },\frac{q}{P} \right) e\bigg( \mp \frac{B\xi}{2\pi} \log (\pi^2 yN) \mp \frac{3B\xi}{2\pi} \log \frac{\xi}{2\pi e} \\&\qquad\qquad\qquad\quad \ + \frac{(\pm B \xi - T )}{2\pi}\log\frac{T\mp B \xi}{2e} + \frac{(\pm B \xi - T')}{2\pi}\log\frac{|T'\mp B \xi|}{2e} \bigg) \mathrm{d} \xi + O_A(T^{-A}), \end{align*} $$

where $W_2^{\pm _1}\left (\xi , \frac {B} { \frac {NX}{PQ} },\frac {q}{P} \right ) = \frac {\epsilon _f^{(1\mp _1 1)/2}}{4\pi } w_{\epsilon _f,J}^{\pm _1}\left (\pm B \xi - T\right ) w_{\epsilon _f,J}^{\pm _1}\left (\pm B \xi -T'\right ) W_1\left (\frac {B^3 q^3 Q^3}{N^{3}X^3}, \frac {B \xi } { \frac {NX}{PQ} } , \frac {q}{P}\right ) \frac {B}{\frac {NX}{PQ} } $ is a $T^{\varepsilon }$ -inert function with compact support in $\mathbb {R}_{>0}^3$ . Let (temporarily)

$$ \begin{align*} h(\xi) =\mp \frac{B\xi}{2\pi} \log (\pi^2 yN) \mp \frac{3B\xi}{2\pi} \log \frac{\xi}{2\pi e} + \frac{(\pm B \xi - T )}{2\pi}\log\frac{T\mp B \xi}{2e} + \frac{(\pm B \xi - T')}{2\pi}\log\frac{|T'\mp B \xi|}{2e}. \end{align*} $$

Then

$$ \begin{align*} h'(\xi) & =\mp \frac{B}{2\pi} \log (\pi^2 yN) \mp \frac{3B}{2\pi} \log \frac{\xi}{2\pi} + \frac{\pm B}{2\pi}\log\frac{T\mp B \xi}{2} + \frac{\pm B}{2\pi}\log\frac{|T'\mp B \xi|}{2} \\ & = \mp \frac{B}{2\pi} \log \frac{yN \xi^3}{2\pi (T\mp B \xi)|T'\mp B \xi|} , \end{align*} $$

and

$$\begin{align*}h"(\xi) = \mp \frac{3B}{2\pi \xi} \pm \frac{B}{2\pi} \frac{ \mp B }{(T\mp B\xi)} \pm \frac{B}{2\pi} \frac{ \mp B }{(T'\mp B\xi)} , \quad h^{(j)}(\xi) \asymp_j \frac{NX}{PQ}, \quad j\geq2. \end{align*}$$

By Lemma 2.7 with

$$\begin{align*}X=1, V=T^{-\varepsilon}, Y=\frac{NX}{PQ}, Q=1, \textrm{ and } R= \frac{NX}{PQ}, \end{align*}$$

we have that $G^{\pm _1}(y) \ll _A T^{-A}$ is negligibly small unless $yN \asymp T |T'|$ . Assume $yN \asymp T |T'|$ . Denote the solution of $h'(\xi )=0$ by $\xi _*$ with $\xi _*\asymp 1$ . Then by Lemma 2.8 with

$$\begin{align*}X= T^{\varepsilon}, t_1=\xi, t_2=B, t_3=q, t_4=yN, X_1=1, X_2=\frac{NX}{PQ}, X_3=P, X_4=T|T'|, \textrm{ and } Y=\frac{NX}{PQ}, \end{align*}$$

we get

(4.7) $$ \begin{align} G^{\pm_1}(y) & = (\pi^2 y)^{it} (Ny)^{1/2} e(h(\xi_*)) W_3^{\pm_1}\left( \frac{B} { \frac{NX}{PQ} },\frac{q}{P},\frac{yN}{T|T'|} \right) + O_A(T^{-A}), \end{align} $$

where $W_3^{\pm _1}$ is a $T^{\varepsilon }$ -inert function with compact support in $\mathbb {R}_{>0}^3$ . Note that the assumptions in Lemma 2.8 hold in this case.

We now simplify the expression of $G^{\pm _1}(y)$ . Note that the solution of $h'(\xi )=0$ , that is, $yN\xi ^3=2\pi T|T'|(1\mp \frac {B}{T}\xi )(1\mp \frac {B}{T'}\xi )$ , can be written as

(4.8) $$ \begin{align} \xi_* = \xi_0 + \xi_1 + \xi_2 + \xi_3+ \cdots + \xi_L + \xi_{L+1}, \end{align} $$

where $L \geq 3$ is a large integer and $\xi _{\ell +1}=o(\xi _{\ell })$ ( $0\leq \ell \leq L$ ) with

$$\begin{align*}\xi_0 = \left(\frac{2\pi T|T'|}{yN}\right)^{1/3}, \quad \xi_1 = \frac{1}{3} \left(\mp \frac{B}{T}\mp \frac{B}{T'}\right) \xi_0^2, \quad \xi_2 = \frac{B^2}{3TT'} \xi_0^3, \end{align*}$$

and $\xi _{\ell }$ ( $2<\ell \leq L$ ) is the solution of

$$ \begin{align*} \xi_0^{-3} \sum_{\substack{0\leq i,j,k\leq \ell \\ i+j+k\leq \ell}} \xi_i\xi_j\xi_k = 1\mp \left(\frac{B}{T}+\frac{B}{T'}\right)\sum_{0\leq i\leq \ell-1} \xi_i + \frac{B^2}{TT'} \sum_{\substack{0\leq i,j\leq \ell-2 \\ i+j\leq \ell-2}} \xi_i\xi_j. \end{align*} $$

Note that

$$\begin{align*}\xi_{\ell} = \frac{\xi_0}{3} \bigg( \mp \left(\frac{B}{T}+\frac{B}{T'}\right) \xi_{\ell-1} + \frac{B^2}{TT'} \sum_{\substack{0\leq i,j\leq \ell-2 \\ i+j= \ell-2}} \xi_i\xi_j - \xi_0^{-3} \sum_{\substack{0\leq i,j,k\leq \ell-1 \\ i+j+k=\ell}} \xi_i\xi_j\xi_k \bigg), \quad \ell \leq L. \end{align*}$$

By induction, we have

$$\begin{align*}\xi_{\ell} = P_{\ell}\left(\frac{B}{T},\frac{B}{T'} \right) \xi_0^{\ell+1} = O_L\left( \frac{B^{\ell}}{|T'|^{\ell}} \right), \quad 0\leq \ell\leq L, \quad \textrm{and} \quad \xi_{L+1} \ll_L \frac{B^{L+1}}{|T'|^{L+1}}, \end{align*}$$

where $P_{\ell }$ is a certain homogeneous polynomial of degree $\ell $ . Note that

$$ \begin{align*} h(\xi_*) & = \mp \frac{B\xi_*}{2\pi} \log \left( \frac{yN \xi_*^3}{2\pi e (T \mp B \xi_*)|T'\mp B\xi_*|} \right) - \frac{T}{2\pi} \log \frac{T\mp B \xi_*}{2e} - \frac{T'}{2\pi} \log \frac{|T'\mp B \xi_*|}{2e}. \end{align*} $$

Note that $\xi _*\asymp 1$ and $B/T=o(1)$ . By the Taylor expansion, we get

$$ \begin{align*} h(\xi_*) & = - \frac{T}{2\pi} \log \frac{T}{2e} - \frac{T'}{2\pi} \log \frac{|T'|}{2e} \pm \frac{B}{2\pi} \xi_* + \frac{1}{2\pi} \sum_{j\geq1} \frac{1}{j} \left(\frac{(\pm B)^{j}}{T^{j-1}}+\frac{(\pm B)^{j}}{T^{\prime j-1}}\right) \xi_*^{j} \\ & = - \frac{T}{2\pi} \log \frac{T}{2e} - \frac{T'}{2\pi} \log \frac{|T'|}{2e} \pm \frac{B}{2\pi} \sum_{0\leq \ell\leq L} Q_{\ell}^{\pm} \left(\frac{B}{T},\frac{B}{T'} \right) \xi_0^{\ell+1} + O_L\left( \frac{B^{L+2}}{|T'|^{L+1}} \right), \end{align*} $$

where $Q_{\ell }^{\pm }$ is a certain homogeneous polynomial of degree $\ell $ . Note that we have $Q_0^{\pm }\left (\frac {B}{T},\frac {B}{T'} \right ) =3$ and $Q_1^{\pm }\left (\frac {B}{T},\frac {B}{T'} \right ) = \mp \frac {1}{2} \left (\frac {B}{T}+\frac {B}{T'}\right ) $ . Hence, by (4.7), we get

$$ \begin{align*} G^{\pm_1}(y) &= (\pi^2 y)^{it} (Ny)^{1/2} e\bigg( - \frac{T}{2\pi} \log \frac{T}{2e} - \frac{T'}{2\pi} \log \frac{|T'|}{2e} \pm \frac{B}{2\pi} \sum_{0\leq \ell\leq L} Q_{\ell}^{\pm} \left(\frac{B}{T},\frac{B}{T'} \right) \xi_0^{\ell+1}\bigg) \\&\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \ \cdot W_3^{\pm_1}\left( \frac{B} { \frac{NX}{PQ} },\frac{q}{P},\frac{yN}{T|T'|} \right) + O_A(T^{-A}). \end{align*} $$

Here, we take $L=L(A)$ to be large enough. This completes the proof of Lemma 4.2 (iii).

By (4.2), (4.3), and Lemma 4.2, we obtain

(4.9) $$ \begin{align} S_{r}^{\pm}(N,X,P) \ll T^{\varepsilon} \sup_{M \asymp \frac{P^2 T|T'|}{N} } |S_{r}^{\pm}(N,X,P,M)| + O(T^{-A}) \end{align} $$

if $T^{\varepsilon } \ll \frac {NX}{PQ}\ll |T'|^{1-\varepsilon }$ , and

(4.10) $$ \begin{align} S_{r}^{\pm}(N,X,P) \ll T^{\varepsilon} \sup_{M \ll \frac{P^2 T^2}{N} + \frac{NX^2}{Q^2} } |S_{r}^{\pm}(N,X,P,M)| + O(T^{-A}) \end{align} $$

if $ \frac {NX}{PQ}\gg |T'|^{1-\varepsilon }$ , where

$$ \begin{align*} S_{r}^{\pm}(N,X,P,M) & : = X \frac{N^{1/2} }{Q}\sum_{\substack{q\sim P}} \;\frac{1}{q} U\left(\frac{q}{P}\right) \; \sideset{}{^{\star}}\sum_{\substack{a\bmod{q} }} \frac{NX}{Q} \sum_{n_1|qr} \sum_{\substack{n_2=1 \\n_2 \asymp \frac{r N^{2}X^3} {n_1^2 Q^3} }}^{\infty} \frac{A(n_2,n_1)}{n_1n_2} S\left(-r\bar{a},\pm n_2;\frac{rq}{n_1}\right) \\&\quad\cdot V\left(\frac{ n_1^2n_2 Q^3}{r N^{2}X^3}\right) \sum_{\pm_1} \sum_{\substack{m\geq1 \\ m\asymp M}} \frac{ \lambda_{f}(m) }{m^{1/2}} \left(\frac{m}{q^2}\right)^{it} e\left(\frac{\pm_1\bar{a} m}{q}\right) W\left( \frac{m}{M}\right) \mathcal{I}^{\pm_1}(n_2,n_1,r,m,q) , \end{align*} $$

where $U,\; V,\; W$ are certain $T^{\varepsilon }$ -inert functions with compact support in $\mathbb {R}_{>0}$ and

(4.11) $$ \begin{align} \mathcal{I}^{\pm_1}(n_2,n_1,r,m,q) = e\left( \pm \frac{B}{2\pi} \sum_{0\leq \ell \leq L} Q_{\ell}^{\pm} \left(\frac{B}{T},\frac{B}{T'} \right) \xi_0^{\ell+1} \right) \end{align} $$

with $B=\frac { N^{1/3} ( n_1^2n_2)^{1/3} }{ r^{1/3} q} $ and $\xi _0=\left (\frac {2\pi q^2 T|T'|}{mN}\right )^{1/3}$ if $T^{\varepsilon } \ll \frac {NX}{PQ}\ll |T'|^{1-\varepsilon }$ , and

(4.12) $$ \begin{align} \mathcal{I}^{\pm_1}(n_2,n_1,r,m,q) = \left( \frac{PQ}{NX}\right)^{1/2} \int_{\mathbb{R}} \left(\frac{ n_1^2n_2}{ mq} \right)^{i\tau} W_1\left(\frac{ n_1^2n_2 Q^3}{r N^{2}X^3}, \frac{\pm \tau} { \frac{NX}{PQ} } ,\frac{q}{P} \right) w^{\pm_1}(\tau) \mathrm{d} \tau \end{align} $$

if $ \frac {NX}{PQ}\gg |T'|^{1-\varepsilon }$ . Here, we have used the Mellin technique to remove the weight function $W^{\pm _1}_3$ to get (4.11) without writing explicitly the dependence on those new parameters.

Changing the order of summations, we get

(4.13) $$ \begin{align} &S_{r}^{\pm}(N,X,P,M) = X \frac{N^{1/2} }{Q} \frac{NX}{Q} \sum_{n_1\ll Pr} \frac{1}{n_1} \sum_{\pm_1} \sum_{\substack{n_2=1 }}^{\infty} \frac{A(n_2,n_1)}{n_2} V\left(\frac{ n_1^2n_2 Q^3}{r N^{2}X^3}\right)\notag\\& \qquad \cdot\sum_{\substack{q\sim P \\ n_1\mid qr}} \;\frac{1}{q}U\left(\frac{q}{P}\right) \sum_{\substack{m\geq1 \\ m\asymp M }} \frac{ \lambda_{f}(m) }{m^{1/2}} \left(\frac{m}{q^2}\right)^{it} W\left(\frac{m}{M}\right) \mathcal{C}^{\pm_1}(n_2,n_1,r,m,q) \mathcal{I}^{\pm_1}(n_2,n_1,r,m,q) , \end{align} $$

where

(4.14) $$ \begin{align} \begin{aligned} \mathcal{C}^{\pm_1}(n_2,n_1,r,m,q) & := \sideset{}{^{\star}}\sum_{\substack{a\bmod{q} }} e\left(\frac{\pm_1\bar{a} m}{q}\right) S\left(-r\bar{a},\pm n_2;\frac{rq}{n_1}\right) \\ & = \; \sideset{}{^{\star}}\sum_{\alpha \bmod rq/n_1} e\left(\frac{\pm n_2\bar\alpha}{rq/n_1}\right) \sideset{}{^{\star}}\sum_{\substack{a\bmod{q} }} e\left(\frac{- n_1\bar{a} \alpha \pm_1\bar{a} m}{q}\right) \\ & = \sum_{d\mid q} d \mu\left(\frac{q}{d}\right) \; \sideset{}{^{\star}}\sum_{\substack{\alpha \bmod rq/n_1 \\ \pm_1 m \equiv n_1\alpha \bmod d}} e\left(\frac{\pm n_2 \bar\alpha}{rq/n_1}\right). \end{aligned} \end{align} $$

Here, we have used the following identity for the Ramanujan sum

$$ \begin{align*} R_q(b) & = \;\sideset{}{^{\star}}\sum_{a\bmod{q}} e\left( \frac{b \overline{a}}{q} \right) = \sum_{d\mid (q,b)} d \mu\left(\frac{q}{d}\right). \end{align*} $$

4.2 The nonoscillating case

If $\frac {NX}{PQ}\ll T^{\varepsilon }$ , then we have $X\ll \frac {PQ} {N}T^{\varepsilon }$ and

$$ \begin{align*} S_{r}^{\pm}(N,X,P) & = \frac{1}{Q}\sum_{\substack{q\sim P}} \int_{\mathbb{R}} V\left(\frac{\pm x}{X}\right) \; \sideset{}{^{\star}}\sum_{\substack{a\bmod{q} }} \sum_{\pm} \sum_{n_1|qr} \sum_{n_2=1}^{\infty} \frac{A(n_2,n_1)}{n_1n_2} \\ & \cdot S\left(-r\bar{a},\pm n_2;\frac{rq}{n_1}\right) \Psi_x^{\pm}\left(\frac{n_1^2n_2}{q^3r}\right) \sum_{\substack{m\geq1}} \lambda_f(m) e\left(\frac{m a}{q}\right) e\left(\frac{m x}{qQ}\right) m^{-it} W\left(\frac{m}{N}\right) \mathrm{d}x. \end{align*} $$

We now apply the Voronoi summation formula (see Lemma 2.4) to the sum over m getting

(4.15) $$ \begin{align} \sum_{\substack{m\geq1}} \lambda_f(m) e\left(\frac{m a}{q}\right) e\left(\frac{m x}{qQ}\right) m^{-it} W\left(\frac{m}{N}\right) = q \sum_{\pm_1} \sum_{m\geq1} \frac{ \lambda_{f}(m) }{m} e\left(\frac{\pm_1\bar{a} m}{q}\right) G^{\pm_1} \left(\frac{m}{q^2} \right), \end{align} $$

where $g(m)= e\left (\frac {m x}{qQ}\right ) m^{-it} W\left (\frac {m}{N}\right )$ and $G^{\pm _1}$ is defined as in (2.8).

Proof. We first consider the case $yN\gg T^{2+\varepsilon }$ . By the same argument as in the proof of Lemma 4.2 (i), we get $G^{\pm _1}(y)\ll y^{-6} T^{-A}$ if $yN\gg T^{2+\varepsilon }$ .

Now, assume $yN\ll T^{2+\varepsilon }$ . As in the proof of Lemma 4.2, we have

$$ \begin{align*} G^{\pm_1}(y) = \frac{\epsilon_f^{(1\mp_1 1)/2}}{4\pi^2} \int_{\mathbb{R}} (\pi^2 y)^{1/2-i\tau+it} \gamma_2^{\pm_1}(-1/2+i\tau-it) \tilde{g}(1/2-i\tau+it) \mathrm{d} \tau. \end{align*} $$

If $\frac {NX}{PQ}\ll T^{\varepsilon }$ , then

$$ \begin{align*} \tilde{g}(1/2-i\tau+it) & = \int_{\mathbb{R}} e\left(\frac{u x}{qQ}\right) u^{-it} W\left(\frac{u}{N}\right) u^{1/2-i\tau+it-1} \mathrm{d} u \\ & = N^{1/2-i\tau} \int_{\mathbb{R}} e\left(\frac{Nx}{qQ} \xi- \frac{1}{2\pi} \tau \log \xi \right) W\left(\xi\right) \xi^{-1/2} \mathrm{d} \xi. \end{align*} $$

By Lemma 2.7 with

$$\begin{align*}X=1, V=T^{-\varepsilon}, Y=R=|\tau| \textrm{ and } Q=1, \end{align*}$$

we have $\tilde {g}(1/2-i\tau +it) \ll |\tau |^{-A}$ if $|\tau |\gg T^{2\varepsilon }$ . By Stirling’s formula, we have

$$ \begin{align*} G^{\pm_1}(y) &= (\pi^2 y)^{it} (Ny)^{1/2} \int_{\mathbb{R}} \int_{\mathbb{R}} (\pi^2 y N\xi)^{-i\tau} \exp\left( i(\tau - T)\log\frac{T-\tau}{2e} + i(\tau-T')\log\frac{|T'-\tau|}{2e} \right) \\&\qquad\cdot w_{\epsilon_f,J}^{\pm_1}(\tau-T) w_{\epsilon_f,J}^{\pm_1}(\tau-T') U\left( \frac{\tau}{T^{2\varepsilon}} \right) \mathrm{d} \tau \; e\left(\frac{Nx}{qQ} \xi\right) W\left(\xi\right) \xi^{-1/2} \mathrm{d} \xi + O(T^{-A}), \end{align*} $$

where U is a fixed compactly supported smooth function satisfying that $U^{(j)}(u) \ll _j 1$ for all $j\geq 0$ , and $U(u)=1$ if $u\in [-1,1]$ . Let (temporarily)

$$\begin{align*}h(\tau) = - \tau \log (\pi^2 y N\xi) + (\tau - T)\log\frac{T-\tau}{2e} + (\tau-T')\log\frac{|T'-\tau|}{2e}. \end{align*}$$

Then, we have

$$\begin{align*}h'(\tau) &= - \log (\pi^2 y N\xi) + \log\frac{T-\tau}{2} + \log\frac{|T'-\tau|}{2} ,\\h"(\tau) &= \frac {-1}{T-\tau} - \frac{1} {T'-\tau} , \quad h^{(j)}(\tau) \ll |T'|^{-j+1}, \quad j\geq2. \end{align*}$$

Note that the weight function $w(\tau ) = w_{\epsilon _f,J}^{\pm _1}(\tau -T) w_{\epsilon _f,J}^{\pm _1}(\tau -T') U\left ( \frac {\tau }{T^{2\varepsilon }} \right )$ satisfies that $w^{(j)}(\tau ) \ll T^{-2j\varepsilon }$ . By Lemma 2.7 with

$$\begin{align*}X=1, V=T^{2\varepsilon}, Y=Q=|T'| \textrm{ and } R=1, \end{align*}$$

we have $G^{\pm _1}(y) \ll T^{-A}$ , unless $yN\asymp T |T'|$ , in which case, we have $G^{\pm _1}(y) \ll (yN)^{1/2} T^{\varepsilon } \ll T^{1/2+\varepsilon }|T'|^{1/2}$ .

By (4.2) and (4.15), we have

$$ \begin{align*} S_{r}^{\pm}(N,X,P) & =\frac{1}{Q}\sum_{\substack{q\sim P}} \int_{\mathbb{R}} V\left(\frac{\pm x}{X}\right) \; \sum_{\pm} \sum_{n_1|qr} \sum_{n_2=1}^{\infty} \frac{A(n_2,n_1)}{n_1n_2} \Psi_x^{\pm}\left(\frac{n_1^2n_2}{q^3r}\right)\\ & \hskip 60pt \cdot q \sum_{\pm_1} \sum_{m\geq1} \frac{ \lambda_{f}(m) }{m} \mathcal{C}^{\pm_1}(n_2,n_1,r,m,q) G^{\pm_1} \left(\frac{m}{q^2} \right) \mathrm{d}x , \end{align*} $$

where $\mathcal {C}^{\pm _1}(n_2,n_1,r,m,q)$ is defined in (4.14). Note that we have

$$\begin{align*}\mathcal{C}^{\pm_1}(n_2,n_1,r,m,q) \ll \sum_{d\mid q} d \; \sideset{}{^{\star}}\sum_{\substack{\alpha \bmod rq/n_1 \\ \pm_1 m \equiv n_1\alpha \bmod d}} 1 \ll rq^{1+\varepsilon}. \end{align*}$$

By Lemmas 4.1 and 4.3, we obtain

$$ \begin{align*} S_{r}^{\pm}(N,X,P) & \ll \frac{N^{\varepsilon}}{Q}\sum_{\substack{q\sim P}} rq X \sum_{n_1|qr} \frac{1}{n_1} \!\sum_{n_2 \ll \frac{rP^3}{Nn_1^2}T^{\varepsilon}} \frac{|A(n_2,n_1)|}{n_2} q\\& \quad \times \sum_{m\asymp \frac{P^2 T |T'|}{N}} \frac{ |\lambda_{f}(m)| }{m} T^{1/2} |T'|^{1/2} + O(T^{-A}). \end{align*} $$

By (2.1) and (2.3), we get

$$ \begin{align*} S_{r}^{\pm}(N,X,P) & \ll N^{\varepsilon} \frac{1}{Q}\sum_{\substack{q\sim P}} rq X \sum_{n_1|qr} \frac{1}{n_1} n_1^{\theta_3 } q T^{1/2} |T'|^{1/2} + O(T^{-A}) \nonumber \\ & \ll N^{\varepsilon} \frac{rP^3 XT^{1/2} |T'|^{1/2}}{Q} + O(T^{-A}). \end{align*} $$

Note that by our assumption, we have $X\ll \frac {PQ} {N}T^{\varepsilon }$ . Hence, we get

(4.16) $$ \begin{align} S_{r}^{\pm}(N,X,P) & \ll N^{\varepsilon} \frac{rP^3 PQ T^{1/2} |T'|^{1/2}}{QN} \ll N^{\varepsilon} \frac{r Q^4 T^{1/2} |T'|^{1/2}}{N} \nonumber \\ & \ll N^{1/2+\varepsilon} \left(T^{7/8} |T'|^{19/40} + T^{57/56} |T'|^{17/56} \right), \end{align} $$

provided $N\ll \frac {T^{3/2+\varepsilon }|T'|^{3/2}}{r^2}$ and $Q=\sqrt {\frac {N}{K}}$ with

(4.17) $$ \begin{align} K\geq \left\{ \begin{array}{ll} T^{3/16}|T'|^{31/80}, & \textrm{if } T'\gg T^{5/6}, \\ T^{13/112}|T'|^{53/112}, & \textrm{if } T^{3/5+\varepsilon} \ll T'\ll T^{5/6}. \end{array} \right. \end{align} $$

7.1 The nonzero cases

If $\mp n_1 q_2 q_3 \pm _1 m'n \neq 0$ , then the innermost sums over $d_3^{\prime }$ , $d_2^{\prime }$ and $q_2^{\prime }$ in $\Omega _1$ are bounded by

(7.5) $$ \begin{align} \ll \sum_{d_3^{\prime}\mid q_3} d_3^{\prime} \sum_{\substack{d_2^{\prime} \ll P/q_1q_3 \\ \mp n_1 q_2q_3 \pm_1 m'n \equiv 0 \bmod d_2^{\prime}d_3^{\prime}}} \frac{P}{q_1q_3 } \ll N^{\varepsilon} \frac{P}{q_1}. \end{align} $$

Hence, we have

$$ \begin{align*} \Omega_{1} \ll\ &N^{\varepsilon} H(N_*)\cdot N_2 \frac{rq_1^3}{n_1} \frac{P}{q_1} \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} q_3 \sum_{d_3\mid q_3} \sum_{d_2 \ll P/q_1q_3} d_2\sum_{\substack{q_2\sim P/q_1q_3 \\ (q_2,n_1r)=1 \\ d_2\mid q_2}} (q_2/d_2,d_3) \\ &\quad\cdot \sum_{\substack{m'\asymp \frac{P^2 T |T'|}{N} }} |c_{m'}|^2 \sum_{\substack{n\asymp N_* \\ q_3\mid n}} \left( \frac{P^2 T |T'|}{Nq_1}\frac{(d_2d_3,n)}{d_2d_3} + 1\right). \end{align*} $$

Note that we have

(7.6) $$ \begin{align} q_3 \sum_{\substack{ n\sim N_* \\ q_3 \mid n}} (d,n) &\ll q_3 \sum_{n\sim N_*/q_3} (d,q_3 n) \leq q_3 \sum_{n\sim N_*/q_3} (d,n) (d,q_3)\notag \\ &\quad\leq (d,q_3) q_3 \sum_{d'\mid d} d' \sum_{n\sim N_*/q_3,\; d'\mid n} 1 \ll d^{\varepsilon} (d,q_3) N_*. \end{align} $$

By (5.1), we have

$$ \begin{align*} \Omega_{1} & \ll N^{\varepsilon} H(N_*)\cdot N_2 N_* \frac{rq_1^3}{n_1} \frac{P}{q_1} \frac{P^2 T |T'|}{N} \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} \sum_{d_3\mid q_3} \\ & \hskip 60pt \cdot \sum_{d_2 \ll P/q_1q_3} d_2 \sum_{\substack{q_2\sim P/q_1q_3 \\ (q_2,n_1r)=1 \\ d_2\mid q_2}} (q_2/d_2,d_3) \left(\frac{P^2 T |T'|}{Nq_1}\frac{ (d_2,q_3) }{d_2} + 1\right) \\ & \ll N^{\varepsilon} H(N_*)\cdot N_2 N_* \frac{rq_1^3}{n_1} \frac{P}{q_1} \frac{P^2 T |T'|}{N} \left(\frac{P^2 T |T'|}{Nq_1}\frac{P}{q_1} + \frac{P^2}{q_1^2} \right). \end{align*} $$

Recall that $N_2 = \frac {r N^{2}X^3} {n_1^2 Q^3} $ . If $\frac {N n_1}{q_1 P |T'|^2} N^{\varepsilon } + \frac {P^2 Q^3 n_1}{q_1 N^2 X^3} N^{\varepsilon } \ll N_* \ll \frac {P Q^2 n_1}{q_1 NX^2} N^{\varepsilon }$ , then by (7.2), we obtain

$$ \begin{align*} \Omega_{1} & \ll N^{\varepsilon} \left( \frac{r N^{2}X^3} {n_1^2 Q^3}\right)^{1/2} \frac{rq_1^3}{n_1} \frac{r^{1/2} P}{ q_1^{1/2} n_1^{1/2}} \frac{P}{q_1} \left(\frac{P Q^2 n_1}{q_1 NX^2}\right)^{1/2} \frac{P^2 T |T'|}{N} \left(\frac{P^2 T |T'|}{Nq_1}\frac{P}{q_1} + \frac{P^2}{q_1^2} \right) \\ & \ll \frac{r^2 P^{15/2} T^2 |T'|^2 X^{1/2}} { n_1^{2} q_1 Q^{1/2} N^{3/2} } + \frac{r^2 P^{13/2} T |T'| X^{1/2}} { n_1^{2} q_1 Q^{1/2} N^{1/2} }. \end{align*} $$

By (5.2) and (6.1), the contribution from $\Omega _{1}$ to $S_{r}^{\pm }(N,X,P) $ is bounded by

$$ \begin{align*} & \ll N^{\varepsilon} \frac {Q} {r P^2 T^{1/2} |T'|^{1/2} X} \left(\frac{r N^{2}X^3} {Q^3} \right)^{1/2} \left( \frac{r P^{15/4} T |T'| X^{1/4}} { Q^{1/4} N^{3/4} } + \frac{r P^{13/4} T^{1/2} |T'|^{1/2} X^{1/4}} { Q^{1/4} N^{1/4} } \right) \\ & \ll N^{\varepsilon} r^{1/2} N^{1/4} Q T^{1/2} |T'|^{1/2} + N^{\varepsilon} r^{1/2} N^{3/4} Q^{1/2}. \end{align*} $$

By $Q=\frac { N^{1/2}}{K^{1/2}}$ and $N\ll \frac {T^{3/2+\varepsilon } |T'|^{3/2}}{r^2}$ , the above is bounded by

(7.7) $$ \begin{align} \ll N^{\varepsilon} N^{1/2} \frac{ T^{7/8} |T'|^{7/8}}{K^{1/2}} + N^{\varepsilon} N^{1/2} \frac{ T^{3/4} |T'|^{3/4}}{K^{1/4}}. \end{align} $$

If $1\ll N_* \ll \frac {N n_1}{q_1 P |T'|^2} N^{\varepsilon } + \frac {P^2 Q^3 n_1}{q_1 N^2 X^3} N^{\varepsilon } $ , then by (7.2), we obtain

$$ \begin{align*} \Omega_{1} & \ll N^{\varepsilon} \frac{r N^{2}X^3} {n_1^2 Q^3} \frac{rq_1^3}{n_1} \frac{P}{q_1} \left( \frac{N n_1}{q_1 P |T'|^2} + \frac{P^2 Q^3 n_1}{q_1 N^2 X^3} \right) \frac{P^2 T|T'|}{N} \left(\frac{P^2 T|T'|}{Nq_1}\frac{P}{q_1} + \frac{P^2}{q_1^2} \right) \\ & \ll N^{\varepsilon} \frac{r^2 N P^5 T|T'| X^3} {n_1^2 q_1 Q^3} \left( \frac{T}{|T'|} + \frac{P^3 Q^3 T|T'|}{N^3 X^3} + \frac{N}{P |T'|^2} + \frac{P^2 Q^3}{N^2 X^3} \right). \end{align*} $$

By (5.2) and (6.1), the contribution from $\Omega _{1}$ to $S_{r}^{\pm }(N,X,P) $ is bounded by

$$ \begin{align*} & \ll N^{\varepsilon} \frac {Q} {r P^2 T^{1/2}|T'|^{1/2} X} \left( \frac{r N^{2}X^3} {Q^3} \right)^{1/2} \frac{r N^{1/2} P^{5/2} T^{1/2}|T'|^{1/2} X^{3/2}} {Q^{3/2}} \\ & \hskip 60pt \cdot \left( \frac{T^{1/2}}{|T'|^{1/2}} + \frac{P^{3/2} Q^{3/2} T^{1/2}|T'|^{1/2}}{N^{3/2} X^{3/2}} + \frac{N^{1/2}}{P^{1/2} |T'|} + \frac{P Q^{3/2}}{N X^{3/2}} \right) \\ & \ll N^{\varepsilon} \frac{r^{1/2} N^{3/2} } {Q^{3/2}}\frac{T^{1/2}}{|T'|^{1/2}} + N^{\varepsilon} r^{1/2} Q^{3/2} T^{1/2}|T'|^{1/2} + N^{\varepsilon} \frac{r^{1/2} N^{2}} {Q^{2} |T'|} + N^{\varepsilon} r^{1/2} N^{1/2} Q. \end{align*} $$

Here, we have used $X\ll T^{\varepsilon }$ . By $Q=\frac { N^{1/2}}{K^{1/2}}$ and $N\ll \frac {T^{3/2+\varepsilon } |T'|^{3/2}}{r^2}$ , the above is bounded by

(7.8) $$ \begin{align} \ll N^{1/2+\varepsilon} \frac{K^{3/4} T^{7/8}}{|T'|^{1/8}} + N^{1/2+\varepsilon} \frac{ T^{7/8}|T'|^{7/8}}{K^{3/4}} + N^{1/2+\varepsilon} \frac{T^{3/4} K} { |T'|^{1/4}} + N^{1/2+\varepsilon} \frac{T^{3/4}|T'|^{3/4}}{K^{1/2}}. \end{align} $$

7.2 The zero case

If $\mp n_1 q_2 q_3 \pm _1 m'n = 0$ , then we have $n\asymp \frac {n_1 q_2 q_3}{m'} \ll \frac {n_1 N}{q_1 PT|T'|}$ . Since $n\asymp N_*$ , we have $\Omega _2=0$ , unless $N_* \ll \frac {n_1 N}{q_1 PT|T'|}$ , in which case, by (7.2), we have $H(N_*)=1$ . In $\Omega _2$ , we should consider the sums over $d_2,\ q_2, \ q_3$ first. Since $d_2\mid q_2$ , we rewrite $q_2$ as $d_2 q_2$ . Hence

$$ \begin{align*} \Omega_{2}\ &\ll N_2 \frac{rq_1^3}{n_1} \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} q_3 \sum_{d_3^{\prime}\mid q_3} \sum_{\substack{d_2^{\prime} \ll P/q_1q_3 }} d_2^{\prime} \sum_{\substack{q_2^{\prime}\sim P/q_1q_3 \\ d_2^{\prime} \mid q_2^{\prime}}} (q_2^{\prime}/d_2^{\prime},d_3^{\prime}) \sum_{\substack{m'\asymp \frac{P^2 T |T'|}{N} }} |c_{m'}|^2 \\ &\quad\cdot \sum_{\substack{1 \ll |n|\ll \frac{n_1 N}{q_1 PT |T'|} \\ q_3\mid n}} \sum_{d_3\mid q_3} d_3 \sum_{\substack{d_2 \ll P/q_1q_3 \\ (d_2,\frac{n}{d_3}) \mid \frac{q_3}{d_3} }} d_2 \left(\frac{P^2 T |T'|}{Nq_1}\frac{(d_2,n/d_3)}{d_2} + 1\right) \sum_{\substack{q_2\sim P/q_1q_3d_2 \\ \mp n_1 d_2 q_2q_3 \pm_1 m'n =0 } } 1. \end{align*} $$

Note that $d_2d_3\leq d_2q_3\ll P/q_1$ and the choices of $(d_2,d_3,q_2)$ are at most $N^{\varepsilon }$ for each fixed $m',n$ as $d_2d_3q_2 \mid d_2 q_2q_3 \mid m'n$ . Note that $ (d_2,\frac {n}{d_3}) \mid \frac {q_3}{d_3}$ gives $(d_2,n/d_3) d_3 \leq q_3$ . Hence

$$ \begin{align*} \Omega_{2}\ \ll N_2 \frac{rq_1^3}{n_1} \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} q_3 \sum_{d_3^{\prime}\mid q_3} &\sum_{\substack{d_2^{\prime} \ll P/q_1q_3 }} d_2^{\prime} \sum_{\substack{q_2^{\prime}\sim P/q_1q_3d_2^{\prime} }} (q_2^{\prime},d_3^{\prime}) \\ &\cdot \sum_{\substack{m'\asymp \frac{P^2 T |T'|}{N} }} |c_{m'}|^2 \sum_{\substack{1 \ll |n|\ll \frac{n_1 N}{q_1 PT |T'|} \\ q_3\mid n}} \left(\frac{P^2 T |T'|}{Nq_1} q_3 + \frac{P}{q_1} \right). \end{align*} $$

By (5.1) and (7.6), we have

$$ \begin{align*} \Omega_{2} \ll N_2 \frac{rq_1^3}{n_1} \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} \sum_{d_3^{\prime}\mid q_3} \sum_{\substack{d_2^{\prime} \ll P/q_1q_3 }} d_2^{\prime} \sum_{\substack{q_2^{\prime}\sim P/q_1q_3d_2^{\prime} }} (q_2^{\prime},d_3^{\prime}) \left( \frac{n_1 N}{q_1 PT |T'|} \right) \left(\frac{P^2 T |T'|}{Nq_1} q_3 + \frac{P}{q_1} \right) \frac{P^2 T |T'|}{N}. \end{align*} $$

Changing the order of summations, we get

$$ \begin{align*} \Omega_{2} & \ll N_2 \frac{rq_1^3}{n_1} \frac{n_1 N}{q_1 PT |T'|} \frac{P^2 T |T'|}{N} \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} \!\!\!\left(\frac{P^2 T |T'|}{Nq_1} q_3 + \frac{P}{q_1} \right) \!\sum_{d_3^{\prime}\mid q_3} \sum_{\substack{d_2^{\prime} \ll P/q_1q_3 }} d_2^{\prime} \sum_{\substack{q_2^{\prime}\sim P/q_1q_3d_2^{\prime} }} \!(q_2^{\prime},d_3^{\prime}) \\ & \ll N^{\varepsilon} N_2 rq_1^2 P \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} \left(\frac{P^2 T |T'|}{Nq_1} q_3 + \frac{P}{q_1} \right) \frac{P^2}{q_1^2q_3^2} \ll N^{\varepsilon} \frac{r^2 N^{2} P^4 X^3} {n_1^2 q_1 Q^3} \left(\frac{P T |T'|}{N} + 1 \right). \end{align*} $$

By (5.2) and (6.1), the contribution from $\Omega _{2}$ to $S_{r}^{\pm }(N,X,P) $ is bounded by

$$ \begin{align*} & \ll N^{\varepsilon} \frac {Q} {r P^2 T^{1/2}|T'|^{1/2} X} \frac{r^{1/2} N X^{3/2}} {Q^{3/2}} \frac{r N P^2 X^{3/2}} { Q^{3/2}} \left(\frac{P^{1/2} T^{1/2}|T'|^{1/2}}{N^{1/2}} + 1 \right) \\ & \ll N^{\varepsilon} \frac {r^{1/2} N^{3/2} } { Q^{3/2} } + N^{\varepsilon} \frac {r^{1/2} N^2 } { Q^{2} T^{1/2}|T'|^{1/2}}. \end{align*} $$

Here, we have used $P\leq Q$ and $X\ll T^{\varepsilon }$ . By $Q=\frac { N^{1/2}}{K^{1/2}}$ and $N\ll \frac {T^{3/2+\varepsilon } |T'|^{3/2}}{r^2}$ , the above is bounded by

(7.9) $$ \begin{align} \ll N^{1/2+\varepsilon } T^{3/8}|T'|^{3/8} K^{3/4} + N^{1/2+\varepsilon } T^{1/4}|T'|^{1/4} K. \end{align} $$

8.1 The nonzero cases

If $\mp n_1 q_2 q_3 \pm _1 m'n \neq 0$ , then by (7.5), we have

$$ \begin{align*} \Omega_{31}\ &\ll N^{\varepsilon} N_2 \frac{rq_1^3}{n_1} \frac{P}{q_1} \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} q_3 \sum_{d_3\mid q_3} \sum_{d_2 \ll P/q_1q_3} d_2\sum_{\substack{q_2\sim P/q_1q_3 \\ (q_2,n_1r)=1 \\ d_2\mid q_2}} (q_2/d_2,d_3) \\ &\quad\cdot \sum_{\substack{|n|\ll \frac{P Q^2 n_1}{q_1 NX^2} N^{\varepsilon} \\ q_3\mid n}} \left(\frac{M}{q_1}\frac{(d_2d_3,n)}{d_2d_3} + 1\right) \sum_{\substack{m'\asymp M}} |c_{m'}|^2. \end{align*} $$

By (5.1) and (7.6), we have

$$ \begin{align*} \Omega_{31} & \ll N^{\varepsilon} N_2 \frac{rq_1^3}{n_1} \frac{P}{q_1} \frac{P Q^2 n_1}{q_1 NX^2} M \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} \sum_{d_3\mid q_3} \sum_{d_2 \ll P/q_1q_3} d_2 \sum_{\substack{q_2\sim P/q_1q_3 \\ (q_2,n_1r)=1 \\ d_2\mid q_2}} (q_2/d_2,d_3) \left(\frac{M}{q_1}\frac{ (d_2,q_3) }{d_2} + 1\right) \\ & \ll N^{\varepsilon} N_2 \frac{rq_1^3}{n_1} \frac{P}{q_1} \frac{P Q^2 n_1}{q_1 NX^2} M \left(\frac{M}{q_1}\frac{P}{q_1} + \frac{P^2}{q_1^2} \right). \end{align*} $$

Recall that $N_2 = \frac {r N^{2}X^3} {n_1^2 Q^3} $ . We obtain

(8.2) $$ \begin{align} \Omega_{31} \ll N^{\varepsilon} \frac{r N^{2}X^3} {n_1^2 Q^3} \frac{rq_1^3}{n_1} \frac{P}{q_1} \frac{P Q^2 n_1}{q_1 NX^2} M \left(\frac{M}{q_1}\frac{P}{q_1} + \frac{P^2}{q_1^2} \right) \ll N^{\varepsilon} \frac{r^2 N P^2 X}{n_1^2 q_1 Q } M \left( M P + P^2 \right). \end{align} $$

By (5.2) and (6.1), the contribution from $\Omega _{31}$ to $S_{r}^{\pm }(N,X,P,M) $ is bounded by

$$ \begin{align*} & \ll N^{\varepsilon} \frac{ Q }{r N^{1/2} P X M^{1/2} } \frac{r^{1/2} N X^{3/2}} {Q^{3/2}} \left( \frac{r N^{1/2} P X^{1/2}}{Q^{1/2}} M^{1/2} \left( \frac{PT}{N^{1/2}}P^{1/2} + \frac{N^{1/2}X}{Q} P^{1/2} + P \right) \right) \\ & \ll N^{\varepsilon} \frac{ r^{1/2} N^{1/2} TP^{3/2} }{ Q } + N^{\varepsilon} \frac{ r^{1/2} N^{3/2} P^{1/2} }{ Q^2 } + N^{\varepsilon}\frac{ r^{1/2} N }{ Q } P. \end{align*} $$

Here, we have used $X\ll T^{\varepsilon }$ and $M\ll \frac {P^2T^2}{N}+\frac {NX^2}{Q^2}$ as in (4.10). Note that by the assumption $\frac {NX}{PQ}\gg |T'|^{1-\varepsilon }$ , we have $P\ll \frac {N}{Q |T'| } T^{\varepsilon }$ . Together with $Q=\frac { N^{1/2}}{K^{1/2}}$ and $N\ll \frac {T^{3/2+\varepsilon } |T'|^{3/2}}{r^2}$ , the above is bounded by

(8.3) $$ \begin{align} & \ll N^{\varepsilon} \frac{ r^{1/2} N^{2} T }{ Q^{5/2} |T'|^{3/2} } + N^{\varepsilon} \frac{ r^{1/2} N^{3/2} }{ Q^{3/2} } + N^{\varepsilon}\frac{ r^{1/2} N^2 }{ Q^2 |T'| } \nonumber \\ & \ll N^{1/2+\varepsilon} \frac{ T^{11/8} K^{5/4} }{|T'|^{9/8}} + N^{1/2+\varepsilon} T^{3/8}|T'|^{3/8} K^{3/4} + N^{1/2+\varepsilon} \frac{ T^{3/4} K}{|T'|^{1/4}}. \end{align} $$

8.2 The zero case

In $\Omega _{32}$ , we should consider the sums over $d_2,\ d_3, \ q_2$ first. Since $d_2\mid q_2$ , we rewrite $q_2$ as $d_2 q_2$ . Hence, we have

$$ \begin{align*} \Omega_{32}\ &\ll N_2 \frac{rq_1^3}{n_1} \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} q_3 \sum_{d_3^{\prime}\mid q_3} \sum_{\substack{d_2^{\prime} \ll P/q_1q_3 }} d_2^{\prime} \sum_{\substack{q_2^{\prime}\sim P/q_1q_3 \\ d_2^{\prime} \mid q_2^{\prime}}} (q_2^{\prime}/d_2^{\prime},d_3^{\prime}) \sum_{\substack{m'\asymp M }} |c_{m'}|^2 \\ &\quad \cdot \sum_{\substack{1 \ll |n|\ll \frac{P Q^2 n_1}{q_1 NX^2} N^{\varepsilon} \\ q_3\mid n}} \sum_{d_3\mid q_3} \sum_{d_2 \ll P/q_1q_3} \left(\frac{M}{q_1} \Big(d_2,\frac{n}{d_3}\Big)d_3 + d_3 d_2 \right) \sum_{\substack{q_2\sim P/q_1q_3d_2 \\ \mp n_1 d_2 q_2q_3 \pm_1 m'n =0 } } 1. \end{align*} $$

Note that $d_2d_3\leq d_2q_3\ll P/q_1$ and the choices of $(d_2,d_3,q_2)$ are at most $N^{\varepsilon }$ for each fixed $m',n$ as $d_2 d_3 q_2 \mid d_2 q_2q_3 \mid m'n$ . Note that $(d_2,\frac {n}{d_3}) \mid \frac {q_3}{d_3} $ implies $(d_2,n/d_3) d_3 \leq q_3$ . Hence

$$ \begin{align*} \Omega_{32} \ll N_2 \frac{rq_1^3}{n_1} \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} \sum_{d_3^{\prime}\mid q_3} \sum_{\substack{d_2^{\prime} \ll P/q_1q_3 }} d_2^{\prime} \sum_{\substack{q_2^{\prime}\sim P/q_1q_3d_2^{\prime} }} (q_2^{\prime},d_3^{\prime}) q_3 \sum_{\substack{1 \ll |n|\ll\frac{P Q^2 n_1}{q_1 NX^2} N^{\varepsilon} \\ q_3\mid n}} \!\!\kern-2pt\left(\frac{M}{q_1} q_3 + \frac{P}{q_1} \right) \kern-2pt\!\sum_{\substack{m'\asymp M}} \kern-1pt|c_{m'}|^2\kern-1pt. \end{align*} $$

By (5.1), we have

$$ \begin{align*} \Omega_{32} & \ll N^{\varepsilon} N_2 \frac{rq_1^3}{n_1} \sum_{\substack{q_3\ll P/q_1 \\ (q_3,n_1r)=1}} \sum_{d_3^{\prime}\mid q_3} \sum_{\substack{d_2^{\prime} \ll P/q_1q_3 }} d_2^{\prime} \sum_{\substack{q_2^{\prime}\sim P/q_1q_3d_2^{\prime} }} (q_2^{\prime},d_3^{\prime}) \frac{P Q^2 n_1}{q_1 NX^2} \left(\frac{M}{q_1} q_3 + \frac{P}{q_1} \right)M \\ & \ll N^{\varepsilon} \frac{r^2 N P^3 X } {n_1^2 q_1 Q} M \left( M + P \right). \end{align*} $$

Note that this bound is the same as the bound for $\Omega _{31}$ in (8.2). Hence, we get the same bound for the contribution from $\Omega _{32}$ to $S_{r}^{\pm }(N,X,P,M) $ .