Summation formulae of multiplicative functions over arithmetic progressions and applications

Abstract

In this paper, we investigate the asymptotic distribution of a class of multiplicative functions over arithmetic progressions without the Ramanujan conjecture. We also apply these results to some interesting arithmetic functions in automorphic context, such as coefficients of automorphic L-functions, coefficients of their Rankin–Selberg.

1 Introduction

Problems concerning the asymptotic distribution of arithmetic functions in arithmetic progressions are very classical in analytic number theory, and appear all over the place. Let q be a positive integer and a be an integer prime to q, and let $\{a_n\}_{n=1}^{\infty }$ be an arithmetic sequence of complex numbers. Define

$$ \begin{align*} \mathcal{S}(x;a,q)=\sum_{\substack{n\leq x\\n\equiv a\,(\mathrm{mod}\,q)}}a_n. \end{align*} $$

One expects the sequence to be generally well distributed in residue classes to modulo q, namely

(1.1) $$ \begin{align} \mathcal{S}(x;a,q)= \frac{1}{\varphi(q)}\sum_{\substack{n\leq x\\ (a,q)=1}}a_n+\text{small error}, \end{align} $$

where $\varphi $ is Euler’s function. For example, if $a_n=\Lambda (n),$ the von Mangoldt function, the Siegel–Walfisz theorem says that for any $q\leq \log ^A x$

$$ \begin{align*} \sum_{\substack{n\leq x\\n\equiv a\,(\mathrm{mod}\,q)}}\Lambda(n)=\frac{x}{\varphi(q)}+O\big(x\exp(-c_A\sqrt{\log x})\big), \end{align*} $$

where A is any real number and $c_A$ is some constant depending only on $A.$ If ${a_n=\tau _k(n),}$ the number of representations of n as the product of k factors, (1.1) holds for $q\leq x^{\theta _k-\varepsilon }$ with

$$\begin{align*}\theta_2=\frac{2}{3}, \quad \theta_3=\frac{1}{2}+\frac{1}{82},\quad \theta_4=\frac{1}{2},\ldots \end{align*}$$

(see the details in [10]). Another example is $a_n=\lambda _f(n),$ the normalized Fourier coefficients of a holomorphic cusp form f, Smith [23] showed that (1.1) holds uniformly for $q\leq x^{\frac {2}{3}}.$ Moreover, Murty gave some interesting remarks on Smith’s work and said “It is likely that the methods of [23] are applicable for coefficients of Dirichlet series attached to automorphic representation of higher $\mathrm {GL}(n,{{\mathbb A}}_{{\mathbb Q}})$ ” at the end of this paper.

Let $d\geq 2$ be an integer, and let $\mathcal {F}(d)$ be the set of all cuspidal automorphic representations $\pi $ of $\mathrm {GL}(d)$ over ${\mathbb Q}$ with trivial central character. Let $q_{\pi }$ denote the arithmetic conductor of $\pi $ . For each $\pi \in \mathcal {F}(d)$ , the corresponding L-function is defined by absolutely convergent Dirichlet series as

$$ \begin{align*} L(s,\pi) =\sum_{n=1}^{\infty}\lambda_{\pi}(n)n^{-s} \end{align*} $$

for $\Re s>1$ . Motivated by the remarks of Murty as above, it is interesting to study the distribution of Dirichlet coefficients $\lambda _{\pi }(n)$ in arithmetic progressions

(1.2) $$ \begin{align} \sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}\lambda_{\pi}(n). \end{align} $$

In general, one needs to replace the congruence $n\equiv a\,(\mathrm {mod}\,q)$ in (1.2) by a character sum of additive or multiplicative characters modulo q. Smith [23] chose to use the additive characters and then investigated the properties of generating series of $\lambda _f(n)\text {e}(an/q)$ including the analytical continuation and functional equation, where $\text {e}(x):=\exp (2\pi i x)$ for any $x\in \mathbb {R}$ . However, for the higher rank case on $\mathrm {GL}(d)$ , the functional equation of Dirichlet series $\sum \limits _{n=1}^{\infty }\lambda _{\pi }(n)\text {e}(an/q)n^{-s} $ is complicated and lacks a little symmetry structure (see [17] for details). Hence, in contrast to the work of Smith, we shall replace the congruence in (1.2) by a character sum of multiplicative characters, and can prove the following result.

Theorem 1.1 If $\pi \in \mathcal {F}(d)$ with $(q,aq_{\pi })=1$ , then we have

$$\begin{align*}\sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}\lambda_{\pi}(n)\ll\left\{ \begin{array}{ll} \tau_{d}(q)x^{1-\frac{1}{d}}, \;& \text{if} q\leq x^{\frac{1}{d}},\\[1.5mm] \tau_{d^2}(q)x^{1-\frac{d+1}{d^2+1}}\log x,\;& \text{if} q\leq x^{\frac{2}{d^2+1}}. \end{array} \right. \end{align*}$$

Assume the generalized Ramanujan conjecture holds for $\pi $ , then we have

$$\begin{align*}\sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}\lambda_{\pi}(n)\ll \tau_{d}(q)\big(q^{\frac{d-1}{2}}\log q+ x^{1-\frac{2}{d+1}}\big) \end{align*}$$

for $q\leq x^{2/(d+1)}.$ The implied constants all depend on $\pi $ only.

Another important arithmetic function is the coefficient $\lambda _{\pi \times \widetilde {\pi }}(n)$ of the Rankin–Selberg L-function $L(s,\pi \times \widetilde {\pi })$ , where $\widetilde {\pi }$ denotes the contragredient of $\pi \in \mathcal {F}(d)$ . This example is also our motivation for using the multiplicative characters to detect the congruence.

Theorem 1.2 If $\pi \in \mathcal {F}(d)$ with $(q,aq_{\pi })=1$ , then we have

$$\begin{align*}\sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}\lambda_{\pi\times \widetilde{\pi}}(n)=\frac{c_{\pi,q}}{\varphi(q)}x+O\Big(\tau_{d^2}(q)q^{\frac{d^2-1}{2}}\log q \Big)+O\Big(\tau_{d^2}(q)x^{\frac{d^2-1}{d^2+1}}\Big) \end{align*}$$

for $q\leq x^{\frac {2}{d^2+1}}$ , where $c_{\pi ,q}$ is defined by $ c_{\pi ,q}=\mathop {\mathrm {Res}}\limits _{s=1} \big (L(s, \pi \times \widetilde {\pi } )\big )\prod \limits _{p|q}L(1, \pi _p\times \widetilde {\pi }_p)^{-1}$ , and the implied constant depends on $\pi $ only.

As in the argument of Theorem 1.1, if the coefficients $\lambda _{\pi }(n)$ of L-functions are not all nonnegative, we can produce a formula for $\sum _{n\leq x}\lambda _{\pi }(n)$ in terms of a sum of $\lambda _{\pi }(n)$ over a short interval. Our next goal is to strengthen Theorem 1.1 for special cases by improving some related estimates over short intervals.

Let k and N be positive integers with k even and N square-free, and $\Gamma _0(N)$ be the group of matrices $\gamma =\begin {pmatrix} a & b \\ c& d \end {pmatrix} \in \mathrm {SL}(2,{\mathbb Z})$ with the condition $c\equiv 0(\text {mod}\, N)$ . Let $H_k^{*}(N)$ denote the set of arithmetically normalized primitive cusp forms of weight k for $\Gamma _0(N)$ which are eigenfunctions of all the Hecke operators. Any $f\in H_k^{*}(N)$ has a Fourier expansion at infinity given by

$$ \begin{align*} f(z)=\sum_{n=1}^\infty \lambda_f(n)n^{\frac{k-1}{2}}\text{e}(nz), \end{align*} $$

where $ \lambda _f(1)=1$ and the eigenvalues $\lambda _f(n)\in \mathbb {R}$ . Deligne’s bound gives

(1.3) $$ \begin{align} |\lambda_f(n)|\leq \tau(n) \end{align} $$

for all $n\geq 1$ , where we put as usual $\tau _2(n)=\tau (n)$ . The eigenvalues $\lambda _f(n)$ enjoy the multiplicative property

$$ \begin{align*} \lambda_f(m)\lambda_f(n)=\sum_{\substack{d|(m,n)\\ (d,N)=1}}\lambda_f\Big(\frac{mn}{d^2}\Big) \end{align*} $$

for all integers $m, n\geq 1.$ In particular, $\lambda _f(n)$ are multiplicative. The Hecke L-function $L(s,f)$ associated with f has the Euler product representation

$$\begin{align*}L(s,f)=\sum_{n\geq 1}\frac{\lambda_f(n)}{n^s}=\prod_p\left(1-\frac{\lambda_f(p)}{p^s}+\frac{\psi_0(p)}{p^{2s}}\right)^{-1}, \end{align*}$$

where $\psi _0$ denotes the principal character modulo N. We rewrite the Euler product as

$$\begin{align*}L(s,f)=\prod_p\left(1-\frac{\alpha_f(p)}{p^s}\right)^{-1}\left(1-\frac{\beta_f(p)}{p^s}\right)^{-1}, \end{align*}$$

where $\alpha _f(p), \beta _f(p)$ are complex numbers with

$$ \begin{align*} \left\{ \begin{array}{ll} \alpha_f(p)=\varepsilon_p p^{-\frac{1}{2}}, \,\beta_f(p)=0, \quad &\text{if }\, p|N, \\ \alpha_f(p)=\overline{\beta_f(p)}, \,|\alpha_f(p)|=|\beta_f(p)|=1, \quad &\text{if }\, p \nmid N, \end{array} \right. \end{align*} $$

and $\varepsilon _p\in \{\pm 1\}$ . For each $d\geq 1$ , we define the twisted dth symmetric power L-function by the degree $d+1$ Euler product

(1.4) $$ \begin{align} L(s,\mathrm{sym}^d f )=\prod_p\prod_{0\leq j\leq d}\left(1-\frac{\alpha_f(p)^{d-j}\beta_f(p)^{j}}{p^s}\right)^{-1}:= \sum_{n\geq 1}\frac{\lambda_{\mathrm{sym}^d f}(n)}{n^s}. \end{align} $$

Note that $L(s,\mathrm {sym}^1f)=L(s,f)$ .

Recently, Newton and Thorne [19, Theorem B] proved that if $d\geq 1$ , then the dth symmetric power lift $\mathrm {sym}^d f$ corresponds to a cuspidal automorphic representation of $\mathrm { GL}(d+1,{{\mathbb A}}_{{\mathbb Q}})$ with trivial central character. Moreover, for each prime $p,$ let $\theta _p\in [0,\pi ]$ be the unique angel such that $\lambda _f(p)=2\cos \theta _p$ . The Sato–Tate conjecture states that the sequence $\{\theta _p\}$ is equidistributed in the interval $[0,\pi ]$ with respect to the measure ${\text {d}}\mu _{ST}:=(2/\pi )\sin ^2\theta {\text {d}}\theta .$ Equivalently, for any continuous function $g\in C([0,\pi ])$ , one has

(1.5) $$ \begin{align} \sum_{p\leq x\atop p\nmid N}g(\theta_p)\sim \Big(\int_{0}^{\pi}g(\theta) {\text{d}}\mu_{ST}\Big)\frac{x}{\log x}\qquad \text{as} \quad x\longrightarrow \infty. \end{align} $$

This is now a theorem of Barnet-Lamb, Geraghty, Harris, and Taylor [1].

For this special arithmetic function $\lambda _{\mathrm {sym}^d f}(n)$ on $\mathrm {GL}_{d+1}$ , we get the following result.

Theorem 1.3 Let $f\in H_k^{*}(N)$ and $\lambda _{\mathrm {sym}^d f}(n)$ be the coefficients of $L(s,\mathrm {sym}^d f )$ . For $(q,aN)=1,$ we have

$$\begin{align*}\sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}\lambda_{\mathrm{sym}^d f}(n)\ll \tau_{d+1}(q)\Big(q^{\frac{d}{2}}(\log q)^{1-\gamma_d}+x^{\frac{d}{d+2}}(\log x)^{-\gamma_d}\Big) \end{align*}$$

for $q\leq x^{\frac {2}{d+2}}$ , where $\gamma _d=1-\frac {4(d+1)}{d(d+2)\pi } \cot \big (\frac {\pi }{2(d+1)}\big )>0.15$ , and the implied constant depends on f and d.

Remark 1.1 For any fixed $f\in H_k^{*}(N)$ and $(q,aN)=1,$ Smith [23] obtained a uniform estimate

$$\begin{align*}\sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}\lambda_{f}(n)\ll \tau(q)x^{\frac{1}{3}}\log x \end{align*}$$

for $q\leq x^{2/3}$ . Compared this with the case $d=1$ in Theorem 1.3, it is obvious that our result is of a smaller size.

2 The main result

All these results in the theorems above are some specific applications of our technical formulae in Theorem 2.1. To state this core result, we need to describe the situation that we consider. Inspired by the series of works [5–8] of Duke and Iwaniec who have developed several techniques for estimating the coefficients of L-functions that satisfy standard functional equations, this paper here is to investigate the average order of a class of multiplicative functions over arithmetic progressions under some similar conditions.

1. (A1) Euler product and Dirichlet series. Let $\mathcal {A} = \{\mathcal {A}_p\}$ be a sequence of square complex matrices of order d indexed by primes, with monic characteristic polynomial $P_p(x)=P_p^{\mathcal {A} }(x)\in {\mathbb C}[x]$ and roots $\alpha _j(p)$ . Then our general L-function $L(s, \mathcal {A})$ will be given by

   (2.1) $$ \begin{align} L(s, \mathcal {A}) = \prod_p \prod_{j=1}^d \Big(1-\frac{\alpha_j(p)}{p^s}\Big)^{-1} =\sum_{n=1}^\infty\frac{a_n}{n^s}, \end{align} $$
   where we assume that the product and the series are absolutely convergent for $\Re (s)> 1$ . Note that $|\alpha _j(p)| \leq p$ for all p, which is implied by the convergence of the Euler product for $\Re s> 1$ .

2. (A2) Analytic continuation. There is some $m = m(\mathcal {A})$ such that $L(s, \mathcal {A})$ can be continued analytically over all of $\mathbb {C}$ except possibly for a pole of order m at $s=1$ .

3. (A3) Functional equation. Let a Gamma factor $\Delta (s)$ be defined by

   $$\begin{align*}\Delta(s)=\prod_{j=1}^d \Gamma_{\mathbb{R}}(s + \mu_j), \end{align*}$$
   where $\Gamma _{\mathbb {R}}(s)=\pi ^{-s/2}\Gamma (s/2)$ , and $\mu _j$ is an arbitrary complex number with $\Re \mu _j>-1$ for each $1\leq j\leq d$ . The complete L-function
   $$\begin{align*}\Lambda(s, \mathcal {A}) := q_{\mathcal {A}}^{\frac{s}{2}} \Delta(s)L(s, \mathcal {A}) \end{align*}$$
   has finite order, and satisfies the functional equation
   $$\begin{align*}\Lambda(1-s, \mathcal {A}) = \omega_{\mathcal {A}} \overline{\Lambda(1-\bar{s}, \mathcal {A})}, \end{align*}$$
   where $q_{\mathcal {A}} $ is a positive integer and $\omega _{\mathcal {A}} $ is a complex number with $|\omega _{\mathcal {A}}| = 1$ , which are called the arithmetic conductor and root number of $\mathcal {A}$ , respectively.

4. (A4) $\mathrm {GL}(1)$ twists. Let $\chi (\mathrm {mod}\,q)$ be a primitive Dirichlet character with $q>1$ and $(q, q_{\mathcal {A}})=1$ . The twisted L-function

   $$ \begin{align*}L(s, \mathcal {A} \otimes \chi)=\prod_{p} \prod_{j=1}^d \Big(1-\frac{\alpha_j(p)\chi(p)}{p^s}\Big)^{-1}=\sum_{n=1}^\infty\frac{a_n\chi(n)}{n^s} \end{align*} $$
   can be analytically continued to be an entire function. Moreover, the complete L-function
   $$\begin{align*}\Lambda(s, \mathcal {A}\otimes \chi) := q_{\mathcal {A}\otimes \chi}^{\frac{s}{2}}\Delta\big(s + \kappa_{\mathrm{sgn(\chi)}}\big)L(s, \mathcal {A}\otimes \chi) \end{align*}$$
   has finite order, and satisfies the functional equation
   (2.2) $$ \begin{align} \Lambda(s, \mathcal {A}\otimes \chi) = \omega_{\mathcal {A}\otimes \chi} \overline{\Lambda(1-\bar{s}, \mathcal {A}\otimes \chi)}, \end{align} $$
   where $q_{\mathcal {A}\otimes \chi }>0$ and $\omega _{\mathcal {A}\otimes \chi } $ is a complex number with $|\omega _{\mathcal {A}\otimes \chi } | = 1$ . We emphasize that the Gamma factor of $\mathcal {A}\otimes \chi $ depends on the parity of $\chi $ , but not on the characters $\chi $ . For $(q, q_{\mathcal {A}})=1$ , we also assume that $q_{\mathcal {A}\otimes \chi } = q_{\mathcal {A}}\, q^{d}$ and the root number $ \omega _{\mathcal {A}\otimes \chi }$ is given by
   $$ \begin{align*} \omega_{\mathcal {A}\otimes\chi}=\eta_{\mathcal {A},\mathrm{sgn(\chi)}}\chi(q_{\mathcal {A}})\Big(\frac{\tau(\chi)}{\sqrt{q}}\Big)^{d}, \end{align*} $$
   where $\eta _{\mathcal {A},\mathrm {sgn(\chi )}}$ with $|\eta _{\mathcal {A},\mathrm {sgn(\chi )}}|=1$ depends on $\mathcal {A}$ and the parity of $\chi $ only, $\tau (\chi )$ is the Gauss sum
   $$\begin{align*}\tau(\chi)=\sum_{b\,(\mathrm{mod}\,q)}\chi(b)\text{e}\Big(\frac{b}{q}\Big). \end{align*}$$

Some hypotheses about the size of the coefficients have to be assumed in order to prove our result. The Ramanujan conjecture (RC) states that for any $\varepsilon>0$ , $a_n\ll n^\varepsilon $ for all $n\geq 1.$ As is well known, RC has been proved only for a limited class of functions (the Hecke L-functions, and the L-functions coming from the cuspidal holomorphic forms for congruence groups, see Deligne [4]), although it is generally believed that all the L-functions appearing in number theory should satisfy RC. For example, it is conjectured to hold for the L-functions associated with cuspidal automorphic representations on $\mathrm {GL}(d)$ . In general, only some rather weak estimates for the coefficients are at our disposal. Hence, it is interesting to consider the possibility of obtaining some results under some weaker assumptions instead of RC. We introduce the following notation: $s_{j,\mathcal {A}}(p)$ denotes the jth elementary symmetric function of the roots $\alpha _1(p),\ldots ,\alpha _d(p)$ , that is,

(2.3) $$ \begin{align} s_{j,\mathcal {A}}(p)=\sum_{1\leq i_1<\cdots<i_j\leq d}\alpha_{i_1}(p)\dots\alpha_{i_j}(p). \end{align} $$

Hypothesis ${\mathbf{H(\theta _d)}}$ : For all primes p with $(p, q_{\mathcal {A}})=1$ , one has

$$\begin{align*}|\alpha_j(p)|\leq p^{\theta_d} \quad \text{and}\quad s_{j,\mathcal {A}}(p)\ll p^{\min\{j,d-j\}\theta_d} \quad \text{for any} \;1\leq j\leq d. \end{align*}$$

Hypothesis ${\mathbf{{S}}}$ : There exists some $b_{\mathcal {A}}>0$ such that the first moment of absolute values of the coefficients satisfies the bound

$$\begin{align*}\sum_{n\leq x}|a_n|\ll x(\log x)^{b_{\mathcal {A}}-1}. \end{align*}$$

Our main result states as follows.

Theorem 2.1 Let $L(s, \mathcal {A})$ be an L-function satisfying the conditions (A1)–(A4) with $d\geq 2$ , and let $(q,a q_{\mathcal {A}})=1$ . Then under Hypothesis $\mathrm {H(\theta _d)}$ with $\theta _d<1-\frac {1}{d}$ and Hypothesis $\mathrm {S}$ , we have

$$ \begin{align*} \sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}a_n=&M_{0}(x;q)+O\Big(\frac{\tau(q)}{q}y(\log x)^{m-1} \Big)+O\Big(\tau_{d}(q)q^{\frac{d-1}{2}}(\log q)^{b_{\mathcal {A}}} \Big)\\ &+O\bigg(\tau_{d}(q)\Big(\frac{qx}{y}\Big)^{\frac{d-1}{2}}(\log x)^{b_{\mathcal {A}}-1}\bigg) +O\Big( \sum_{\substack{x<n\leq x+O(y) \\n\equiv a\,(\mathrm{mod}\,q)}}|a_n|\Big), \end{align*} $$

where y is an arbitrary real number with $0<y<x$ , $M_{0}(x;q)$ is defined by

$$\begin{align*}M_{0}(x;q)=\frac{1}{\varphi(q)}\mathop{Res}\limits_{s=1} \Big(\frac{1}{s} L(s, \mathcal {A}\otimes \chi_0 )x^{s}\Big). \end{align*}$$

In addition, if $a_n\geq 0,$ we have

$$ \begin{align*} \sum_{\substack{n\leq x\\ n\equiv a\,({\mathrm{mod}}\,q)}}a_n=M_{0}(x;q)+O\Big(\tau_{d}(q)q^{\frac{d-1}{2}}(\log q)^{b_{{\mathcal {A}}}} \Big)+O\Big(\tau_{d}(q)x^{\frac{d-1}{d+1}}(\log x)^{\max\{b_{\mathcal{A}},m\}-1}\Big). \end{align*} $$

We note that the implied constants above depend on $\mathcal {A}$ , including the degree d, the parameters $\mu _j$ and the arithmetic conductor $q_{\mathcal {A}}$ of $\mathcal {A}$ .

Under Hypothesis $\mathrm {H(\theta _d)}$ with $1-\frac {1}{d}\leq \theta _{d}<1$ and Hypothesis $\mathrm {S}$ , the above two assertions hold provided that $\tau _{d}(q)$ is replaced by $\tau _{d+1}(q)$ in the error terms.

Remark 2.1 Chandrasekharan and Narasimhan [3] established these results for $q=1$ . Under some additional assumptions on functional equations for additive twists of L-functions, Smith [21] investigated the analogous problem as in Theorem 2.1 for some positive integers q. However, the lack of a good symmetry structure for these functional equations could increase the difficulty of applications, such as in [21, 22]. We here take full advantage of multiplicative twists of L-function in this aspect.

In the modern sense, one may apply the Voronoï formula of $a_n$ to study its distribution over arithmetic progressions. However, the corresponding formulae are intricate and constrained for most of our interest objects $a_n$ , such as general divisor functions, coefficients of automorphic L-functions and their Rankin–Selberg convolutions.

The paper is organized as follows. In Section 3, we state a few background results we shall need, including a fact in multiplicative number theory, and some properties about general L-functions. In Sections 4, we prove Theorem 2.1. In order to apply this theorem to the automorphic context, we introduce some related knowledge on automorphic L-functions and their Rankin–Selberg in Section 5. Finally, in Section 6, we explore all various of applications and give the proofs of Theorems 1.1–1.3.

3 Preliminaries

In this section, we present the results and tools needed in our proofs.

The common tool in complex analysis is the method of contour integration, which could give a direct link between the summation associated with an arithmetic function and the corresponding Dirichlet series. The following lemma is a standard contour integration (see, for example, [14, Lemma 1]).

Lemma 3.1 If k is any positive integer and $c>0,$ then

$$ \begin{align*} \frac{1}{2\pi i} \int_{(c)}\frac{x^s}{s(s+1)\dots (s+k)}\mathrm{d} s= \left\{ \begin{array}{ll} \frac{1}{k!}(1-\frac{1}{x})^k, & \text{if}\ x \geq 1,\\ 0, & \text{if}\ 0 \leq x \leq 1. \end{array} \right. \end{align*} $$

Now we start to recall and show some uniform estimates for various analytic quantities related to an individual L-function. It turns out that most results for the L-function are expressed conveniently in terms of the analytic conductor. Put

$$\begin{align*}q_{\infty}(s)=\prod_{j=1}^{d}(|s + \mu_j|+3). \end{align*}$$

Then the analytic conductor $q_{\mathcal {A}\otimes \chi }(s)$ is defined by (see, for example, [11, equation (5.6)])

$$\begin{align*}q_{\mathcal {A}\otimes \chi}(s)=q_{\mathcal {A}\otimes \chi}q_{\infty}(s)=q_{\mathcal {A}\otimes \chi}\prod_{j=1}^{d}(|s + \mu_j|+3). \end{align*}$$

We first state the approximate functional equation, which expresses $L(s, \mathcal {A} \otimes \chi )$ in the critical strip as a sum of two Dirichlet series.

Lemma 3.2 Let $\chi (\mathrm {mod}\,q)$ be a primitive Dirichlet character with $q>1$ and ${(q, q_{\mathcal {A}})=1}$ . For $0\leq \Re s\leq 1,$ there exists a smooth function $V_s$ such that

$$\begin{align*}L(s, \mathcal {A} \otimes \chi) = \sum_{n=1}^\infty \frac{a_n\chi(n)}{n^s}V_s\Big(\frac{n}{X\sqrt{q_{\mathcal {A}\otimes \chi}}}\Big)+\omega_{\mathcal {A}\otimes \chi}(s)\sum_{n=1}^\infty \frac{\overline{a_n}\, \overline{\chi}(n)}{n^{1-s}}V_{1-s}\Big(\frac{n X}{\sqrt{q_{\mathcal {A}\otimes \chi}}}\Big), \end{align*}$$

where $ X$ is an arbitrary positive real number, and

$$\begin{align*}\omega_{\mathcal {A}\otimes \chi}(s)=\omega_{\mathcal {A}\otimes \chi}q_{\mathcal {A}\otimes \chi}^{\frac{1}{2}-s}\frac{\Delta(1-s+\kappa_{\mathrm{sgn(\chi)}})}{\Delta(s+\kappa_{\mathrm{sgn(\chi)}})}. \end{align*}$$

The function $V_s$ and its partial derivatives $V_s^{(k)}\; (k = 1,2,\ldots )$ satisfy, for any $C> 0$ , the following uniform growth estimates at $0$ and $\infty $ :

$$\begin{align*}V_s(x)=\left\{ \begin{array}{ll} 1+O\Big(\big(\frac{x}{q_{\infty}(s)}\big)^C\Big) \\ [1.5mm] O\Big(\big(1+\frac{x}{q_{\infty}(s)}\big)^{-C}\Big), \end{array} \right.\quad V_s^{(k)}(x)=O\Big(\big(1+\frac{x}{q_{\infty}(s)}\big)^{-C}\Big), \end{align*}$$

where the implied constants depend only on $C, k,$ and d.

Proof This follows from [11, Theorem 5.3 and Proposition 5.4] in the same manner.

Lemma 3.3 Let $\chi $ is any Dirichlet character $(\mathrm {mod}\,q)$ with $(q,q_{\mathcal {A}})=1$ , and let ${s=\sigma +it.}$ Then we have, for $-\varepsilon \leq \sigma \leq 1+\varepsilon $ and $|t|\geq 1$ ,

$$\begin{align*}L(s, \mathcal {A} \otimes \chi) \ll_{\mathcal {A}} (q|t|)^{d(1-\sigma)+\varepsilon}. \end{align*}$$

Proof Assume $\chi (\mathrm {mod}\,q)$ is induced by a primitive character $\chi _1(\mathrm {mod}\,r)$ , then

$$\begin{align*}L(s, \mathcal {A} \otimes \chi)=L(s, \mathcal {A}\otimes \chi_1)\prod_{p|\frac{q}{r}} \prod_{j=1}^d \Big(1-\frac{\alpha_j(p)\chi_1(p)}{p^s}\Big). \end{align*}$$

Recall that $|\alpha _j(p)|\leq p$ in Condition (A1). Thus, we have, for $-\varepsilon \leq \sigma \leq 1+\varepsilon $ and $|t|\geq 1$ ,

$$\begin{align*}\prod_{p|\frac{q}{r}} \prod_{j=1}^d \Big(1-\frac{\alpha_j(p)\chi_1(p)}{p^s}\Big)\ll\prod_{p|\frac{q}{r}}\Big(1+p^{1-\sigma}\Big)^d\leq \big(\frac{q}{r}\big)^{d(1-\sigma)+\varepsilon}. \end{align*}$$

Moreover, the convexity bound of $L(s, \mathcal {A}\otimes \chi _1)$ states

$$\begin{align*}L(s, \mathcal {A}\otimes \chi_1)\ll q_{\mathcal {A}\otimes \chi_1}(s)^{\frac{1-\sigma}{2}+\varepsilon}\ll_{\mathcal {A}} (r|t|)^{\frac{d(1-\sigma)}{2}+\varepsilon} \end{align*}$$

for $-\varepsilon \leq \sigma \leq 1+\varepsilon $ and $|t|\geq 1$ (see [11, equation (5.20)]). Finally, combining these results above, we conclude Lemma 3.3.

4 Proof of Theorem 2.1

For technical convenience, one usually works with the weighted sum

(4.1) $$ \begin{align} A_\varrho(x;q,a)=\frac {1}{\Gamma(\varrho+1)}\sideset{}{^{\prime}}\sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}a_n(x-n)^\varrho, \end{align} $$

where $(q, aq_{\mathcal {A}})=1$ , $\varrho $ is a sufficiently large integer, and the symbol $\prime $ indicates that the last term has to be multiplied by $1/2$ if $\varrho =0$ and $x=n$ . Detecting the congruence condition in (4.1) by the multiplicative characters $\chi (\mathrm {mod}\,q),$ we obtain the identity

$$ \begin{align*} \sideset{}{^{\prime}}\sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}a_n(x-n)^\varrho=\frac{1}{\varphi(q)}\sum_{\chi(\mathrm{mod}\,q) }\overline{\chi}(a){\underset{n\le x}{\sum\nolimits^{\prime}}}a_n\chi(n)(x-n)^\varrho. \end{align*} $$

Each character $\chi (\mathrm {mod}\,q)$ can be induced by a primitive character $\chi (\mathrm {mod}\,r)$ with $r|q$ . Note that the character for $\chi (\mathrm {mod}\,q)$ with the case $r=1$ is principle. Thus, we get

(4.2) $$ \begin{align} \begin{aligned} \nonumber \Gamma(\varrho+1)A_\varrho(x;q,a)=&{\frac{1}{\varphi(q)}}\sum_{r|q }\; \sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r) }\overline{\chi}(a) \sideset{}{^{\prime}}\sum_{\substack{n\leq x\\(n,q/r)=1}}a_n\chi(n)(x-n)^\varrho\\=&{\frac{1}{\varphi(q)}}\sum_{r|q}\sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r) }\overline{\chi}(a)\sum_{h| (q/r)}\mu(h)\chi(h)h^\varrho {\underset{n\le x/h}{\sum\nolimits^{\prime}}}a_{hn}\chi(n)\Big({\frac{x}{h}}-n\Big)^\varrho\\\nonumber =&{\frac{1}{\varphi(q)}}\sum_{hr|q}\mu(h)h^\varrho \sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r) }\overline{\chi}(a)\chi(h) {\underset{n\le x/h}{\sum\nolimits^{\prime}}}a_{hn}\chi(n)\Big({\frac{x}{h}}-n\Big)^\varrho, \end{aligned} \end{align} $$

where the formula

$$\begin{align*}\sum_{d|n}\mu (d)=\left\{ \begin{array}{ll} 1, & \text{if} n=1, \\ 0, & \text{otherwise,} \end{array} \right. \end{align*}$$

is used to relax the coprimality condition $(n,q/r)=1$ above.

The transformation of the innermost sum over n requires factoring the arithmetic function $a_{hn}$ . To this end, we exploit the Euler product for $L(s, \mathcal {A})$ . Write

$$ \begin{align*} L(s, \mathcal {A}) = \prod_p \prod_{j=1}^d (1-\alpha_j(p)/p^s)^{-1} :=\prod_p L(s, \mathcal {A}_p). \end{align*} $$

With the notation $s_{j,\mathcal {A}}(p)$ as in (2.3), the reciprocal of the local L-function can be given by

$$\begin{align*}L(s,\mathcal {A}_p)^{-1}=1-s_{1,\mathcal {A}}(p)p^{-s}+s_{2,\mathcal {A}}(p)p^{-2s}+\cdots+(-1)^d s_{d,\mathcal {A}}(p)p^{-ds}.\end{align*}$$

Thus, we have

$$\begin{align*}\Big(1-s_{1,\mathcal {A}}(p)p^{-s}+s_{2,\mathcal {A}}(p)p^{-2s}+\cdots+(-1)^d s_{d,\mathcal {A}}(p)p^{-d s}\Big)\Big(\sum_{\nu=0}^{\infty}a_{p^{\nu}}p^{-\nu s}\Big)=1.\end{align*}$$

Hence for all $\nu \in {\mathbb Z}$ , we obtain the recursive relation

$$\begin{align*}a_{p^{\nu}}-s_{1,\mathcal {A}}(p)a_{p^{\nu-1}}+s_{2,\mathcal {A}}(p)a_{p^{\nu-2}}+\cdots+(-1)^d s_{d,\mathcal {A}}(p)a_{p^{\nu-d}}=\delta_{0\nu}\end{align*}$$

subject to the convention that $a_{p^{\nu }}=0$ for negative $\nu .$ Notice that h is square-free. Now if we suppose $h=\prod p$ , we get by the recursion and multiplicativity

$$ \begin{align*} \begin{aligned} \sum_{n=1}^{\infty}a_{hn}n^{-s}=&\prod_{p| h}\Big(\sum_{\nu=0}^{\infty}a_{p^{\nu+1}}p^{-\nu s}\Big)\prod_{p\nmid h}L(s,\mathcal {A}_p)\\ =&L(s,\mathcal {A})\prod_{p| h}\Big(L(s,\mathcal {A}_p)^{-1}\sum_{\nu=0}^{\infty}\frac{a_{p^{\nu+1}}}{p^{\nu s}}\Big)\\ =&L(s,\mathcal {A})\prod_{p| h}\Big(s_{1,\mathcal {A}}(p)-s_{2,\mathcal {A}}(p)p^{-s}+\cdots+(-1)^{d-1}s_{d,\mathcal {A}}(p)p^{-(d-1)s}\Big). \end{aligned} \end{align*} $$

Hence, it is clear that $a_{hn}$ factors as follows:

(4.3) $$ \begin{align} a_{hn}=\sum_{cm=n}a(h,c)a_m, \end{align} $$

where $a(h,c)$ is defined for $c|h^{d-1}$ by

$$ \begin{align*}a(h,c)=\sum_{h=h_0h_1\cdots h_{d-1}}\prod_{j=0}^{d-1}\prod_{p|h_j}(-1)^{j}s_{j+1,\mathcal {A}}(p)\end{align*} $$

with $h_0,h_1,\ldots ,h_{d-1}$ mutually coprime such that

$$\begin{align*}c=\Big(\prod_{p|h_1}p\Big)\Big(\prod_{p|h_2}p\Big)^2\dots\Big(\prod_{p|h_{d-1}}p\Big)^{d-1}. \end{align*}$$

Using the above formulas and Hypothesis $\mathrm {H(\theta _d)}$ , one can show that

(4.4) $$ \begin{align} a(h,c)\ll h^{\frac{d\theta_d}{2}+\varepsilon}. \end{align} $$

Inserting the identity (4.3) into the innermost sum over n in the last line of (4.2), we get

$$ \begin{align*} \frac {1}{\Gamma(\varrho+1)}{\underset{n\le x/h}{\sum\nolimits^{\prime}}}a_{hn}\chi(n)\Big(\frac{x}{h}-n\Big)^\varrho=\sum_{c|h^{d-1}}a(h,c)c^\varrho\chi(c)B_{\varrho}\Big(\frac{x}{ch},\chi\Big), \end{align*} $$

where

$$\begin{align*}B_{\varrho}(y,\chi)=\frac {1}{\Gamma(\varrho+1)} {\underset{m\le y}{\sum\nolimits^{\prime}}}a_{m}\chi(m)(y-m)^\varrho. \end{align*}$$

Next, we turn to evaluate the summation $B_{\varrho }(y,\chi )$ . By condition $\mathrm {(A1)}$ , it is known that $L(s, \mathcal {A} \otimes \chi )$ converges absolutely for $\Re s\geq 1+\varepsilon $ . Then it follows from Lemma 3.1 that

$$\begin{align*}B_{\varrho}(y,\chi)=\frac{1}{2\pi i}\int_{(1+\varepsilon)}\frac{\Gamma(s)}{\Gamma(\varrho+1+s)} L(s, \mathcal {A} \otimes \chi)y^{\varrho+s}{\text{d}} s, \end{align*}$$

where $\varrho $ is a sufficiently large integer compared to d. Using the analytic properties $\mathrm {(A2), (A5)}$ of $L(s, \mathcal {A} \otimes \chi )$ and the bound in Lemma 3.3, we could move the line of integration to $\Re s= -\varepsilon <0$ , change the variable from s to $1-s$ and apply the functional equation (2.2) to get

(4.5) $$ \begin{align} B_{\varrho}(y,\chi)=\delta_{r1} \mathop{\mathrm{Res}}\limits_{s=1} \Big(\frac{\Gamma(s)}{\Gamma(\varrho+1+s)} L(s, {\mathcal{A}} )y^{\varrho+s}\Big)+\frac{1}{\Gamma(\varrho+1)}L(0, {\mathcal {A}} \otimes \chi)y^{\varrho}+ E_{\varrho}(y,\chi), \end{align} $$

where $\delta _{r1}$ denotes the diagonal symbol of Kronecker and

$$\begin{align*}E_{\varrho}(y,\chi)=\frac{\omega_{\mathcal {A}\otimes \chi} }{2\pi i}\int_{(1+\varepsilon)}\frac{\Gamma(1-s)\Delta\big(s+\kappa_{\mathrm{sgn(\chi)}}\big)}{\Gamma(\varrho+2-s)\Delta\big(1-s + \kappa_{\mathrm{ sgn(\chi)}}\big)}y^{\varrho+1-s}q_{\mathcal {A}\otimes \chi}^{s-\frac{1}{2}} L(s,\overline{\mathcal {A}}\otimes \overline{\chi}){\text{d}} s. \end{align*}$$

Denote the contributions of these three terms on the right-hand side of (4.5) to the sum $A_\varrho (x;q,a)$ by $M_\varrho (x;q)$ , $H_\varrho (x;q)$ and $S_\varrho (x;q)$ , respectively. This is to say

(4.6) $$ \begin{align} A_\varrho(x;q,a)=M_\varrho(x;q)+H_\varrho(x;q)+S_\varrho(x;q), \end{align} $$

where

(4.7) $$ \begin{align} \nonumber M_{\varrho}(x;q)=&{\frac{1}{\varphi(q)}}\sum_{h|q}\mu(h) \sum_{c|h^{d-1}}a(h,c) \mathop{\mathrm{Res}}\limits_{s=1} \Big(\frac{\Gamma(s)}{\Gamma(\varrho+1+s)} L(s, \mathcal {A} )\Big(\frac{x}{ch}\Big)^{\varrho+s}\Big),\\H_{\varrho}(x;q)=&\frac{1}{\Gamma(\varrho+1)\varphi(q)}\sum_{hr|q}\mu(h) \sum_{c|h^{d-1}}a(h,c) \sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r) }\chi(\overline{a}ch)L(0, \mathcal {A} \otimes \chi)x^{\varrho},\\\nonumber S_\varrho(x;q)=&\frac{1}{\varphi(q)}\sum_{hr|q}\mu(h) \sum_{c|h^{d-1}}a(h,c) (ch)^\varrho \sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r) }\chi(\overline{a}ch) E_{\varrho}\Big(\frac{x}{ch},\chi\Big). \end{align} $$

We introduce the difference operator

$$ \begin{align*} \Delta_y^\varrho F(x)=\sum_{v=0}^\varrho(-1)^{\varrho-v}C_\varrho^v F(x+vy), \end{align*} $$

where y is a positive parameter less than x and $C_\varrho ^v$ denotes the binomial coefficient. If F has $\varrho $ derivatives, then one has

(4.8) $$ \begin{align} \Delta_y^\varrho F(x)=\int_{x}^{x+y}{\text{d}} t_1\int_{t_1}^{t_1+y}{\text{d}} t_2\ldots\int_{t_{\varrho-1}}^{t_{\varrho-1}+y}F^{(\varrho)}(t_{\varrho}){\text{d}} t_{\varrho}, \end{align} $$

where $F^{(\varrho )}$ is the $\varrho $ th derivative of F.

We first apply the operator $\Delta _y^\varrho $ to $A_\varrho (x;q,a)$ and obtain

$$\begin{align*}\Delta_y^\varrho A_\varrho(x;q,a)=\sideset{}{^{\prime}}\sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}} \kern-1.2pt a_n\frac {\Delta_y^\varrho(x-n)^\varrho}{\Gamma(\varrho+1)}+\sum_{v=0}^\varrho(-1)^{\varrho-v}C_\varrho^v \sideset{}{^{\prime}}\sum_{\substack{x<n\leq x+vy \\ n\equiv a\,(\mathrm{mod}\,q)}} \kern-1pt a_n (x+vy-n)^\varrho. \end{align*}$$

Since

$$\begin{align*}\frac {1}{\Gamma(\varrho+1)}\Delta_y^\varrho (x-n)^\varrho=y^\varrho, \end{align*}$$

we get

(4.9) $$ \begin{align} \Delta_y^\varrho A_\varrho(x;q,a)= y^\varrho A_0(x;q,a)+O_\varrho\bigg(y^\varrho \sum_{\substack{x<n\leq x+\varrho y \\ n \equiv a\,(\mathrm{mod}\,q)}}|a_n|\bigg). \end{align} $$

Furthermore, if $a_n\geq 0,$ then $A_0(x;q,a)$ is monotone. Thus, it follows from (4.8) that

(4.10) $$ \begin{align} \Delta_y^\varrho A_\varrho(x-\varrho y;q,a) \leq y^\varrho A_0(x;q,a)\leq \Delta_y^\varrho A_\varrho(x;q,a). \end{align} $$

Next, we shall apply the operator $\Delta _y^\varrho $ to $M_\varrho (x;q)$ , $H_\varrho (x;q)$ and $S_\varrho (x;q)$ , separately. From now on, we assume that the implied constant in the notation $\ll $ or O is allowed to depend on $\mathcal {A}, \varrho $ for convenience.

4.1 Computation of $\Delta _y^\varrho S_\varrho (x;q)$

By the Dirichlet series expression of $L(s,\overline {\mathcal {A}}\otimes \overline {\chi }))$ , we can rewrite $E_{\varrho }(y,\chi )$ as

(4.11) $$ \begin{align} E_{\varrho}(y,\chi)= \omega_{\mathcal {A}\otimes \chi}q_{\mathcal {A}\otimes \chi}^{\varrho+\frac{1}{2}}\sum_{n=1}^{\infty}\frac{\overline{a_n}\,\overline{\chi}(n)}{n^{1+\varrho}}J\left( \frac{ny}{q_{\mathcal {A}\otimes \chi}}\right), \end{align} $$

where

$$\begin{align*}J(x)=\frac{1}{2\pi i}\int_{(c)}\frac{\Gamma(1-s)\Delta\big(s+\kappa_{\mathrm{sgn(\chi)}}\big)}{\Gamma(\varrho+2-s)\Delta\big(1-s + \kappa_{\mathrm{sgn(\chi)}}\big)}x^{\varrho+1-s}{\text{d}} s. \end{align*}$$

We shall deal with the integral $J(x)$ by means of the following result (see [3, equations (4.5) and (4.11)] or [14, Theorem 3]).

Lemma 4.1 With the notation as before, suppose $d\geq 2$ . Let $0\leq \varrho \in {\mathbb Z}$ and $c\in \mathbb {R}$ . Then for suitable choices c and $\varrho $ , we have

$$\begin{align*}J(x)=O\Big(x^{\frac{1}{2}+\left(1-\frac{1}{d}\right)\varrho-\frac{1}{2d}}\Big)\quad \text{and} \quad J^{(\varrho)}(x)=O\Big(x^{\frac{1}{2}-\frac{1}{2d}}\Big). \end{align*}$$

Combining (4.11) with the expression of $S_\varrho (x;q)$ in (4.7), we conclude

$$ \begin{align*} \begin{aligned} S_\varrho(x;q)=&\frac{1}{\varphi(q)}\sum_{hr|q}\mu(h) \sum_{c|h^{d-1}}a(h,c) (ch)^\varrho\sum_{n=1}^{\infty}\frac{\overline{a_n}}{n^{1+\varrho}}\sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r) }\chi(\overline{an}ch) \omega_{\mathcal {A}\otimes \chi}q_{\mathcal {A}\otimes \chi}^{\varrho+\frac{1}{2}}J\left( \frac{nx}{chq_{\mathcal {A}\otimes \chi}}\right). \end{aligned} \end{align*} $$

Recall that

$$\begin{align*}q_{\mathcal {A}\otimes \chi}=q_{\mathcal {A}}\,r^{d}\quad \text{and} \quad \omega_{\mathcal {A}\otimes \chi}= \eta_{\mathcal {A},\mathrm{sgn(\chi)}}\chi(q_{\mathcal {A}})\Big(\frac{\tau(\chi)}{\sqrt{r}}\Big)^{d} \end{align*}$$

for $(r, q_{\mathcal {A}})=1$ . Since the $\eta _{\mathcal {A},\mathrm {sgn(\chi )}}$ and $J(x)$ depend on the parity of $\chi ,$ but not on the character itself, we need to break up the sum over $\chi $ separately into even and odd characters. We put

$$\begin{align*}K_{\pm}(a,r) =\frac{1}{2}\sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r)}(1\pm\chi(-1))\overline{\chi}(a) \Big(\frac{\tau(\chi)}{\sqrt{r}}\Big)^{d}. \end{align*}$$

Moreover, we display the dependence by writing $J_{+}$ and $J_{-}$ , respectively, in place of J. Thus, we have

$$ \begin{align*} \begin{aligned} S_{\varrho}(x;q)=&\frac{1}{\varphi(q)}\sum_{hr|q}\mu(h) \sum_{c|h^{d-1}}a(h,c)\, (chq_{\mathcal {A}}r^{d})^\varrho \,(q_{\mathcal {A}}r^{d})^{\frac{1}{2}} \\ &\times \sum_{\pm}\eta_{\mathcal {A},\mathrm{sgn(\chi)}}\sum_{n=1}^{\infty}\frac{\overline{a_n}}{n^{1+\varrho}}K_{\pm}(an\overline{chq_{\mathcal {A}}},r)J_{\pm}\left( \frac{nx}{chq_{\mathcal {A}}r^{d}}\right). \end{aligned} \end{align*} $$

Lemma 4.2 Let $K_{\pm }(a,r)$ be as above with $(a,r)=1$ . Then we have

$$\begin{align*}|K_{\pm}(a,r)|\leq \varphi(r)r^{-\frac{1}{2}}\tau_{d}(r). \end{align*}$$

Proof It is clear that

(4.12) $$ \begin{align} K_{\pm}(a,r)=\frac{1}{2}\left( K(a,r)\pm K(-a,r)\right), \end{align} $$

where

$$\begin{align*}K(a,r) =\sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r)}\overline{\chi}(a) \Big(\frac{\tau(\chi)}{\sqrt{r}}\Big)^{d}. \end{align*}$$

In fact, $K(a,r)$ appears in a long list of literature, such as the series works of Duke and Iwaniec about estimating coefficients of L-functions (see [5–9]), the work of Luo, Rudnick, and Sarnak on the Selberg conjecture [16] and the work of Luo about nonvanishing of $\mathrm {GL}(d) L$ -functions [15]. It plays a key role in making these remarkable achievements.

As in the proof of [9], by the definition of Gauss sum, we infer that

$$\begin{align*}r^{\frac{d}{2}}K(a,r) =\sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r)}\overline{\chi}(a) \left(\sum_{b(\mathrm{mod}\,r)}\chi(b)\text{e}\left(\frac{b}{r}\right)\right)^{d}. \end{align*}$$

Changing the order of summation and using the relation [11, equation (3.8)]

$$\begin{align*}\sideset{}{^*}\sum_{\chi \bmod r}\chi(m)=\sum_{l|(m-1,r)}\varphi(l)\mu\left(\frac{r}{l}\right)\end{align*}$$

when $(r,m)=1,$ we get

$$ \begin{align*} \begin{split} r^{\frac{d}{2}}K(a,r) =&\sum_{lk=r}\varphi(l)\mu(k)\sideset{}{^*}\sum_{\substack{b_1,\ldots,b_{d}(\mathrm{mod}\,r) \\ b_1\cdots b_{d}\equiv a(\mathrm{mod}\,l)}} \text{e}\left(\frac{b_1+\cdots +b_{d}}{r}\right)\\ =&\sum_{\substack{lk=r\\ (l,k)=1}}\varphi(l)\mu(k)^{d+1}\sideset{}{^*}\sum_{\substack{b_1,\ldots,b_{d}(\mathrm{mod}\,l) \\ b_1\cdots b_{d}\equiv a(\mathrm{mod}\,l)}} \text{e}\left(\frac{(b_1+\cdots +b_{d})\overline{k}}{r}\right). \end{split} \end{align*} $$

Note that the innermost sum is the generalized Kloosterman sum for which Deligne [4] has established the bound $\tau _{d}(l)l^{\frac {d-1}{2}}.$ Employing Deligne’s bound, we directly have

$$\begin{align*}|r^{\frac{d}{2}}K(a,r)|\leq \sum_{lk=r}\varphi(l)\tau_{d}(l)l^{\frac{d-1}{2}}\leq \varphi(r)r^{\frac{d-1}{2}}\tau_{d}(r) \sum_{k|r}\frac{1}{\varphi(k) k^{\frac{d-1}{2}}}\ll \varphi(r)r^{\frac{d-1}{2}}\tau_{d}(r), \end{align*}$$

which implies this lemma from (4.12).

We continue to compute $\Delta _y^\varrho S_\varrho (x;q)$ . Now we apply the operator $\Delta _y^\varrho $ to $S_{\varrho }(x;q)$ and obtain from Lemma 4.2 that

(4.13) $$ \begin{align} \begin{aligned} \Delta_y^\varrho S_{\varrho}(x;q)\ll &\frac{1}{\varphi(q)}\sum_{hr|q}|\mu(h)|\sum_{c|h^{d-1}}|a(h,c)| (chr^{d})^\varrho r^{\frac{d}{2}} \\ &\times \sum_{\pm}\sum_{n=1}^{\infty}\frac{|a_n|}{n^{1+\varrho}}\big|K_{\pm}(an\overline{chq_{\mathcal {A}}},r)\big|\left|\Delta_y^\varrho J_{\pm}\left( \frac{nx}{chq_{\mathcal {A}}r^{d}}\right)\right|\\ \ll &\frac{1}{\varphi(q)}\sum_{hr|q}|\mu(h)|\sum_{c|h^{d-1}}|a(h,c)| (chr^{d})^\varrho \varphi(r)r^{\frac{d-1}{2}}\tau_{d}(r)\\ &\times \sum_{\pm}\sum_{n=1}^{\infty}\frac{|a_n|}{n^{1+\varrho}}\left|\Delta_y^\varrho J_{\pm}\left( \frac{nx}{chq_{\mathcal {A}}r^{d}}\right)\right|. \end{aligned} \end{align} $$

By definition of the operator $\Delta _y^\varrho $ and Lemma 4.1, one easily has

$$\begin{align*}\Delta_y^\varrho J_{\pm}(x)=\left\{ \begin{array}{ll} O(|J_{\pm}(x)|)=O\Big(x^{\frac{1}{2}+\left(1-\frac{1}{d}\right)\varrho-\frac{1}{2d}}\Big), \\ O(y^\varrho |J_{\pm}^{(\varrho)}(x)|)=O\Big(y^\varrho x^{\frac{1}{2}-\frac{1}{2d}}\Big). \end{array} \right. \end{align*}$$

Thus, we have

$$ \begin{align*} \Delta_y^\varrho J_{\pm}\left( \frac{nx}{chq_{\mathcal {A}}r^{d}}\right)\ll_{\mathcal {A}} \min\left\{\left( \frac{nx}{ch r^{d}}\right)^{\frac{1}{2}+\left(1-\frac{1}{d}\right)\varrho-\frac{1}{2d}},\; \left( \frac{ny}{ch r^{d}}\right)^\varrho \left( \frac{nx}{ch r^{d}}\right)^{\frac{1}{2}-\frac{1}{2d}}\right\}. \end{align*} $$

We divide the innermost summation in (4.13) into two parts by the parameter $z>0$ , which shall be chosen later. For any $\varepsilon>0,$ under Hypothesis $\mathrm {S}$ , we get

$$ \begin{align*} \begin{aligned} \sum_{n>z}\frac{|a_n|}{n^{1+\varrho}}\left|\Delta_y^\varrho J_{\pm}\left( \frac{nx}{chq_{\mathcal {A}}r^{d}}\right)\right|\ll& \sum_{n>z}\frac{|a_n|}{n^{1+\varrho}}\left( \frac{nx}{ch r^{d}}\right)^{\frac{1}{2}+\left(1-\frac{1}{d}\right)\varrho-\frac{1}{2d}}\\ \ll&\left( \frac{x}{ch r^{d}}\right)^{\frac{1}{2}+\left(1-\frac{1}{d}\right)\varrho-\frac{1}{2d}}z^{\frac{1}{2}-\frac{\varrho}{d}-\frac{1}{2d}}(\log z)^{b_{\mathcal {A}}-1},\\ \end{aligned} \end{align*} $$

and

$$ \begin{align*} \begin{aligned} \sum_{n\leq z}\frac{|a_n |}{n^{1+\varrho}}\left|\Delta_y^\varrho J_{\pm}\left( \frac{nx}{chq_{\mathcal {A}}r^{d}}\right)\right|\ll& \sum_{n\leq z}\frac{|a_n|}{n^{1+\varrho}}\left( \frac{ny}{ch r^{d}}\right)^\varrho \left( \frac{nx}{ch r^{d}}\right)^{\frac{1}{2}-\frac{1}{2d}}\\ \ll& \left( \frac{y}{ch r^{d}}\right)^\varrho \left( \frac{xz}{ch r^{d}}\right)^{\frac{1}{2}-\frac{1}{2d}}(\log z)^{b_{\mathcal {A}}-1}.\\ \end{aligned} \end{align*} $$

On taking $z=\frac {chr^{d}x^{d-1}}{y^{d}},$ we have

$$ \begin{align*} \sum_{n=1}^{\infty}\frac{|a_n|}{n^{1+\varrho}}\left|\Delta_y^\varrho J_{\pm}\left( \frac{nx}{chq_{\mathcal {A}}r^{d}}\right)\right|\ll \left( \frac{y}{ch r^{d}}\right)^\varrho \left(\frac{x}{y}\right)^{\frac{d-1}{2}}(\log x)^{b_{\mathcal {A}}-1}. \end{align*} $$

Inserting this into (4.13) and applying the estimate (4.4) yield

(4.14) $$ \begin{align} \begin{aligned} \Delta_y^\varrho S_{\varrho}(x;q)\ll &\left(\frac{x}{y}\right)^{\frac{d-1}{2}}\frac{y^\varrho (\log x)^{b_{\mathcal {A}}-1}}{\varphi(q)}\sum_{hr|q}|\mu(h)| \sum_{c|h^{d-1}}|a(h,c) | \varphi(r)r^{\frac{d-1}{2}}\tau_{d}(r)\\ \ll&\left(\frac{x}{y}\right)^{\frac{d-1}{2}}\frac{y^\varrho (\log x)^{b_{\mathcal {A}}-1}}{\varphi(q)}\sum_{hr=q} h^{\frac{d\theta_d}{2}+\varepsilon}\varphi(r)r^{\frac{d-1}{2}}\tau_{d}(r). \end{aligned} \end{align} $$

It is easy to deduce

$$ \begin{align*} \Delta_y^\varrho S_{\varrho}(x;q) \ll\left(\frac{qx}{y}\right)^{\frac{d-1}{2}}y^\varrho (\log x)^{b_{\mathcal {A}}-1}\tau_{d}(q) \end{align*} $$

when $\theta _{d}<1-\frac {1}{d},$ and

$$ \begin{align*} \Delta_y^\varrho S_{\varrho}(x;q) \ll\left(\frac{qx}{y}\right)^{\frac{d-1}{2}}y^\varrho (\log x)^{b_{\mathcal {A}}-1}\tau_{d+1}(q) \end{align*} $$

when $1-\frac {1}{d}\leq \theta _{d}<1.$

4.2 Computation of $\Delta _y^\varrho H_\varrho (x;q)$

Lemma 4.3 Let $(r, aq_{\mathcal {A}})=1.$ Then we have

$$\begin{align*}\sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r) }\chi(a)L(0, \mathcal {A} \otimes \chi)\ll \varphi(r)r^{\frac{d-1}{2}} \tau_{d}(r)(\log r)^{b_{\mathcal {A}}}. \end{align*}$$

Proof By the approximate functional equation in Lemma 3.2 with $X=r^{-d/3}$ , we have

$$\begin{align*}L(0, \mathcal {A} \kern1.3pt{\otimes}\kern1pt \chi) = \sum_{n\leq r^{d/6+\varepsilon}}\kern-1pt a_n\chi(n)V_0\Big(\frac{n}{q_{\mathcal {A}}^{1/2} r^{d/6}}\kern-0.5pt\Big)+\omega_{\mathcal {A}\otimes \chi}(0)\kern-0.1pt \sum_{n\leq r^{5d/6+\varepsilon}} \kern-1pt\frac{\overline{a_n}\, \overline{\chi}(n)}{n}V_{1}\Big(\frac{n}{q_{\mathcal {A}}^{1/2} r^{5d/6}}\kern-0.5pt\Big)+O(r^{-2020}\kern-0.3pt). \end{align*}$$

We average the approximate functional equation over all primitive characters $(\mathrm {mod}\,r).$ Thus, the sum

$$\begin{align*}\sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r) }\chi(a)L(0, \mathcal {A} \otimes \chi) \end{align*}$$

is decomposed into two parts $T_1$ and $T_2$ with negligible error $O(r^{-2019})$ . Since $L(s, \mathcal {A})$ is absolutely convergent for $\Re s>1$ , we get

(4.15) $$ \begin{align} T_1=r \sum_{n\leq r^{d/6+\varepsilon}}|a_n|\ll r^{\frac{d}{6}+1+\varepsilon}. \end{align} $$

To treat the contribution of $T_2$ , we first note that $\omega _{\mathcal {A}\otimes \chi }(s)$ and $V_s$ depend on the parity of $\chi $ , but not on the characters $\chi .$ Similar to the previous argument for $S_{\varrho }(x;q)$ , we break up the sum $T_2$ over $\chi $ separately into even and odd characters, and then get

$$ \begin{align*} \begin{aligned} T_2=&\sum_{\substack{n\leq r^{5d/6+\varepsilon}\\ (n,r)=1}} \frac{\overline{a_n}}{n} \sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r) }\chi(a\overline{n})\omega_{{\mathcal {A}}\otimes \chi}(0)V_{1}\Big(\frac{n}{q_{{\mathcal A}}^{1/2} r^{5d/6}}\Big) \\ \ll & r^{\frac{d}{2}} \sum_{\pm} \sum_{\substack{n\leq r^{5d/6+\varepsilon}\\ (n,r)=1}} \frac{|a_n|}{n}|K_{\pm}(n\overline{aq_{\mathcal {A}}},r)|. \end{aligned} \end{align*} $$

Using Hypothesis $\mathrm {S}$ and Lemma 4.2, we therefore have

(4.16) $$ \begin{align} T_2 \ll \varphi(r)r^{\frac{d-1}{2}} \tau_{d}(r)(\log r)^{b_{\mathcal {A}}}. \end{align} $$

Collecting (4.15) and (4.16), Lemma 4.3 immediately follows.

If the operator $\Delta _y^\varrho $ acts on $H_{\varrho }(x;q)$ , then we obtain from Lemma 4.3 that

$$ \begin{align*} \begin{aligned} \Delta_y^\varrho H_{\varrho}(x;q)=&\frac{1}{\Gamma(\varrho+1)\varphi(q)}\sum_{hr|q}\mu(h) \sum_{c|h^{d-1}}a(h,c) y^{\varrho} \sideset{}{^*}\sum_{\chi(\mathrm{mod}\,r) }\chi(\overline{a}ch)L(0, \mathcal {A} \otimes \chi)\\ \ll &\frac{y^{\varrho}}{\varphi(q)}\sum_{hr|q}|\mu(h)| \sum_{c|h^{d-1}}|a(h,c)| \varphi(r)r^{\frac{d-1}{2}} \tau_{d}(r)(\log r)^{b_{\mathcal {A}}}. \end{aligned} \end{align*} $$

Similar to the previous estimate for (4.14), we get

$$ \begin{align*} \Delta_y^\varrho H_{\varrho}(x;q) \ll y^\varrho q^{\frac{d-1}{2}}\tau_{d}(q)(\log q)^{b_{\mathcal {A}}} \end{align*} $$

when $\theta _{d}<1-\frac {1}{d},$ and

$$ \begin{align*} \Delta_y^\varrho H_{\varrho}(x;q) \ll y^\varrho q^{\frac{d-1}{2}}\tau_{d+1}(q)(\log q)^{b_{\mathcal {A}}} \end{align*} $$

when $1-\frac {1}{d} \leq \theta _{d}< 1.$

4.3 Computation of $\Delta _y^\varrho M_\varrho (x;q)$

By the relation (4.3), we have

$$ \begin{align*} M_{\varrho}(x;q)=\frac{1}{\varphi(q)}\mathop{\mathrm{Res}}\limits_{s=1} \Big(\frac{\Gamma(s)}{\Gamma(\varrho+1+s)} L(s, \mathcal {A}\otimes \chi_0 )x^{\varrho+s}\Big), \end{align*} $$

where $\chi _0$ is the principle character $(\mathrm {mod}\,q)$ . Let $\mathcal {C_\varepsilon }$ be a cycle with a center at $s=1$ and a radius of $\varepsilon $ . Then $M_{\varrho }(x;q)$ can also be written as

$$\begin{align*}M_{\varrho}(x;q)=\frac{1}{\varphi(q)}\frac{1}{2\pi i}\int_{\mathcal{C_\varepsilon}}\frac{\Gamma(s)}{\Gamma(\varrho+1+s)} L(s, \mathcal {A}\otimes \chi_0 )x^{\varrho+s} {\text{d}} s. \end{align*}$$

In dealing with $\Delta _y^\varrho M_{\varrho }(x;q)$ , the identity (4.8) immediately implies

$$\begin{align*}\Delta_y^\varrho M_{\varrho}(x;q)= \int_{x}^{x+y} {\text{d}} t_1\int_{t_1}^{t_1+y} {\text{d}} t_2\cdots\int_{t_{\varrho-1}}^{t_{\varrho-1}+y}M_{0}(t_{\varrho};q){\text{d}} t_{\varrho}. \end{align*}$$

By introducing the change of variables $t_j\mapsto y\, v_j+t_{j-1}$ for $1\leq j\leq \varrho $ with $t_0=x$ , we have

$$\begin{align*}\Delta_y^\varrho M_{\varrho}(x;q)=y^\varrho \int_{0}^{1}\cdots\int_{0}^{1}M_{0}(x+y(v_1+\cdots+v_{\varrho});q) {\text{d}} v_1\cdots {\text{d}} v_{\varrho}. \end{align*}$$

Then the first mean value theorem for integrals implies that

$$\begin{align*}\Delta_y^\varrho M_{\varrho}(x;q)=y^\varrho M_{0}(x+\xi y;q) \end{align*}$$

for some $0<\xi <\varrho .$ From the differential form of the mean value theorem, we have

$$\begin{align*}\Delta_y^\varrho M_{\varrho}(x;q)=y^\varrho M_{0}(x;q)+\xi y^{\varrho+1}M_{0}^{'}(x+\xi_1 y;q) \end{align*}$$

for some $0<\xi _1<\xi $ , where $M_{0}^{'}(x;q)$ is the derivative of $M_{0}(x;q)$ given by

$$ \begin{align*} M_{0}^{'}(x;q)=\frac{1}{\varphi(q)}\frac{1}{2\pi i}\int_{\mathcal{C_\varepsilon}} L(s, \mathcal {A}\otimes \chi_0 )x^{s-1} {\text{d}} s. \end{align*} $$

We can rewrite $L(s, \mathcal {A} \otimes \chi _0)$ as

$$\begin{align*}L(s, \mathcal {A} \otimes \chi_0)=G_q(s,\mathcal {A}) L(s, \mathcal {A}), \end{align*}$$

where

$$ \begin{align*} G_q(s,\mathcal {A})=\prod_{p|q} \prod_{j=1}^d \Big(1-\frac{\alpha_j(p)}{p^s}\Big). \end{align*} $$

For any $j\geq 0,$ we obtain from general Leibniz rule that

$$\begin{align*}\frac{q}{\varphi(q)}G_q^{(j)}(1,\mathcal {A})\ll (\log q)^{j} \prod_{p|q}\Big(1+\frac{1}{p^{1-\theta_d}}\Big)^{d}\Big(1-\frac{1}{p}\Big)^{-1}\ll \tau(q)(\log q)^{j} \end{align*}$$

if $\theta _{d}< 1$ . The residue theorem then yields

$$ \begin{align*} \begin{aligned} M_{0}^{'}(x;q)=&\frac{1}{\varphi(q)}\mathop{\mathrm{Res}}\limits_{s=1} \Big(G_q(s,\mathcal {A}) L(s, \mathcal {A})x^{s-1}\Big)\\ \ll &\frac{1}{q}(|G_q^{(m-1)}(1,\mathcal {A})|+|G_q(1,\mathcal {A})|(\log qx)^{m-1})\\ \ll &\frac{\tau(q)(\log qx)^{m-1}}{q}. \end{aligned} \end{align*} $$

Thus, we have

$$ \begin{align*} \Delta_y^\varrho M_{\varrho}(x;q)=y^\varrho \left(M_{0}(x;q)+O\left(\frac{\tau(q)}{q}y(\log x)^{m-1}\right)\right). \end{align*} $$

At last, we just note that these terms do not exist when the pole order m of $L(s,\mathcal {A})$ at $s=1$ equals zero, which means that $L(s,\mathcal {A})$ is an entire function.

4.4 The finishing touches

We first assume $\theta _{d}<1-\frac {1}{d}$ . Applying the operator $\Delta _y^\varrho $ to both sides of (4.6), we have

$$\begin{align*}\Delta_y^\varrho A_\varrho(x;q,a)=\Delta_y^\varrho M_\varrho(x;q)+\Delta_y^\varrho H_\varrho(x;q)+\Delta_y^\varrho S_\varrho(x;q). \end{align*}$$

Collecting these estimates of $\Delta _y^\varrho M_\varrho (x;q)$ , $\Delta _y^\varrho H_\varrho (x;q)$ and $\Delta _y^\varrho S_\varrho (x;q)$ as in Sections 4.1–4.3, it follows that

(4.17) $$ \begin{align} \nonumber \frac{\Delta_y^\varrho A_\varrho(x;q,a)}{y^\varrho}=&M_{0}(x;q)+O\left({\frac{\tau(q)}{q}}y(\log qx)^{m-1} \right)+O\left(q^{\frac{d-1}{2}}\tau_{d}(q)(\log q)^{b_{\mathcal {A}}} \right)\\ &+O\left(\tau_{d}(q)\left({\frac{qx}{y}}\right)^{\frac{d-1}{2}}(\log x)^{b_{\mathcal {A}}-1}\right). \end{align} $$

Thus, we conclude the first assertion of Theorem 2.1 from (4.9).

In addition $a_n\geq 0,$ the differential form of the mean value theorem gives

$$ \begin{align*} M_{0}(x;q)-M_{0}(x-\varrho y;q)\ll&y\max_{\xi\ll 1}|M_{0}^{'}(x+\xi y;q)|\\\ll&{\frac{\tau(q)}{q}}y(\log qx)^{m-1}. \end{align*} $$

From the estimates (4.17), it is easy to derive that

$$\begin{align*}\Delta_y^\varrho A_\varrho(x-\varrho y;q,a)\quad \text{and} \quad \Delta_y^\varrho A_\varrho(x;q,a) \end{align*}$$

are equal to

(4.18) $$ \begin{align} \nonumber &M_{0}(x;q)+O\left(\frac{\tau(q)}{q}y(\log qx)^{m-1} \right)+O\left(q^{\frac{d-1}{2}}\tau_{d}(q)(\log q)^{b_{\mathcal {A}}} \right)\\ &+O\left(\tau_{d}(q)\left(\frac{qx}{y}\right)^{\frac{d-1}{2}}(\log x)^{b_{\mathcal {A}}-1}\right). \end{align} $$

Using the inequalities (4.10), we then infer $A_0(x;q,a)$ also asymptotically equals (4.18). On taking $y=q x^{\frac {d-1}{d+1}},$ we finally derive

$$ \begin{align*} &A_0(x;q,a)\\&\quad=M_{0}(x;q)+O\left(q^{\frac{d-1}{2}}\tau_{d}(q)(\log q)^{b_{\mathcal {A}}} \right)+O\left(\tau_{d}(q)x^{\frac{d-1}{d+1}}(\log x)^{\max\{b_{\mathcal {A}},m\}-1}\right)\hspace{-0.5pt}, \end{align*} $$

which completes the proof of the second assertion in Theorem 2.1.

If $1-\frac {1}{d} \leq \theta _{d}<1,$ we get analogous conclusions, where the only difference is that the divisor function $\tau _{d}(q)$ in the error terms is replaced by $\tau _{d+1}(q).$

5 Background on automorphic L-functions and their Rankin–Selberg

We are mainly interested in some arithmetic functions arising from cuspidal automorphic representations. So we recall and show some standard facts about L-functions related to cuspidal automorphic representations in this section. We refer the reader to [24, Section 2] for a more detailed overview.

5.1 Standard L-functions

For $\pi =\otimes _{p}\pi _{p}\in \mathcal {F}(d)$ with $d\geq 2$ , the standard L-function $L(s,\pi )$ associated with $\pi $ is of the form

$$\begin{align*}L(s,\pi)=\prod_{p<\infty} L(s,\pi_p)=\sum_{n=1}^{\infty}\frac{\lambda_{\pi}(n)}{n^s}. \end{align*}$$

The Euler product and Dirichlet series converge absolutely for $\Re (s)>1$ . For each (finite) prime p, the inverse of the local factor $L(s,\pi _p)$ is a polynomial in $p^{-s}$ of degree $\leq d$

$$\begin{align*}L(s,\pi_p)^{-1}=\prod_{j=1}^{d}\Big(1-\frac{\alpha_{j,\pi}(p)}{p^{s}}\Big) \end{align*}$$

for suitable complex numbers $\alpha _{j,\pi }(p)$ . With this convention, we have $\alpha _{j,\pi }(p)\neq 0$ for all j whenever $p\nmid q_{\pi }$ , and it might be the case that $\alpha _{j,\pi }(p)=0$ for some j when $p\mid q_{\pi }$ , where $q_{\pi }$ is the arithmetic conductor of $\pi $ . At the archimedean place of ${\mathbb Q}$ , there are d complex Langlands parameters $\mu _{j,\pi }$ from which we define

$$\begin{align*}L(s,\pi_{\infty}) =\prod_{j=1}^{d}\Gamma_{\mathbb{R}}(s+\mu_{j,\pi}). \end{align*}$$

For all primes p, it is known that there exists a constant

(5.1) $$ \begin{align} \theta_d\in\Big[0,\, \frac{1}{2}-\frac{1}{d^2+1}\Big] \end{align} $$

such that

$$ \begin{align*} | \alpha_{j,\pi}(p)|\leq p^{\theta_d}\qquad\text{and}\qquad \Re(\mu_{j,\pi})\geq -\theta_d \end{align*} $$

for all j. Furthermore, for any unramified prime p and any $1\leq j\leq d$ , one has

(5.2) $$ \begin{align} p^{-\theta_d}\leq | \alpha_{j,\pi}(p)|\leq p^{\theta_d}\qquad\text{and}\qquad |\Re(\mu_{j,\pi})|\leq \theta_d. \end{align} $$

The generalized Ramanujan conjectures assert that $\theta _d$ may be taken as $0$ .

With all the local factors defined as above, we can turn to the functional equation. The contragredient $\widetilde {\pi }$ of $\pi \in \mathcal {F}(d)$ is also an irreducible cuspidal automorphic representation in $\mathcal {F}(d)$ . Thus, we have

$$ \begin{align*} \big\{\alpha_{j,\widetilde{\pi}}(p):\ 1\leq j\leq d\big\}=\big\{\overline{ \alpha_{j,\pi}(p)}:\ 1\leq j\leq d \big\} \end{align*} $$

for each $p< \infty $ , and

$$ \begin{align*} \big\{\mu_{j,\widetilde{\pi}}:\ 1\leq j\leq d \big\}=\big\{\overline{\mu_{j,\pi}}:\ 1\leq j\leq d \big\}. \end{align*} $$

Define the completed L-function

$$\begin{align*}\Lambda(s,\pi) = q_{\pi}^{s/2}L(s,\pi)L(s,\pi_{\infty}). \end{align*}$$

Thus, $\Lambda (s,\pi )$ extends to an entire function. Moreover, $\Lambda (s,\pi ) $ is bounded in vertical strips and satisfies a functional equation of the form

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

where $\omega _\pi $ is a complex number of modulus $1$ .

5.2 Rankin–Selberg L-functions

Now we turn to the Rankin–Selberg L-functions. Let $\pi =\otimes _p \pi _{p}\in \mathcal {F}(d)$ and ${\pi '=\otimes _p \pi _{p}'\in \mathcal {F}(d')}$ . The Rankin–Selberg L-function $L(s,\pi \times \pi ')$ associated with $\pi $ and $\pi '$ is of the form

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

The Euler product and Dirichlet series converge absolutely for $\Re (s)>1$ . For each (finite) prime p, the inverse of the local factor $L(s,\pi _p\times \pi _{p}')$ is a polynomial in $p^{-s}$ of degree $\leq dd'$

(5.3) $$ \begin{align} L(s,\pi_p\times\pi_{p}')^{-1}=\prod_{j=1}^{d}\prod_{j'=1}^{d'}\Big(1-\frac{\alpha_{j,j',\pi\times\pi'}(p) }{p^{s}}\Big) \end{align} $$

for suitable complex numbers $\alpha _{j,j',\pi \times \pi '}(p)$ . With $\theta _d$ as in (5.1), we have the pointwise bound

(5.4) $$ \begin{align} |\alpha_{j,j',\pi\times\pi'}(p)|\leq p^{\theta_{d}+\theta_{d'}}. \end{align} $$

If $p\nmid q_{\pi }$ or $p\nmid q_{\pi '}$ , then we have the equality of sets

(5.5) $$ \begin{align} \big\{\alpha_{j,j',\pi\times\pi'}(p):\ j\leq d,\, j'\leq d' \big\} =\big\{ \alpha_{j,\pi}(p)\alpha_{j',\pi'}(p):\ j\leq d,\, j'\leq d'\big\}. \end{align} $$

At the archimedean place of ${\mathbb Q}$ , there are $dd'$ complex Langlands parameters $\mu _{j,j', \pi \times \pi '}$ from which we define

$$\begin{align*}L(s,\pi_\infty\times\pi_{\infty}')=\pi^{-\frac{dd's}{2}}\prod_{j=1}^{d}\prod_{j'=1}^{d'}\Gamma\Big(\frac{s+\mu_{j,j', \pi\times\pi'}}{2}\Big). \end{align*}$$

These parameters satisfy the equality

$$\begin{align*}\big\{\mu_{j, j', \widetilde{\pi}\times\widetilde{\pi}'}\big\}=\big\{\overline{\mu_{j,j', \pi\times\pi'}}\big\} \end{align*}$$

for $1\leq j\leq d,\, 1\leq j'\leq d'$ and the pointwise bound

(5.6) $$ \begin{align} \Re(\mu_{j,j', \pi\times\pi'})\geq-\theta_{d}-\theta_{d'}. \end{align} $$

The complete L-function

$$\begin{align*}\Lambda(s,\pi\times\pi')=q_{\pi\times\pi'}^{s/2}L(s,\pi\times\pi')L(s,\pi_\infty\times\pi_{\infty}') \end{align*}$$

has a meromorphic continuation and is bounded (away from its poles) in vertical strips. Under our normalization on the central characters, $\Lambda (s,\pi \times \pi ')$ is entire if and only if $\widetilde {\pi }\not =\pi '$ . Moreover, $\Lambda (s,\pi \times \pi ')$ satisfies the functional equation

$$ \begin{align*} \Lambda(s,\pi\times\pi')=\omega_{\pi\times\pi'}\Lambda(1-s,\widetilde{\pi}\times\widetilde{\pi}'), \end{align*} $$

where $\omega _{\pi \times \pi '}$ is a complex number of modulus 1.

Finally, we recall some estimates for $\pi '=\widetilde {\pi }$ . It is known from [13, Lemma 3.1] that

(5.7) $$ \begin{align} |\lambda_{\pi}(n)|^2\leq \lambda_{\pi\times\widetilde{\pi}}(n) \end{align} $$

hold for all positive integer n. Moreover, $L(s,\pi \times \widetilde {\pi })$ extends to the complex plane with a simple pole at $s = 1.$ Hence, Landau’s lemma [2, Theorem 3.2] gives

(5.8) $$ \begin{align} \sum_{n\leq x} \lambda_{\pi\times\widetilde{\pi}}(n)=c_{\pi}x+O_{\pi}\Big(x^{\frac{d^2-1}{d^2+1}} \Big) \end{align} $$

for some constant $c_\pi>0$ .

5.3 Twists

Let $\chi (\mathrm {mod}\,q)$ be a primitive Dirichlet character with $(q,q_\pi )=1$ . As is well known, $\chi $ corresponds to a Hecke character of the idele class group $\mathbb {A}^{\times }/{\mathbb Q}^{\times }$ trivial on $\mathbb {R}_{+}^\times $ , so $\chi $ is of the form $\chi =\otimes _p \chi _p$ .

We apply the Rankin–Selberg theory described above to the following situation: Fix $\pi $ in $\mathcal {F}(d)$ with $m\geq 2$ , and let $\chi $ be a primitive Dirichlet character modulo q. Take $\pi '=\chi $ . The twisted L-function is given by

$$ \begin{align*} L(s, \pi\otimes \chi) = \sum_{n=1}^\infty \frac{\lambda_{\pi}(n)\chi(n)}{ n^{s}}. \end{align*} $$

The corresponding complete L-function

$$ \begin{align*}\Lambda(s,\pi\otimes \chi)= (q_{\pi}q^d)^{s/2}L(s,\pi_{\infty}\times \chi_{\infty})L(s, \pi\otimes \chi)\end{align*} $$

has an analytic continuation to the whole complex plane and satisfies the following functional equation:

$$ \begin{align*}\Lambda(s,\pi\otimes \chi)=\omega_{\pi\otimes\chi}\Lambda(1-s,\widetilde{\pi}\otimes \overline{\chi}),\end{align*} $$

where $L(s,\pi _{\infty }\times \chi _{\infty })$ is given by

$$\begin{align*}L(s,\pi_{\infty}\otimes \chi_{\infty})=\prod_{j=1}^{d}\Gamma_{\mathbb{R}}\big(s+\mu_{j,\pi\otimes\chi}\big). \end{align*}$$

Similarly, if we take $\pi '=\widetilde {\pi }(\chi ):=\widetilde {\pi }\otimes \chi $ , then we have

$$\begin{align*}L(s,\pi\times\widetilde{\pi}(\chi))=\sum_{n=1}^{\infty}\frac{\lambda_{\pi\times\widetilde{\pi}}(n)\chi(n)}{n^s}. \end{align*}$$

The complete L-function

$$ \begin{align*}\Lambda(s,\pi\times\widetilde{\pi}(\chi))= (q_{\pi\times\widetilde{\pi}}q^{2d})^{s/2}L(s,\pi_{\infty}(\chi_{\infty})\times\widetilde{\pi}_{\infty})L(s, \pi\times\widetilde{\pi}(\chi))\end{align*} $$

has an analytic continuation to the whole complex plane and satisfies the following functional equation:

$$ \begin{align*}\Lambda(s,\pi\times\widetilde{\pi}(\chi))=\omega_{\pi\times\widetilde{\pi}(\chi)}\Lambda(1-s,\pi\times \widetilde{\pi}(\overline{\chi})),\end{align*} $$

where

$$\begin{align*}L(s,\pi_{\infty}\times\widetilde{\pi}_{\infty}(\chi_{\infty}))=\prod_{j=1}^{d}\prod_{j'=1}^{d}\Gamma_{\mathbb{R}}\big(s+\mu_{j,j',\pi\times\widetilde{\pi}(\chi)}\big). \end{align*}$$

Due to the work of Müller and Speh [18, proof of Lemma 3.1], all local Langlands parameters $\mu _{j,\pi \otimes \chi }$ and $\mu _{j,j',\pi \times \widetilde {\pi }(\chi )}$ depend on $\pi $ and the parity of $\chi $ at most (see also [24, proof of Lemma 2.1]). Moreover, the relatively explicit expressions of $ \omega _{\pi \otimes \chi }$ and $\omega _{\pi \times \widetilde {\pi }(\chi )}$ are required. We adopt the argument of Barthel–Ramakrishnan [2, Proposition 4.1] or Luo–Rudnick–Sarnak [16, Lemma 2.1] and show the following result.

Lemma 5.1 Let $\pi \in \mathcal {F}(d)$ , and let $\chi (\mathrm {mod}\,q)$ be a primitive Dirichlet character with $(q,q_{\pi })=1$ . Then we have

$$\begin{align*}\omega_{\pi\otimes\chi}=\eta_{\pi, \mathrm{sgn}(\chi)}\chi(q_{\pi}) \tau(\chi)^{d} q^{-\frac{d}{2}}, \end{align*}$$

where $\eta _{\pi , \mathrm {sgn}(\chi )}$ depends on $\pi $ and the parity of $\chi $ only, and $|\eta _{\pi , \mathrm {sgn}(\chi )}|=1.$

Proof Let the $\epsilon $ -factor be defined by

$$\begin{align*}L(s,\pi_{\infty}\otimes \chi_{\infty})L(s, \pi\otimes \chi)=\epsilon(s,\pi\otimes\chi)L(1-s,\pi_{\infty}\otimes \chi_{\infty})L(1-s, \pi\otimes \chi). \end{align*}$$

By the functional equation, the relation between the $\epsilon $ -factor and the root number is

$$\begin{align*}\epsilon(s,\pi\otimes\chi)= (q_{\pi}q^d)^{\frac{1}{2}-s}\omega_{\pi\otimes\chi}. \end{align*}$$

Moreover, it can be written as a product of local factors by fixing an additive character $\psi =\prod _{p\leq \infty }\psi _p$ :

(5.9) $$ \begin{align} \epsilon(s,\pi\otimes\chi)= \prod_{p\leq \infty}\epsilon(s,\pi_p\otimes\chi_p,\psi_p). \end{align} $$

If $p\nmid q_{\pi }q,$ where $\pi _p$ and $\chi _p$ are both unramified, then

(5.10) $$ \begin{align} \epsilon(s,\pi_p\otimes\chi_p,\psi_p)=1. \end{align} $$

Suppose that $p^{r(\chi _p)}\parallel q$ , in which case $\chi _p$ is ramified with conductor $p^{r(\chi _p)}$ . By assumption, $\pi _p$ is the canonical component of $\pi _{q}=\operatorname {Ind}\left (\mathrm {GL}_{d}, B; \mu _{1}, \ldots , \mu _{d}\right )$ where B is the Borel subgroup of $\mathrm {GL}_{m}$ and $\mu _{j}(x)=|x|^{u_{j}}$ are unramified characters. Then $\pi _{q}\otimes \chi _p=\operatorname {Ind}\left (\mathrm {GL}_{d}, B; \chi \mu _{1}, \ldots , \chi \mu _{d}\right )$ . Thus, we have

$$ \begin{align*} \begin{aligned} \epsilon(s,\pi_p\otimes\chi_p,\psi_p)&=\prod_{j=1}^{d} \epsilon(s,\mu_{j}\otimes\chi_p,\psi_p) \\ &=\prod_{j}^{d} \epsilon\left(s, \mu_{j} \chi_p, \psi_{p}\right) \\ &=\prod_{j}^{d} \epsilon\left(s+u_{j}, \chi_p, \psi_{p}\right)\hspace{-0.5pt}, \end{aligned} \end{align*} $$

where the abelian $\epsilon $ -factor (for $\chi $ primitive) is given by

$$ \begin{align*}\epsilon\left(s, \chi, \psi_{q}\right)=\tau(\chi) p^{-{r(\chi_p)}s}. \end{align*} $$

Since $\epsilon (s,\pi _p,\psi _p)=1$ and the central character of $\pi $ is trivial, which means that $\sum _{j=1}^mu_{j}=0$ , we have

(5.11) $$ \begin{align} \begin{aligned} \epsilon(s,\pi_p\otimes\chi_p,\psi_p) &=\prod_{j}^{d} \tau(\chi) p^{-{r(\chi_p)}(s+u_{j})} \\ &=\tau\left(\chi, \psi_{p}\right)^{d} p^{-dr(\chi_p)s}\epsilon(s,\pi_p,\psi_p). \end{aligned} \end{align} $$

Suppose that $p^{r(\pi _p)}\parallel q_{\pi }$ , in which case $\chi _p$ is unramified given by $\chi _{p}(x)=|x|^{v_{p}}$ . With this given, we have

(5.12) $$ \begin{align} \begin{aligned} \epsilon(s,\pi_p\otimes\chi_p,\psi_p)&= \epsilon(s+v_p,\pi_p,\psi_p)\\ &=\omega_{\pi_p}p^{r(\pi_p)\left(\frac{1}{2}-s-v_{p}\right)} \\ &=\chi\left(p^{r(\pi_p)}\right) \epsilon(s,\pi_p,\psi_p), \end{aligned} \end{align} $$

Consider the archimedean place. It is known from [12] that $\epsilon (s,\pi _\infty ,\psi _\infty )$ and $ \epsilon (s,\pi _\infty \otimes \chi _\infty ,\psi _\infty )$ are constants, hence equal to the corresponding values at $s=1/2$ . Since $\chi _{\infty }(x)=\mathrm {sgn}(x)|x|^{v_{\infty }}$ , the constant $\epsilon (s,\pi _p\otimes \chi _p,\psi _p)$ depends only on $\pi $ and the parity of $\chi $ .

Finally, inserting (5.10), (5.11) and (5.12) into (5.9), we get

(5.13) $$ \begin{align} \begin{aligned} \epsilon(s,\pi\otimes\chi)= &\Bigg(\prod_{p|q}\tau\left(\chi, \psi_{p}\right)^{d} p^{-dr(\chi_p)s}\epsilon(s,\pi_p,\psi_p)\Bigg) \Bigg(\prod_{p|q_{\pi}}\chi\left(p^{r(\pi_p)}\right) \epsilon(s,\pi_p,\psi_p)\Bigg)\\ &\times\frac{\epsilon_\infty(\frac{1}{2},\pi_\infty\otimes\chi_\infty,\psi_\infty)}{\epsilon_\infty(\frac{1}{2},\pi_\infty,\psi_\infty)}\epsilon_\infty(s,\pi_\infty,\psi_\infty) \\ =&c_{\pi, \mathrm{sgn}(\chi)}\chi(q_{\pi}) \tau(\chi)^{m} q^{-d s} \epsilon(s, \pi), \end{aligned} \end{align} $$

where $c_{\pi , \mathrm {sgn}(\chi )}:=\epsilon _\infty (1/2,\pi _\infty \otimes \chi _\infty ,\psi _\infty )/\epsilon _\infty (1/2,\pi _\infty ,\psi _\infty )$ is a constant depending on $\pi $ and the parity of $\chi $ only. Thus, the relation (5.13) of $\epsilon $ -factors gives

$$ \begin{align*} \begin{aligned} \omega_{\pi\otimes\chi}= &\Bigg(\prod_{p|q}\tau\left(\chi, \psi_{p}\right)^{d} p^{-dr(\chi_p)s}\epsilon(s,\pi_p,\psi_p)\Bigg) \Bigg(\prod_{p|q_{\pi}}\chi\left(p^{r(\pi_p)}\right) \epsilon(s,\pi_p,\psi_p)\Bigg)\\ &\times\frac{\epsilon_\infty(\frac{1}{2},\pi_\infty\otimes\chi_\infty,\psi_\infty)}{\epsilon_\infty(\frac{1}{2},\pi_\infty,\psi_\infty)}\epsilon_\infty(s,\pi_\infty,\psi_\infty) \\ =&c_{\pi, \mathrm{sgn}(\chi)}\chi(q_{\pi}) \tau(\chi)^{d} q^{-\frac{d}{2}} \omega_{\pi}, \end{aligned} \end{align*} $$

which implies $|c_{\pi , \mathrm {sgn}(\chi )}|=1$ in turn. On putting $\eta _{\pi , \mathrm {sgn}(\chi )}=c_{\pi , \mathrm {sgn}(\chi )}\omega _{\pi }$ , we complete the proof of this lemma.

Similar to Lemma 5.1, we can also show the following lemma.

Lemma 5.2 Let $\pi \in \mathcal {F}(d)$ be a cuspidal automorphic representation of $\mathrm {GL}(d)$ of conductor $q_{\pi }$ with trivial central character, and $\chi (\mathrm {mod}\,q)$ be a primitive Dirichlet character with $(q,q_{\pi })=1$ . Then we have

$$\begin{align*}\omega_{\pi\times\widetilde{\pi}(\chi)}=\eta_{\pi\times\widetilde{\pi}, \mathrm{sgn}(\chi)}\chi(q_{\pi\times\widetilde{\pi}}) \tau(\chi)^{d^2} q^{-\frac{d^2}{2}}, \end{align*}$$

where $\eta _{\pi \times \widetilde {\pi }, \mathrm {sgn}(\chi )}$ depends on $\pi $ and the parity of $\chi $ only, and $|\eta _{\pi \times \widetilde {\pi }, \mathrm {sgn}(\chi )}|=1.$

6 Applications of Theorems 2.1

6.1 Proof of Theorem 1.2

From the discussion in Section 5, we see that the Rankin–Selberg L-function ${L(s,\pi \times \widetilde {\pi })}$ satisfies Conditions (A1)–(A3) with $m=1$ , and its twisted L-function ${L(s,\pi \otimes \chi )}$ satisfies Condition (A4), where the later follows from Lemma 5.2.

Next, we discuss the sizes of various types for the coefficients $\lambda _{\pi \times \widetilde {\pi }}(n)$ . The asymptotic formula (5.8) yields Hypothesis $\mathrm {S}$ with $b_{\pi \times \widetilde {\pi }}=1$ . Since the central character of $\pi $ is trivial, one has

$$\begin{align*}s_{d,\pi}(p)=\alpha_{1,\pi}(p)\alpha_{2,\pi}(p)\cdots\alpha_{d,\pi}(p)=1 \end{align*}$$

for all primes p with $(p,q_\pi )=1.$ Then it follows from (5.2) and (5.5) that

$$\begin{align*}|\alpha_{j,j',\pi\times\pi'}(p)|\leq p^{2\theta_{d}}, \quad \; s_{j,\pi\times \widetilde{\pi}}(p)\ll p^{2\min\{j,d^2-j\}\theta_d} \end{align*}$$

for any prime p with $(p,q_\pi )=1$ and any $1\leq j\leq d^2$ , which implies Hypothesis $\mathrm {H(\theta _{d^2})}$ with $ \theta _{d^2}=2\theta _d\leq 1-\frac {2}{d^2+1}<1-\frac {1}{d^2}$ . Therefore, we can apply Theorem 2.1 to the nonnegative coefficients $\lambda _{\pi \times \widetilde {\pi }}(n)$ , and then obtain

$$ \begin{align*} \sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}\lambda_{\pi\times \widetilde{\pi}}(n)=&M_{0}(x;q)+O_{\pi}\Big(\tau_{d^2}(q)q^{\frac{d^2-1}{2}}\log q \Big)+O_{\pi}\Big(\tau_{d^2}(q)x^{\frac{d^2-1}{d^2+1}}\Big), \end{align*} $$

where the main term is given by

$$\begin{align*}M_{0}(x;q)=\frac{1}{\varphi(q)}\mathop{\mathrm{Res}}\limits_{s=1} \Big(\frac{1}{s} L\big(s, \pi\times \widetilde{\pi}(\chi_0) \big)x^{s}\Big). \end{align*}$$

Since

$$\begin{align*}L\big(s, \pi\times \widetilde{\pi}(\chi_0) \big)=L(s, \pi\times \widetilde{\pi})\prod_{p|q}L(s, \pi_p\times \widetilde{\pi}_p)^{-1} \end{align*}$$

and $L(s, \pi \times \widetilde {\pi })$ has a simple pole at $s = 1,$ we have

$$\begin{align*}M_{0}(x;q)=\frac{1}{\varphi(q)}\mathop{\mathrm{Res}}\limits_{s=1} \big(L(s, \pi\times \widetilde{\pi} )\big)\prod_{p|q}L(1, \pi_p\times \widetilde{\pi}_p)^{-1}x. \end{align*}$$

This completes the proof of Theorem 1.2.

6.2 Proof of Theorem 1.1

Similar to the argument in Section 6.1, we can apply Theorem 2.1 to the coefficients $\lambda _{\pi }(n)$ . By applying the Cauchy–Schwarz inequality, (5.7) and (5.8), we get

(6.1) $$ \begin{align} \sum_{\substack{x<n\leq x+y\\ n\equiv a\,(\mathrm{mod}\,q)}}|\lambda_{\pi}(n)|\ll_{\pi} \Big(\frac{xy}{q}\Big)^{1/2} \end{align} $$

for any $q\leq y\leq x$ , which yields Hypothesis $\mathrm {S}$ with $b_\pi =1$ . Since $L(s,\pi )$ is entire, the main term and the first error term do not exist when applying Theorem 2.1. Thus, we obtain

(6.2) $$ \begin{align} \sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}\lambda_{\pi}(n)\ll_{\pi} \tau_{d}(q)q^{\frac{d-1}{2}}\log q+ \tau_{d}(q)\Big(\frac{qx}{y}\Big)^{\frac{d-1}{2}} +\sum_{\substack{x<n\leq x+O(y) \\ n \equiv a\,(\mathrm{mod}\,q)}}|\lambda_{\pi}(n)|. \end{align} $$

Inserting the bound (6.1) and taking $y=qx^{1-\frac {2}{d}}$ , we get the first bound

$$\begin{align*}\sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}\lambda_{\pi}(n)\ll_{\pi} \tau_{d}(q)x^{1-\frac{1}{d}} \end{align*}$$

for $q\leq x^{\frac {1}{d}}$ .

Moreover, it follows from Theorem 1.2 that

$$\begin{align*}\sum_{\substack{x<n\leq x+y\\ n\equiv a\,(\mathrm{mod}\,q)}}\lambda_{\pi\times \widetilde{\pi}}(n)\ll_{\pi} \frac{c_{\pi,q}}{\varphi(q)}y+O\Big(\tau_{d^2}(q)q^{\frac{d^2-1}{2}}\log q \Big)+O\Big(\tau_{d^2}(q)x^{\frac{d^2-1}{d^2+1}}\Big) \end{align*}$$

for $q\leq x^{\frac {2}{d^2+1}}$ . By (5.3) and (5.4), the constant $c_{\pi ,q}$ satisfies

$$\begin{align*}c_{\pi,q}\ll_{\pi} \prod_{p|q}\big(1+p^{-\frac{2}{d^2+1}+\varepsilon}\big)^{d^2} \ll \tau(q). \end{align*}$$

Note that $q/\varphi (q)\ll \log q$ . Further, we get from (5.7) that

$$\begin{align*}\sum_{\substack{x<n\leq x+y\\ n\equiv a\,(\mathrm{mod}\,q)}}|\lambda_{\pi}(n)|\ll \tau_{d^2}(q)\log x\left(\frac{y}{q}+\sqrt{\frac{y}{q}} \cdot x^{\frac{1}{2}-\frac{1}{d^2+1}}\right) \end{align*}$$

for $q\leq x^{\frac {2}{d^2+1}}$ . On taking $y=qx^{1-\frac {2d}{d^2+1}}$ , the estimate (6.2) gives the second bound

$$\begin{align*}\sum_{\substack{n\leq x\\ n\equiv a\,(\mathrm{mod}\,q)}}\lambda_{\pi}(n)\ll \tau_{d^2}(q)x^{1-\frac{d+1}{d^2+1}}\log x \end{align*}$$

for $q\leq x^{\frac {2}{d^2+1}}$ .

Assume the Ramanujan conjecture holds for $\pi $ , the Brun–Titchmarsh inequality (see Shiu [20, Theorem 1]) yields

(6.3) $$ \begin{align} \sum_{\substack{x<n\leq x+y\\ n\equiv a\,(\mathrm{mod}\,q)}}|\lambda_{\pi}(n)|\leq \frac{y}{\varphi(q)\log x}\exp\Big(\sum_{\substack{p\leq x\\ p\nmid q}}\frac{|\lambda_{\pi}(p)|}{p}\Big) \end{align} $$

provided that $q\leq y^{1-\varepsilon }$ and $x^\varepsilon \leq y\leq x$ . By Mertens’ theorem and the prime number theorem for Rankin–Selberg L-function $L(s,\pi \times \widetilde {\pi })$ (see [13, p. 630]), one has

$$\begin{align*}\sum_{p\leq x}\frac{|\lambda_{\pi}(p)|}{p}\ll \Big(\sum_{p\leq x}\frac{1}{p}\Big)^{\frac{1}{2}}\Big(\sum_{p\leq x} \frac{\lambda_{\pi\times \widetilde{\pi}}(p)}{p}\Big)^{\frac{1}{2}}\ll \log\log x. \end{align*}$$

Inserting this estimate into (6.3), we obtain

$$\begin{align*}\sum_{\substack{x<n\leq x+y\\ n\equiv a\,(\mathrm{mod}\,q)}}|\lambda_{\pi}(n)|\ll \frac{y}{\varphi(q)} \end{align*}$$

provided that $q\leq y^{1-\varepsilon }$ and $x^\varepsilon \leq y\leq x$ . Substitute this into (6.2) and taking $y=qx^{1-\frac {2}{d+1}}$ , the last assertion follows.

6.3 Proof of Theorem 1.3

We begin with evaluating the summation about $\lambda _{\mathrm {sym}^d f}(n)$ in a short interval.

Lemma 6.1 Let $f\in H_k^{*}(N)$ and $\lambda _{\mathrm {sym}^d f}(n)$ be the coefficients of $L(s,\mathrm {sym}^d f )$ . For $(q,aN)=1,$ we have

$$\begin{align*}\sum_{\substack{x<n\leq x+y \\ n \equiv a\,(\mathrm{mod}\,q)}}|\lambda_{\mathrm{sym}^d f}(n)|\ll \frac{y}{\varphi(q)(\log x)^{\gamma_d}} \end{align*}$$

provided that $q\leq y^{1-\varepsilon }$ and $x^\varepsilon \leq y\leq x$ , where $\gamma _d=1-\frac {4(d+1)}{d(d+2)\pi } \cot \left (\frac {\pi }{2(d+1)}\right )$ and ${0.15<\gamma _d<0.19.}$

Proof Let

$$\begin{align*}U_d(\cos\theta_p)=\frac{\sin((d+1)\theta_p)}{\sin\theta_p} \end{align*}$$

be the d-th Chebyshev polynomial of the second type. One can easily check via (1.4) that

$$\begin{align*}\lambda_{\mathrm{sym}^d f}(p) = U_d(\cos\theta_p),\qquad p\nmid N. \end{align*}$$

By the Sato–Tate conjecture (1.5) and a straightforward calculation of Maple, we get

$$ \begin{align*} \begin{aligned} \sum_{\substack{p\leq x\\ p\nmid q}}|\lambda_{\mathrm{sym}^d f}(p)|\leq& \sum_{\substack{p\leq x\\ p\nmid N}}|\lambda_{\mathrm{sym}^d f}(p)|+O(1)\\ \sim& \Big(\int_{0}^{\pi} \frac{|\sin((d+1)\theta)|}{\sin\theta} {\text{d}} \mu_{ST}\Big)\frac{x}{\log x}\\ \sim& \frac{4(d+1)}{d(d+2)\pi} \cot \left(\frac{\pi}{2(d+1)}\right)\frac{x}{\log x}. \end{aligned} \end{align*} $$

Hence, we derive by partial summation and substituting this into (6.3) that

$$\begin{align*}\sum_{\substack{x<n\leq x+y \\ n \equiv a\,(\mathrm{mod}\,q)}}|\lambda_{\mathrm{sym}^d f}(n)|\ll \frac{y}{\varphi(q)(\log x)^{\gamma_d} }, \end{align*}$$

where $\gamma _d=1-\frac {4(d+1)}{d(d+2)\pi } \cot \left (\frac {\pi }{2(d+1)}\right)$ . It is clear that $\gamma _d$ is strictly increasing. Thus, for any $d\geq 1$ , we have

$$\begin{align*}0.15<1-\frac{8}{3\pi}=\gamma_1\leq \gamma_d\leq \lim\limits_{d\rightarrow \infty}\gamma_d=1-\frac{8}{\pi^2}<0.19.\\[-42pt] \end{align*}$$

Finally, the proof of Theorem 1.3 is completed if we combine the first assertion of Theorem 2.1 with Lemma 6.1, the choice $y=qx^{\frac {d}{d+2}}$ and the fact $q/\varphi (q)\leq \tau (q)$ .