3.1 Standard L-functions

Let $m \ge 1$ be an integer. Fix $\pi =\otimes _{p} \pi _{p} \in \mathfrak {E}_{m}$ , where p ranges over all primes. The standard function $L(s, \pi )$ is given by the Dirichlet series

(3.1) $$ \begin{align} \begin{aligned} L(s, \pi) &=\prod_p L(s,\pi_{p})=\sum_{n=1}^{\infty} \frac{\lambda_\pi(n)}{n^s},\end{aligned} \end{align} $$

with both the series and the product converging absolutely for $\mathrm {Re}(s)>1$ . For each p, the local factor $L(s,\pi _{p})$ is defined in terms of Satake parameters $\{\alpha _{1,\pi }(p),\dots ,\alpha _{m,\pi }(p)\}$ by

$$ \begin{align*}L(s,\pi_{p}):= \prod_{j=1}^m\left(1-\frac{\alpha_{j, \pi}(p)}{p^s}\right)^{-1}=\sum_{i=0}^{\infty} \frac{\lambda_\pi\left(p^i\right)}{p^{is}}. \end{align*} $$

Let $\mathfrak {q}_{\pi }$ denote the conductor of $\pi $ . For $1\le j\le m$ , we have $\alpha _{j,\pi }(p)\neq 0$ when $p\nmid \mathfrak {q}_{\pi }$ . When $p\mid \mathfrak {q}_{\pi }$ , the case might be that $\alpha _{j,\pi }(p)=0$ . At the archimedean place of $\mathbb {Q}$ , there exist m complex Langlands parameters $\mu _{\pi }(j)$ , from which we define the Dirichlet series

(3.2) $$ \begin{align} L_{\infty}(s, \pi):=\pi^{-m s / 2} \prod_{j=1}^m \Gamma\left(\frac{s+\mu_\pi(j)}{2}\right). \end{align} $$

Let $\tilde {\pi }$ denote the contragredient representation of $\pi \in \mathfrak {E}_m$ , which is also an irreducible cuspidal automorphic representation in $\mathfrak {E}_m$ . For each $p \le \infty $ , we have the following equations

$$ \begin{align*}\big\{\alpha_{j, \tilde{\pi}}(p):1\le j\le m\big\}=\big\{\overline{\alpha_{j, \pi}(p)}:1\le j\le m\big\} \end{align*} $$

and

$$ \begin{align*}\big\{\mu_{\tilde{\pi}}(j):1\le j\le m\big\}=\big\{\overline{\mu_\pi(j)}:1\le j\le m\big\}. \end{align*} $$

For a non-trivial $\pi $ , the complete L-function is defined by

$$ \begin{align*}\Lambda(s, \pi):=\mathfrak{q}_{\pi}^{\frac{s}{2}} L(s, \pi) L_{\infty}(s, \pi). \end{align*} $$

The complete L-function $\Lambda (s, \pi )$ has an analytic continuation to the entire complex plane, and is of finite order. Moreover, it satisfies the functional equation

$$ \begin{align*}\Lambda(s, \pi)=W_{\pi} \Lambda(1-s, \tilde{\pi}), \end{align*} $$

where $W_{\pi }$ is the root number (a complex number of magnitude $1$ ). For all primes p and $1 \le j \le m$ , it follows from [Reference Kim20] $(2 \le m \le 4)$ and [Reference Luo, Rudnick and Sarnak23] $(m \ge 5)$ that the best known bound is

$$ \begin{align*}\left|\alpha_{j, \pi}(p)\right| \le p^{\theta_{m}} \quad \text { and } \quad -\mathrm{Re}(\mu_{\pi}(j))\le \theta_{m}, \end{align*} $$

where

(3.3) $$ \begin{align} \theta_{2}=\frac{7}{64}, \quad \theta_{3}=\frac{5}{14}, \quad \theta_{4}=\frac{9}{22}, \quad \theta_{m}=\frac{1}{2}-\frac{1}{m^{2}+1}\ (m \ge 5). \end{align} $$

The GRC predicts that

$$ \begin{align*}\theta_{m}=0\end{align*} $$

for all $m\ge 1.$ The analytic conductor of $\pi $ is defined by

(3.4) $$ \begin{align} C(\pi, t):=\mathfrak{q}_{\pi} \prod_{j=1}^{m}\left(1+\left|i t+\mu_{\pi}(j)\right|\right), \quad C(\pi):=C(\pi, 0). \end{align} $$

Taking the logarithmic derivative for $L(s, \pi )$ , we obtain that for $\mathrm {Re}(s)>1$ ,

$$ \begin{align*}-\frac{L^{\prime}}{L}(s, \pi)=\sum_{p} \sum_{k=1}^{\infty} \frac{a_{\pi} (p^k) \log p}{p^{k s}}=\sum_{n=1}^{\infty} \frac{a_\pi(n) \Lambda(n)}{n^s}, \end{align*} $$

where

(3.5) $$ \begin{align} a_\pi\left(p^k\right)=\sum_{j=1}^m \alpha_{j, \pi}(p)^k. \end{align} $$

Proof Let y be a parameter satisfying the inequality $1 \le y \le x$ , will be determined later. Denote $\phi (z)$ as the function with support on the interval $[0, x+y]$ , satisfying the conditions $\phi (z)=1$ for $1 \le z \le x$ , and $|\phi (z)| \le 1$ elsewhere. By the Cauchy–Schwarz inequality, (3.15) and the Prime Number theorem in short intervals [Reference Huxley17], we can achieve a smoother expression by writing

(3.8) $$ \begin{align} \sum_{n\le x} \Lambda(n)a_{\pi}(n)=\sum_n \Lambda(n)a_{\pi}(n) \phi(n)+O(\sqrt{x y}), \quad x^{7/12}\ll y\ll x. \end{align} $$

For $0 \le z \le x+y$ , we take

$$ \begin{align*}\phi(z)=\min \left(\frac{z}{y}, 1,1+\frac{x-z}{y}\right) \end{align*} $$

and set $\phi (z)=0$ for $z>x+y$ . The Mellin transform of $\phi (z)$ satisfies

$$ \begin{align*}\hat{\phi}(s)=\int_0^{x+y} \phi(z) z^{s-1} \mathrm{d} z \ll \frac{x^\sigma}{|s|} \min \left(1, \frac{x}{|s| y}\right) \end{align*} $$

for $s=\sigma +i t, \frac {1}{2} \le \sigma \le 2$ . The sum on the right-hand side of (3.8) can be expressed as

$$ \begin{align*}\sum_n \Lambda(n)a_{\pi}(n) \phi(n)=\frac{1}{2 \pi i} \int_{(2)}-\frac{L^{\prime}}{L}(s, \pi) \hat{\phi}(s) \mathrm{d} s, \end{align*} $$

where the integral converges absolutely. Our operations will be carried out within the region defined by

$$ \begin{align*}\mathcal{Z}:=\left\{s=\sigma+i t \mid \sigma= \delta+\varepsilon\right\}. \end{align*} $$

All the zeros of $L(s, \pi )$ are at least distance $\varepsilon $ from $\mathcal {Z}$ . Therefore, it follows from [Reference Iwaniec and Kowalski18, Equation (5.27)] and [Reference Iwaniec and Kowalski18, Equation (5.28)] that for $s \in \mathcal {Z}$ ,

$$ \begin{align*}-\frac{L^{\prime}}{L}(s, \pi) \ll \varepsilon^{-1}(\log C(\pi)+m\log(|t|+3)). \end{align*} $$

By moving the integration from the line $\mathrm {Re}(s)=2$ to $\mathcal {Z}$ and combining the above inequality, we apply Cauchy’s theorem to derive the following:

$$ \begin{align*} \sum_n \Lambda(n)a_{\pi}(n) \phi(n)&=\frac{1}{2 \pi i} \int_{\mathcal{Z}}-\frac{L^{\prime}}{L}(s, \pi) \hat{\phi}(s) \mathrm{d}s\nonumber \\ &\ll \varepsilon^{-1}\int_{0}^{x/y}\frac{x^\sigma}{t+1} ( \log C(\pi)+m\log(t+3))\mathrm{d} t\nonumber\\ &\quad +\varepsilon^{-1}\int_{x/y}^{\infty}\frac{x^{\sigma+1}}{t^2y} ( \log C(\pi)+m\log(t+3))\mathrm{d} t \nonumber\\ &\ll \varepsilon^{-1} x^{\delta+\varepsilon} (m \log \mathfrak{q}_{\pi}+m\log(x/y+3))\log (x/y). \end{align*} $$

We take $y=x\exp (-\sqrt {\log x})$ into the above inequality and combine it with (3.8), thereby obtaining (3.6).

Under GRC, the difference is that we can use [Reference Iwaniec and Kowalski18, Equation (5.53)] to obtain that

$$ \begin{align*}\sum_{n\le x} \Lambda (n)a_{\pi}(n) =\sum_{|\gamma| \le T} \frac{x^\rho-1}{\rho}+O\left(\frac{x}{T}(\log x) \log \left(x^m C(\pi)\right)\right),\end{align*} $$

where $\rho =\beta +i \gamma $ runs over the zeros of $L(s, \pi )$ in the critical strip of height up to T with $1 \le T \le x$ . The implied constant is absolute. Using [Reference Iwaniec and Kowalski18, Equation (5.27)] and choosing $T=x^{1-\delta }(\log x)\log (x^{m}C(\pi ))$ , we find that

$$ \begin{align*}\sum_{n\le x} \Lambda(n)a_{\pi}(n) \ll x^{\delta}(\log x)^{2}(\log \log (x^{m} C(\pi)))^{2}.\\[-40pt] \end{align*} $$

3.2 Rankin–Selberg L-functions

Let $\pi ^{\prime }=\otimes _{p} \pi _{p}^{\prime } \in \mathfrak {E}_{m^{\prime }}$ $(m^{\prime } \ge 1)$ and $\pi =\otimes _{p} \pi _{p} \in \mathfrak {E}_{m}$ $(m \ge 1)$ . The Rankin–Selberg L-function $L\left (s, \pi \times \pi ^{\prime }\right )$ associated with $\pi $ and $\pi ^{\prime }$ is of the form

$$ \begin{align*}L\left(s, \pi \times \pi^{ \prime}\right)=\prod_{p} L\left(s, \pi_{p} \times \pi_{p}^{\prime }\right)=\sum_{n=1}^{\infty} \frac{\lambda_{\pi \times \pi^{\prime}}(n)}{n^{s}} \end{align*} $$

for $\mathrm {Re}(s)>1$ . For each (finite) prime p, the inverse of the local factor $L\left (s, \pi _{p} \times \pi _{p}^{\prime }\right )$ is defined as a polynomial in $p^{-s}$ of degree not exceeding $ m m^{ \prime }$ :

(3.9) $$ \begin{align} L\left(s, \pi_{p} \times \pi_{p}^{ \prime}\right)^{-1}:=\prod_{j=1}^{m} \prod_{j^{ \prime}=1}^{m^{ \prime}}\left(1-\frac{\alpha_{j, j^{\prime}, \pi \times \pi^{ \prime}}(p)}{p^{s}}\right), \end{align} $$

where $\alpha _{j, j^{ \prime }, \pi \times \pi ^{ \prime }}(p)$ are suitable complex numbers. Given $\theta _{m}$ and $\theta _{m^{\prime }}$ as in (3.3), we have the pointwise bound

(3.10) $$ \begin{align} \left|\alpha_{j, j^{ \prime}, \pi \times \pi^{ \prime}}(p)\right| \le p^{\theta_{m}+\theta_{m^{ \prime}}} \le p^{1-\frac{1}{m m^{ \prime}}}. \end{align} $$

If $p \nmid \mathfrak {q}_{\pi } \mathfrak {q}_{\pi ^{ \prime }}$ , we have

(3.11) $$ \begin{align} \bigg\{\alpha_{j, j^{ \prime}, \pi \times \pi^{ \prime}}(p): j \le m, j^{ \prime} \le m^{ \prime}\bigg\}=\bigg\{\alpha_{j, \pi}(p) \overline{\alpha_{j^{\prime }, \pi^{ \prime}}(p)}: j \le m, j^{ \prime} \le m^{\prime }\bigg\}. \end{align} $$

At the archimedean place of $\mathbb {Q}$ , there are $m m^{\prime }$ complex Langlands parameters $\mu _{\pi \times \pi ^{\prime }}\left (j, j^{\prime }\right )$ from which we define the Dirichlet series

$$ \begin{align*}L_{\infty}\left(s, \pi \times \pi^{\prime}\right):=\pi^{-\frac{m m^{\prime} s}{2}} \prod_{j=1}^{m} \prod_{j^{\prime}=1}^{m^{\prime}} \Gamma\left(\frac{s+\mu_{\pi \times \pi^{\prime}}(j, j^{\prime})}{2}\right). \end{align*} $$

These parameters satisfy the pointwise bound

$$ \begin{align*}\mathrm{Re}\left(\mu_{\pi \times \pi^{\prime}}\left(j, j^{\prime}\right)\right) \ge-\theta_{m}-\theta_{m^{\prime}}. \end{align*} $$

Let $\mathfrak {q}_{\pi \times \pi ^{\prime }}$ denote the conductor of $\pi \times \pi ^{\prime }$ . As with $L(s, \pi \times \pi ^{\prime })$ , we define the analytic conductor $C\left (\pi \times \pi ^{\prime }\right )$ by

(3.12) $$ \begin{align} C\left(\pi \times \pi^{\prime}, t\right):=\mathfrak{q}_{\pi \times \pi^{\prime}} \prod_{j=1}^{m} \prod_{j^{\prime}=1}^{m^{\prime}}\left(1+\left|i t+\mu_{\pi \times \pi^{\prime}}\left(j, j^{\prime}\right)\right|\right), \ C\left(\pi \times \pi^{\prime}\right):=C\left(\pi \times \pi^{\prime}, 0\right). \end{align} $$

The combined work of [Reference Bushnell and Henniart4] alongside [Reference Soundararajan and Thorner30, Appendix] shows that

(3.13) $$ \begin{align} C\left(\pi \times \pi^{\prime}, t\right) \ll C\left(\pi \times \pi^{\prime}\right)(1+|t|)^{m^{\prime} m}, \quad C\left(\pi \times \pi^{\prime}\right) \le e^{O\left(m^{\prime} m\right)} C(\pi)^{m^{\prime}} C\left(\pi^{\prime}\right)^{m}. \end{align} $$

Let $r_{\pi \times \pi ^{\prime }} \in \{0,1\}$ be the order of the pole of $L\left (s, \pi \times \pi ^{\prime }\right )$ at $s=1$ . We have $r_{\pi \times \pi ^{\prime }}=1$ if and only if $\pi ^{\prime }=\tilde {\pi }$ . The function $\Lambda \left (s, \pi \times \pi ^{\prime }\right )=(s(s-1))^{r_{\pi \times \pi ^{\prime }}} \mathfrak {q}_{\pi \times \pi ^{\prime }}^{s / 2} L\left (s, \pi \right .$ $\left .\times \pi ^{\prime }\right ) L_{\infty }\left (s, \pi \times \pi ^{\prime }\right )$ is entire of order one. There exists a complex number $W\left (\pi \times \pi ^{\prime }\right )$ of modulus one such that

$$ \begin{align*}\Lambda\left(s, \pi \times \pi^{\prime}\right)=W\left(\pi \times \pi^{\prime}\right) \Lambda\left(1-s, \tilde{\pi} \times \tilde{\pi}^{\prime}\right).\end{align*} $$

Furthermore, $L(s, \pi \times \tilde {\pi })$ extends to the complex plane with a simple pole at $s=1$ , hence

(3.14) $$ \begin{align} \sum_{n\le x} \lambda_{\pi \times \widetilde{\pi}}(n) \sim x \mathrm{Res}_{s=1} L(s, \pi \times \widetilde{\pi}) \ll x, \end{align} $$

as a consequence of a standard Tauberian argument. Taking the logarithmic derivative for $L(s, \pi \times \pi ^{\prime })$ , we obtain that, for $\mathrm {Re}(s)>1$ ,

$$ \begin{align*}-\frac{L^{\prime}}{L}(s, \pi\times\pi^{\prime})=\sum_{p} \sum_{k=1}^{\infty} \frac{a_{\pi\times\pi^{\prime}} (p^k) \log p}{p^{k s}}=\sum_{n=1}^{\infty} \frac{a_{\pi\times\pi^{\prime}}(n) \Lambda(n)}{n^s},\end{align*} $$

where

$$ \begin{align*}a_{\pi\times\pi^{\prime}}\left(p^k\right)=\sum_{j=1}^m \sum_{j^{\prime}=1}^{m^{\prime}} \alpha_{j, j^{\prime}, \pi \times \pi^{\prime}}(p)^k. \end{align*} $$

Building upon Shahidi’s non-vanishing result for $L(s, \pi \times \tilde {\pi })$ at $\mathrm {Re}(s)=1$ (see [Reference Shahidi29]), then

(3.15) $$ \begin{align} \sum_{n \le x}\Lambda(n)\left|a_{\pi}(n)\right|{}^{2}\ll\sum_{n \le x} \Lambda(n) a_{\pi \times \widetilde{\pi}}(n) \sim x. \end{align} $$

3.3 Twists

Let $\chi $ be a primitive Dirichlet character with conductor q. Fix $\pi =\otimes _{p} \pi _{p} \in \mathfrak {E}_{m}$ with $m\ge 2$ . By [Reference Cogdell5], the twisted L-function $L(s,\pi \otimes \chi )$ equals

$$ \begin{align*}\sum_{n=1}^{\infty} \frac{\lambda_{\pi \otimes \chi}(n) }{n^s} =\prod_p \prod_{j=1}^m \left(1-\frac{\alpha_{j, \pi \otimes \chi }(p)}{p^s}\right)^{-1}. \end{align*} $$

Furthermore, by (3.11), if $p \nmid q$ , then

$$ \begin{align*}\begin{aligned} \left\{\alpha_{j, \pi\otimes\chi }(p): 1 \le j \le m\right\} =\left\{ \chi(p)\alpha_{j, \pi }(p): 1 \le j \le m\right\}. \end{aligned} \end{align*} $$

Hence, for $\mathrm {Re}(s)>1$ , we have

$$ \begin{align*}\begin{aligned} \sum_{n=1}^{\infty} \frac{\chi(n)\lambda_{\pi }(n) }{n^s} &=\prod_p \prod_{j=1}^m \left(1-\frac{\chi(p)\alpha_{j, \pi }(p)}{p^s}\right)^{-1}=L(s, \pi\otimes\chi ) \prod_{p \mid q} \prod_{j=1}^m\left(1-\frac{\alpha_{j, \pi \otimes \chi}(p)}{p^s}\right). \end{aligned} \end{align*} $$

We use the following zero-free region along with standard contour integration techniques to prove Proposition 2.1.

Lemma 3.2 Let $\pi \in \mathfrak {E}_{m}^{b}$ and $Q\ge 3$ . There exists a constant $c_\pi>0$ , depending effectively on $\pi $ , such that for all primitive Dirichlet characters $\chi (\bmod \thinspace q)$ with $q \le Q$ , with at most one exception, the L-function $L(s, \pi \otimes \chi )$ is non-zero in the region

$$ \begin{align*}\mathrm{Re}(s) \ge 1-\frac{c_\pi}{\log (Q(3+|\mathrm{Im}(s)|))}. \end{align*} $$

If the exceptional character $\chi _1$ exists, then $\chi _1$ is quadratic. $L\left (s, \pi \otimes \chi _1\right )$ has exactly one zero $\beta _1$ in this region, and $\beta _1$ is both real and simple. For all $\varepsilon _{0}>0$ , there exists an ineffective constant $c_\pi (\varepsilon _{0})>0$ depending on $\pi $ and $\varepsilon _0$ such that $\beta _1 \le 1-$ $c_\pi (\varepsilon _{0}) Q^{-\varepsilon _{0}}$ .

4.1 Reduction

Define the sum

$$ \begin{align*}\sum_{n \le x}\Lambda(n)g(n) e(n^{k} \alpha), \end{align*} $$

where $ g(n)$ is an arithmetic function. According to the Dirichlet approximation theorem, for any $\alpha \in [0,1)$ , there exist integers a and q such that $\alpha $ can be expressed as (2.4), where Q is a parameter to be chosen later. Combining (2.4) with partial summation, the above sum is bounded by

(4.1) $$ \begin{align} \left(1+\frac{x^{k}}{q Q}\right) \max _{1 \le t \le x}\left|\sum_{n \le t}\Lambda(n) g(n) e\left(\frac{an^{k}}{q}\right)\right|. \end{align} $$

Due to the orthogonality of characters, we find that

(4.2) $$ \begin{align} \begin{aligned} \sum_{n \le t}\Lambda(n) g(n) e\left(\frac{a n^{k}}{q}\right) &=\sum_{d \mid q} \sum_{\substack{h=1 \\ (h, q / d)=1}}^{q} e\left(\frac{a d^{k} h^{k}}{q}\right) \sum_{\substack{l \le t / d \\ l \equiv h(\!\!\bmod q / d)}}\Lambda(d l) g(d l) \\ &=\sum_{d \mid q} \sum_{h=1 }^{q/d} e\left(\frac{a d^{k} h^{k}}{q}\right) \sum_{l \le t / d}\Lambda(d l) g(d l)\frac{1}{\varphi(q / d)} \sum_{\substack{\chi(\!\!\bmod q / d)\\ (h, q / d)=1}} \bar{\chi}(h) \chi(l)\\& \ll \sum_{d \mid q} \frac{1}{\varphi(q / d)} \max _{\chi\left(\!\!\bmod q / d\right)}\bigg|\sum_{l \le t / d} \Lambda(d l)g(d l) \chi(l)\bigg| \\ & \quad \times \sum_{\chi\left(\!\!\bmod q / d\right)}\bigg|\sum_{\substack{h=1 \\ (h, q / d)=1}}^{q / d} \bar{\chi}(h) e\left(\frac{ad^k h^k }{q}\right)\bigg|, \end{aligned} \end{align} $$

where $\varphi $ is the Euler function. By the Cauchy–Schwarz inequality and Parseval’s identity, the innermost sum over $\chi $ is bounded by

(4.3) $$ \begin{align} & \varphi\left(q / d\right)^{1/2}\left(\sum_{\chi\left(\!\!\bmod q / d\right)}\bigg|\sum_{\substack{h=1 \\ (h, q / d)=1}}^{q / d} \bar{\chi}(h) e\left(\frac{ad^k h^k }{q}\right)\bigg|^2\right)^{1/2} \nonumber\\ =\ & \varphi\left(q / d\right)^{1/2}\left(\sum_{\substack{h_1=1 \\ \left(h_1, q / d\right)=1}}^{q / d} \sum_{\substack{h_2=1 \\ \left(h_2, q / d\right)=1}}^{q / d} e\left(\frac{ad^k\left(h_1^k-h_2^k\right) }{q}\right) \sum_{\substack{\chi\left(\!\!\bmod q / d\right)}} \bar{\chi}\left(h_1\right) \chi\left(h_2\right)\right)^{1/2} \nonumber\\=\ & \varphi\left(q / d\right)\left(\sum_{\substack{h_1=1 \\ \left(h_1, q / d\right)=1}}^{q / d} \sum_{\substack{h_2=1 \\ h_2 \equiv h_1\left(\!\!\bmod q / d\right)}}^{q / d}e\left(\frac{ad^k\left(h_1^k-h_2^k\right)}{q}\right)\right) ^{1/2}=\varphi\left(q / d\right)^{3/2}. \end{align} $$

Inserting the estimates (4.2) and (4.3) into (4.1), then

(4.4) $$ \begin{align} \sum_{n \le x}\Lambda(n) g(n) e\left(n^{k} \alpha\right) \ll \left(1+\frac{x^{k}}{q Q}\right) \max _{1 \le t \le x}\left\{ q^{1 / 2} \sum_{d \mid q} \frac{1}{d^{1 / 2}} \max _{\chi (\!\!\bmod q / d)}\bigg|\sum_{l \le t / d}\Lambda(d l) g(d l) \chi(l)\bigg|\right\}. \end{align} $$

4.2 Proof of Proposition 2.1

It follows from (3.1) and (3.5) that

(4.5) $$ \begin{align} \sum_{p\le x}\lambda_{\pi}(p)e(p^{k}\alpha)\log p=\sum_{n \le x}\Lambda(n)a_{\pi}(n) e(n^{k} \alpha)-\sum_{p^{k_1}\le x, k_1\ge 2 } a_{\pi}(p^{k_1})e(p^{k_1}\alpha)\log p. \end{align} $$

Using the Cauchy–Schwarz inequality, (3.14) and (3.15), the left side of the above equation is equal to

(4.6) $$ \begin{align} \sum_{n \le x}\Lambda(n)a_{\pi}(n) e(n^{k} \alpha)+O(x^{3/4}). \end{align} $$

Let $g(n)=a_{\pi }(n)$ in (4.4). Then the above summation is

(4.7) $$ \begin{align} \ll \left(1+\frac{x^{k}}{q Q}\right) \max _{1 \le t \le x}\left\{ q^{1 / 2} \sum_{d \mid q} \frac{1}{d^{1 / 2}} \max _{\chi (\bmod q / d)}\bigg|\sum_{l \le t / d}\Lambda(d l) a_{\pi}(d l) \chi(l)\bigg|\right\}. \end{align} $$

Let $l=l_{1} l_{2} $ with $l_{1} \mid d^{\infty }$ and $\left (l_{2}, l_{1}d\right )=1.$ By the multiplicative property of $a_{\pi }(dl)$ , the formula in the brace is

(4.8) $$ \begin{align} \ll &q^{1 / 2} \sum_{d \mid q,\ d\neq 1} \frac{1}{d^{1 / 2}} \max _{\chi(\bmod q / d)}\bigg|\sum_{l_{1} \mid d^{\infty},\ l_{1} \le t/d}\Lambda(dl_1) a_{\pi}(dl_{1})\chi(l_{1})\bigg|\nonumber\\ &+q^{1 / 2}\max _{\chi(\bmod q)}\bigg|\sum_{l_{2} \le t} \Lambda(l_{2}) a_{\pi}\left(l_{2}\right) \chi\left(l_{2}\right)\bigg|. \end{align} $$

Due to the property of $\Lambda (n)$ and the upper bound of $|a_{\pi }(n)|$ , the first item of (4.8) is

(4.9) $$ \begin{align} \ll q^{1 / 2}t^{1/2} \sum_{p^{k} \mid q,\ k\ge 1} \frac{1}{p^{k / 2}}\bigg|\sum_{p^{k_1}\le t/p^{k},\ k_1\ge 1}\log p\bigg| \ll (qt)^{1/2}(\log q) \log t. \end{align} $$

If $\chi ^{\prime }(\bmod \thinspace q^{\prime })$ is the primitive Dirichlet character that induces $\chi (\bmod \thinspace q)$ , then $q^{\prime }|q$ . For the second item of (4.8), by (3.3), we get

(4.10) $$ \begin{align} \left|\sum_{l_{2} \le t }\Lambda(l_{2}) a_{\pi}\left(l_{2}\right) \chi\left(l_{2}\right)-\sum_{l_{2} \le t } \Lambda(l_{2}) a_{\pi \otimes \chi^{\prime}}(l_{2})\right| \le \sum_{p \mid q} \sum_{p^k \le t, \ k \ge 1}\left|a_\pi\left(p^k\right)\right| \log p \ll_{\varepsilon}t^{\theta_m+\varepsilon}. \end{align} $$

Without loss of generality, assume that $\chi ^{\prime }$ is the exceptional character $\chi _{1}$ in Lemma 3.2, and let $\beta ^{\prime }$ be the corresponding exceptional zero. It follows from (3.13) and (3.12) that the modulus of the conductor of $\pi \otimes \chi ^{\prime }$ is smaller than $\mathfrak {q}_{\pi }q^{\prime m}$ . We apply [Reference Iwaniec and Kowalski18, Equation (5.52)] to the twisted L-function $L(s,\pi \otimes \chi ^{\prime })$ and conclude that

$$ \begin{align*}\sum_{l_{2} \le t } \Lambda(l_{2}) a_{\pi \otimes \chi^{\prime}}(l_{2})=-\frac{t^{\beta^{\prime}}}{\beta^{\prime}}+O\bigg(\mathfrak{q}_{\pi}^{\frac{1}{2}}q^{\prime\frac{m}{2}}t \exp \left(-\frac{c_\pi}{2} \sqrt{\log t}\right)\bigg). \end{align*} $$

Since Lemma 3.2 gives the bound $\beta ^{\prime } \le 1-c_\pi (\varepsilon _0) q^{-\varepsilon _0}$ , where $\varepsilon _0>0$ is a fixed constant chosen later, for all $q^{\prime }\le q\le (\log t)^{A}$ , we have

$$ \begin{align*} \sum_{l_{2} \le t } \Lambda(l_{2}) a_{\pi \otimes \chi^{\prime}}(l_{2}) \ll_{\pi} t\exp \left(-c_\pi(\varepsilon_0)(\log t) q^{-\varepsilon_0}\right)+ tq^{\frac{m}{2}} \exp \left(-\frac{c_\pi}{4} \sqrt{\log t}\right). \end{align*} $$

Choose $\varepsilon _0=1/(2A)$ . Then inserting (4.7)–(4.10) and the above inequality into (4.6), we get

$$ \begin{align*}\sum_{p\le x}\lambda_{\pi}(p)e(p^{k}\alpha)\log p \ll_{\pi,A} \bigg(1+\frac{x^{k}}{qQ}\bigg)q^{\frac{m+1}{2}}x \exp \left(-c \sqrt{\log x}\right), \end{align*} $$

where $c=\min \left \{\frac {c_\pi }{8},\frac {c_{\pi }(1/(2A))}{2}\right \}$ .

4.3 Proof of Proposition 2.2

Under Hypothesis S, Lemma 3.1 gives that

$$ \begin{align*}\sum_{l_{2} \le t } \Lambda(l_{2}) a_{\pi \otimes \chi^{\prime}}(l_{2}) \ll mt^{\delta+\varepsilon} (\log t)^{2} \log (\mathfrak{q}_{\pi}q^{m}).\end{align*} $$

Inserting (4.7)–(4.10) and the above inequality into (4.6), we get

$$ \begin{align*}\sum_{p\le x}\lambda_{\pi}(p)e(p^{k}\alpha)\log p\ll m^{2}\bigg(1+\frac{x^{k}}{qQ}\bigg)q^{\frac{1}{2}}x^{\delta+\varepsilon}(\log x\mathfrak{q}_{\pi})^{3}+x^{3/4}. \end{align*} $$

5.1 Preliminaries and lemmas

The initial step of the Hardy–Littlewood circle method is to divide the unit interval into the major arcs and the minor arcs. The major arcs $\mathfrak {M}$ and minor arcs $\mathfrak {m}$ are defined as follows:

(5.1) $$ \begin{align} \ \ \ \ \ \ \ \ \mathfrak{M}=\bigcup_{q\le P}\bigcup_{1\le a\le q \atop (a, q)=1}\mathfrak{M}(q, a), \ \ \ \mathfrak{M}(q, a)=\Big[\frac{a}{q}-\frac{1}{qQ},\frac{a}{q}+\frac{1}{qQ}\Big], \ \ \ \ \ \mathfrak{m}=\big[1/Q,1+1/Q\big]\setminus \mathfrak{M}, \end{align} $$

where $1<2P<Q$ , $PQ=N$ , P, Q are parameters to be chosen later. Let

(5.2) $$ \begin{align} T_{k,1}(N, \alpha)=\sum_{p_{1} \le N^{1/k}} \lambda_{\pi}(p_{1}) e(p_{1}^{k} \alpha), \quad H_{k, u^{\prime}}(N, \alpha)=\sum_{p_{u^{\prime}} \le N^{1/k}} e(p_{u^{\prime}}^{k} \alpha) \quad (2 \le u^{\prime}\le u). \end{align} $$

Using Parseval’s identity, we write

(5.3) $$ \begin{align} \sum_{(p_1,p_2,\dots,p_u)\in\mathcal{J}_{k,u}(N)}\lambda_{\pi}(p_{1}) &=\int_0^1 T_{k,1}(N, \alpha) \prod_{u^{\prime}=2}^{u} H_{k,u^{\prime}}(N, \alpha)e(-N \alpha) \mathrm{d} \alpha\nonumber\\ &=\int_{\mathfrak{M}} T_{k,1}(N, \alpha) \prod_{u^{\prime}=2}^{u} H_{k,u^{\prime}}(N, \alpha)e(-N \alpha) \mathrm{d} \alpha \nonumber\\ &\quad+\int_{\mathfrak{m}} T_{k,1}(N, \alpha) \prod_{u^{\prime}=2}^{u} H_{k,u^{\prime}}(N, \alpha)e(-N \alpha) \mathrm{d} \alpha. \end{align} $$

To control the integral on the minor arcs, we need certain estimates concerning exponential sums over primes.

Lemma 5.1 Suppose that $\alpha $ is a real number and there exist integers a and q such that $\alpha $ satisfies

(5.4) $$ \begin{align} \alpha=\frac{a}{q}+\lambda, \quad 1 \le a \le q, \quad(a, q)=1, \quad|\lambda| \le \frac{1}{q^2}. \end{align} $$

1. (1) For $k=1$ ,

   $$ \begin{align*}\sum_{m\le x}\Lambda(m)e(m^{k}\alpha)\ll x (\log x)^3\left(\frac{1}{q}+\frac{1}{x^{2 / 5}}+\frac{q}{x}\right)^{1 / 2}. \end{align*} $$

2. (2) For all $k \ge 2$ and $\varepsilon _1>0$ ,

   $$ \begin{align*}\sum_{m\le x}\Lambda(m)e(m^{k}\alpha) \ll x^{1+\varepsilon_1}\left(\frac{1}{q}+\frac{1}{x^{1 / 2}}+\frac{q}{x^k}\right)^{1 / K^2}, \end{align*} $$
   where $K=2^{k-1}$ .

Sometimes, for $k\ge 2$ , it is better to have the factor $x^{\varepsilon _1}$ above replaced by some power of $\log x$ . This can be done by a simple modification.

Lemma 5.2 Let $k \ge 2$ . Suppose that $\alpha $ is a real number and there exist integers a and q such that $\alpha $ satisfies (5.4). Then

$$ \begin{align*}\sum_{m\le x}\Lambda(m)e(m^{k}\alpha) \ll x (\log x)^{c^{\prime}}\left(\frac{1}{q}+\frac{1}{x^{1 / 2}}+\frac{q}{x^k}\right)^{1 / (2 K^2)}, \end{align*} $$

where $c^{\prime }$ is a positive constant that depends only on k, and K is defined in Lemma 5.1.

Lemma 5.3 Let $k\ge 1$ , $\gamma _k=1/2+\log k/\log 2$ . Suppose that $\alpha $ is a real number and there exist integers a and q such that $\alpha $ satisfies (5.4). Then

$$ \begin{align*}\sum_{m\le x}\Lambda(m)e(m^{k}\alpha)\ll (d(q))^{\gamma_k}\left\{x^{1 / 2} \sqrt{q\left(1+|\lambda| x^k\right)}+x^{4 / 5}+\frac{x}{\sqrt{q\left(1+|\lambda| x^k\right)}}\right\} (\log x)^{c^{\prime\prime}}, \end{align*} $$

where $d(n)$ is the divisor function and $c^{\prime \prime }$ is an absolute positive constant.

5.2 Proof of Theorem 1.1

5.2.1 Major arcs

As for $r_{\pi }(N)$ , we use the orthogonality to derive that

(5.5) $$ \begin{align} \begin{aligned} \int_0^1\left|H_{1, u^{\prime}}(N, \alpha)\right|{}^2 \mathrm{d}\alpha \le \sum_{p_{u^{\prime}_{1}}\le N}\sum_{p_{u^{\prime}_{2}}\le N} \int_0^1 e((p_{u^{\prime}_{1}}-p_{u^{\prime}_{2}})\alpha)\mathrm{d}\alpha \le \sum_{p\le N}1\ll N/\log N. \end{aligned} \end{align} $$

Let $P\le (\log N)^{100B}$ , where P be chosen later. By applying the Cauchy–Schwarz inequality, the above inequality, partial summation and Proposition 2.1,

(5.6) $$ \begin{align} \int_{\mathfrak{M}} & T_{1,1}(N, \alpha) H_{1, 2}(N, \alpha)H_{1, 3}(N, \alpha) e(-N \alpha)\mathrm{d} \alpha \nonumber\\ \quad & \ll\left(\max _{\alpha \in\mathfrak{M}}\left|T_{1,1}(N, \alpha)\right|\right)\left\{\int_0^1\left|H_{1,2}(N, \alpha)\right|{}^2 \mathrm{d} \alpha\right\}^{1 / 2} \left\{\int_0^1\left|H_{1,3}(N, \alpha)\right|{}^2 \mathrm{d} \alpha\right\}^{1 / 2}\nonumber \\ & \ll_{\pi,B}P^{\frac{m+1}{2}}N^{2} \exp \left(- c\sqrt{\log N}\right), \end{align} $$

where c is the constant in Proposition 2.1.

Regarding $w_{\pi }(N)$ , we apply [Reference Blomer and Granville3, Theorem 3] to give

(5.7) $$ \begin{align} \int_0^1\left|H_{2,2}(N, \alpha)H_{2,3}(N, \alpha)\right|{}^2 \mathrm{d} \alpha \le \sum_{p_2^{2}+p_3^{2}=p_2^{\prime2}+p_3^{\prime2}\atop p_2, p_3 ,p_{2}^{\prime },p_3^{\prime}\le N^{1/2} }1\ll N\log N. \end{align} $$

Let $P\le (\log N)^{100B}$ , where P be chosen later. It follows from the Cauchy–Schwarz inequality, the above inequality, partial summation and Proposition 2.1 that

(5.8) $$ \begin{align} \int_{\mathfrak{M}} T_{2,1}(N, \alpha)&H_{2,2}(N, \alpha)H_{2,3}(N, \alpha)H_{2,4}(N, \alpha)H_{2,5}(N, \alpha)e(-N \alpha)\mathrm{d} \alpha\nonumber\\ \quad \ll&\left(\max _{\alpha \in\mathfrak{M}}\left|T_{2,1}(N, \alpha)\right|\right)\left\{\int_0^1\left|H_{2,2}(N, \alpha)H_{2,3}(N, \alpha)\right|{}^2 \mathrm{d} \alpha\right\}^{1 / 2}\nonumber\\ &\quad \times\left\{\int_0^1\left|H_{2,4}(N, \alpha)H_{2,5}(N, \alpha)\right|{}^2 \mathrm{d} \alpha\right\}^{1 / 2}\nonumber\\ \quad \ll&_{\pi,B}P^{\frac{m+1}{2}}N^{\frac{3}{2}} \exp \left(- \frac{c}{2}\sqrt{\log N}\right), \end{align} $$

where c is the constant in Proposition 2.1.

5.2.2 Minor arcs

By the Polya–Vinogradov inequality and [Reference Iwaniec and Kowalski18, Equation (5.48)],

(5.9) $$ \begin{align} \int_0^1\left|T_{k,1}(N, \alpha)\right|{}^2 \mathrm{d} \alpha &\le \sum_{p_{1}\le N^{1/k}}\sum_{p_{2}\le N^{1/k}}\lambda_{\pi}(p_{1}) \bar{\lambda}_{\pi}(p_{2}) \int_0^1 e((p_{1}^{k}-p_{2}^{k})\alpha) \mathrm{d} \alpha\nonumber\\ &\le \sum_{p\le N^{1/k}}|\lambda_{\pi}(p)|^{2 }\ll N^{1/k}m^{2}(\log(N\mathfrak{q}_{\pi}))^{2}. \end{align} $$

Following from the Cauchy–Schwarz inequality, (5.5) and the above inequality, we have

(5.10) $$ \begin{align} \int_{\mathfrak{m}} & T_{1,1}(N, \alpha) H_{1, 2}(N, \alpha)H_{1, 3}(N, \alpha) e(-N \alpha)\mathrm{d} \alpha\nonumber \\ \quad & \le\left(\max _{\alpha \in\mathfrak{m}}\left|H_{1,2}(N, \alpha)\right|\right)\left\{\int_0^1\left|T_{1,1}(N, \alpha)\right|{}^2 \mathrm{d} \alpha\right\}^{1 / 2}\left\{\int_0^1\left|H_{1,3}(N, \alpha)\right|{}^2 \mathrm{d} \alpha\right\}^{1 / 2} \nonumber \\ & \ll \left(\max _{\alpha \in\mathfrak{m}}\left|H_{1,2}(N, \alpha)\right|\right)Nm\log(N\mathfrak{q}_{\pi}). \end{align} $$

Using the first inequality in Lemma 5.1 and partial summation, we get

$$ \begin{align*} \max _{\alpha \in\mathfrak{m}}\left|H_{1,2}(N, \alpha)\right|\ll (Q^{1/2}N^{1/2}+P^{-1/2}N+N^{4/5})(\log N)^{3}. \end{align*} $$

Inserting this bound into (5.10), then

(5.11) $$ \begin{align} \begin{aligned} \int_{\mathfrak{m}} & T_{1,1}(N, \alpha) H_{1,2}(N, \alpha) H_{1,3}(N, \alpha) e(-N \alpha) \mathrm{d} \alpha \ll N^{2}m\log(N\mathfrak{q}_{\pi})(P^{-1/2}+N^{-1/5})(\log N)^{3}. \end{aligned} \end{align} $$

For $w_{\pi }(N)$ , we obtain that

(5.12) $$ \begin{align} \int_0^1\left|T_{2,1}(N, \alpha)H_{2,3}(N, \alpha)\right|{}^2 \mathrm{d} \alpha& \ll \sum_{p_1^{2}+p_3^{2}=p_1^{\prime2}+p_3^{\prime2}\atop p_1, p_3, p_{1}^{\prime }, p_3^{\prime }\le N^{1/2}}|\lambda_{\pi}(p_1)\lambda_{\pi}(p_1^{\prime})|\nonumber\\ &\ll\sum_{p_1\le N^{1/2}}|\lambda_{\pi}(p_{1})|^{2} \sum_{p_1^{\prime}\le N^{1/2}} \sum_{p_1^{2}-p_1^{\prime2}=(p_3^{\prime}+p_3)(p_3^{\prime}-p_3)\atop p_3, p_3^{\prime }\le N^{1/2}}1\nonumber\\ &\ll Nm^{2}\log^{2}(N\mathfrak{q}_{\pi})\log N. \end{align} $$

Treat $w_{\pi }(N)$ in the same way as the estimate for $r_{\pi }(N)$ , to obtain

(5.13) $$ \begin{align} &\int_{\mathfrak{m}} T_{2,1}(N, \alpha)H_{2,2}(N, \alpha)H_{2,3}(N, \alpha)H_{2,4}(N, \alpha)H_{2,5}(N, \alpha)e(-N \alpha)\mathrm{d} \alpha\nonumber\\ & \le\left(\max _{\alpha \in\mathfrak{m}}\left|H_{2,2}(N, \alpha)\right|\right)\left\{\int_0^1\left|T_{2,1}(N, \alpha)H_{2,3}(N, \alpha)\right|{}^2 \mathrm{d} \alpha\right\}^{1 / 2}\nonumber\\&\quad\times\left\{\int_0^1\left|H_{2,4}(N, \alpha)H_{2,5}(N, \alpha)\right|{}^2 \mathrm{d} \alpha\right\}^{1 / 2} \nonumber\\ & \ll \left(\max _{\alpha \in\mathfrak{m}}\left|H_{2,2}(N, \alpha)\right|\right)Nm\log(N\mathfrak{q}_{\pi})\log N. \end{align} $$

In accordance with Lemma 5.2 and by partial summation, we give the following upper bound

$$ \begin{align*}\max _{\alpha \in\mathfrak{m}}\left|H_{2,2}(N, \alpha)\right|\ll N^{1/2}(\log N)^{c^{\prime}}\left(\frac{1}{P}+\frac{1}{N^{1/4}}+\frac{Q}{N}\right)^{1/8}. \end{align*} $$

Inserting the above estimate into (5.13),

(5.14) $$ \begin{align} &\int_{\mathfrak{m}} T_{2,1}(N, \alpha)H_{2,2}(N, \alpha)H_{2,3}(N, \alpha)H_{2,4}(N, \alpha)H_{2,5}(N, \alpha)e(-N \alpha)\mathrm{d} \alpha \nonumber\\ &\ll N^{3/2}m\log(N\mathfrak{q}_{\pi})(\log N)^{c^{\prime}+1}\bigg(P^{-1 / 8} +N^{-1/32}\bigg). \end{align} $$

5.2.3 Choosing parameters

For $r_{\pi }(N)$ , we choose $P=(\log N)^{2B+8}$ . Then by (5.6), (5.11) and (5.3), we have

$$ \begin{align*}r_\pi(N)=\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)} \lambda_{\pi}\left(p_1\right) \ll_{\pi, B} N^{2}(\log N)^{-B}. \end{align*} $$

For $w_{\pi }(N)$ , we choose $P=(\log N)^{8B+8c^{\prime }+16}$ . It follows from (5.8), (5.14), and (5.3) that

$$ \begin{align*}w_\pi(N)=\sum_{(p_1,p_2,p_3,p_4,p_5)\in\mathcal{J}_{2,5}(N)} \lambda_{\pi}\left(p_1\right) \ll_{\pi, B} N^{3/2}(\log N)^{-B}.\end{align*} $$

5.3 Proof of Theorem 1.2

We use Proposition 2.2 to improve the upper bounds in (5.6) and (5.8) respectively. Then

(5.15) $$ \begin{align} \begin{aligned} \int_{\mathfrak{M}} & T_{1,1}(N, \alpha) H_{1, 2}(N, \alpha)H_{1, 3}(N, \alpha) e(-N \alpha)\mathrm{d} \alpha \ll Pm^2 N^{1+\delta+\varepsilon} (\log ( N\mathfrak{q}_{\pi}))^3+N^{7/4}/\log N \end{aligned} \end{align} $$

and

(5.16) $$ \begin{align} \int_{\mathfrak{M}}& T_{2,1}(N, \alpha)H_{2,2}(N, \alpha)H_{2,3}(N, \alpha)H_{2,4}(N, \alpha)H_{2,5}(N, \alpha)e(-N \alpha)\mathrm{d} \alpha\nonumber\\ &\ll Pm^2 N^{1+\delta/2+\varepsilon} (\log ( N\mathfrak{q}_{\pi}))^3\log N+N^{11/8}\log N. \end{align} $$

For $r_\pi (N)$ under Hypothesis S, we choose $P=N^{\frac {2}{3}(1-\delta )}$ in (5.11) and (5.15) to get

$$ \begin{align*}r_\pi(N) \ll m^2 N^{5/3+\delta/3+\varepsilon}(\log ( N\mathfrak{q}_{\pi}))^3. \end{align*} $$

For $w_\pi (N)$ under Hypothesis S, put $P=N^{\frac {1}{3}(1-\delta )}, \ Q=N/P$ and $P^{\prime }=N^{1 / 3}, \ Q^{\prime }=N/P^{\prime }$ . On the minor arcs, one notes that each $\alpha \in \mathfrak {m}$ can be written as (5.1) with $P<q \le Q$ . By Dirichlet’s rational approximation, each $\alpha \in \mathfrak {m}$ can also be written as (2.4) with $1 \le q \le Q^{\prime }$ and $|\lambda | \le 1 / q Q^{\prime }$ . Let $\mathfrak {n}$ be the subset of $\alpha \in \mathfrak {m}$ such that $P^{\prime }<q \le Q^{\prime }$ and $|\lambda | \le 1 / qQ^{\prime }$ . We now apply the second inequality of Lemma 5.1 and partial summation to get

(5.17) $$ \begin{align} \max _{\alpha \in\mathfrak{n}}\left|H_{2,2}(N, \alpha)\right| \ll N^{1/2+\varepsilon}\left(\frac{1}{P^{\prime}}+\frac{1}{N^{1 / 4}}+\frac{Q^{\prime}}{N}\right)^{1 / 4} \ll N^{7/16+\varepsilon}. \end{align} $$

For $\alpha \in \mathfrak {m} \backslash \mathfrak {n}$ , we have

$$ \begin{align*}P<q \le P^{\prime}, \quad|\lambda| \le \frac{1}{q Q^{\prime}} \end{align*} $$

or

$$ \begin{align*}q \le P, \quad \frac{1}{q Q}<|\lambda| \le \frac{1}{q Q^{\prime}}. \end{align*} $$

In either case, there holds

$$ \begin{align*}N^{\frac{1}{6}(1-\delta)}\ll \sqrt{\min \left(P, \frac{N}{Q}\right)} \ll \sqrt{q\left(1+|\lambda| N\right)} \ll \sqrt{P^{\prime}+\frac{N}{Q^{\prime}}} \ll N^{1 /6}. \end{align*} $$

Then Lemma 5.3 gives

(5.18) $$ \begin{align} \max _{\alpha \in \mathfrak{m} \backslash \mathfrak{n}}\left|H_{2,2}(N, \alpha)\right| & \ll\left\{N^{1 / 4} \sqrt{q\left(1+|\lambda| N\right)}+N^{2 / 5}+\frac{N^{1/2}}{\sqrt{q\left(1+|\lambda|N\right)}}\right\} d(q)^{2} (\log N)^{c^{\prime\prime}} \nonumber\\ & \ll N^{5 / 12+\varepsilon} +N^{\frac{1}{3}+\frac{\delta}{6}+\varepsilon}. \end{align} $$

Combing (5.17) and (5.18), we derive that

(5.19) $$ \begin{align} \max _{\alpha \in \mathfrak{m}}\left|H_{2,2}(N, \alpha)\right| \ll N^{\frac{7}{16}+\varepsilon} +N^{\frac{1}{3}+\frac{\delta}{6}+\varepsilon}. \end{align} $$

It follows from (5.19), (5.13), (5.16), and (5.3) that

$$ \begin{align*}w_{\pi}(N)\ll m^{2}(N^{\frac{23}{16}+\varepsilon}+N^{\frac{4}{3}+\frac{\delta}{6}+\varepsilon})(\log ( N\mathfrak{q}_{\pi}))^3. \end{align*} $$

This finishes the proof of Theorem 1.2.

6.1 The modular L-functions

Fix a non-CM form $f\in H_{k_{1}}^*(N_1)$ . Let

$$ \begin{align*}f(z)=\sum_{n=1}^{\infty} \lambda_f(n) n^{(k_{1}-1) / 2} e(n z) \end{align*} $$

be its normalized Fourier expansion at the cusp $\infty $ . For each prime p, let $\theta _{p} \in [0, \pi ]$ be the unique angle such that $\lambda _{f}(p)=2 \cos \theta _{p}$ . The modular L-function $L(s, f)$ associated with f has the Euler product representation

$$ \begin{align*}L(s, f)=\sum_{n=1}^{\infty} \frac{\lambda_{f}(n)}{n^{s}}=\prod_{p}\left(1-\frac{\lambda_{f}(p)}{p^{s}}+\frac{\chi_{0}(p)}{p^{2 s}}\right)^{-1}, \quad \mathrm{Re}(s)>1, \end{align*} $$

where $\chi _{0}$ is the trivial Dirichlet character modulo $N_1$ . Let’s rewrite the Euler product as

$$ \begin{align*}L(s, f)=\prod_{p \mid q}\left(1-\frac{\left(-\lambda_{p} p^{-\frac{1}{2}}\right)}{p^{s}}\right) \prod_{p \nmid q} \prod_{j=0}^{1}\left(1-\frac{e^{i(2 j-1) \theta_{p}}}{p^{s}}\right)^{-1}, \quad \mathrm{Re}(s)>1, \end{align*} $$

where $\lambda _{p} \in \{-1,1\}$ is the eigenvalue of the Atkin–Lehner operator $|_{k_1} W\left (Q_{p}\right )$ . According to Deligne [Reference Deligne7], for any prime p, there are two numbers $\alpha _{f}(p)$ and $\beta _{f}(p)$ such that

(6.1) $$ \begin{align} \alpha_{f}(p)=e^{i \theta_{p}},\quad \beta_{f}(p)=e^{-i \theta_{p}} \end{align} $$

and

$$ \begin{align*}\lambda_{f}(p)=\alpha_{f}(p)+\beta_{f}(p) \end{align*} $$

with $\theta _{p} \in [0, \pi ]$ . Deligne proved the Ramanujan–Petersson conjecture, which states that

$$ \begin{align*}\left|\lambda_{f}(n)\right| \le d(n). \end{align*} $$

For each $j \ge 1$ and $\mathrm {Re}(s)>1$ , the jth symmetric power L-function attached to f is defined by

(6.2) $$ \begin{align} L\left(s, \mathrm{sym}^{j} f\right)=\sum_{n=1}^{\infty} \frac{\lambda_{\mathrm{sym}^{j} f}(n)}{n^{s}}=\prod_{p \mid q} \prod_{m=0}^{j}\left(1-\frac{\alpha_{ \mathrm{sym}^{j}f,m }(p)}{p^{s}}\right)^{-1} \prod_{p \nmid q} \prod_{m=0}^{j}\left(1-\frac{e^{i(2 m-j) \theta_{p}}}{p^{s}}\right)^{-1}. \end{align} $$

The values $\alpha _{ \mathrm {sym}^{j}f,m }(p)$ can be determined by using [Reference Soundararajan and Thorner30, Appendix] and [Reference David, Gafni, Malik, Prabhu and Turnage-Butterbaugh6, Appendix]. Note that $L\left (s,\mathrm {sym}^{1} f\right )=L(s, f)$ and $L\left (s,\mathrm {sym}^{0} f\right )=\zeta (s)$ . It is well known that $\lambda _{\mathrm {sym}^{j} f}(n)$ is a real multiplicative function. Via (6.2), one easily verify that

(6.3) $$ \begin{align} \lambda_{\mathrm{sym}^{j} f}(p)=\frac{\sin(n+1)\theta_p}{\sin \theta_p}=U_{j}\left(\cos \theta_{p}\right), \quad p \nmid N_1. \end{align} $$

Let $\Gamma _{\mathbb {R}}(s):=\pi ^{-s/2}\Gamma (s/2)$ and $\Gamma _{\mathbb {C}}(s)=\Gamma _{\mathbb {R}}(s) \Gamma _{\mathbb {R}}(s+1)$ . With these definitions, we obtain that

(6.4) $$ \begin{align} L\left(s,\left(\mathrm{sym}^j f\right)_{\infty}\right)= \begin{cases}q_{\mathrm{sym}^j f} \prod_{m=1}^{(j+1) / 2} \Gamma_{\mathbb{C}}\left(s+\left(m-\frac{1}{2}\right)(k-1)\right), & \text { if } j \text { is odd}, \\ q_{\mathrm{sym}^j f} \Gamma_{\mathbb{R}}(s+r) \prod_{m=1}^{j / 2} \Gamma_{\mathbb{C}}(s+m(k-1)), & \text { if } j \text { is even},\end{cases} \end{align} $$

where $q_{\mathrm {sym}^{j} f}$ is a suitable integer, and $r=0$ if $j \equiv 0\ (\!\bmod \ 4)$ , and $r=1$ if $j \equiv 2\ (\!\bmod \ 4)$ . Let $C(\mathrm {sym}^{j}f)$ be the analytic conductor of $L(s, \mathrm {sym}^{j}f)$ . For any primitive Dirichlet character $\chi $ , we give a proof that the twist L-function $L(s, \mathrm {sym}^{j}f\otimes \chi )$ has no Landau–Siegel zeros.

Lemma 6.1 Let Q be a large positive number and $\chi $ be a primitive Dirichlet character modulo q with $q\le Q$ . Define

$$\begin{align*}y_{f} = \begin{cases} 0 & \text{if } f \text{ has squarefree level or } f \text{ corresponds}\\ &\text{with a non-CM elliptic curve over } \mathbb{Q}, \\ 2 & \text{otherwise}. \end{cases} \end{align*}$$

There exists a constant $c_{9}>0$ such that if $n \ge 1$ , then $L\left (s, \operatorname {sym}^n f \otimes \chi \right ) \neq 0$ for

$$ \begin{align*} \mathrm{Re}(s) \ge 1-\frac{c_{9}}{n^{2+y_{f}} \log (nk_1N_1Q (3+|\operatorname{Im}(s)|))}. \end{align*} $$

6.2 The Sato–Tate conjecture

Serre’s modular version of the Sato–Tate conjecture posits that if f is non-CM, then the sequence $\left \{\theta _{p}\right \}$ is uniformly distributed in the interval $[0, \pi ]$ with respect to the measure $\mathrm {d} \mu _{S T}:=(2 / \pi ) \sin ^{2} \theta \mathrm {d} \theta $ . More explicitly, let $I_{0}$ be a subinterval of $[-2,2]$ , and for a positive real number x, define the set

$$ \begin{align*} N_{I_{0}}(f, x):=\#\left\{p \le x: \mathrm{gcd}(p, N_1)=1, \lambda_f(p) \in I_{0}\right\}. \end{align*} $$

The Sato–Tate conjecture states that for a fixed non-CM form $f \in H_{k_{1}}^*(N_1)$ , we have

$$ \begin{align*} \lim _{x \rightarrow \infty} \frac{N_{I_{0}}(f, x)}{\pi(x)}=\int_{I_{0}} \mu_{\infty}(t) \mathrm{d} t, \end{align*} $$

where

$$ \begin{align*}\mu_{\infty}(t):= \begin{cases} \frac{1}{\pi} \sqrt{1-\frac{t^2}{4}}, & \text { if } t \in[-2,2], \\ 0, & \text { otherwise. } \end{cases} \end{align*} $$

If $\chi _I$ is the indicator function of the interval $I=[\alpha , \beta ]\subseteq [0,\pi ]$ , then

$$ \begin{align*}\sum_{\substack{p \le x \\ \lambda_{f}(p) \in [2\cos\beta, 2\cos\alpha]}}1=\sum_{p \le x} \chi_I\left(\theta_p\right). \end{align*} $$

Thorner [Reference Thorner32] approximate $\chi _I$ with a differentiable function using the following construction formula in [Reference Vinogradov37].

Lemma 6.2 Let R be a positive integer, and let $a, b, \delta ^{\prime } \in \mathbb {R}$ satisfy

$$ \begin{align*}0<\delta^{\prime}<1 / 2, \quad \delta^{\prime} \le b-a \le 1- \delta^{\prime}. \end{align*} $$

Then there exists an even periodic function $g(y)$ with period 1 satisfying

1. (1) $g(y)=1$ when $y \in \left [a+\frac { \delta ^{\prime }}{2} , b-\frac { \delta ^{\prime }}{2}\right ]$ ,

2. (2) $g(y)=0$ when $y \in \left [b+\frac { \delta ^{\prime }}{2}, 1+a-\frac { \delta ^{\prime }}{2}\right ]$ ,

3. (3) $0 \le g(y)\le 1$ when y is in the rest of the interval $\left [a-\frac { \delta ^{\prime }}{2} , 1+a-\frac { \delta ^{\prime }}{2} \right ]$ , and $g(y)$ has the Fourier expansion

   $$ \begin{align*}g(y)=b-a+\sum_{n=1}^{\infty}\left(a_n \cos (2 \pi n y)+b_n \sin (2 \pi n y)\right), \end{align*} $$
   where for all $n \ge 1$ ,
   $$ \begin{align*}\left|a_n\right|,\left|b_n\right| \le \min \left\{2(b-a), \frac{2}{n \pi}, \frac{2}{n \pi}\left(\frac{R}{\pi n \delta^{\prime}}\right)^R\right\}. \end{align*} $$

6.3 Proof of Theorem 1.4

If $\chi _I$ is the indicator function of the interval $I=[\alpha , \beta ]\subseteq [0,\pi ]$ , then

$$ \begin{align*}\# \mathcal{J}_{1,3,f, I}(N)=\sum_{\substack{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N) \\ \lambda_{f}(p_1) \in [2\cos\beta, 2\cos\alpha]}} 1=\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)} \chi_I\left(\theta_{p_1}\right). \end{align*} $$

Let $g(\theta _{p_1})$ be defined as in Lemma 6.2 with $a=\frac {\alpha }{2 \pi }-\frac {\delta ^{\prime }}{2}$ and $b=\frac {\beta }{2 \pi }+\frac {\delta ^{\prime }}{2}$ . Thorner [Reference Thorner32] constructed $g^{+}(\theta _{p_1}; I, \delta ^{\prime })=g\Big (\frac {\theta _{p_1}}{2 \pi }\Big )+g\Big (-\frac {\theta _{p_1}}{2 \pi }\Big )$ as a pointwise upper bound for $\chi _I(\theta _{p_1})$ . By repeating this construction with $a=\frac {\alpha }{2 \pi }+\frac {\delta ^{\prime }}{2}$ , and $b=\frac {\beta }{2 \pi }-\frac {\delta ^{\prime }}{2}$ , we can get a lower bound $g^{-}(\theta _{p_1}; I, \delta ^{\prime })$ for $\chi _I(\theta _{p_1})$ . Thus, according to Lemma 6.2, we can express $g^{\pm }(\theta _{p_1}; I, \delta ^{\prime })$ in terms of the basis of Chebyshev polynomials of the second kind $\{U_n(\cos (\theta _{p_1})) \}_{n=0}^{\infty }$ as follows:

(6.5) $$ \begin{align} g^{\pm}(\theta_{p_1}; I, \delta^{\prime})=a_0^{\pm}(I, \delta^{\prime})-a_2^{\pm}(I, \delta^{\prime})+\sum_{n=1}^{\infty}\left(a_n^{\pm}(I, \delta^{\prime})-a_{n+2}^{\pm}(I, \delta^{\prime})\right) U_n(\cos (\theta_{p_1})), \end{align} $$

where $a_n^{\pm }(I, \delta ^{\prime })$ represents the nth Fourier coefficient in the cosine expansion of $g^{\pm }(\theta _{p_1}; I, \delta ^{\prime })$ . Furthermore, we derive

(6.6) $$ \begin{align} \begin{aligned} \left|a_0^{\pm}(I, \delta^{\prime})-a_2^{\pm}(I, \delta^{\prime})-\mu_{S T}(I)\right| & \ll \delta^{\prime} , \end{aligned} \end{align} $$

and for $n\ge 1$ ,

(6.7) $$ \begin{align} \left|a_n^{\pm}(I, \delta^{\prime})\right|,\left|a_{n+2}^{\pm}(I, \delta^{\prime})\right| \le \min \left\{2(b-a), \frac{2}{n \pi}, \frac{2}{n \pi}\left(\frac{R}{\pi n \delta^{\prime}}\right)^R\right\}.\end{align} $$

Following the method in [Reference Thorner31], when summing $g^{\pm }\left (\theta _{p_1}; I, \delta ^{\prime }\right )$ over $\mathcal {J}_{1, 3}(N)$ , we can interchange the order of summation by choosing R to ensure absolute convergence. Thus by (6.5), we obtain the following expression:

$$ \begin{align*}\begin{aligned} \sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)} g^{\pm}(\theta_{p_1}; I, \delta^{\prime}) &=\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)}(a_0^{\pm}(I, \delta^{\prime})-a_2^{\pm}(I, \delta^{\prime}))\nonumber\\ &\quad+\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)}\bigg(\sum_{n=1}^{\infty}\left(a_n^{\pm}(I, \delta^{\prime})-a_{n+2}^{\pm}(I, \delta^{\prime})\right) U_n(\cos (\theta_{p_1}))\bigg)\nonumber\\ &=(a_0^{\pm}(I, \delta^{\prime})-a_2^{\pm}(I, \delta^{\prime}))\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)}1\nonumber\\ &\quad+\sum_{n=1}^{\infty}\left(a_n^{\pm}(I, \delta^{\prime})-a_{n+2}^{\pm}(I, \delta^{\prime})\right)\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)}U_n(\cos (\theta_{p_1})).\end{aligned} \end{align*} $$

It follows from (6.6), (6.7), and (6.3) and the above equation that

(6.8) $$ \begin{align} \begin{aligned} \sum_{(p_1,p_2,p_3) \in\mathcal{J}_{1,3}(N)} \chi_I\left(\theta_{p_1}\right) =(\mu_{\mathrm{ST}}(I)+O(\delta^{\prime}))\bigg(\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)} 1\bigg)+O(A), \end{aligned} \end{align} $$

where

(6.9) $$ \begin{align} A=\sum_{n=\delta^{\prime-1}}^{\infty}\frac{4}{n \pi}\left(\frac{R}{\pi n \delta^{\prime}}\right)^R \sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)}\lambda_{\mathrm{sym}^{n}f}(p_1)+\sum_{n=1}^{\delta^{^\prime-1}} \frac{2}{n\pi}\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)}\lambda_{\mathrm{sym}^{n}f}(p_1). \end{align} $$

Let $\pi _{f} \in \mathfrak {E}_{2}$ correspond to f. It is known from [Reference Newton and Thorne25] and [Reference Newton and Thorne26] that $\mathrm {sym}^{j} f$ corresponds to $\mathrm {sym}^{j}\pi _{f} \in \mathfrak {E}_{j+1}$ . Utilizing the method outlined in the proof of Theorem 1.2, (3.7) with the zero-free region in Lemma 6.1, and [Reference Thorner33, Equation (6.5)], if the level of f is squarefree, we derive the inequality

$$ \begin{align*}\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)} \lambda_{\mathrm{sym}^{n}f}(p_1)\ll (n+1) N^{2} \exp\bigg(-\frac{c_{9}\log N}{n^2 \log (nk_1N_1\log N )} \bigg). \end{align*} $$

Based on Lemma 6.2, we choose $\delta ^{\prime -1}=(\log N)^{1/2} / \log (k_1 N_1 \log N)$ , and $R=2$ in (6.9), then

$$ \begin{align*}A\ll \frac{N^{2} (\log N)^{1/2}}{\log (k_1 N_1 \log N)}\exp\bigg(-\frac{c_{9}\log^{2} (k_1 N_1 \log N)}{ \log ( k_1N_1 \log^{2} N )} \bigg). \end{align*} $$

Inserting this upper bound of A into (6.8), we have

$$ \begin{align*}\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)} \chi_I\left(\theta_{p_1}\right) =\mu_{\mathrm{ST}}(I)\left(\# \mathcal{J}_{1,3}(N)\right)+O\left(\frac{\log (k_1 N_1 \log N)}{ (\log N)^{1/2} }\left(\# \mathcal{J}_{1,3}(N)\right)\right). \end{align*} $$

Similar to the discussion above, if the level of f is not squarefree, we choose ${\delta ^{\prime -1}=(\log N)^{1/4} / \log (k_1 N_1 \log N)}$ and $R=2$ in (6.9) to obtain

$$ \begin{align*}\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)} \chi_I\left(\theta_{p_1}\right) =\mu_{\mathrm{ST}}(I)\left(\# \mathcal{J}_{1,3}(N)\right)+O\left(\frac{\log (k_1 N_1 \log N)}{ (\log N)^{1/4} }\left(\# \mathcal{J}_{1,3}(N)\right)\right). \end{align*} $$

Under Hypothesis S, and using the method outlined in the proof of Theorem 1.2, (3.7), and [Reference Thorner33, Equation (6.5)], we derive the inequality:

$$ \begin{align*}\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)} \lambda_{\mathrm{sym}^{n}f}(p_1)\ll (n+1) N^{5/3+\delta/3}(\log N)^3\log (k_1N_1n). \end{align*} $$

In (6.9), we choose $\delta ^{\prime }=RN^{\delta /6-1/6}\log ( k_1N_1N)$ and $R=2$ based on Lemma 6.2, then

$$ \begin{align*}A \ll N^{11/6+\delta/6}(\log N)^{3}. \end{align*} $$

Substituting this upper bound of A into (6.8), we have

$$ \begin{align*}\sum_{(p_1,p_2,p_3)\in\mathcal{J}_{1,3}(N)} \chi_I\left(\theta_{p_1}\right) =\mu_{\mathrm{ST}}(I)\left(\# \mathcal{J}_{1,3}(N)\right)+O(N^{11/6+\delta/6}(\log N)^{3}). \end{align*} $$