abconsymbol=c

# A Bombieri–Vinogradov theorem for higher rank groups

Yujiao Jiang  Yujiao Jiang
School of Mathematics and Statistics
Shandong University
Weihai
Shandong 264209
China<EMAIL_ADDRESS>,  Guangshi Lü  Guangshi Lü
School of Mathematics
Shandong University
Jinan
Shandong 250100
China<EMAIL_ADDRESS>,  Jesse Thorner  Jesse Thorner
Department of Mathematics
University of Illinois
Urbana
IL 61801
United States<EMAIL_ADDRESS>and  Zihao Wang  Zihao Wang
School of Mathematics
Shandong University
Jinan
Shandong 250100
China<EMAIL_ADDRESS>
###### Abstract.

We establish a result of Bombieri–Vinogradov type for the Dirichlet
coefficients at prime ideals of the standard $L$-function associated to a
self-dual cuspidal automorphic representation $\pi$ of $\operatorname{GL}_{n}$
over a number field $F$ which is not a quadratic twist of itself. Our result
does not rely on any unproven progress towards the generalized Ramanujan
conjecture or the nonexistence of Landau–Siegel zeros. In particular, when
$\pi$ is fixed and not equal to a quadratic twist of itself, we prove the
first unconditional Siegel-type lower bound for the twisted $L$-values
$|L(1,\pi\otimes\chi)|$ in the $\chi$-aspect, where $\chi$ is a primitive
quadratic Hecke character over $F$. Our result improves the levels of
distribution in other works that relied on these unproven hypotheses. As
applications, when $n=2,3,4$, we prove a $\mathrm{GL}_{n}$ analogue of the
Titchmarsh divisor problem and a nontrivial bound for a certain
$\mathrm{GL}_{n}\times\mathrm{GL}_{2}$ shifted convolution sum.

###### Key words and phrases:

Bombieri–Vinogradov theorem, automorphic $L$-functions, Siegel’s zero.

###### 2010 Mathematics Subject Classification:

11F66, 11M41

###### Contents

  1. 1 Introduction
  2. 2 Properties of $L$-functions
  3. 3 Preliminary reductions and a generalized Vaughan identity
  4. 4 Zero-free regions
  5. 5 Auxiliary estimates
  6. 6 Large-sieve type estimates
  7. 7 Type I sums: Proof of Theorem 1.3
  8. 8 A bilinear form
  9. 9 Proof of Theorem 3.1
  10. 10 An arithmetic application: Proof of Corollary 1.4

##  1\. Introduction

The distribution of primes in arithmetic progressions attracts a lot of
attention among mathematicians. Let $a,q$ be two integers such that $(a,q)=1$.
We denote by $\pi(x)$ the number of primes $p\leqslant x$ and by $\pi(x;q,a)$
the number of primes $p\leqslant x$ satisfying $p\equiv
a\,(\operatorname{mod}{q})$. Dirichlet’s theorem indicates the following

$\pi(x;q,a)\sim\frac{\pi(x)}{\varphi(q)},$

where $\varphi$ is Euler’s totient function. Later, after Siegel’s result on
the location of exceptional zero of Dirichlet $L$-functions, Walfisz proved
that for all $\alpha>0$, there exists an ineffective constant $c_{\alpha}>0$
such that if $q\leqslant(\log x)^{\alpha}$, then

$\pi(x;q,a)=\frac{\pi(x)}{\varphi(q)}+O\big{(}x\exp(-{c_{\alpha}}(\log
x)^{1/2})\big{)}.$

When the modulus $q$ gets larger, this problem becomes much more difficult. If
the generalized Riemann hypothesis (GRH) holds, then

$\pi(x;q,a)=\frac{\pi(x)}{\varphi(q)}+O\big{(}x^{\frac{1}{2}}\log qx\big{)}$

holds for $q\geqslant x^{1/2-\varepsilon}$. However, such a hypothesis is very
far from being proved.

The celebrated Bombieri–Vinogradov theorem in some sense shows that GRH holds
on average. To be precise, let $A$ be any positive real number, there exists
$B=B(A)>0$ such that for $Q\leqslant x^{\frac{1}{2}}(\log x)^{-B}$,

(1.1) $\sum_{q\leqslant Q}\max_{(a,q)=1}\max_{y\leqslant
x}\Big{|}\pi(y,q,a)-\frac{\pi(y)}{\varphi(q)}\Big{|}\ll_{A}\frac{x}{(\log
x)^{A}}.$

This can be viewed as a fine substitute for the GRH in many applications. The
theorem was originally proved using zero density estimates. After the work of
Bombieri and Vinogradov, different proofs of this theorem are given by
Gallagher [8] and Vaughan [33].

There are a lot of higher-rank analogues of the classical Bombieri–Vinogradov
theorem. Firstly, by means of Gallagher’s method, Grupp [10] obtained under a
certain condition concerning Siegel’s zeros of $\operatorname{GL}_{2}$
automorphic $L$-functions,

$\sum_{q\leqslant x^{2/9}(\log
x)^{-B}}\max_{(a,q)=1}\Big{|}\sum_{\begin{subarray}{c}n\leqslant x\\\ n\equiv
a\,(\operatorname{mod}{q})\end{subarray}}\Lambda(m)\tau(m)m^{-\frac{11}{2}}\Big{|}\ll_{A}\frac{x}{(\log
x)^{A}},$

where $\Lambda(m)$ is the von Mangoldt function and $\tau(m)$ is the Ramanujan
$\tau$-function. Later, Perelli [26] used the generalized Vaughan identity for
$\operatorname{GL}_{2}$ automorphic $L$-functions and unconditionally proved
the mean-value theorem with a level of distribution $2/5$ instead of $2/9$.
Actually, Perelli’s approach still works for any holomorphic cusp form on
$\operatorname{SL}_{2}(\mathbb{Z})$. Recently, Acharya [1] and the first two
authors [17] improved independently the level to $1/2$ for any holomorphic or
Maass cusp form on $\operatorname{SL}_{2}(\mathbb{Z})$. For any automorphic
form $\pi$ on higher-rank group $\operatorname{SL}_{n}(\mathbb{Z})$ with
$n\geqslant 3$, let $\lambda_{\pi}(m)$ to be the $m$-th Dirichlet coefficient
of the associated $L$-function $L(s,\pi)$, one can also show a result of
Bombieri–Vinogradov type

(1.2) $\sum_{q\leqslant
Q}\max_{(a,q)=1}\Big{|}\sum_{\begin{subarray}{c}m\leqslant x\\\ m\equiv
a\,(\operatorname{mod}{q})\end{subarray}}\Lambda(m)\lambda_{\pi}(m)\Big{|}\ll_{A,\pi}\frac{x}{(\log
x)^{A}}.$

For instance, the first two authors [17] established (1.2) with
$Q=x^{\frac{2}{n+1}}(\log x)^{-B}$ under the generalized Ramanujan conjecture
(GRC) and a certain condition concerning Siegel’s zeros of the twisted
$L$-functions $L(s,\pi\otimes\chi)$. Wong [34] showed (1.2) with
$Q=x^{\min\\{\frac{1}{n-2},\frac{1}{2}\\}-\varepsilon}$ under two similar
conditions. The main tools of Jiang and Lü are the generalized Vaughan
identity and the distribution of $\lambda_{\pi}(m)$ in arithmetic
progressions, while that of Wong is Gallagher’s technique as in [8].

In this paper, we will explore further the possibility of Vaughan’s method and
show an unconditional result for higher-rank groups in a number field. We
refer the reader to Section 2 for the detailed introduction to the notation.
Let $\mathbb{A}_{F}$ be the ring of adeles over a number field $F$, and let
$\mathfrak{F}_{n}$ be the set of cuspidal automorphic representations of
$\mathrm{GL}_{n}(\mathbb{A}_{F})$ with unitary central character, normalized
such that the central character is trivial on the diagonally embedded copy of
the positive reals. Given $\pi\in\mathfrak{F}_{n}$, let $\mathfrak{q}_{\pi}$
be the conductor of $\pi$, $L(s,\pi)$ be the associated standard $L$-function,
and $\widetilde{\pi}\in\mathfrak{F}_{n}$ be the contragredient representation.
We write $\lambda_{\pi}(\mathfrak{n})$ to be the $\mathfrak{n}$-th Dirichlet
coefficient of $L(s,\pi)$, where $\mathfrak{n}$ is an integral ideal in $F$.
Let $\mathrm{N}=\mathrm{N}_{F/\mathbb{Q}}$ to be the numerical norm. As the
classical Bombieri–Vinogradov theorem (1.1), we will consider estimates of
large sieve type associated to $\lambda_{\pi}(\mathfrak{p})$ with certain
congruence condition. We denote by $\mathrm{Cl}^{+}(\mathfrak{m})$ the narrow
class group modulo $\mathfrak{m}$. Let $h(\mathfrak{m})$ be the cardinality of
$\mathrm{Cl}^{+}(\mathfrak{m})$ and
$\varphi_{F}(\mathfrak{m}):=\mathrm{N}\mathfrak{m}\prod_{\mathfrak{p}|\mathfrak{m}}(1-\mathrm{N}\mathfrak{p}^{-1})$.

Our arguments require that if $\pi\in\mathfrak{F}_{n}$, then
$\pi=\widetilde{\pi}$. This self-duality implies that
$\lambda_{\pi}(\mathfrak{n})\in\mathbb{R}$ for all $\mathfrak{n}$. Also, we
require that for all $\mathfrak{m}\subseteq\mathcal{O}_{F}$ and all nontrivial
primitive quadratic Hecke characters of $\mathrm{Cl}^{+}(\mathfrak{m})$, we
have $\pi\neq\pi\otimes\chi$. We let $\mathfrak{F}_{n}^{\flat}$ denote the set
of all $\pi\in\mathfrak{F}_{n}$ satisfying these two hypotheses. We prove the
following result.

######  Theorem 1.1.

Fix $\pi\in\mathfrak{F}_{n}^{\flat}$. If $A>0$,
$B=2^{\frac{n[F:\mathbb{Q}]}{4}}(6A+12n+34)+2n-4$, and
$\eta=\max\\{2,\frac{n}{2}\\}$, then

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{p}\leqslant y\\\
\mathfrak{p}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{p})\Big{|}\ll_{A}\frac{x}{(\log
x)^{A}},$

where $x\geqslant 2$ and $Q=x^{\frac{1}{\eta}}(\log x)^{-B}$. The implied
constant is ineffective.

######  Remark.

  1. 1.

The congruence condition “$\mathfrak{p}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})$” is defined in Section 2.4. This generalizes
the usual notion of congruences on the integers to the integral ideals of $F$.
In particular, if $F=\mathbb{Q}$, then the bound in Theorem 1.1 becomes

$\sum_{q\leqslant Q}\max_{\gcd(a,q)=1}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}p\leqslant y\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\lambda_{\pi}(p)\Big{|}\ll_{A}\frac{x}{(\log
x)^{A}},$

where $p$ (resp. $a$ and $q$) are rational primes (resp. rational integers).

  2. 2.

If we adjust Theorem 1.1 so that we sum over $\mathfrak{m}$ satisfying
$(\mathfrak{m},\mathfrak{q}_{\pi})=\mathcal{O}_{F}$, then we may obtain a
similar result with the same level of distribution. Our result would then hold
for all self-dual $\pi$ since the condition $\pi\neq\pi\otimes\chi$
automatically holds.

  3. 3.

The weight $\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}$ is introduced
by Huxley in [14] to cancel the contribution coming from the unit group (see
(2.11)).

  4. 4.

Note that the analogue of Elliott-Halberstam conjecture will predict that
$\eta=1+\varepsilon$, and the GRH for automorphic $L$-functions will trivially
give that $\eta=2$. Since the arithmetic conductor of $L(s,\pi\otimes\chi)$
might be quite large, it is hard to achieve any of them by our argument.

To handle the contribution when $\mathrm{N}\mathfrak{m}$ is smaller than a
power of $\log x$, we need to prove an analogue of the Siegel–Walfisz theorem
for the Dirichlet coefficients of $-\frac{L^{\prime}}{L}(s,\pi)$. One of the
novelties in our work which allows us to prove such a result without recourse
to unproven hypotheses is a new Siegel-type lower bound for
$|L(1,\pi\otimes\chi)|$ when $\chi$ is a primitive quadratic Hecke character
and $\pi\in\mathfrak{F}_{n}$ (not necessarily self-dual) is not a quadratic
twist of itself.

######  Theorem 1.2.

Fix $\pi\in\mathfrak{F}_{n}$, and suppose that $\pi\neq\pi\otimes\nu$ for all
primitive quadratic Hecke characters $\nu$. Let
$\chi\,(\mathrm{mod}{\,\mathfrak{q}})$ be a primitive quadratic Hecke
character. For all $\varepsilon>0$, there exists an ineffective constant
$c_{\pi}^{\prime}(\varepsilon)>0$ such that $|L(1,\pi\otimes\chi)|\geqslant
c_{\pi}^{\prime}(\varepsilon)\mathrm{N}\mathfrak{q}^{-\varepsilon}$.

######  Remark.

Over $\mathbb{Q}$, Theorem 1.2 was claimed by Molteni in his PhD thesis, but
there is a serious deficiency in his argument. Since this deficiency has
occurred in several different papers (even before Molteni’s), we detail the
deficiency and address it in Section 4.

To handle the contribution when $\mathrm{N}\mathfrak{m}$ is larger than a
power of $\log x$, we require a modification of Vaughan’s approach to the
Bombieri–Vinogradov theorem. The problem of estimating

$\sideset{}{}{\sum}_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
X}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{p}\leqslant y\\\
\mathfrak{p}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{p})\Big{|}$

is equivalent to that of handling

$\sideset{}{}{\sum}_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})\Big{|}$

with a harmless error, where $\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})$
is the coefficient of $-\frac{L^{\prime}}{L}(s,\pi)$. We derive a generalized
Vaughan identity, which gives an expression for
$\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})$, and then apply it to
decompose the above object into the Type I sum

(1.3) $\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{n})\Big{|}$

and the Type II sum

(1.4) $\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}{\underset{\mathrm{N}\mathfrak{ln}\leqslant
y\atop\mathfrak{ln}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})}{\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}}}a(\mathfrak{l})b(\mathfrak{n})\Big{|}.$

Note that (1.4) is actually bilinear form with $L,N$ in suitable ranges, and
$a(\mathfrak{n}),b(\mathfrak{n})$ are arithmetic functions related to $\pi$.

In our setting, a strong bound for (1.3) is already new. Since it is useful in
contexts beyond that of Theorem 1.1, we state it as its own theorem.

######  Theorem 1.3.

Fix $\pi\in\mathfrak{F}_{n}$. If $A>0$, $B=2^{n[F:\mathbb{Q}]/4}(2A+16)+2n-5$,
and $\eta=\max\\{\frac{n}{2},2\\}$, then

$\sideset{}{}{\sum}_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{n})\Big{|}\ll_{\pi}\frac{x}{(\log
x)^{A}},$

where $x\geqslant 2$ and $Q=x^{\frac{1}{\eta}}(\log x)^{-B}$.

######  Remark.

The value of $\eta$ in Theorem 1.1 is totally determined by the value of
$\eta$ in Theorem 1.3. We improve previous results because we notice that more
cancellation can be obtained by summing over the modulus, which is absent in
the work of [17, 33]. In fact, Vaughan [33] directly used the Pólya-Vinogradov
inequality, and the first two authors [17] used the Voronoï formula on ${\rm
GL}(n)$ to treat the sum of $\lambda_{\pi}(\mathfrak{n})$ over a single
arithmetic progression.

We estimate (1.4) through bilinear sum methods, proving a general result
similar to [16, Theorem 17.4]. However, the important condition of [16,
Theorem 17.4] is that one of these two arithmetic functions $a(\mathfrak{n})$
and $b(\mathfrak{n})$ satisfies a Siegel–Walfisz hypothesis. In our situation,
we need to verify that both $\lambda_{\pi}(\mathfrak{n})$ and
$\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})$ satisfy a Siegel–Walfisz
hypothesis. This hypothesis is straightforward to verify for
$\lambda_{\pi}(\mathfrak{n})$, and as mentioned above, we verify this
hypothesis for $\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})$ as a corollary
of Theorem 1.2. Note that there is a cumbersome cut-off condition in (1.4). To
handle this, we adopt a trick of Vaughan in [33].

Our upper bound for (1.4) involves the second moments of some arithmetic
functions of length $x$, whose magnitudes need to be of order $O(x(\log
x)^{c})$ for some computable constant $c$. If GRC holds for $\pi$, then the
desired upper bound follows from elementary estimate of divisor functions. In
[17], the first two authors bounded these arithmetic functions under
Hypothesis H of Rudnick and Sarnak [27]. This mild conjecture is implied by
GRC and is only known to hold for few cases. In order to circumvent this
additional assumption, we instead bound them by some Dirichlet convolutions of
$\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})$ through the dual Pieri rule
and a combinatorial lemma of Soundrarajan. The desired upper bound then
follows from the Rankin–Selberg theory.

As in the classical case, Theorem 1.1 is a fruitful result. As an application,
we will give one analogue of Titchmarsh’s divisor problem on
$\operatorname{GL}_{n}$ over $\mathbb{Q}$ with $2\leqslant n\leqslant 4$. Let
$d(m)$ be the usual divisor function. It is known that $d(m)$ are Fourier
coefficients of $\frac{\partial}{\partial s}E(z,s)$ at $s=\frac{1}{2}$, where
$E(z,s)$ is the Eisenstein series for $\operatorname{SL}_{2}(\mathbb{Z})$.
Thus, the following result may be also viewed as the shifted convolution sum
at primes for $\operatorname{GL}_{n}\times\operatorname{GL}_{2}$.

######  Corollary 1.4.

Let $2\leqslant n\leqslant 4$, and fix $\pi\in\mathfrak{F}_{n}^{\flat}$. If
$x\geqslant 2$ and $\pi$ is defined over $\mathbb{Q}$, then

$\sum_{p\leqslant x}\lambda_{\pi}(p)d(p-1)\ll_{\pi}\frac{x(\log\log
x)^{\frac{3}{2}}}{\sqrt{\log x}},$

where the implied constant depends on $\pi$.

######  Remark.

The case with $n=2$ is known by the work of Acharya [1]. Under GRC, the first
two authors [17] handled the cases with $n=2,3$ and obtained a stronger upper
bound than that in Corollary 1.4.

If we use Theorem 1.3 instead of Theorem 1.1, then the argument leading to
Corollary 1.4 produces a corresponding shifted convolution bound over the
integers.

######  Corollary 1.5.

Let $2\leqslant n\leqslant 4$, and fix $\pi\in\mathfrak{F}_{n}^{\flat}$. If
$x\geqslant 2$ and $\pi$ is defined over $\mathbb{Q}$, then

$\sum_{m\leqslant x}\lambda_{\pi}(m)d(m-1)\ll_{\pi}x(\log\log
x)^{\frac{3}{2}},$

where the implied constant depends on $\pi$.

Finally, we show that Corollary 1.4 and Corollary 1.5 do in fact provide non-
trivial estimates. We first recall an elementary result (see [17, page 937],
for example)

(1.5) $\sum_{q\leqslant
x}\frac{1}{\varphi(q)}=\frac{\zeta(2)\zeta(3)}{\zeta(6)}\log x+O(1).$

Assuming the Riemann hypothesis for all of the twisted $L$-functions
$L(s,\pi\times(\widetilde{\pi}\otimes\chi))$ as well as GRC, it follows from
[16, Theorem 5.15] that

$\sum_{\begin{subarray}{c}p\leqslant x\\\ p\equiv
1\,(\operatorname{mod}{q})\end{subarray}}|\lambda_{\pi}(p)|^{2}\log
p=\frac{x}{\varphi(q)}+O(x^{\frac{1}{2}}(\log qx)^{2}).$

We average over the modulus $q\leqslant x^{1/3}$ and obtain from (1.5) and
partial summation that

$\sum_{q\leqslant x^{1/3}}\sum_{\begin{subarray}{c}p\leqslant x\\\ p\equiv
1\,(\operatorname{mod}{q})\end{subarray}}|\lambda_{\pi}(p)|^{2}\asymp x.$

One can easily verify that $|\lambda_{\pi}(p)|\gg|\lambda_{\pi}(p)|^{2}$ under
GRC, so the above estimate gives

$\sum_{p\leqslant x}|\lambda_{\pi}(p)|d(p-1)\gg\sum_{p\leqslant
x}|\lambda_{\pi}(p)|^{2}d(p-1)\gg\sum_{q\leqslant
x^{1/3}}\sum_{\begin{subarray}{c}p\leqslant x\\\ p\equiv
1\,(\operatorname{mod}{q})\end{subarray}}|\lambda_{\pi}(p)|^{2}\gg x.$

This means that there exists some cancellation in the sequence
$\\{\lambda_{\pi}(p)d(p-1)\\}$, where $p$ runs over all primes.

For Corollary 1.5, we argue as follows. Firstly, we recall an interesting
result in [32]: “ _Let a multiplicative function $f(m)\geqslant 0$ satisfy the
following conditions: (i) $f(m)\geqslant 0$; $f(p^{r})\leqslant A^{r}$ for
some $A>0$; (ii) $f(m)\ll m^{\varepsilon}$ for any $\varepsilon>0$; (iii)
$\sum_{p\leqslant x}f(p)\log p\geqslant\alpha x$ with some $\alpha>0$, then
one has the asymptotic formula_

$\sum_{m\leqslant x}f(m)d(m-1)=C_{f}\sum_{m\leqslant x}f(m)\log x(1+o(1))$

_for some constant $C_{f}$ depending on $f$._” Next, suppose that GRC holds,
we then obtain from [18, p. 595] that

(1.6) $\sum_{p\leqslant x}\left|\lambda_{\pi}(p)\right|\log
p\geqslant\Big{(}\frac{1}{m}+o(1)\Big{)}x,$

and

(1.7) $\sum_{m\leqslant x}\left|\lambda_{\pi}(m)\right|\gg\frac{x}{(\log
x)^{1-\frac{1}{n}}}.$

One can easily check that with the help of (1.6), the above conditions
(i)-(iii) hold for $f(m)=|\lambda_{\pi}(m)|$ under GRC. Hence, we could get

(1.8) $\sum_{m\leqslant x}|\lambda_{\pi}(m)|d(m-1)=C\sum_{m\leqslant
x}|\lambda_{\pi}(m)|\log x(1+o(1))$

for some constant $C$ depending on $\pi$. Combining (1.7) with (1.8), we have

$\sum_{m\leqslant x}|\lambda_{\pi}(m)|d(m-1)\gg x(\log x)^{\frac{1}{n}},$

which implies that Corollary 1.5 do give a non-trivial upper bound.

##  2\. Properties of $L$-functions

###  2.1. Conventions

For a parameter $\delta$, we use the notation $f\ll_{\delta}g$ or
$f=O_{\delta}(g)$ to denote that there exists a constant $c\geqslant 0$,
depending at most on $\delta$ and $\pi$, such that such that $|f|\leqslant cg$
in a range that will be clear in context.

###  2.2. Automorphic $L$-functions

Let $F$ be a number field with discriminant $D_{F}$ and $d=[F:\mathbb{Q}]$.
Let $\mathcal{O}_{F}$ be the ring of integers in $F$. For each place $v$ of
$F$, denote by $F_{v}$ the completion of $F$ with respect to $v$ and by
$O_{v}$ the local ring of integers. The prime ideals
$\mathfrak{p}\subseteq\mathcal{O}_{F}$ and the nonarchimedean places $v$ are
in bijective correspondence. So we may write $\mathfrak{p}$ interchangeably
with nonarchimedean places. Each $\pi\in\mathfrak{F}_{n}$ is a restricted
tensor product $\bigotimes_{v}\pi_{v}$ of smooth admissible representations of
$\mathrm{GL}_{n}(F_{v})$ such that $\pi_{v}$ is unramified for almost all
finite places $v$. Let $\mathfrak{q}_{\pi}$ be the conductor of $\pi$, which
has the property that $\pi_{\mathfrak{p}}$ is ramified if and only if
$\mathfrak{p}|\mathfrak{q}_{\pi}$.

For each prime ideal $\mathfrak{p}$, the standard local $L$-function is
defined in terms of Satake parameters
$A_{\pi}(\mathfrak{p})=\\{\alpha_{1,\pi}(\mathfrak{p}),\dots,\alpha_{n,\pi}(\mathfrak{p})\\}$
by

(2.1)
$L(s,\pi_{\mathfrak{p}}):=\prod_{j=1}^{n}(1-\alpha_{j,\pi}(\mathfrak{p})\mathrm{N}\mathfrak{p}^{-s})^{-1},$

where $\mathrm{N}=\mathrm{N}_{F/\mathbb{Q}}$ is the absolute norm over
$\mathbb{Q}$. For $\mathfrak{p}\nmid\mathfrak{q}_{\pi}$, we have
$\alpha_{j,\pi}(\mathfrak{p})\neq 0$ for all $i\in\\{1,\ldots,n\\}$. However,
it might be the case that $\alpha_{i,\pi}(\mathfrak{p})=0$ for some $j$ when
$\mathfrak{p}|\mathfrak{q}_{\pi}$. The standard (finite) $L$-function is
defined to be

(2.2)
$L(s,\pi)=\prod_{\mathfrak{p}}L(s,\pi_{\mathfrak{p}}):=\prod_{\mathfrak{p}}\sum_{k=0}^{\infty}\frac{\lambda_{\pi}(\mathfrak{p}^{k})}{\mathrm{N}\mathfrak{p}^{ks}}=\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\lambda_{\pi}(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}$

for $\mathop{\mathrm{Re}}s>1$, where the product is over all prime ideals
$\mathfrak{p}$ and the sum is over all integral ideals $\mathfrak{n}$. We can
see that $\lambda_{\pi}(\mathfrak{n})$ is multiplicative, that is
$\lambda_{\pi}(\mathfrak{n}_{1}\mathfrak{n}_{2})=\lambda_{\pi}(\mathfrak{n}_{1})\lambda_{\pi}(\mathfrak{n}_{2})$
for coprime integral ideals $\mathfrak{n}_{1}$ and $\mathfrak{n}_{2}$. We can
also write $\lambda_{\pi}(\mathfrak{n})$ in terms of Satake parameters

$\lambda_{\pi}(\mathfrak{p}^{k})=\sum_{m_{1}+\cdots+m_{n}=k}\prod_{j=1}^{n}\alpha_{j,\pi}^{m_{j}}(\mathfrak{p})$

and extend it to all integral ideals $\mathfrak{n}$ by multiplicativity.
Taking logarithmic derivatives in (2.2), we can see that for
$\mathop{\mathrm{Re}}s>1$,

$-\frac{L^{\prime}}{L}(s,\pi)=\sum_{\mathfrak{p}}\sum_{k=1}^{\infty}\frac{a_{\pi}(\mathfrak{p}^{k})\log\mathrm{N}\mathfrak{p}}{{\mathrm{N}\mathfrak{p}}^{ks}}=\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}},$

where

$\Lambda_{F}(\mathfrak{n}):=\left\\{\begin{aligned}
&\log\mathrm{N}\mathfrak{p}\;\;\text{ if }\mathfrak{n}=\mathfrak{p}^{k}\text{
for some }k\in\mathbb{N},\\\ &0\,\;\;\;\quad\quad\text{
otherwise,}\end{aligned}\right.$

and $a_{\pi}(\mathfrak{p}^{k})=\sum_{j=1}^{n}\alpha_{j,\pi}(\mathfrak{p})^{k}$
. We set $a_{\pi}(\mathfrak{n})=0$ if $\mathfrak{n}$ is not a prime ideal
power. Note that $a_{\pi}(\mathfrak{p})=\lambda_{\pi}(\mathfrak{p})$. We write
$\mu_{\pi}(\mathfrak{n})$ to be the coefficients of Dirichlet series
$L(s,\pi)^{-1}$, namely

(2.3)
$L(s,\pi)^{-1}=\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\mu_{\pi}(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}$

for $\mathop{\mathrm{Re}}s>1$. Then it can be is given by

(2.4) $\mu_{\pi}(\mathfrak{n})=\begin{dcases}0&\mbox{if
$\mathfrak{p}^{n+1}|\mathfrak{n}$ for some prime $\mathfrak{p}$,}\\\
\prod\limits_{\begin{subarray}{c}\mathfrak{p}^{\ell}\|\mathfrak{n}\\\
\ell\leqslant n\end{subarray}}(-1)^{\ell}\sum\limits_{1\leqslant
j_{1}<\cdots<j_{\ell}\leqslant
n}\alpha_{j_{1},\pi}(\mathfrak{p})\cdots\alpha_{j_{\ell},\pi}(\mathfrak{p})&\mbox{otherwise.}\end{dcases}$

Clearly, $\mu_{\pi}(\mathfrak{n})$ is multiplicative.

Now suppose $v$ is an archimedean place of $F$ (denoted $v|\infty$), so
$F_{v}=\mathbb{R}$ or $\mathbb{C}$. Denote $\Gamma(s)$ to be the usual gamma
function and define

$\Gamma_{v}(s):=\left\\{\begin{array}[]{ll}\pi^{-s/2}\Gamma(s/2)&\text{ if
}F_{v}=\mathbb{R},\\\ 2(2\pi)^{-s}\Gamma(s)&\text{ if
}F_{v}=\mathbb{C}.\end{array}\right.$

For each archimedean place $v$, there exists $n$ Langlands parameters
$\mu_{1,\pi}(v),\ldots,\mu_{n,\pi}(v)$ from which we define

$L(s,\pi_{v})=\prod_{j=1}^{n}\Gamma_{v}(s+\mu_{j,\pi}(v)).$

If we denote

$L_{\infty}(s,\pi)=\prod_{v|\infty}L(s,\pi_{v}),$

then for nontrivial $\pi$, the complete $L$-function defined by

$\Lambda(s,\pi)=(D_{F}^{n}\mathrm{N}\mathfrak{q}_{\pi})^{\frac{s}{2}}L(s,\pi)L_{\infty}(s,\pi)$

extends to an entire function of order 1 and is bounded in the vertical strip.
Luo, Rudnick, and Sarnak [21] and Müller and Speh [25] proved that there
exists $\theta_{n}\in[0,\frac{1}{2}-\frac{1}{n^{2}+1}]$ such that we have the
uniform bounds

(2.5)
$|\alpha_{j,\pi}(\mathfrak{p})|\leqslant{\mathrm{N}\mathfrak{p}}^{\theta_{n}}\quad\text{
and }\quad-\operatorname{Re}(\mu_{j,\pi}(v))\leqslant\theta_{n}.$

The generalized Ramanujan conjecture (GRC) predicts that $\theta_{n}=0$.

We denote by $\widetilde{\pi}$ the contragradient representation of $\pi$
which is also an irreducible cuspidal automorphic representation with unitary
central character. One can show that
$\mathfrak{q}_{\widetilde{\pi}}=\mathfrak{q}_{\pi}$. We can also define the
$L$-function associated to $\widetilde{\pi}$ in the same fashion. We have the
equalities of sets
$\\{\overline{\alpha_{j,\pi}(\mathfrak{p})}\\}_{j=1}^{n}=\\{\alpha_{j,\widetilde{\pi}}(\mathfrak{p})\\}_{j=1}^{n}$
and
$\\{\overline{\mu_{j,\pi}(v)}\\}_{j=1}^{n}=\\{\mu_{j,\widetilde{\pi}}(v)\\}_{j=1}^{n}$.
There exists a complex number $\varepsilon(\pi)$ of modulus 1 such that

$\Lambda(s,\pi)=\varepsilon(\pi)\Lambda(1-s,\widetilde{\pi}).$

Now we define the analytic conductor of $\pi$. We write
$q(\pi):=D_{F}^{n}\mathrm{N}\mathfrak{q}_{\pi}$ for the arithmetic conductor,
and the analytic conductor is defined by

$C(\pi,t):=q(\pi)\prod_{v|\infty}\prod_{j=1}^{n}(3+|it+\mu_{j,\pi}(v)|^{d_{v}}):=q(\pi)q_{\infty}(\pi,t),$

where $d_{v}=1$ if $F_{v}=\mathbb{R}$ and $d_{v}=2$ if $F_{v}=\mathbb{C}$ This
is an important parameter to describe $L(s,\pi)$. For example, the convexity
bound, the zero-free region and second moment estimates can be described in
terms of analytic conductor (see sections below).

###  2.3. Rankin–Selberg $L$-functions

Let $\pi=\bigotimes_{v}\pi_{v}\in\mathfrak{F}_{n}$ and
$\pi^{\prime}=\bigotimes_{v}\pi_{v}^{\prime}\in\mathfrak{F}_{n^{\prime}}$. The
Rankin–Selberg $L$-function at a finite place $\mathfrak{p}$ is defined to be

(2.6)
$L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}^{\prime})=\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}(1-\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})\mathrm{N}\mathfrak{p}^{-s})^{-1},$

where $\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})$ are suitable
complex numbers. For a finite place $\mathfrak{p}$ such that either
$\pi_{\mathfrak{p}}$ or $\pi_{\mathfrak{p}}^{\prime}$ is unramified, we have
the equality of sets
$\\{\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})\\}=\\{\alpha_{j,\pi}(\mathfrak{p})\alpha_{j^{\prime},\pi^{\prime}}(\mathfrak{p})\\}$.
We also define the (finite) Rankin–Selberg $L$-function to be

(2.7)
$L(s,\pi\times\pi^{\prime})=\prod_{\mathfrak{p}}L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}^{\prime}):=\prod_{\mathfrak{p}}\sum_{k=0}^{\infty}\frac{\lambda_{\pi\times\pi^{\prime}}(\mathfrak{p}^{k})}{\mathrm{N}\mathfrak{p}^{ks}}=\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\lambda_{\pi\times\pi^{\prime}}(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}$

for $\mathop{\mathrm{Re}}s>1$, where the product is over all prime ideals
$\mathfrak{p}$ and the sum is over nonzero integral ideals $\mathfrak{n}$.

For each archimedean place $v$, the local $L$-factor at $v$ is

$L(s,\pi_{v}\times\pi_{v}^{\prime})=\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}\Gamma_{v}(s+\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v))$

for suitable complex numbers $\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)$.
Define

$L_{\infty}(s,\pi\times\pi^{\prime})=\prod_{v|\infty}L(s,\pi_{v}\times\pi_{v}^{\prime}).$

When $v$ is a place such that both $\pi_{v}$ and $\pi_{v}^{\prime}$ are
unramified, then we have the equality of sets
$\\{\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)\\}=\\{\mu_{j,\pi}(v)+\mu_{j^{\prime},\pi^{\prime}}(v)\\}$.
By our normalization of the central characters, we have
$L(s,\pi\times\pi^{\prime})$ has a pole at $s=1$ with order
$r_{\pi\times\pi^{\prime}}=1$ if and only if
$\pi^{\prime}\simeq\widetilde{\pi}$, and $r_{\pi\times\pi^{\prime}}=0$
otherwise. We can also associate an arithmetic conductor
$q(\pi\times\pi^{\prime})$ to $\pi\times\pi^{\prime}$, so the complete
Rankin–Selberg $L$-function is defined by

$\Lambda(s,\pi\times\pi^{\prime}):=(s(s-1))^{r_{\pi\times\pi^{\prime}}}q(\pi\times\pi^{\prime})^{\frac{s}{2}}L(s,\pi\times\pi^{\prime})L_{\infty}(s,\pi\times\pi^{\prime}).$

It is entire of order 1 and satisfies the following functional equation

$\Lambda(s,\pi\times\pi^{\prime})=\varepsilon(\pi\times\pi^{\prime})\Lambda(1-s,\widetilde{\pi}\times\widetilde{\pi}^{\prime}),$

where $\varepsilon(\pi\times\pi^{\prime})$ is a complex number of modulus 1.
It follows from the explicit description of the numbers
$\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})$ and
$\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)$ in [30] and [30, Appendix]
yields the bounds

(2.8)
$|\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})|\leqslant{\mathrm{N}\mathfrak{p}}^{\theta_{n}+\theta_{n^{\prime}}}\quad\text{
and
}\quad-\operatorname{Re}(\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v))\leqslant\theta_{n}+\theta_{n^{\prime}}.$

We also define the analytic conductor $C(\pi\times\pi^{\prime},t)$ by

$C(\pi\times\pi^{\prime},t)=q(\pi\times\pi^{\prime})\prod_{v|\infty}\prod_{j=1}^{n}\prod_{j=1}^{n^{\prime}}(3+|it+\mu_{i,j,\pi\times\pi^{\prime}}(v)|^{d_{v}}):=q(\pi\times\pi^{\prime})q_{\infty}(\pi\times\pi^{\prime},t)$

for $d_{v}$ as above. An important inequality about conductors (see [3]) is

(2.9) $C(\pi\times\pi^{\prime},t)\ll
C(\pi,0)^{n^{\prime}}C(\pi^{\prime},0)^{n}(3+|t|)^{nn^{\prime}d}.$

We are especially interested in the case where $\pi^{\prime}=\widetilde{\pi}$.
In this case the Rankin–Selberg $L$-function $L(s,\pi\times\widetilde{\pi})$
has non-negative Dirichlet coefficients
$\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})$ (see Lemma 5.2 for
instance). Moreover, $L(s,\pi\times\widetilde{\pi})$ extends to the complex
plane with a simple pole at $s=1$. Hence, it follows from a standard Tauberian
argument that

(2.10) $\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})\sim
x\mathop{\mathrm{Res}}_{s=1}L(s,\pi\times\widetilde{\pi})\ll x.$

###  2.4. $\mathrm{GL}_{1}$-twists

Let $F$ be a number field. By a modulus $m$ of $F$, we mean a function

$m:\\{\text{all places of }F\\}\rightarrow\mathbb{Z}$

such that

  1. (1)

for all nonarchimedean places $v$, we have $m(v)\geqslant 0$, with $m(v)=0$
for all but finitely many $v$.

  2. (2)

if $v$ is a real archimedean place, then $m(v)=0$ or 1.

  3. (3)

if $v$ is a complex archimedean palce, then $m(v)=0$.

For a modulus $m$, we write

$U_{m}(v):=\begin{cases}(F_{v}^{\times})^{m(v)+1}&\mbox{if $v$ is
archimedean,}\\\ 1+v^{m(v)}&\mbox{if $v$ is nonarchimedean and $m(v)\neq
0$,}\\\ O_{v}^{\times}&\mbox{if $v$ is nonarchimedean and
$m(v)=0$.}\end{cases}$

Thus, in each case, $U_{m}(v)$ is a neighbourhood of 1 in $F_{v}^{\times}$.
Note that $m(v)=0$ for all but finitely many nonarchimedean $v$, so

$U_{m}:=\prod_{v}U_{m}(v)$

is an open subset of the idele group $\mathbb{A}_{F}^{\times}$, where the
product is over all places $v$ of $F$. For any modulus $m$, we can define the
ray class group modulo $m$ to be
$\mathrm{Cl}(m):=\mathbb{A}_{F}^{\times}/F^{\times}U_{m}$. By a narrow class
group modulo an integral ideal $\mathfrak{a}$, we mean that it is defined by
the modulus

$m_{\mathfrak{a}}(v):=\begin{cases}\mathrm{ord}_{v}(\mathfrak{a})&\mbox{if $v$
is nonarchimedean},\\\ 1&\mbox{if $v$ is real archimedean,}\\\ 0&\mbox{if $v$
is complex archimedean.}\end{cases}$

where $\mathrm{ord}_{v}$ is the additive valuation with respect to $v$. We
define
$\mathrm{Cl}^{+}(\mathfrak{a}):=\mathbb{A}_{F}^{\times}/F^{\times}U_{m_{\mathfrak{a}}}$,
which is a finite group, and whose cardinality is denoted by
$h(\mathfrak{a})$. Later we may also use $\mathfrak{a}$ to denote this modulus
for the simplicity of notation. If
$(\mathfrak{b},\mathfrak{a})=\mathcal{O}_{F}$, one can use the map
$\mathfrak{b}\mapsto\prod_{\mathfrak{p}}\varpi_{\mathfrak{p}}^{\mathrm{ord}_{\mathfrak{p}}(\mathfrak{b})}\times\prod_{v|\infty}1\mod
F^{\times}U_{m_{\mathfrak{a}}}$ to projects $\mathfrak{b}$ to
$\mathrm{Cl}^{+}(\mathfrak{a})$, where $\varpi_{\mathfrak{p}}$ is any fixed
choice of uniformizer in $F_{\mathfrak{p}}$. So by
“$\mathfrak{b}\equiv\mathfrak{c}$ in $\mathrm{Cl}^{+}(\mathfrak{a})$”, we mean
that both $\mathfrak{b}$ and $\mathfrak{c}$ are coprime with $\mathfrak{a}$
and they have the same image under this map.

One may also define the ray class group in terms of ideals. Let $J_{F}$ be the
group of fractional ideals in $F$. If $S$ is a finite set of prime ideals in
$F$, we denote by $J_{F}^{S}$ the subgroup of $J$ generated by the prime
ideals not in $S$. Define

$F^{S}=\\{x\in F:(x)\in J_{F}^{S}\\}=\\{x\in F:v_{\mathfrak{p}}(x)=0\text{ for
all finite }\mathfrak{p}\in S\\}.$

Given a modulus $m$, we denote by $F_{m,1}$ the set consisting of elements
$a\in F^{\times}$ satisfying

$\Big{\\{}\begin{aligned} \mathrm{ord}_{v}(a-1)&\geqslant m(v)&&\,\,\text{all
nonarchimedean $v$ with $m(v)>0$,}\\\ a_{v}&>0&&\text{ all real archimedean
$v$ with $m(v)>0$,}\end{aligned}$

where $a_{v}$ is the image of $a$ in $F_{v}$. If $S(m)=\\{\text{prime ideals
}\mathfrak{p}\colon m(\mathfrak{p})>0\\}$, then the ray class group modulo $m$
can also be defined by $J^{S(m)}/F_{m,1}$. By [22, Theorem 1.7, Chapter 5], we
have the following exact sequence

(2.11) $0\rightarrow\mathcal{O}_{F}^{\times}/(\mathcal{O}_{F}^{\times}\cap
F_{m,1})\rightarrow
F^{S(m)}/F_{m,1}\rightarrow\mathrm{Cl}(m)\rightarrow\mathrm{Cl}_{F}\rightarrow
0,$

where $\mathrm{Cl}_{F}$ is the class group of $F$. Moreover, we have the
following isomorphism

$F^{S(m)}/F_{m,1}\simeq\prod_{\begin{subarray}{c}v|\infty\text{ real}\\\
m(v)>0\end{subarray}}\\{\pm
1\\}\times\prod_{\begin{subarray}{c}\mathfrak{p}\\\
m(\mathfrak{p})>0\end{subarray}}(\mathcal{O}_{F}/\mathfrak{p}^{m(\mathfrak{p})})^{\times}.$

As a result, if we define
$\varphi_{F}(\mathfrak{m})=\mathrm{N}\mathfrak{m}\prod_{\mathfrak{p}|\mathfrak{m}}(1-\frac{1}{\mathrm{N}\mathfrak{p}})$
to be Euler’s totient function in $F$, then

$h(\mathfrak{m})=h\cdot
2^{r}\cdot\varphi_{F}(\mathfrak{m})\cdot|\mathcal{O}_{F}^{\times}/(\mathcal{O}_{F}^{\times}\cap
F_{m_{\mathfrak{m}},1})|^{-1},$

where $h$ is the class number of $F$ and $r$ is the number of real embeddings
of $F$. One can show that
$\varphi_{F}(\mathfrak{m})\gg\mathrm{N}\mathfrak{m}/\log\mathrm{N}\mathfrak{m}$.

For any character $\chi$ on $\mathrm{Cl}^{+}(\mathfrak{a})$, there is a
unitary Hecke character which is also denoted by $\chi=\prod_{v}\chi_{v}$ such
that $\chi(\mathfrak{p})=\chi_{\mathfrak{p}}(\varpi_{\mathfrak{p}})$ if
$\mathfrak{p}\nmid\mathfrak{q}_{\chi}$. One can see that the conductor of
$\chi$ divides $\mathfrak{a}$. We say that $\chi$ is primitive modulo
$\mathfrak{a}$ if $\mathfrak{q}_{\chi}=\mathfrak{a}$. Now, for any
$\pi\in\mathfrak{F}_{n}$, one has $\pi\otimes\chi\in\mathfrak{F}_{n}$. By [5],
the standard $L$-function associated with $\pi\otimes\chi$ equals

$L(s,\pi\otimes\chi)=\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\lambda_{\pi\otimes\chi}(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}.$

For a prime $\mathfrak{p}\nmid(\mathfrak{q}_{\pi},\mathfrak{q}_{\chi})$, we
have
$\\{\alpha_{j,\pi\otimes\chi}(\mathfrak{p})\\}=\\{\alpha_{j,\pi}(\mathfrak{p})\alpha_{\chi}(\mathfrak{p})\\}$.
Recall that
$\alpha_{\chi}(\mathfrak{p})=\chi_{\mathfrak{p}}(\varpi_{\mathfrak{p}})$ for
any uniformizer $\varpi_{\mathfrak{p}}$ in $F_{\mathfrak{p}}$ if
$\mathfrak{p}\nmid\mathfrak{q}_{\chi}$. We set $\chi(\mathfrak{p})=0$ for
$\mathfrak{p|\mathfrak{q}_{\chi}}$, then by discussion above we have

(2.12)
$\lambda_{\pi\otimes\chi}(\mathfrak{n})=\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})\quad\text{when
}(\mathfrak{n},\mathfrak{q}_{\chi})=\mathcal{O}_{F},$

and when $\chi$ is primitive, we have for $\textup{Re}(s)>1$ the identity

(2.13)
$\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}=\prod_{\mathfrak{p}}\prod_{j=1}^{m}\Big{(}1-\frac{\alpha_{j,\pi}(\mathfrak{p})\chi(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}^{-1}=L(s,\pi\otimes\chi)\prod_{\mathfrak{p}|\mathfrak{q}_{\chi}}\prod_{j=1}^{n}\Big{(}{1-\frac{\alpha_{j,\pi\otimes\chi}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}}\Big{)}.$

##  3\. Preliminary reductions and a generalized Vaughan identity

We will prove the following theorem.

######  Theorem 3.1.

Let $\pi\in\mathfrak{F}_{n}^{\flat}$, $\eta=\max\\{2,\frac{n}{2}\\}$, $A>0$,
and $B=2^{n[F:\mathbb{Q}]/4}(6a+12n+54)$. If $x\geqslant 3$ and
$Q=x^{\frac{1}{\eta}}(\log x)^{-B}$, then

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{p}\leqslant y\\\
\mathfrak{p}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}a_{\pi}(\mathfrak{n})\Lambda_{F}(\mathfrak{n})\Big{|}\ll_{A}\frac{x}{(\log
x)^{A}}.$

Assuming Theorem 3.1, we prove Theorem 1.1.

###### Proof of Theorem 1.1.

We will argue that Theorem 3.1 implies that

(3.1) $\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{p}\leqslant y\\\
\mathfrak{p}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{p})\log\mathrm{N}\mathfrak{p}\Big{|}\ll_{A}\frac{x}{(\log
x)^{A}}.$

The desired result will then follow by partial summation. To see that Theorem
3.1 implies (3.1), note that if $\mathfrak{p}$ is a prime ideal, then
$a_{\pi}(\mathfrak{p})\Lambda_{F}(\mathfrak{p})=\lambda_{\pi}(\mathfrak{p})\log\mathrm{N}\mathfrak{p}$.
To estimate the contribution from higher powers of prime ideals, we observe
that by (2.5), we have

$\displaystyle\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{p}^{k}\leqslant
y\\\ \mathfrak{p}^{k}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\\\ k\geqslant
2\end{subarray}}a_{\pi}(\mathfrak{p}^{k})\log\mathrm{N}\mathfrak{p}\ll_{\varepsilon}y^{\theta_{n}+\varepsilon}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{p}^{k}\leqslant
y\\\ \mathfrak{p}^{k}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\\\ k\geqslant 2\end{subarray}}1.$

So the contribution of these terms to the average in Theorem 3.1 is

$\displaystyle\ll\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}y^{\theta_{n}+\varepsilon}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{p}^{k}\leqslant
y\\\ \mathfrak{p}^{k}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\\\ k\geqslant
2\end{subarray}}1\ll\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\max_{\mathrm{N}\mathfrak{a}\leqslant
x}x^{\theta_{n}+\varepsilon}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{p}^{k}\leqslant
x\\\ \mathfrak{p}^{k}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\\\ k\geqslant 2\end{subarray}}1$
$\displaystyle\ll
x^{\theta_{n}+\varepsilon}\max_{\mathrm{N}\mathfrak{a}\leqslant
x}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{p}^{k}\leqslant x\\\ k\geqslant
2\end{subarray}}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{m}\leqslant Q\\\
\mathfrak{p}^{k}\equiv\mathfrak{a}\text{ in }\mathrm{Cl}^{+}(\mathfrak{m})\\\
(\mathfrak{m},\mathfrak{p}^{k})=(\mathfrak{m},\mathfrak{a})=\mathcal{O}_{F}\end{subarray}}1,$

by the convention that $\sum_{\mathfrak{b}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})}1=0$ if
$(\mathfrak{a},\mathfrak{m})\neq\mathcal{O}_{F}$. We now argue that

(3.2) $\max_{\mathrm{N}\mathfrak{a}\leqslant
x}\max_{\begin{subarray}{c}\mathrm{N}\mathfrak{p}^{k}\leqslant x\\\ k\geqslant
2\end{subarray}}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{m}\leqslant Q\\\
\mathfrak{p}^{k}\equiv\mathfrak{a}\text{ in }\mathrm{Cl}^{+}(\mathfrak{m})\\\
(\mathfrak{m},\mathfrak{p}^{k})=(\mathfrak{m},\mathfrak{a})=\mathcal{O}_{F}\end{subarray}}1\ll_{\varepsilon}x^{\varepsilon}.$

Once we establish this, it follows from the prime ideal theorem that

(3.3) $\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}a_{\pi}(\mathfrak{n})\Lambda_{F}(\mathfrak{n})-\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{p}\leqslant
y\\\ \mathfrak{p}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{p})\log\mathrm{N}\mathfrak{p}\Big{|}\ll_{\varepsilon}x^{\frac{1}{2}+\theta_{n}+\varepsilon}.$

Since $x^{\frac{1}{2}+\theta_{n}+\varepsilon}\ll_{A}x(\log x)^{-A}$, this
finishes the passage from powers of prime to prime ideals.

In order to prove (3.2), we begin with the fact that if
$\mathfrak{b}\equiv\mathfrak{a}$ in $\mathrm{Cl}^{+}(\mathfrak{m})$ with
$(\mathfrak{m},\mathfrak{b})=(\mathfrak{m},\mathfrak{a})=\mathcal{O}_{F}$ and
$\mathrm{N}\mathfrak{b}$, $\mathrm{N}\mathfrak{a}\leqslant x$, then there
exists $\omega\in F^{\times}$ such that $\omega$ is totally positive,
$\mathfrak{p}|\mathfrak{m}$ implies
$\mathrm{ord}_{\mathfrak{p}}(\omega-1)\geqslant\mathrm{ord}_{\mathfrak{p}}(\mathfrak{m})$,
and $(\omega)=\mathfrak{b}\mathfrak{a}^{-1}$. Let $h$ be the class number of
$F$, then $\omega^{h}=ba^{-1}$ for some nonzero $a,b\in\mathcal{O}_{F}$ with
$\mathfrak{b}^{h}=(b)$ and $\mathfrak{a}^{h}=(a)$. Since
$\mathrm{ord}_{\mathfrak{p}}(a)=0$ for $\mathfrak{p}|\mathfrak{m}$,
$\mathrm{ord}_{\mathfrak{p}}(b-a)=\mathrm{ord}_{\mathfrak{p}}(b-a)-\mathrm{ord}_{\mathfrak{p}}(a)=\mathrm{ord}_{\mathfrak{p}}(\omega^{h}-1)=\mathrm{ord}_{\mathfrak{p}}(\omega-1)+\mathrm{ord}_{\mathfrak{p}}(\omega^{h-1}+\cdots+1)\geqslant\mathrm{ord}_{\mathfrak{p}}(\mathfrak{m})$
if $\mathfrak{p}|\mathfrak{m}$. Hence $\mathfrak{m}|(b-a)$. Recall that if
$a\in F^{\times}$, then $\prod_{v}|a|_{v}=1$. As a result,
$\prod_{v|\infty}|a|_{v}\leqslant x^{h}$ and $\prod_{v|\infty}|b|_{v}\leqslant
x^{h}$. All of the conjugates of an algebraic integer are algebraic integers,
so their absolute values have a uniform lower bound depending only on $F$. We
can see from this fact that $|a|_{v},|b|_{v}\ll x^{h}$ for all $v|\infty$.
Hence $N(b-a)=\prod_{v|\infty}|b-a|_{v}\ll x^{h[F:\mathbb{Q}]}$. One can check
that there are at most $\tau_{[F:\mathbb{Q}]}(m)$ integral ideals with norm
$m$, where $\tau_{[F:\mathbb{Q}]}(m)$ is the $m$-th Dirichlet coefficient of
$\zeta(s)^{[F:\mathbb{Q}]}$. Since
$\tau_{[F:\mathbb{Q}]}(m)\ll_{\varepsilon}m^{\varepsilon}$, the innermost sum
can be bounded by $\tau_{[F:\mathbb{Q}]}(\mathrm{N}(b-a))$, which is therefore
$\ll_{\varepsilon}x^{\varepsilon}$, as desired. ∎

Our proof of Theorem 3.1 partially follows the approach in Chapter 9 of [7];
see also [31]. As stated in [7], what we need is a combinatorial identity for
sums over primes to produce a bilinear form to which the large sieve
inequality can be applied. We choose to use a generalized version of Vaughan
identity. Define

$M(s)=\sum_{\mathrm{N}\mathfrak{n}\leqslant
X}\frac{\mu_{\pi}(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}},\qquad
N(s)=\sum_{\mathrm{N}\mathfrak{n}\leqslant
Y}\frac{\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}.$

######  Lemma 3.2.

Let $X\geqslant 1$ and $Y\geqslant 1$. If $\mathfrak{n}$ is an integral ideal
with $\mathrm{N}\mathfrak{n}>Y$, then we have

$\displaystyle\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})$
$\displaystyle=\sum_{\begin{subarray}{c}\mathfrak{n=ab}\\\
\mathrm{N}\mathfrak{b}\leqslant
X\end{subarray}}\mu_{\pi}(\mathfrak{b})\lambda_{\pi}(\mathfrak{a})\log\mathrm{N}\mathfrak{a}-{\underset{\begin{subarray}{c}\mathfrak{n=abc}\\\
\mathrm{N}\mathfrak{b}\leqslant X,\mathrm{N}\mathfrak{c}\leqslant
Y\end{subarray}}{\sum\sum}}\lambda_{\pi}(\mathfrak{a})\mu_{\pi}(\mathfrak{b})\Lambda_{F}(\mathfrak{c})a_{\pi}(\mathfrak{c})$
$\displaystyle+{\underset{\begin{subarray}{c}\mathfrak{n=abc}\\\
\mathrm{N}\mathfrak{b}>X,\mathrm{N}\mathfrak{c}>Y\end{subarray}}{\sum\sum}}\lambda_{\pi}(\mathfrak{a})\mu_{\pi}(\mathfrak{b})\Lambda_{F}(\mathfrak{c})a_{\pi}(\mathfrak{c}).$

###### Proof.

If $\textup{Re}(s)>1$, then we have the identity

$\displaystyle\frac{L^{\prime}}{L}(s,\pi)$ $\displaystyle=$ $\displaystyle
L^{\prime}(s,\pi)M(s)+L(s,\pi)M(s)N(s)$
$\displaystyle+\Big{(}\frac{L^{\prime}}{L}(s,\pi)+N(s)\Big{)}\Big{(}1-L(s,\pi)M(s)\Big{)}-N(s).$

Once we identify the coefficients of $\mathrm{N}\mathfrak{n}^{-s}$ on each
side, we obtain the desired result. ∎

We apply Lemma 3.2 with $X=Y<y$ for the $L$-function $L(s,\pi)$ and find that

$\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})=S_{1}+S_{2}-S_{3}+S_{4},$

where

(3.4) $\displaystyle S_{1}$
$\displaystyle:=\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant X\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n}),$
(3.5) $\displaystyle S_{2}$
$\displaystyle:=\sum_{\mathrm{N}\mathfrak{b}\leqslant
X}\mu_{\pi}(\mathfrak{b})\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{c}\leqslant
y/\mathrm{N}\mathfrak{b}\\\ \mathfrak{c}\equiv\mathfrak{ab^{-1}}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{c})\log\mathrm{N}\mathfrak{c},$
(3.6) $\displaystyle S_{3}$
$\displaystyle:=\sum_{\mathrm{N}\mathfrak{b}\leqslant
X}\sum_{\mathrm{N}\mathfrak{c}\leqslant
X}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{d}\leqslant
y/\mathrm{N}\mathfrak{bc}\\\ \mathfrak{d}\equiv\mathfrak{a(bc)^{-1}}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{d})\mu_{\pi}(\mathfrak{b})\Lambda_{F}(\mathfrak{c})a_{\pi}(\mathfrak{c})$
$\displaystyle=\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
X^{2}\end{subarray}}\Big{(}\sum_{\begin{subarray}{c}\mathfrak{bc=n}\\\
\mathrm{N}\mathfrak{b}\leqslant X,\mathrm{N}\mathfrak{c}\leqslant
X\end{subarray}}\mu_{\pi}(\mathfrak{b})\Lambda_{F}(\mathfrak{c})a_{\pi}(\mathfrak{c})\Big{)}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{d}\leqslant
y/\mathrm{N}\mathfrak{n}\\\ \mathfrak{d}\equiv\mathfrak{an^{-1}}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{d}),$
(3.7) $\displaystyle S_{4}$
$\displaystyle:=\sum_{\mathrm{N}\mathfrak{b}>X}\sum_{\mathrm{N}\mathfrak{c}>X}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{d}\leqslant
y/\mathrm{N}\mathfrak{bc}\\\ \mathfrak{d}\equiv\mathfrak{a(bc)^{-1}}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{d})\mu_{\pi}(\mathfrak{b})\Lambda_{F}(\mathfrak{c})a_{\pi}(\mathfrak{c})$
$\displaystyle=\sum_{X<\begin{subarray}{c}\mathrm{N}\mathfrak{n}<y/X\end{subarray}}\Big{(}\sum_{\begin{subarray}{c}\mathfrak{bd=n}\\\
\mathfrak{b}>X\end{subarray}}\mu_{\pi}(\mathfrak{b})\lambda_{\pi}(\mathfrak{d})\Big{)}\sum_{\begin{subarray}{c}X<\mathrm{N}\mathfrak{c}\leqslant
y/\mathrm{N}\mathfrak{n}\\\ \mathfrak{c}\equiv\mathfrak{an^{-1}}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\Lambda_{F}(\mathfrak{c})a_{\pi}(\mathfrak{c}).$

After applying the identity with suitable parameters $X$ and $Y$, we are going
to estimate sums involving $S_{1}$, $S_{2}$, $S_{3}$, and $S_{4}$. Note that
their definitions depend on $y$, $\pi$, $\mathfrak{a}$, and the parameter for
truncation $X$ in Vaughan’s identity. The estimates of $S_{i}$ rely on Theorem
1.3 and Lemma 8.2 (see below). We only demonstrate in this section the outline
of the proof assuming Theorem 1.3 and Lemma 8.2. We estimate the sum involving
$S_{2}$ because it is the most typical one. That is, we need to estimate

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}|S_{2}|.$

It behooves us to decompose $S_{2}$ as $S_{2}^{\prime}+S_{2}^{\prime\prime}$,
where

$S_{2}^{\prime}:=\sum_{\mathrm{N}\mathfrak{b}\leqslant
H}\mu_{\pi}(\mathfrak{b})\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{c}\leqslant
y/\mathrm{N}\mathfrak{b}\\\ \mathfrak{c}\equiv\mathfrak{ab^{-1}}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{c})\log\mathrm{N}\mathfrak{c},\quad
S_{2}^{\prime\prime}:=\sum_{H<\mathrm{N}\mathfrak{b}\leqslant
X}\mu_{\pi}(\mathfrak{b})\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{c}\leqslant
y/\mathrm{N}\mathfrak{b}\\\ \mathfrak{c}\equiv\mathfrak{ab^{-1}}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{c})\log\mathrm{N}\mathfrak{c}$

for some parameter $H<X$. Theorem 1.3 and partial summation give that

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}|S_{2}^{\prime}|\ll\frac{Hx}{(\log x)^{A}}$

for some $B$ depending on $A$. For $S_{2}^{\prime\prime}$, we use Lemma 8.2
and the remark below it to obtain

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}|S_{2}^{\prime\prime}|\ll(\log
xQ)^{5}\Big{(}Q\sqrt{x}+\sqrt{Xx}+\frac{x}{\sqrt{H}}+\frac{x}{(\log
x/X)^{A}}\Big{)}.$

Lemma 8.2 requires a Siegel–Walfisz condition for the sequence
$\\{\lambda_{\pi}(\mathfrak{c})\log\mathrm{N}\mathfrak{c}\\}$, which we prove
in Section 9. Moreover, the result relies on the $\ell^{2}$-estimates of
$\mu_{\pi}(\mathfrak{b})$ and
$\lambda_{\pi}(\mathfrak{c})\log\mathrm{N}\mathfrak{c}$, which is also given
in Section 9 based on the inequalities in Section 5. For the sum involving
$S_{3}$ and $S_{4}$ can be treated similarly, but we also require a
Siegel–Walfisz condition for the sequence
$\\{a_{\pi}(\mathfrak{c})\Lambda_{F}(\mathfrak{c})\\}$ in $S_{4}$. This
condition, given by Corollary 4.8, is proved in Section 4; it relies on
Theorem 1.2.

Now it remains to prove Theorem 1.3 and Lemma 8.2, which are proved in
Sections 7 and 8, respectively. Note that Lemma 8.2 is not of the form as in
[7]. For the proof of Lemma 8.2, we employ the trick of Fourier transform as
Vaughan did in [33]. Sections 5 and 6 supply several important estimates for
our proofs of Theorem 1.3 and Lemma 8.2.

##  4\. Zero-free regions

Let $\pi\in\mathfrak{F}_{n}$. We let $\mathbbm{1}\in\mathfrak{F}_{1}$ denote
the trivial representation, whose $L$-function is the Dedekind zeta function
$\zeta_{F}(s)$. Recall that
$\mathfrak{F}_{n}^{\flat}\subseteq\mathfrak{F}_{n}$ is the subset consisting
of $\pi\in\mathfrak{F}_{n}$ such that $\pi=\widetilde{\pi}$ and
$\pi\neq\pi\otimes\chi$ for all nontrivial quadratic primitive Hecke
characters $\chi$. In this section, we prove a zero-free region for
$L(s,\pi\otimes\chi)$ which is comparable to that of Dirichlet characters,
including the first unconditional $\mathrm{N}\mathfrak{q}$-aspect bound on a
possible Landau–Siegel zero. We then use this zero-free region along with
standard contour integration techniques to prove an analogue of the
Siegel–Walfisz theorem for the Dirichlet coefficients
$\lambda_{\pi}(\mathfrak{p})$. We now present the main result of this section.

######  Theorem 4.1.

Let $Q\geqslant 3$ and $\pi\in\mathfrak{F}_{n}^{\flat}$. There exists a
constant $c_{\pi}>0$, depending effectively on $\pi$, such that for all
primitive Hecke characters $\chi\,(\mathrm{mod}{\,\mathfrak{q}})$ with
$\mathrm{N}\mathfrak{q}\leqslant Q$ with at most one exception, the
$L$-function $L(s,\pi\otimes\chi)$ is nonzero in the region

$\textup{Re}(s)\geqslant 1-\frac{c_{\pi}}{\log(Q(3+|\textup{Im}(s)|))}.$

If the exceptional character $\chi_{1}$ exists, then

  * •

$\chi_{1}$ is quadratic.

  * •

$L(s,\chi_{1})$ has exactly one zero $\beta_{1}$ in this region, and
$\beta_{1}$ is both real and simple.

  * •

For all $\varepsilon>0$, there exists an ineffective constant
$c_{\pi}(\varepsilon)>0$ such that $\beta_{1}\leqslant
1-c_{\pi}(\varepsilon)Q^{-\varepsilon}$.

###  4.1. Preliminaries for the zero-free region

We begin with a standard zero-free region.

######  Lemma 4.2.

Let $\pi\in\mathfrak{F}_{n}$, and let $\chi\,(\mathrm{mod}{\,\mathfrak{q}})$
be a primitive Hecke character. There exists an effectively computable
constant $\Cl[abcon]{ZFR_{c}onst}=\Cr{ZFR_{c}onst}(\pi)>0$ such that
$L(s,\pi\otimes\chi)\neq 0$ in the region

$\textup{Re}(s)\geqslant
1-\frac{\Cr{ZFR_{c}onst}}{\log(\mathrm{N}\mathfrak{q}(3+|\textup{Im}(s)|))}$

with the possible exception of one real zero $\beta_{1}<1$ when
$\pi\otimes\chi$ is self-dual. When $\pi=\widetilde{\pi}$, the exceptional
zero can only exist when $\chi$ is primitive, nontrivial, and quadratic.

###### Proof.

When $|\textup{Im}(s)|\neq 0$ or $\pi\otimes\chi$ is not self-dual, then the
result follows from [12, Theroem A.1] with $\pi$ (respectively $\pi^{\prime}$)
therein replaced by $\pi\otimes\chi$ (respectively $\mathbbm{1}$). When
$\textup{Im}(s)=0$ and $\pi\otimes\chi$ is self-dual, then by [12, Theorem
A.1], there exists at most one zero $\beta_{1}<1$ in the stated region, while
the nonvanishing of $L(1,\pi\otimes\chi)$ follows from [19, Theorem A.1]. If
$\pi=\widetilde{\pi}$, then $\pi\otimes\chi$ is self-dual if and only if
$\chi$ is real and primitive. When $\pi=\widetilde{\pi}$ and $\chi$ is
trivial, then by [19, Theorem A.1], there exists effectively computable
constant $\Cl[abcon]{brumleyzfrtriv}=\Cr{brumleyzfrtriv}(\pi)>0$ such that if
$1-\Cr{brumleyzfrtriv}\leqslant s<1$, then $L(s,\pi)\neq 0$. This exhausts all
cases once $\Cr{ZFR_{c}onst}$ is made suitably large (in an effective manner
depending at most on $\pi$). ∎

Next, we quantify the idea that exceptional zeros are rare.

######  Lemma 4.3.

Let $\pi\in\mathfrak{F}_{n}$. Among the primitive quadratic Hecke characters
$\chi\,(\mathrm{mod}{\,\mathfrak{q}})$ with $\mathrm{N}\mathfrak{q}\leqslant
Q$, at most one, say $\chi_{1}$, has the property that $L(s,\pi\otimes\chi)$
has a real zero $\beta_{1}$ in the interval

$1-\frac{\Cr{ZFR_{c}onst}}{\log Q}\leqslant s<1.$

###### Proof.

This follows from [11, Theorem A]. We may take $\Cr{ZFR_{c}onst}$ to be the
same as in Lemma 4.2 once $\Cr{ZFR_{c}onst}$ is made suitably small (in a
manner that depends at most on $\pi$). ∎

###  4.2. Preceding literature

Siegel proved that if $\chi\,(\mathrm{mod}{\,q})$ is a primitive nontrivial
quadratic Dirichlet character, then for all $\varepsilon>0$, there exists an
ineffective constant $c(\varepsilon)>0$ such that
$L(1,\chi)>c(\varepsilon)q^{-\varepsilon}$. All known proofs except for one by
Bombieri [2, Théorème 15] use the fact that if $\chi\,(\mathrm{mod}{\,q})$ and
$\chi^{\prime}\,(\mathrm{mod}{\,q^{\prime}})$ are distinct primitive
nontrivial quadratic Dirichlet characters, and $\chi^{\prime\prime}$ is he
primitive Dirichlet character that induces $\chi\chi^{\prime}$, then there
exists a Dirichlet series $F(s)$, depending explicitly on $\chi$ and
$\chi^{\prime}$, such that $F(s)$ has

  1. (i)

a pole of odd order $r\geqslant 1$ at $s=1$,

  2. (ii)

nonnegative Dirichlet coefficients, and

  3. (iii)

an analytic continuation to suitable region past $\textup{Re}(s)=1$ (e.g.,
$\mathbb{C}-\\{1\\}$),

  4. (iv)

and a residue at $s=1$ that has $L(1,\chi)$ as a factor with integral
multiplicity at least one.

To study $L(1,\chi)$, the most natural choice of $F(s)$ is
$\zeta(s)L(s,\chi)L(s,\chi^{\prime})L(s,\chi^{\prime\prime})$. A pole of odd
order $r$ at $s=1$ is important; under the above hypotheses, the residue
$R_{F}$ of $F(s)$ at $s=1$ satisfies $R_{F}>0$, and as $s\to 1$ along the real
line, we have

(4.1) $F(s)\sim\frac{R_{F}}{s-1}.$

If $0<\varepsilon<1$ and there exists
$\chi^{\prime}\,(\mathrm{mod}{\,q^{\prime}})$ such that
$L(s_{0},\chi^{\prime})=0$ for some $s_{0}\in(1-\varepsilon,1)$, then
$F(s_{0})\leqslant 0$. On the other hand, if no such $\chi^{\prime}$ exists,
then by (4.1), we have that $F(s_{0})\leqslant 0$ for some
$s_{0}\in(1-\varepsilon/2,1)$. Therefore, for all $0<\varepsilon<1$, there
exists $s_{0}\in(1-\varepsilon,1)$ and
$\chi^{\prime}\,(\mathrm{mod}{\,q^{\prime}})$, both depending only on
$\varepsilon$, such that $F(s_{0})\leqslant 0$. Davenport’s book [6, Ch. 20]
is a standard source; it gives Estermann’s proof, which requires this argument
as a key step.

Let $\pi\in\mathfrak{F}_{n}$, and let $\chi\,(\mathrm{mod}{\,\mathfrak{q}})$
and $\chi^{\prime}\,(\mathrm{mod}{\,\mathfrak{q}^{\prime}})$ be distinct
nontrivial primitive quadratic Hecke characters. Let $\psi$ be the primitive
character that induces $\chi^{\prime}\chi$ (whose conductor necessarily
divides $\mathfrak{q}\mathfrak{q}^{\prime}$). The possible existence of the
exceptional real zero of $L(s,\pi\otimes\chi)$ in Lemma 4.2 was eliminated by
Hoffstein and Ramakrishnan [11, Theorem B] under the assumption of automorphy
for certain Rankin–Selberg convolutions depending on $\pi$. When their
automorphy hypothesis is not known to be satisfied, it is unclear how to
construct a Dirichlet series $L(s)$ depending on $\chi$ and $\chi^{\prime}$
with nonnegative Dirichlet coefficients, an analytic continuation, a pole of
odd order at $s=1$, and a residue at $s=1$ that has $L(1,\pi\otimes\chi)$ as a
factor with integral multiplicity at least one.

If we allow for a pole of even order at $s=1$, then we can construct an $L(s)$
satisfying properties (ii)-(iv) above. Let $\pi\in\mathfrak{F}_{n}$, and
suppose that $\pi\neq\pi\otimes\nu$ for all primitive nontrivial quadratic
Hecke characters over $F$. Define $\Pi=\mathbbm{1}\boxplus\pi$ and
$(\Pi\times\widetilde{\Pi})_{\chi}=\chi\boxplus\pi\otimes\chi\boxplus\widetilde{\pi}\otimes\chi\boxplus\pi\times(\widetilde{\pi}\otimes\chi)$.
Consider the representation

(4.2)
$\Pi^{\star}=(\Pi\times\widetilde{\Pi})\boxplus(\Pi\times\widetilde{\Pi})_{\chi}\boxplus(\Pi\times\widetilde{\Pi})_{\chi^{\prime}}\boxplus(\Pi\times\widetilde{\Pi})_{\psi}$

along with its $L$-function

(4.3) $\displaystyle L(s,\Pi^{\star})$
$\displaystyle=\zeta_{F}(s)L(s,\chi)L(s,\chi^{\prime})L(s,\psi)L(s,\pi)L(s,\widetilde{\pi})L(s,\pi\otimes\chi)L(s,\widetilde{\pi}\otimes\chi)$
$\displaystyle\cdot
L(s,\pi\otimes\chi^{\prime})L(s,\widetilde{\pi}\otimes\chi^{\prime})L(s,\pi\otimes\psi)L(s,\widetilde{\pi}\otimes\psi)L(s,\pi\times\widetilde{\pi})L(s,\pi\times(\widetilde{\pi}\otimes\chi))$
$\displaystyle\cdot
L(s,\pi\times(\widetilde{\pi}\otimes\chi^{\prime}))L(s,\pi\times(\widetilde{\pi}\otimes\psi)).$

Our twist hypothesis for $\pi$ ensures that $L(s,\Pi^{\star})$ is holomorphic
on $\mathbb{C}-\\{1\\}$ with a pole of order two at $s=1$. This auxiliary
$L$-function was suggested by Molteni [23, p. 141] in a special case, with
(4.3) providing a natural generalization. Instead of providing full details
for how to prove a Siegel-type lower bound for $|L(1,\pi\otimes\chi)|$ using
$L(s,\Pi^{\star})$, Molteni references a “standard approach to Siegel-type
theorems” in a paper by Golubeva and Fomenko [9]. However, in [9, pp. 87-88],
Golubeva and Fomenko only say that Estermann’s proof of Siegel’s theorem (the
version in Davenport [6, Ch. 20]) applies to (4.2) (with
$\pi\in\mathfrak{F}_{2}$) “after fairly tedious calculations.”

As stated above, (4.3) satisfies properties (ii)-(iv), but not (i), since
$L(s,\Pi^{\star})$ has a pole of order two at $s=1$. In this situation, a
Siegel-type lower bound does not follow from a direct generalization of the
arguments in [6, Ch. 20], or any other argument that proves Siegel’s theorem
using the above auxiliary $L$-function
$\zeta(s)L(s,\chi)L(s,\chi^{\prime})L(s,\chi^{\prime\prime})$. Since
$L(s,\Pi^{\star})$ has a pole of order 2 at $s=1$ and nonnegative Dirichlet
coefficients, it follows that there exists a constant $R_{\Pi^{\star}}>0$ such
that as $s\to 1$ along the reals, we have

$L(s,\Pi^{\star})\sim\frac{R_{\Pi^{\star}}}{(s-1)^{2}}.$

Consequently, there exists $s^{\prime}<1$ such that $L(s,\Pi^{\star})>0$ for
all $s\in[s^{\prime},1)$. Therefore, it is no longer true that for all
$0<\varepsilon<1$, there exists $s_{0}\in(1-\varepsilon,1)$ and
$\chi^{\prime}\,(\mathrm{mod}{\,\mathfrak{q}^{\prime}})$ such that
$L(s_{0},\Pi^{\star})\leqslant 0$. An identical error can also be found in
[15].

In summary, a separate approach is needed in order to produce a lower bound
for $|L(1,\pi\otimes\chi)|$ when $\varepsilon$ is so small that
$L(s_{0},\Pi^{\star})>0$ for all $s_{0}\in(1-\varepsilon,1)$. Such an approach
is provided by the following lemma. This leads to a correction and substantial
generalization of the works in [23, 9, 15].

######  Lemma 4.4.

Let $\chi\,(\mathrm{mod}{\,\mathfrak{q}})$ be a primitive quadratic Hecke
character, and let $\pi\in\mathfrak{F}_{n}$. If $L(s,\pi\otimes\chi)\neq 0$ in
the region

$\textup{Re}(s)\geqslant
1-\frac{\Cr{ZFR_{c}onst}}{\log(\mathrm{N}\mathfrak{q}(3+|\textup{Im}(s)|))},$

then there exist constants $\Cl[abcon]{XLi1}=\Cr{XLi1}(\pi)>0$ and
$\Cl[abcon]{XLi2}=\Cr{XLi2}(\pi)>0$ such that

$|L(1,\pi\otimes\chi)|\geqslant\Cr{XLi1}\exp(-\Cr{XLi2}\sqrt{\log\mathrm{N}\mathfrak{q}}).$

###### Proof.

This follows from work of Li [20, Corollary 7]. While the proofs in [20] are
performed over $F=\mathbb{Q}$, an extension over number fields follows mutatis
mutandis. ∎

###  4.3. An extension of Siegel’s theorem

Let $\Pi=\pi\boxplus\mathbbm{1}$, and define the numbers
$a_{\Pi\times\widetilde{\Pi}}(\mathfrak{p}^{k})$ by the Dirichlet series
identity

$\sum_{\mathfrak{p}}\sum_{k=1}^{\infty}\frac{a_{\Pi\times\widetilde{\Pi}}(\mathfrak{p}^{k})}{k\mathrm{N}\mathfrak{p}^{ks}}=\log
L(s,\Pi\times\widetilde{\pi}),\qquad\textup{Re}(s)>1.$

######  Lemma 4.5.

Let $\pi\in\mathfrak{F}_{n}$, let $\chi\,(\mathrm{mod}{\,\mathfrak{q}})$ and
$\chi^{\prime}\,(\mathrm{mod}{\,\mathfrak{q}^{\prime}})$ be primitive
quadratic Hecke characters, and let $\psi$ be the primitive Hecke character
inducing $\chi^{\prime}\chi$. Let $\Pi=\mathbbm{1}\boxplus\pi$, and recall the
definition of $\Pi^{\star}$ in (4.2). There exists an entire function
$\mathcal{H}(s)=\mathcal{H}_{\pi}(s,\chi,\chi^{\prime})$ such that

$\sum_{\mathfrak{p}\nmid\mathfrak{q}\mathfrak{q}^{\prime}\mathfrak{q}_{\pi}}\sum_{k=1}^{\infty}\frac{a_{\Pi\times\widetilde{\Pi}}(\mathfrak{p}^{k})}{k\mathrm{N}\mathfrak{p}^{ks}}(1+\chi(\mathfrak{p}^{k}))(1+\chi^{\prime}(\mathfrak{p}^{k}))=\log(L(s,\Pi^{\star})\mathcal{H}(s)).$

The Dirichlet coefficients $\lambda^{\star}(\mathfrak{n})$ of
$L(s,\Pi^{\star})\mathcal{H}(s)$ are nonnegative, and
$\lambda^{\star}(\mathcal{O}_{F})=1$. We have the bounds
$|\mathcal{H}(1)|,|\mathcal{H}^{\prime}(1)|\ll_{\varepsilon}(\mathrm{N}\mathfrak{q}^{\prime}\mathfrak{q})^{\varepsilon}$
for all $\varepsilon>0$. Finally, if $t\in\mathbb{R}$, then
$|\mathcal{H}(\frac{1}{2}+it)|\ll(\mathrm{N}\mathfrak{q}^{\prime}\mathfrak{q})^{3n^{2}/2}$.

###### Proof.

We determine $\mathcal{H}(s)$ explicitly using the local calculations in [21,
Lemma 2.1]:

$\displaystyle\mathcal{H}(s)=\prod_{\mathfrak{p}\nmid\mathfrak{q}\mathfrak{q}^{\prime}\mathfrak{q}_{\pi}}$
$\displaystyle\Big{\\{}\Big{[}\prod_{j=1}^{n}\Big{(}1-\frac{\alpha_{j,\pi\otimes\chi}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\Big{(}1-\frac{\alpha_{j,\widetilde{\pi}\otimes\chi}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\Big{]}\Big{[}\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n}\Big{(}1-\frac{\alpha_{j,j^{\prime},\pi\times(\widetilde{\pi}\otimes\chi)}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\Big{]}$
$\displaystyle\cdot\Big{[}\prod_{j=1}^{n}\Big{(}1-\frac{\alpha_{j,\pi\otimes\chi^{\prime}}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\Big{(}1-\frac{\alpha_{j,\widetilde{\pi}\otimes\chi^{\prime}}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\Big{]}\Big{[}\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n}\Big{(}1-\frac{\alpha_{j,j^{\prime},\pi\times(\widetilde{\pi}\otimes\chi^{\prime})}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\Big{]}$
$\displaystyle\cdot\Big{[}\prod_{j=1}^{n}\Big{(}1-\frac{\alpha_{j,\pi\otimes\psi}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\Big{(}1-\frac{\alpha_{j,\widetilde{\pi}\otimes\psi}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\Big{]}\Big{[}\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n}\Big{(}1-\frac{\alpha_{j,j^{\prime},\pi\times(\widetilde{\pi}\otimes\psi)}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\Big{]}\Big{\\}}.$

The claimed bounds of $|\mathcal{H}(1)|$, $|\mathcal{H}^{\prime}(1)|$, and
$|\mathcal{H}(\frac{1}{2}+it)|$ follow from (2.5) and (2.8) along with the
bound
$\\#\\{\mathfrak{p}\colon\mathfrak{p}|\mathfrak{n}\\}\ll(\log\log\mathrm{N}\mathfrak{n})^{-1}\log\mathrm{N}\mathfrak{n}$.

The nonnegativity of $a_{\Pi\times\widetilde{\Pi}}(\mathfrak{p}^{k})$ follows
from the proof of [11, Lemma a]. The nonnegativity of
$(1+\chi(\mathfrak{p}^{k}))(1+\chi^{\prime}(\mathfrak{p}^{k}))$ follows from
the fact that $\chi$ and $\chi^{\prime}$ are quadratic. Thus, the
nonnegativity of the Dirichlet coefficients $\lambda^{\star}(\mathfrak{n})$
follows by exponentiation. ∎

######  Lemma 4.6.

Let $\pi\in\mathfrak{F}_{n}$, and let $\chi$, $\chi^{\prime}$ and $\psi$ be as
in Lemma 4.5. Suppose that $\pi\neq\pi\otimes\nu$ for all
$\nu\in\\{\chi,\chi^{\prime},\psi\\}$. Recall the definition of $\Pi^{\star}$
from (4.2). If $\beta\in(0,1)$, $x\geqslant 3$, and $\varepsilon>0$, then

$\mathop{\mathrm{Res}}_{s=1-\beta}L(s+\beta,\Pi^{\star})\mathcal{H}(s+\beta)x^{s}\Gamma(s)\ll_{\chi^{\prime},\beta,\varepsilon}|L(1,\pi\otimes\chi)|\mathrm{N}\mathfrak{q}^{\varepsilon}x^{1-\beta+\varepsilon}.$

###### Proof.

Define
$H(s)=L(s,\Pi^{\star})\mathcal{H}(s)L(s,\pi\otimes\chi)^{-1}L(s,\widetilde{\pi}\otimes\chi)^{-1}\zeta_{F}(s)^{-1}L(s,\pi\times\widetilde{\pi})^{-1}$.
The hypothesis that $\pi\neq\pi\otimes\nu$ for all
$\nu\in\\{\chi,\chi^{\prime},\psi\\}$ ensures that $H(s)$ is entire. In a
neighbourhood of $s=1$, we have the Laurent expansions

$\zeta_{F}(s)=\frac{\kappa_{F}}{s-1}+\kappa_{F}^{\prime}+O(s-1),\qquad
L(s,\pi\times\widetilde{\pi})=\frac{\kappa_{\pi}}{s-1}+\kappa_{\pi}^{\prime}+O(s-1).$

In view of these definitions, the residue
$\mathop{\mathrm{Res}}_{s=1-\beta}L(s+\beta,\Pi^{\star})\mathcal{H}(s+\beta)x^{s}\Gamma(s)$
equals

$\displaystyle x^{1-\beta}\Gamma(1-\beta)L(1,\pi\otimes\chi)\Big{(}$
$\displaystyle
H(1)L(1,\widetilde{\pi}\otimes\chi)\kappa_{F}\kappa_{\pi}\Big{(}\log
x+\frac{\Gamma^{\prime}(1-\beta)}{\Gamma(1-\beta)}\Big{)}$
$\displaystyle+H(1)L(1,\widetilde{\pi}\otimes\chi)(\kappa_{\pi}\kappa_{F}^{\prime}+\kappa_{F}\kappa_{\pi}^{\prime})$
$\displaystyle+\kappa_{F}\kappa_{\pi}(L(1,\widetilde{\pi}\otimes\chi)H^{\prime}(1)+L^{\prime}(1,\pi\otimes\chi)H(1))$
$\displaystyle+H(1)\kappa_{F}\kappa_{\pi}\frac{L(1,\widetilde{\pi}\otimes\chi)}{L(1,\pi\otimes\chi)}L^{\prime}(1,\pi\otimes\chi)\Big{)}.$

Because $\chi$ is quadratic, we have
$L(1,\widetilde{\pi}\otimes\chi)=\overline{L(1,\pi\otimes\chi)}$, so the ratio
$\frac{L(1,\widetilde{\pi}\otimes\chi)}{L(1,\pi\otimes\chi)}$ has modulus 1.
Therefore, the lemma will follow from the following estimates

$|L(1,\widetilde{\pi}\otimes\chi)|,~{}|L^{\prime}(1,\widetilde{\pi}\otimes\chi)|,~{}|H(1)|,~{}|H^{\prime}(1)|\ll_{\chi^{\prime},\varepsilon}\mathrm{N}\mathfrak{q}^{\varepsilon}.$

The first three estimates follow directly from [20, Theorem 2]. For the last
inequality, it suffices to know that if $\nu\in\\{\chi,\chi^{\prime},\psi\\}$,
then
$|L^{\prime}(1,\pi\times(\widetilde{\pi}\otimes\nu))|\ll_{\chi^{\prime},\varepsilon}\mathrm{N}\mathfrak{q}^{\varepsilon}$.
By Cauchy’s integral formula for derivatives, we have

$L^{\prime}(1,\pi\times(\widetilde{\pi}\otimes\nu))=\frac{1}{2\pi
i}\int_{\Omega}\frac{L(s,\pi\times(\widetilde{\pi}\otimes\nu))}{(s-1)^{2}}ds\ll(\log\mathrm{N}\mathfrak{q})\max_{z\in\Omega}|L(s,\pi\times(\widetilde{\pi}\otimes\nu))|,$

where $\Omega$ is the circle of radius
$O_{\chi^{\prime}}(\frac{1}{\log\mathrm{N}\mathfrak{q}})$ centered at $s=1$.
By [20, Theorem 2] and the Phragmén-Lindelöf principle, we have
$|H^{\prime}(1)|\ll_{\chi^{\prime},\varepsilon}\mathrm{N}\mathfrak{q}^{\varepsilon}$
to finish the proof. ∎

We now perform an auxiliary computation using Lemmata 4.5 and 4.6. If
$x\geqslant 3$ and $\beta\in(0,1)$, then we compute

$\frac{1}{2}\leqslant
e^{-1/x}\leqslant\sum_{\mathfrak{n}}\frac{\lambda^{\star}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{\beta}}e^{-\mathrm{N}\mathfrak{n}/x}=\frac{1}{2\pi
i}\int_{3-i\infty}^{3+i\infty}L(s+\beta,\Pi^{\star})\mathcal{H}(s+\beta)x^{s}\Gamma(s)ds.$

Once we push the contour to the line $\textup{Re}(s)=\frac{1}{2}-\beta$, the
contour integral equals

$\mathop{\mathrm{Res}}_{s=1-\beta}L(s+\beta,\Pi^{\star})\mathcal{H}(s+\beta)x^{s}\Gamma(s)+L(\beta,\Pi^{\star})\mathcal{H}(\beta)+\frac{1}{2\pi
i}\int_{\frac{1}{2}-\beta-i\infty}^{\frac{1}{2}-\beta+i\infty}L(s+\beta,\Pi^{\star})\mathcal{H}(s+\beta))x^{s}\Gamma(s)ds.$

By Lemma 4.5, we have for any $x\geqslant 3$ and $\beta\in(0,1)$ the bound

(4.4)
$\frac{1}{2}\leqslant\mathop{\mathrm{Res}}_{s=1-\beta}L(s+\beta,\Pi^{\star})\mathcal{H}(s+\beta)x^{s}\Gamma(s)+L(\beta,\Pi^{\star})\mathcal{H}(\beta)+O_{\chi^{\prime},\beta,\varepsilon}(\mathrm{N}\mathfrak{q}^{2(n+1)^{2}+\varepsilon}x^{\frac{1}{2}-\beta}).$

###### Proof of Theorem 1.2.

It suffices to let $\mathrm{N}\mathfrak{q}$ be large (with respect to $\pi$)
and $(\log\mathrm{N}\mathfrak{q})^{-1}<\varepsilon<\frac{1}{2}$. If there
exists at most one primitive quadratic nontrivial Hecke character $\nu$ such
that $L(s,\pi\otimes\nu)=0$ for some $s\in(1-\frac{\varepsilon}{4},1)$, then
the desired result follows from Lemmata 4.2 and 4.4 once we make $\Cr{XLi1}$
and $\Cr{XLi2}$ sufficiently small (depending at most on $\pi$ and the sole
exceptional character, if it exists). For the rest of the proof, we may assume
that there exist two distinct primitive quadratic Hecke characters $\nu_{1}$
and $\nu_{2}$ such that both $L(s,\pi\otimes\nu_{1})$ and
$L(s,\pi\otimes\nu_{2})$ vanish somewhere in the interval
$(1-\frac{\varepsilon}{4},1)$.

Subject to this hypothesis, we can choose
$\chi^{\prime}\in\\{\nu_{1},\nu_{2}\\}-\\{\chi\\}$. If we choose
$\beta\in(1-\frac{\varepsilon}{4},1)$ to be a point at which
$L(s,\pi\otimes\chi^{\prime})$ vanishes, then we may conclude that for all
$0<\varepsilon<\frac{1}{2}$, there exist
$\chi^{\prime}\,(\mathrm{mod}{\,\mathfrak{q}^{\prime}})$ and
$\beta\in(1-\frac{\varepsilon}{4},1)$ (depending at most on $\varepsilon$ and
$\pi$) such that $L(\beta,\Pi^{\star})=0$. With these choices of
$\chi^{\prime}\,(\mathrm{mod}{\,\mathfrak{q}^{\prime}})$ and $\beta$, the
bound (4.4) reduces to

$\frac{1}{2}\leqslant\mathop{\mathrm{Res}}_{s=1-\beta}L(s+\beta,\Pi^{\star})\mathcal{H}(s)x^{s}\Gamma(s)+O_{\varepsilon}(\mathrm{N}\mathfrak{q}^{2(n+1)^{2}+\varepsilon}x^{\frac{1}{2}-\beta}).$

Since $\beta\in(1-\frac{\varepsilon}{4},1)$, it follows from Lemma 4.6 that

$1\ll_{\varepsilon}|L(1,\pi\otimes\chi)|\mathrm{N}\mathfrak{q}^{\varepsilon}x^{\frac{\varepsilon}{2}}+\mathrm{N}\mathfrak{q}^{2(n+1)^{2}+\varepsilon}x^{\frac{\varepsilon}{2}-\frac{1}{2}}.$

We choose $x=\mathrm{N}\mathfrak{q}^{4(n+1)^{2}}|L(1,\pi\otimes\chi)|^{-2}$.
Note that there exist effectively computable constants
$\Cl[abcon]{lb1}=\Cr{lb1}(\pi)>0$ and $\Cl[abcon]{lb2}=\Cr{lb2}(\pi)>0$ such
that
$|L(1,\pi\otimes\chi)|\leqslant\Cr{lb1}\exp(\Cr{lb2}\sqrt{\log\mathrm{N}\mathfrak{q}})$
by [20, Theorem 3]. Therefore, since we have assumed that
$\mathrm{N}\mathfrak{q}$ is large, we have that $x\geqslant 3$. We achieve the
desired result by solving for $|L(1,\pi\otimes\chi)|$, and rescaling
$\varepsilon$ in terms of $n$ alone. ∎

######  Corollary 4.7.

Let $\pi\in\mathfrak{F}_{n}$, and suppose that $\pi\neq\pi\otimes\nu$ for all
primitive quadratic Hecke characters $\nu$. Let
$\chi\,(\mathrm{mod}{\,\mathfrak{q}})$ be a primitive quadratic Hecke
character. For all $\varepsilon>0$, there exists an ineffective constant
$c_{\pi}(\varepsilon)>0$ such that $L(s,\pi\otimes\chi)\neq 0$ for $s\geqslant
1-c_{\pi}(\varepsilon)\mathrm{N}\mathfrak{q}^{-\varepsilon}$.

###### Proof.

Suppose that $L(s,\pi\otimes\chi)$ has a real exceptional zero $\beta_{1}$ in
the region given by Lemma 4.2. By the mean value theorem and Lemma 1.2, there
exists

$\sigma\in\Big{[}1-\frac{\Cr{ZFR_{c}onst}}{\log(3\mathrm{N}\mathfrak{q})},1\Big{]}$

such that
$|L^{\prime}(\sigma,\pi\otimes\chi)|(1-\beta_{1})=|L(1,\pi\otimes\chi)|\geqslant
c_{\pi}^{\prime}(\frac{\varepsilon}{2})\mathrm{N}\mathfrak{q}^{-\varepsilon/2}$.
Therefore, we have

$\beta_{1}\leqslant
1-\frac{c_{\pi}^{\prime}(\frac{\varepsilon}{2})}{\mathrm{N}\mathfrak{q}^{\varepsilon/2}|L^{\prime}(\sigma,\pi\otimes\chi)|}.$

The upper bound
$|L^{\prime}(\sigma,\pi\otimes\chi)|\ll_{\varepsilon}\mathrm{N}\mathfrak{q}^{\varepsilon/2}$
follows from [20, Corollary 6] for all $\sigma$ in our range, and the result
follows. ∎

###### Proof of Theorem 4.1.

This follows from Lemmata 4.2 and 4.3 and Corollary 4.7. ∎

###  4.4. An estimate of Siegel–Walfisz type

We apply our zero-free region in Theorem 4.1 to prove the following result.

######  Corollary 4.8.

Let $\pi\in\mathfrak{F}_{n}^{\flat}$ and
$(\mathfrak{m},\mathfrak{a})=\mathcal{O}_{F}$. For all $A>0$, there exists an
ineffective constant $\Cl[abcon]{SW}=\Cr{SW}(\pi,F,A)>0$ such that for
$\mathrm{N}\mathfrak{m}\leqslant(\log x)^{A}$, we have

(4.5) $\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant x\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})\ll_{A}x\exp(-\Cr{SW}\sqrt{\log
x}).$

###### Proof.

Using the orthogonality of characters, we find that

$\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant x\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})\ll_{A}(\log
x)^{A}\max_{\mathrm{N}\mathfrak{m}\leqslant(\log
x)^{A}}\max_{{\psi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}}}\Big{|}\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})\psi(\mathfrak{n})\Big{|}.$

If $\chi\,(\mathrm{mod}{\,\mathfrak{q}})$ is the primitive Hecke character
that induces $\psi\,(\mathrm{mod}{\,\mathfrak{m}})$, then (2.5) and our
constraint that $\mathrm{N}\mathfrak{m}\leqslant(\log x)^{A}$ implies that for
all $\varepsilon>0$, we have

$\Big{|}\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\Lambda_{F}(\mathfrak{n})a_{\pi}(\mathfrak{n})\psi(\mathfrak{n})-\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\Lambda_{F}(\mathfrak{n})a_{\pi\otimes\chi}(\mathfrak{n})\Big{|}\leqslant\sum_{\mathfrak{p}|\mathfrak{m}\mathfrak{q}_{\pi}}\sum_{\begin{subarray}{c}k\geqslant
1\\\ \mathrm{N}\mathfrak{p}^{k}\leqslant
x\end{subarray}}|a_{\pi}(\mathfrak{p}^{k})|\log
N\mathfrak{p}\ll_{\varepsilon}x^{\theta_{n}+\varepsilon}.$

Without loss of generality, assume that $\chi$ is the exceptional character
$\chi_{1}$ in Theorem 4.1, and let $\beta_{1}$ be the corresponding
exceptional zero. We proceed as in [16, Theorem 5.13] and conclude that

$\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\Lambda_{F}(\mathfrak{n})a_{\pi\otimes\chi_{1}}(\mathfrak{n})=-\frac{x^{\beta_{1}}}{\beta_{1}}+O\Big{(}\mathrm{N}\mathfrak{m}^{\frac{n}{2}}x\exp(-\tfrac{c_{\pi}}{2}\sqrt{\log
x})+\sum_{x<\mathrm{N}\mathfrak{n}\leqslant
x+\frac{x}{\exp(\frac{1}{3}\sqrt{\log
x})}}\Lambda_{F}(\mathfrak{n})|a_{\pi}(\mathfrak{n})|\Big{)}.$

Since $2|a_{\pi}(\mathfrak{n})|\leqslant
1+a_{\pi\times\widetilde{\pi}}(\mathfrak{n})$ [30, Theorem A.1] and
$L(1+it,\pi\times\widetilde{\pi})\neq 0$ for all $t\in\mathbb{R}$ [19,
Appendix], we find that $\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}|a_{\pi}(\mathfrak{n})|\Lambda_{F}(\mathfrak{n})\ll x$. Therefore, by the
Cauchy–Schwarz inequality and the bound $\mathrm{N}\mathfrak{m}\leqslant(\log
x)^{A}$, it follows that

$\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\Lambda_{F}(\mathfrak{n})a_{\pi\otimes\chi_{1}}(\mathfrak{n})=-\frac{x^{\beta_{1}}}{\beta_{1}}+O(x\exp(-\min\\{\tfrac{c_{\pi}}{4},\tfrac{1}{6}\\}\sqrt{\log
x})).$

Theorem 4.1 gives the bound $\beta_{1}\leqslant
1-c_{\pi}(\varepsilon)\mathrm{N}\mathfrak{m}^{-\varepsilon}$ for any fixed
$\varepsilon>0$. Since $\mathrm{N}\mathfrak{m}\leqslant(\log x)^{A}$ for some
fixed $A>0$, we set $\varepsilon=1/(2A)$ so that

$x^{\beta_{1}}\ll x\exp(-c_{\pi}(\tfrac{1}{2A})(\log
x)\mathrm{N}\mathfrak{m}^{-\varepsilon})\ll
x\exp(-c_{\pi}(\tfrac{1}{2A})\sqrt{\log x}).$

We put
$c=\frac{1}{10}\min\\{\frac{c_{\pi}}{3},c_{\pi}(\frac{1}{2A}),\frac{1}{6}\\}$,
and the desired result follows. ∎

##  5\. Auxiliary estimates

We begin with a combinatorial lemma due to Soundrarajan [29]. It is useful to
prove some inequalities.

######  Lemma 5.1.

Let $b(1),b(2),\ldots$ be a sequence of complex numbers. Define the sequence
$c(0)=1,c(1),c(2),\ldots$ by means of the formal identity

$\exp\big{(}\sum_{k=1}^{\infty}\frac{b(k)}{k}x^{k}\big{)}=\sum_{m=0}^{\infty}c(m)x^{m}.$

Define the sequence $C(0)=1,C(1),C(2),\ldots$ by means of the formal identity

$\exp\big{(}\sum_{k=1}^{\infty}\frac{|b(k)|^{2}}{k}x^{k}\big{)}=\sum_{m=0}^{\infty}C(m)x^{m}.$

Then $|c(m)|^{2}\leqslant C(m)$ for all $m$.

Now we here introduce several arithmetic inequalities, which will be used
below. The first one is about the coefficients of logarithmic derivatives of
$L$-functions (see the appendix by Brumley in [30])

(5.1) $|a_{\pi}(\mathfrak{n})|^{2}\leqslant
a_{\pi\times\widetilde{\pi}}(\mathfrak{n})$

for any $\mathfrak{n}\subset\mathcal{O}_{F},$ where
$a_{\pi\times\widetilde{\pi}}(\mathfrak{n})$ is given by

$-\frac{L^{\prime}}{L}(s,\pi\times\widetilde{\pi})=\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\Lambda_{F}(\mathfrak{n})a_{\pi\times\widetilde{\pi}}(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}.$

Using Shahidi’s non-vanishing result of $L(s,\pi\times\widetilde{\pi})$ at
$\operatorname{Re}s=1$ (see [28]), one has

(5.2) $\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\Lambda_{F}(\mathfrak{n})a_{\pi\times\widetilde{\pi}}(\mathfrak{n})\sim x.$

Similar to (5.1), we can prove that the corresponding inequality holds when we
replace $a_{\pi}(\mathfrak{n})$ and
$a_{\pi\times\widetilde{\pi}}(\mathfrak{n})$ by $\lambda_{\pi}(\mathfrak{n})$
and $\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})$, respectively.

######  Lemma 5.2.

With the above notation, we have
$|\lambda_{\pi}(\mathfrak{n})|^{2}\leqslant\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})$
for all integral ideals $\mathfrak{n}$.

###### Proof.

See [18, Lemma 3.1]. ∎

######  Lemma 5.3.

With the notation as above, let
$b_{F,\pi}(\mathfrak{n})=\sum\limits_{\mathfrak{a}\mathfrak{b}=\mathfrak{n}}|\lambda_{\pi}(\mathfrak{a})\mu_{\pi}(\mathfrak{b})|$.
If $(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}$, then

(5.3) $\displaystyle b_{F,\pi}(\mathfrak{n})^{2}$
$\displaystyle\leqslant\underbrace{(\lambda_{\pi\times\widetilde{\pi}}*\cdots*\lambda_{\pi\times\widetilde{\pi}})}_{\text{$4(n+1)$
terms}}(\mathfrak{n}),$ (5.4) $\displaystyle|\mu_{\pi}(\mathfrak{n})|^{2}$
$\displaystyle\leqslant\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n}),$
(5.5) $\displaystyle d_{F}(\mathfrak{n})|\mu_{\pi}(\mathfrak{n})|^{2}$
$\displaystyle\leqslant\underbrace{(\lambda_{\pi\times\widetilde{\pi}}*\cdots*\lambda_{\pi\times\widetilde{\pi}})}_{\text{$n+1$
terms}}(\mathfrak{n}),$

where
$\underbrace{(\lambda_{\pi\times\widetilde{\pi}}*\cdots*\lambda_{\pi\times\widetilde{\pi}})}_{\text{$k$
terms}}(\mathfrak{n})=\sum\limits_{\mathfrak{n}_{1}\cdots\mathfrak{n}_{k}=\mathfrak{n}}\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n}_{1})\cdots\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n}_{k})$,
and $d_{F}(\mathfrak{n})$ is the divisor function on $F$.

###### Proof.

From the facts that $\lambda_{\pi}(\mathfrak{n})$, $\mu_{\pi}(\mathfrak{n})$
and $d_{F}(\mathfrak{n})$ are all multiplicative, we know that
$b_{F,\pi}(\mathfrak{n})^{2},|\mu_{\pi}(\mathfrak{n})|^{2},d_{F}(\mathfrak{n})|\mu_{\pi}(\mathfrak{n})|^{2}$
are also multiplicative. It suffices to show the corresponding inequalities
hold at $\mathfrak{n}=\mathfrak{p}^{k}$ for any $k\geqslant 0$ and
$\mathfrak{p}\nmid\mathfrak{q}_{\pi}.$

(1) For the first inequality, we actually show a slightly stronger inequality
at prime ideal powers as follows:

(5.6) $b_{F,\pi}(\mathfrak{p}^{k})^{2}\leqslant
4(n+1)\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{p}^{k})\quad\text{for}\quad\mathfrak{p}\nmid\mathfrak{q}_{\pi}.$

For a set $\\{\alpha_{1},\ldots,\alpha_{n}\\}$, we define the polynomial
$e_{l}(\alpha_{1},\ldots,\alpha_{n})$ by

$e_{l}(\alpha_{1},\ldots,\alpha_{n})=\sum_{1\leqslant
j_{1}<\cdots<j_{l}\leqslant
n}\alpha_{j_{1}}\alpha_{j_{2}}\cdots\alpha_{j_{l}}.$

The polynomial $e_{l}$ is called the $l$-th elementary symmetric polynomial.
If $l=0$, then $e_{l}(x_{1},\ldots,x_{n})\equiv 1$. By convention,
$e_{l}(\alpha_{1},\ldots,\alpha_{n})=0$ for $l>n$. By (2.4), we know that

(5.7) $\mu_{\pi}(\mathfrak{p}^{l})=(-1)^{l}e_{l}(A_{\pi}(\mathfrak{p})).$

A partition $\lambda=(\lambda(i))_{i=1}^{\infty}$ is a sequence of
nonincreasing nonnegative integers
$\lambda(1)\geqslant\lambda(2)\geqslant\cdots$ with only finitely many nonzero
entries. For a partition $\lambda$, let $\ell(\lambda)$ be the number of
nonzero $\lambda(i)$, and let $|\lambda|=\sum_{i=1}^{\infty}\lambda(i)$. For a
set $\\{\alpha_{1},\ldots,\alpha_{n}\\}$ and a partition $\lambda$ with
$\ell(\lambda)\leqslant n$, let $s_{\lambda}(\alpha_{1},\ldots,\alpha_{n})$ be
the Schur polynomial
$\det[(\alpha_{i}^{\lambda(j)+n-j})_{ij}]/\det[(\alpha_{i}^{n-j})_{ij}]$
associated to $\lambda$. If $|\lambda|=0$, then
$s_{\lambda}(\alpha_{1},\ldots,\alpha_{n})\equiv 1$. By convention, if
$\ell(\lambda)>n$, then $s_{\lambda}(\alpha_{1},\ldots,\alpha_{n})\equiv 0$.
Cauchy’s identity [4, Chapter 38], tells us that

$L(s,\pi_{\mathfrak{p}})=\prod_{j=1}^{n}\Big{(}1-\frac{\alpha_{j,\pi}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}^{-1}=\sum_{k=0}^{\infty}\frac{s_{(k,0,0,\ldots)}(A_{\pi}(\mathfrak{p}))}{\mathrm{N}\mathfrak{p}^{ks}}$

and, for $\mathfrak{p}\nmid\mathfrak{q}_{\pi}$,

$L(s,\pi_{\mathfrak{p}}\times\widetilde{\pi}_{\mathfrak{p}})=\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n}\Big{(}1-\frac{\alpha_{j,\pi}(\mathfrak{p})\overline{\alpha_{j^{\prime},\pi}(\mathfrak{p})}}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}^{-1}=\sum_{\lambda}\frac{s_{\lambda}(A_{\pi}(\mathfrak{p}))\overline{s_{\lambda}(A_{\pi}(\mathfrak{p}))}}{\mathrm{N}\mathfrak{p}^{s|\lambda|}},$

where the sum ranges over all partitions. Then we have

(5.8)
$\lambda_{\pi}(\mathfrak{p}^{k})=s_{(k,0,0,\ldots)}(A_{\pi}(\mathfrak{p})),\qquad\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{p}^{k})=\sum_{|\lambda|=k}|s_{\lambda}(A_{\pi}(\mathfrak{p}))|^{2}.$

Thus, by (5.7) and (5.8), the dual Pieri rule [4, Theorem 40.4] yields that

$\displaystyle b_{F,\pi}(\mathfrak{p}^{k})$
$\displaystyle=\sum_{l=0}^{\min\\{k,n\\}}\big{|}e_{l}(A_{\pi}(\mathfrak{p}))s_{(k-l,0,0,\ldots)}(A_{\pi}(\mathfrak{p}))\big{|}$
$\displaystyle\leqslant$
$\displaystyle\big{|}s_{(k,0,0,\ldots)}(A_{\pi}(\mathfrak{p}))\big{|}+\big{|}s_{({\tiny\underbrace{1,1,\ldots,1}_{\text{$k$
terms}}},0,0,\ldots)}(A_{\pi}(\mathfrak{p}))\big{|}$
$\displaystyle+\sum_{l=1}^{\min\\{k,n\\}-1}(\big{|}s_{({\tiny\underbrace{k-l+1,1,\ldots,1}_{\text{$l$
terms}}},0,0,\ldots)}(A_{\pi}(\mathfrak{p}))\big{|}+\big{|}s_{({\tiny\underbrace{k-l,1,\ldots,1}_{\text{$l+1$
terms}}},0,0,\ldots)}(A_{\pi}(\mathfrak{p}))\big{|}).$

By the Cauchy–Schwarz inequality and (5.8), we then have

$\displaystyle|b_{F,\pi}(\mathfrak{p}^{k})|^{2}$
$\displaystyle\leqslant(n+1)\Big{(}\big{|}s_{(k,0,0,\ldots)}(A_{\pi}(\mathfrak{p}))\big{|}^{2}+\big{|}s_{({\tiny\underbrace{1,1,\ldots,1}_{\text{$k$
terms}}},0,0,\ldots)}(A_{\pi}(\mathfrak{p}))\big{|}^{2}\Big{)}$
$\displaystyle+2(n+1)\sum_{l=1}^{\min\\{k,n\\}-1}\Big{(}\big{|}s_{({\tiny\underbrace{k-l+1,1,\ldots,1}_{\text{$l$
terms}}},0,0,\ldots)}(A_{\pi}(\mathfrak{p}))\big{|}^{2}+\big{|}s_{({\tiny\underbrace{k-l,1,\ldots,1}_{\text{$l+1$
terms}}},0,0,\ldots)}(A_{\pi}(\mathfrak{p}))\big{|}^{2}\Big{)}$
$\displaystyle\leqslant
4(n+1)\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{p}^{k}).$

This completes the proof of (5.6), which further implies the inequality (5.3).

(2) For the second one, We see from (2.1) and (2.6) that

$\log
L(s,\pi_{\mathfrak{p}})=\sum_{j=1}^{n}\sum_{k=1}^{\infty}\frac{\alpha_{j,\pi}(\mathfrak{p})^{k}}{k\mathrm{N}\mathfrak{p}^{ks}}=\sum_{k=1}^{\infty}\frac{a_{\pi}(\mathfrak{p}^{k})}{k\mathrm{N}\mathfrak{p}^{ks}}$

and

$\log
L(s,\pi_{\mathfrak{p}}\times\widetilde{\pi}_{\mathfrak{p}})=\sum_{1\leqslant
j,j^{\prime}\leqslant
n}\sum_{k=1}^{\infty}\frac{(\alpha_{j,\pi}(\mathfrak{p})\overline{\alpha_{j^{\prime},\pi}(\mathfrak{p})})^{k}}{k\mathrm{N}\mathfrak{p}^{ks}}=\sum_{k=1}^{\infty}\frac{|a_{\pi}(\mathfrak{p}^{k})|^{2}}{k\mathrm{N}\mathfrak{p}^{ks}}$

for $\mathfrak{p}\nmid\mathfrak{q}_{\pi}$. Comparing these with (2.3) and
(2.7), we have

$\exp\Big{(}-\sum_{k=1}^{\infty}\frac{a_{\pi}(\mathfrak{p}^{k})}{k\mathrm{N}\mathfrak{p}^{ks}}\Big{)}=\sum_{k=0}^{\infty}\frac{\mu_{\pi}(\mathfrak{p}^{k})}{\mathrm{N}\mathfrak{p}^{ks}}$

and

$\exp\Big{(}\sum_{k=1}^{\infty}\frac{|a_{\pi}(\mathfrak{p}^{k})|^{2}}{k\mathrm{N}\mathfrak{p}^{ks}}\Big{)}=\sum_{k=0}^{\infty}\frac{\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{p}^{k})}{\mathrm{N}\mathfrak{p}^{ks}}.$

By Lemma 5.1, we then get

(5.9)
$|\mu_{\pi}(\mathfrak{p}^{k})|^{2}\leqslant\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{p}^{k})$

for any $k\geqslant 0$ and $\mathfrak{p}\nmid\mathfrak{q}_{\pi}$. By
multiplicativity, the inequality (5.4) follows.

(3) By (2.4) and (5.9), we have

$d_{F}(\mathfrak{p}^{k})|\mu_{\pi}(\mathfrak{p}^{k})|^{2}\leqslant(n+1)\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{p}^{k})\quad\text{for
any}\;k\geqslant 0\;\text{and }\mathfrak{p}\nmid\mathfrak{q}_{\pi}.$

So the inequality (5.5) follows. ∎

######  Lemma 5.4.

Let $b_{F,\pi}(\mathfrak{n})$ be defined as in Lemma 5.3. Then we have

$\sum_{\mathrm{N}\mathfrak{n}\leqslant x}b_{F,\pi}(\mathfrak{n})^{2}\ll x(\log
x)^{4n+3},\qquad\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}|\mu_{\pi}(\mathfrak{n})|^{2}\ll
x,\qquad\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}d_{F}(\mathfrak{n})|\mu_{\pi}(\mathfrak{n})|^{2}\ll x(\log x)^{n}.$

###### Proof.

In order to prove three upper bound estimates in a unified way, we introduce
an arithmetic function $f(\mathfrak{n})$ satisfying the following two
conditions:

  1. (1)

$0\leqslant
f(\mathfrak{n})\ll\underbrace{(\lambda_{\pi\times\widetilde{\pi}}*\cdots*\lambda_{\pi\times\widetilde{\pi}})}_{\text{$k$
terms}}(\mathfrak{n})$ for all
$(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}$ and some $k\geqslant 1$;

  2. (2)

$f(\mathfrak{n})\ll\mathrm{N}\mathfrak{n}^{\delta}$ for any integral ideal
$\mathfrak{n}$ and some $\delta<1$.

It is clear that the generating series of
$\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})*\cdots*\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})$
is exactly $L(s,\pi\times\widetilde{\pi})^{k}$. Using the analytic properties
of the Rankin–Selberg $L$-function $L(s,\pi\times\widetilde{\pi})$ and the
Tauberian theorem, we find that there exists a constant
$\Cl[abcon]{leading}=\Cr{leading}(\pi,k)>0$ such that

$\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\underbrace{(\lambda_{\pi\times\widetilde{\pi}}*\cdots*\lambda_{\pi\times\widetilde{\pi}})}_{\text{$k$
terms}}(\mathfrak{n})\sim\Cr{leading}x(\log x)^{k-1}.$

Now, it follows that

$\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant x\\\
(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}f(\mathfrak{n})\ll
x(\log x)^{k-1}.$

Moreover, for any $\mathfrak{n}\subset\mathcal{O}_{F},$ one has
$\mathfrak{n}=\mathfrak{n}_{1}\mathfrak{n}_{2}$ with
$\mathfrak{n}_{1}|\mathfrak{q}_{\pi}^{\infty}$ and
$(\mathfrak{n}_{2},\mathfrak{q}_{\pi})=\mathcal{O}_{F}$, where
$\mathfrak{n}_{1}|\mathfrak{q}_{\pi}^{\infty}$ means that
$\mathfrak{p}|\mathfrak{n}_{1}$ implies $\mathfrak{p}|\mathfrak{q}_{\pi}$ for
any $\mathfrak{p}$. Thus, we have

(5.10) $\displaystyle\sum_{\mathrm{N}\mathfrak{n}\leqslant x}f(\mathfrak{n})$
$\displaystyle=\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}_{1}\leqslant
x\\\
\mathfrak{n}_{1}\mid\mathfrak{q}_{\pi}^{\infty}\end{subarray}}f(\mathfrak{n}_{1})\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}_{2}\leqslant
x/\mathrm{N}\mathfrak{n}_{1}\\\
(\mathfrak{n}_{2},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}f(\mathfrak{n}_{2})$
$\displaystyle\ll x(\log
x)^{k-1}\prod_{\mathfrak{p}|\mathfrak{q}_{\pi}}\Big{(}1+O\Big{(}\frac{1}{\mathrm{N}\mathfrak{p}^{1-\delta}}\Big{)}\Big{)}\ll
x(\log x)^{k-1}.$

By (2.5), it is easy to see that
$b_{F,\pi}(\mathfrak{n})^{2},|\mu_{\pi}(\mathfrak{n})|^{2}$ and
$d_{F}(\mathfrak{n})|\mu_{\pi}(\mathfrak{n})|^{2}$ are all
$\ll\mathrm{N}\mathfrak{n}^{2\theta_{n}+\varepsilon}$. Note that
$2\theta_{n}\leqslant 1-\frac{2}{n^{2}+1}$. Together this with Lemma 5.3, we
can take $f(\mathfrak{n})$ to be
$b_{F,\pi}(\mathfrak{n})^{2},|\mu_{\pi}(\mathfrak{n})|^{2}$ and
$d_{F}(\mathfrak{n})|\mu_{\pi}(\mathfrak{n})|^{2}$ with $k=4(n+1),1$ and
$n+1$, respectively. Thus, the estimate (5.10) yields this lemma. ∎

##  6\. Large-sieve type estimates

In this section we obtain large sieve estimates for Dirichlet polynomials and
$L$-functions. The main tool is the large sieve inequality for number fields
introduced by Huxley [13].

######  Lemma 6.1.

Let $c(\mathfrak{n})$ be any complex coefficients and define the Dirichlet
polynomial to be

$D(s,\chi)=\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\frac{c(\mathfrak{n})\chi(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}.$

Then we have

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{\mathrm{N}\mathfrak{m}}{\varphi_{F}(\mathfrak{m})}\sideset{}{{}^{*}}{\sum}_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}}|D(s,\chi)|^{2}\ll(Q^{2}+x)\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}\Big{|}\frac{c(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{s}}\Big{|}^{2},$

where the second sum is over all primitive narrow ideal class character modulo
$\mathfrak{m}$ and the implied constant depends on the number field.

To obtain large sieve inequality for $L$-functions, we use the approximation
functional equation to approximate the $L$-functions in the critical strip by
Dirichlet series. We state the approximate functional equation as in [16,
Theorem 5.3].

######  Lemma 6.2.

Let $X>0$ and $\pi\in\mathfrak{F}_{n}$. Then for
$\mathop{\mathrm{Re}}s\in(0,1)$, we have

$L(s,\pi)=\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\lambda_{\pi}(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}V_{s}\Big{(}\frac{X\mathrm{N}\mathfrak{n}}{\sqrt{q(\pi)}}\Big{)}+\varepsilon(s,\pi)\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\overline{\lambda_{\pi}(\mathfrak{n})}}{{\mathrm{N}\mathfrak{n}}^{1-s}}V_{1-s}\Big{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{q(\pi)}}\Big{)},$

where

$\varepsilon(s,\pi)=\varepsilon(\pi)q(\pi)^{\frac{1}{2}-s}\frac{L_{\infty}(1-s,\widetilde{\pi})}{\L_{\infty}(s,\pi)}.$

Moreover, for any $A>0$, $V_{s}(y)$ is a function satisfying the following
estimate

$V_{s}(y)\ll_{A}\Big{(}1+\frac{y}{\sqrt{q_{\infty}(\pi,\operatorname{Im}s)}}\Big{)}^{-A}.$

We shall use these two lemmata to deduce the second moments of twisted
automorphic $L$-functions.

######  Proposition 6.3.

For any real number $t$ and $Q>0$, we have

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\sideset{}{{}^{*}}{\sum}_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}}|L(\tfrac{1}{2}+it,\pi\otimes\chi)|^{2}\ll(Q^{2}+Q^{\frac{n}{2}}(3+|t|)^{\frac{n[F:\mathbb{Q}]}{2}})(\log
Q(3+|t|))^{2},$

where the second sum is over all primitive narrow ideal class character modulo
$\mathfrak{m}$. The implied constant depends on $F$ and $\pi$.

###### Proof.

By $q(\pi\otimes\chi)\sim Q_{1}$, we mean that
$Q_{1}<q(\pi\otimes\chi)\leqslant 2Q_{1}$. Since $\chi$ is a character modulo
$\mathfrak{m}$ and $\mathrm{N}\mathfrak{m}\leqslant Q$, it follows from (2.9)
that $q(\pi\otimes\chi)\ll Q^{n}$. As a result,

(6.1) $\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}\end{subarray}}|L(\tfrac{1}{2}+it,\pi\otimes\chi)|^{2}\ll(\log
Q)\max_{Q_{1}\ll Q^{n}}\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}\\\
q(\pi\otimes\chi)\sim
Q_{1}\end{subarray}}|L(\tfrac{1}{2}+it,\pi\otimes\chi)|^{2}.$

Since $\pi\otimes\chi\in\mathfrak{F}_{n}$ and
$(\mathfrak{m},\mathfrak{q}_{\pi})=\mathcal{O}_{F}$, we obtain from (2.13) and
Lemma 6.2 that

$\displaystyle|L(\tfrac{1}{2}+it,\pi\otimes\chi)|^{2}$
$\displaystyle\ll\Big{|}\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\lambda_{\pi\otimes\chi}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\Big{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{q(\pi\otimes\chi)}}\Big{)}\Big{|}^{2}$
$\displaystyle+\Big{|}\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\overline{\lambda_{\pi\otimes\chi}(\mathfrak{n})}}{\mathrm{N}\mathfrak{n}^{1/2-it}}V_{\frac{1}{2}-it}\Big{(}\frac{X\mathrm{N}\mathfrak{n}}{\sqrt{q(\pi\otimes\chi)}}\Big{)}\Big{|}^{2}:=|D_{1}(X)|^{2}+|D_{2}(X)|^{2}$

for all $X>0$. As a result,

$|L(\tfrac{1}{2}+it,\pi\otimes\chi)|^{2}\ll\int_{1}^{2}|D_{1}(X)|^{2}\frac{dX}{X}+\int_{1}^{2}|D_{2}(X)|^{2}\frac{dX}{X}.$

Note that the conductor $q(\pi\otimes\chi)\sim Q_{1}$. We denote
$X_{1}=(Q_{1}/q(\pi\otimes\chi))^{\frac{1}{2}}$. We perform a change of
variable $X\mapsto XX_{1}$ for the first integral, while $X\mapsto
XX_{1}^{-1}$ for the second one. Consequently,

(6.2) $\displaystyle|L(\tfrac{1}{2}+it,\pi\otimes\chi)|^{2}$
$\displaystyle\ll\int_{X_{1}}^{2X_{1}}\Big{|}\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\lambda_{\pi\otimes\chi}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\bigg{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\bigg{)}\Big{|}^{2}\frac{dX}{X}$
$\displaystyle+\int_{X_{1}^{-1}}^{2X_{1}^{-1}}\Big{|}\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\overline{\lambda_{\pi\otimes\chi}(\mathfrak{n})}}{\mathrm{N}\mathfrak{n}^{1/2-it}}V_{\frac{1}{2}-it}\bigg{(}\frac{X\mathrm{N}\mathfrak{n}}{\sqrt{Q_{1}}}\bigg{)}\Big{|}^{2}\frac{dX}{X}.$

For any $A>0$, we can see that
$V_{s}(\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}})\ll\frac{(Q(3+|t|)^{[F:\mathbb{Q}]})^{\frac{nA}{2}}}{\mathrm{N}\mathfrak{n}^{A}}$
when $\mathrm{N}\mathfrak{n}>({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}}$. For
any sufficiently small $\varepsilon>0$, we can choose $A=A(\varepsilon)$ in
Lemma 6.2 to such that

$\sum_{\mathrm{N}\mathfrak{n}>({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}+\varepsilon}}\frac{\lambda_{\pi\otimes\chi}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\bigg{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\bigg{)}\ll
1.$

When
$\mathrm{N}\mathfrak{n}\leqslant({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}+\varepsilon}$,
we shall make use of the large sieve inequality. For this, we first apply
(2.5) and the relation (2.12), and then get

$\sum_{\mathrm{N}\mathfrak{n}\leqslant({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}+\varepsilon}}\frac{\lambda_{\pi\otimes\chi}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\bigg{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\bigg{)}\\\
=\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}+\varepsilon}\\\
(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}\frac{\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\bigg{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\bigg{)}+O(N\mathfrak{q}_{\pi}^{\varepsilon}).$

Accordingly, the contribution from the first term on the right-hand side of
(6.2) to the double sum on the right-hand side of (6.1) is

(6.3) $\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}\\\
q(\pi\otimes\chi)\sim
Q_{1}\end{subarray}}\int_{X_{1}}^{2X_{1}}\Big{|}\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\lambda_{\pi\otimes\chi}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\bigg{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\bigg{)}\Big{|}^{2}\frac{dX}{X}\\\
\ll\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}\\\
q(\pi\otimes\chi)\sim
Q_{1}\end{subarray}}\int_{X_{1}}^{2X_{1}}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}+\varepsilon}\\\
(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}\frac{\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\bigg{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\bigg{)}\Big{|}^{2}\frac{dX}{X}+Q^{2}\\\
\ll Q^{2}+\int_{X_{1}}^{2X_{1}}\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}\\\
q(\pi\otimes\chi)\sim
Q_{1}\end{subarray}}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}}\\\
(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}\frac{\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\bigg{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\bigg{)}\Big{|}^{2}\frac{dX}{X}\\\
+(\log QT)\max_{\frac{n}{2}\leqslant\frac{\log
M}{\log(Q(3+|t|)^{[F:\mathbb{Q}]})}\leqslant\frac{n}{2}+\varepsilon}\int_{X_{1}}^{2X_{1}}\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}\\\
q(\pi\otimes\chi)\sim
Q_{1}\end{subarray}}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\sim
M\\\
(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}\frac{\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\bigg{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\bigg{)}\Big{|}^{2}\frac{dX}{X}.$

Moreover,
$V_{\frac{1}{2}+it}\big{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\big{)}$
is bounded by $O(1)$ when
$\mathrm{N}\mathfrak{n}\leqslant({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}}$.
Thus, by Lemma 6.1, Lemma 5.2 and (2.10), we have

(6.4) $\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}\\\
q(\pi\otimes\chi)\sim
Q_{1}\end{subarray}}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}}\\\
(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}\frac{\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\bigg{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\bigg{)}\Big{|}^{2}$
$\displaystyle\ll(Q^{2}+Q^{\frac{n}{2}}(3+|t|)^{\frac{n[F:\mathbb{Q}]}{2}})\log
Q(3+|t|).$

For
$({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}}\leqslant\mathrm{N}\mathfrak{n}\leqslant({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}+\varepsilon}$,
we take $A=1$ in Lemma 6.2 to have
$V_{\frac{1}{2}+it}(\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}})\ll\frac{({Q(3+|t|)^{[F:\mathbb{Q}]}})^{\frac{n}{2}}}{\mathrm{N}\mathfrak{n}}$.
Similar to the estimate (6.4), we get

(6.5) $\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}\\\
q(\pi\otimes\chi)\sim
Q_{1}\end{subarray}}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\sim
M\\\
(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}\frac{\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\bigg{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\bigg{)}\Big{|}^{2}$
$\displaystyle\ll(Q^{2}+M)Q^{n}T^{n[F:\mathbb{Q}]}\sum_{\mathrm{N}\mathfrak{n}\sim
M}\frac{|\lambda_{\pi}(\mathfrak{n})|^{2}}{{\mathrm{N}\mathfrak{n}^{3}}}$
$\displaystyle\ll(Q^{2}+M)\frac{({Q(3+|t|)^{[F:\mathbb{Q}]}})^{n}}{M^{2}}.$

Inserting (6.4) and (6.5) into (6.3), we have

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}\\\
q(\pi\otimes\chi)\sim
Q_{1}\end{subarray}}\int_{X_{1}}^{2X_{1}}\Big{|}\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\lambda_{\pi\otimes\chi}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1/2+it}}V_{\frac{1}{2}+it}\bigg{(}\frac{\mathrm{N}\mathfrak{n}}{X\sqrt{Q_{1}}}\bigg{)}\Big{|}^{2}\frac{dX}{X}\ll(Q^{2}+Q^{\frac{n}{2}}(3+|t|)^{\frac{n[F:\mathbb{Q}]}{2}})\log
Q(3+|t|).$

We could treat the dual sum similarly and derive the contribution from the
first term on the right-hand side of (6.2) is also bounded by
$O((Q^{2}+Q^{\frac{n}{2}}(3+|t|)^{\frac{n[F:\mathbb{Q}]}{2}})\log Q(3+|t|))$
Then this proposition follows. ∎

##  7\. Type I sums: Proof of Theorem 1.3

For technical convenience, one usually works with the weighted sum

(7.1)
$\psi_{\rho}(y,\pi,\mathfrak{m},\mathfrak{a}):=\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
y\\\ \mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{n})\Big{(}1-\frac{\mathrm{N}\mathfrak{n}}{y}\Big{)}^{\rho},$

where $(\mathfrak{m},\mathfrak{a})=\mathcal{O}_{F}$, $\rho\geqslant 0$. We
want to prove a Bombieri–Vinogradov theorem with the smooth weight for
$\lambda_{\pi}(\mathfrak{n})$. To be precise, we have

######  Lemma 7.1.

Let $\eta=\max\\{2,\frac{n}{2}\\}$,
$\rho=\lfloor\frac{n[F:\mathbb{Q}]}{4}\rfloor+1$, where $\lfloor\cdot\rfloor$
denotes the usual floor function. If $A$ is any positive number, then we have

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\psi_{\rho}(y,\pi,\mathfrak{m},\mathfrak{a})\Big{|}\ll\frac{x}{(\log
x)^{A}},$

where $Q=x^{\frac{1}{\eta}}(\log x)^{-B}$ with $B=A+2n+3$.

###### Proof.

Detecting the congruence condition in (7.1) by the multiplicative characters
$\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})},$ we obtain the identity

(7.2) $\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{n})\Big{(}1-\frac{\mathrm{N}\mathfrak{n}}{y}\Big{)}^{\rho}=\frac{1}{h(\mathfrak{m})}\sum_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}}\overline{\chi}(\mathfrak{a})\sum_{\mathrm{N}\mathfrak{n}\leqslant
y}\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})\Big{(}1-\frac{\mathrm{N}\mathfrak{n}}{y}\Big{)}^{\rho}.$

We shall treat the innermost sum on the right-hand side of (7.2) by the
technique of standard contour integration, which could give a direct link
between the summation associated to an arithmetic function and the
corresponding Dirichlet series. If $\rho$ is any positive integer and $c>0,$
then we have the Mellin inversion formula

(7.3) $\frac{1}{2\pi
i}\int_{(c)}\frac{x^{s}}{s(s+1)\cdots(s+\rho)}ds=\left\\{\begin{array}[]{ll}\frac{1}{\rho!}(1-\frac{1}{x})^{\rho}&\hbox{if
$x\geqslant 1$},\\\ 0&\hbox{if $0\leqslant x\leqslant 1$.}\end{array}\right.$

Then it follows from (7.3) that

$\sum_{\mathrm{N}\mathfrak{n}\leqslant
y}\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})\Big{(}1-\frac{\mathrm{N}\mathfrak{n}}{y}\Big{)}^{\rho}=\frac{1}{2\pi
i}\int_{(1+\varepsilon)}\frac{\Gamma(s)}{\Gamma(\rho+1+s)}\Big{(}\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}\Big{)}y^{s}ds.$

If $\chi$ is induced by a primitive character
$\chi_{1}(\text{mod}\;\mathfrak{m}^{\prime}),$ then
$\mathfrak{m}^{\prime}|\mathfrak{m}$. We further get from (2.13) that

(7.4)
$\sum_{\mathfrak{n}\subset\mathcal{O}_{F}}\frac{\lambda_{\pi}(\mathfrak{n})\chi(\mathfrak{n})}{{\mathrm{N}\mathfrak{n}}^{s}}=\Big{(}\prod_{\mathfrak{p}}\prod_{j=1}^{n}\Big{(}1-\frac{\alpha_{j,\pi}(\mathfrak{p})\chi_{1}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}^{-1}\Big{)}\prod_{\mathfrak{p}|\mathfrak{m}}\prod_{j=1}^{n}\Big{(}1-\frac{\alpha_{j,\pi}(\mathfrak{p})\chi_{1}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\\\
=L(s,\pi\otimes\chi_{1})\Big{(}\prod_{\mathfrak{p}|\mathfrak{m}^{\prime}}\prod_{j=1}^{n}\Big{(}1-\frac{\alpha_{j,\pi\otimes\chi}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\Big{)}\prod_{\mathfrak{p}|\mathfrak{m}}\prod_{j=1}^{n}\Big{(}1-\frac{\alpha_{j,\pi}(\mathfrak{p})\chi_{1}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}.$

Due to the estimate (2.5), for any $\varepsilon>0,$

$\Big{(}\prod_{\mathfrak{p}|\mathfrak{m}^{\prime}}\prod_{j=1}^{n}\Big{(}1-\frac{\alpha_{j,\pi\otimes\chi}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\Big{)}\prod_{\mathfrak{p}|\mathfrak{m}}\prod_{j=1}^{n}\Big{(}1-\frac{\alpha_{j,\pi}(\mathfrak{p})\chi_{1}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)}\ll
d_{F}(\mathfrak{m})^{2n}$

at $\mathop{\mathrm{Re}}s=1/2$. Applying the analytic properties of
$L(s,\pi\otimes\chi_{1})$ and then moving the line of integration to
$\mathop{\mathrm{Re}}(s)=1/2$. Thus, by the residue theorem, it is bounded by

$d_{F}(\mathfrak{m})^{2n}y^{\frac{1}{2}}\int_{(1/2)}|L(s,\pi\otimes\chi_{1})|\frac{|ds|}{|s|^{\rho+1}}.$

Gathering these estimates, we then have

(7.5) $\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{n})\Big{(}1-\frac{\mathrm{N}\mathfrak{n}}{y}\Big{)}^{\rho}\Big{|}$
$\displaystyle\ll
x^{\frac{1}{2}}\int_{(1/2)}\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{d_{F}(\mathfrak{m})^{2n}}{\varphi_{F}(\mathfrak{m})}\sum_{\mathfrak{m^{\prime}}|\mathfrak{m}}\;\sideset{}{{}^{*}}{\sum}_{\chi_{1}\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m^{\prime}})}}|L(s,\pi\otimes\chi_{1})|\frac{|ds|}{|s|^{\rho+1}}$
$\displaystyle\ll
x^{\frac{1}{2}}\int_{(1/2)}\sum_{\mathrm{N}\mathfrak{r}\leqslant
Q}\frac{d_{F}(\mathfrak{r})^{2n}}{\varphi_{F}(\mathfrak{r})}\sum_{\mathrm{N}\mathfrak{m^{\prime}}\leqslant
Q/\mathrm{N}\mathfrak{r}}\frac{d_{F}(\mathfrak{m^{\prime}})^{2n}}{\varphi_{F}(\mathfrak{m^{\prime}})}\;\sideset{}{{}^{*}}{\sum}_{\chi_{1}\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m^{\prime}})}}|L(s,\pi\otimes\chi_{1})|\frac{|ds|}{|s|^{\rho+1}},$

where the trivial inequalities $d_{F}(\mathfrak{bc})\leqslant
d_{F}(\mathfrak{b})d_{F}(\mathfrak{c})$ and
$\varphi_{F}(\mathfrak{bc})\geqslant\varphi_{F}(\mathfrak{b})\varphi_{F}(\mathfrak{c})$.
are used in the last step. By Proposition 6.3, the Cauchy–Schwarz inequality
and the elementary estimate $\sum_{\mathrm{N}\mathfrak{n}\leqslant
x}d_{F}(\mathfrak{n})^{k}\ll x(\log x)^{2^{k}-1}$ for any positive integer
$k$, we have

$\displaystyle\sum_{\mathrm{N}\mathfrak{m^{\prime}}\leqslant
Q/\mathrm{N}\mathfrak{r}}\frac{d_{F}(\mathfrak{m^{\prime}})^{2n}}{\varphi_{F}(\mathfrak{m^{\prime}})}\sideset{}{{}^{*}}{\sum}_{\chi_{1}\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m^{\prime}})}}|L(s,\pi\otimes\chi_{1})|$
$\displaystyle\ll(\log Q)^{2}\max_{R\leqslant
Q/\mathrm{N}\mathfrak{r}}\frac{1}{R}\sum_{\mathrm{N}\mathfrak{m^{\prime}}\sim
R}d_{F}(\mathfrak{m^{\prime}})^{2n}\sideset{}{{}^{*}}{\sum}_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}}|L(s,\pi\otimes\chi)|$
$\displaystyle\ll(\log Q)^{2n+2}\max_{R\leqslant
Q/\mathrm{N}\mathfrak{r}}\Big{(}\sum_{\mathrm{N}\mathfrak{m}\sim
R}\sum_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}}|L(s,\pi\otimes\chi)|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\ll\Big{(}\frac{Q}{\mathrm{N}\mathfrak{r}}+\big{(}\frac{Q}{\mathrm{N}\mathfrak{r}}\big{)}^{\frac{n}{4}}\Big{)}|s|^{\frac{n[F:\mathbb{Q}]}{4}}(\log
Q|s|)^{2n+3}.$

The lemma follows once we insert this estimate into (7.5). ∎

###### Proof of Theorem 1.3.

Now we turn to proof of Theorem 1.3. To do this, we follow the method of [24,
Lemma 2], which is originally due to [8]. Let $z=z(y)=\frac{y}{(\log
y)^{A/2}}$ with the parameter $A$ as in Lemma 7.1. It is easy to check that

(7.6)
$\displaystyle\int_{y-z}^{y}t^{\rho-1}\psi_{\rho-1}(t,\pi,\mathfrak{m},\mathfrak{a})dt=\frac{y^{\rho}}{\rho}\psi_{\rho}(y,\pi,\mathfrak{m},\mathfrak{a})-\frac{(y-z)^{\rho}}{\rho}\psi_{\rho}(y-z,\pi,\mathfrak{m},\mathfrak{a}).$

We can rewrite this integral on the left-hand side of (7.6) as

(7.7)
$zy^{\rho-1}\psi_{\rho-1}(y,\pi,\mathfrak{m},\mathfrak{a})-\int_{y-z}^{y}\Big{(}y^{\rho-1}\psi_{\rho-1}(y,\pi,\mathfrak{m},\mathfrak{a})-t^{\rho-1}\psi_{\rho-1}(t,\pi,\mathfrak{m},\mathfrak{a})\Big{)}dt.$

It is obvious that the integrand of (7.7) equals to

(7.8) $\displaystyle\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
t\\\ \mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{n})\Big{(}(y-\mathrm{N}\mathfrak{n})^{\rho-1}-(t-\mathrm{N}\mathfrak{n})^{\rho-1}\Big{)}+\sum_{\begin{subarray}{c}t<\mathrm{N}\mathfrak{n}\leqslant
y\\\ \mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}\lambda_{\pi}(\mathfrak{n})(y-\mathrm{N}\mathfrak{n})^{\rho-1}$
$\displaystyle\ll
zy^{\rho-2}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}|\lambda_{\pi}(\mathfrak{n})|$

for $y-z\leqslant t\leqslant y$. Then it follows from (7.6)–(7.8) that

$\displaystyle\psi_{\rho-1}(y,\pi,\mathfrak{m},\mathfrak{a})=$
$\displaystyle\frac{y}{z\rho}\psi_{\rho}(y,\pi,\mathfrak{m},\mathfrak{a})-\frac{(y-z)^{\rho}}{zy^{\rho-1}\rho}\psi_{\rho}(y-z,\pi,\mathfrak{m},\mathfrak{a})+O\Big{(}\frac{z}{y}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
y\\\ \mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}|\lambda_{\pi}(\mathfrak{n})|\Big{)}.$

As a result, we can estimate the sum involving
$\psi_{\rho-1}(y,\pi,\mathfrak{m},\mathfrak{a})$ as follows

(7.9) $\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\psi_{\rho-1}(y,\pi,\mathfrak{m},\mathfrak{a})\Big{|}$
$\displaystyle\ll(\log x)^{\frac{A}{2}}\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}\psi_{\rho}(y,\pi,\mathfrak{m},\mathfrak{a})\Big{|}$
$\displaystyle+\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\,(\log
y)^{-\frac{A}{2}}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}|\lambda_{\pi}(\mathfrak{n})|.$

Note that the contribution of the term involving
$\psi_{\rho}(y-z,\pi,\mathfrak{m},\mathfrak{a})$ can be dominated by the first
term of the right-hand side of (7.9), since we have taken the maximum over
$y\leqslant x$.

We first treat the last term on the right-hand side of (7.9). By Lemma 5.2 and
the Rankin–Selberg theory, we can get
$\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
x\end{subarray}}|\lambda_{\pi}(\mathfrak{n})|\leqslant
x^{\frac{1}{2}}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
x\end{subarray}}|\lambda_{\pi}(\mathfrak{n})|^{2}\ll x$. As a result,

(7.10) $\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\,(\log
y)^{-\frac{A}{2}}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}|\lambda_{\pi}(\mathfrak{n})|$
$\displaystyle\ll$ $\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{x^{\frac{1}{3}}\leqslant
y\leqslant x}\,(\log
y)^{-\frac{A}{2}}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant y\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}|\lambda_{\pi}(\mathfrak{n})|+x^{\frac{5}{6}}$
$\displaystyle\ll$ $\displaystyle(\log
x)^{-\frac{A}{2}}\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
x\\\ \mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}|\lambda_{\pi}(\mathfrak{n})|+x^{\frac{5}{6}}$

since $Q\ll X^{\frac{1}{2}}$. Using the Cauchy–Schwarz inequality and Lemma
6.1, we obtain

$\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
x\\\ \mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}|\lambda_{\pi}(\mathfrak{n})|$
$\displaystyle\ll(\log Q)^{4}\max_{CR\leqslant
Q}\frac{1}{CR}\sum_{\mathrm{N}\mathfrak{r}\sim
R}\sum_{\mathrm{N}\mathfrak{c}\sim
C}\sideset{}{{}^{*}}{\sum}_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{c})}}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
x\\\
(\mathfrak{n},\mathfrak{cr})=\mathcal{O}_{F}\end{subarray}}|\lambda_{\pi}(\mathfrak{n})|\chi(\mathfrak{n})\Big{|}$
$\displaystyle\ll(\log Q)^{4}\max_{CR\leqslant
Q}\frac{1}{R}\sum_{\mathrm{N}\mathfrak{r}\sim
R}\Big{(}\sum_{\mathrm{N}\mathfrak{c}\sim
C}\sideset{}{{}^{*}}{\sum}_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{c})}}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
x\\\
(\mathfrak{n},\mathfrak{cr})=\mathcal{O}_{F}\end{subarray}}|\lambda_{\pi}(\mathfrak{n})|\chi(\mathfrak{n})\Big{|}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\ll(\log Q)^{4}(Q+x^{\frac{1}{2}})x^{\frac{1}{2}}\ll x(\log
x)^{4}.$

Substituting this into (7.10), we bound the contribution of the last term on
the right-hand side of (7.9) by $\ll x(\log x)^{4-\frac{A}{2}}$.

By Lemma 7.1, we see that the first term in the right-hand side of (7.9) is
$O(x(\log x)^{-\frac{A}{2}})$. Thus, we conclude from the above that

$\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
X}\Big{|}\psi_{\rho-1}(y,\pi,\mathfrak{m},\mathfrak{a})\Big{|}\ll\frac{x}{(\log
x)^{\frac{A}{2}-4}}$

for $Q=X^{\frac{1}{\eta}}(\log x)^{-B}$ where
$\eta=\max\\{2,\frac{n}{2}\\},B=A+2n+3$ and the implied constant depends on
$\pi$,$F$ and $A$. Repeating this process $\rho$ times yields

$\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
X}\Big{|}\psi_{0}(y,\pi,\mathfrak{m},\mathfrak{a})\Big{|}\ll\frac{X}{(\log
X)^{\frac{A+8}{2^{\rho}}-8}}.$

Finally, we transform parameter $\frac{A+8}{2^{\rho}}-8$ to $A$, finishing the
proof. ∎

##  8\. A bilinear form

######  Lemma 8.1.

Let $\mathcal{A}=\\{a(\mathfrak{l})\\}$ and
$\mathcal{B}=\\{b(\mathfrak{n})\\}$ be two sequences of complex numbers with
$\mathcal{B}$ satisfying the following Siegel–Walfisz hypothesis: For
$(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}$ and
$\mathrm{N}\mathfrak{m}\leqslant(\log N)^{A}$ for any $A>0$,

(8.1) $\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant N\\\
\mathfrak{n}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})\end{subarray}}b(\mathfrak{n})\ll\frac{(\mathcal{B}(N)N)^{\frac{1}{2}}}{(\log
N)^{9A}}.$

Then we have

(8.2) $\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\Big{|}{\underset{\mathfrak{ln}\equiv\mathfrak{a}\text{
in }\mathrm{Cl}^{+}(\mathfrak{m})}{\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}}}a(\mathfrak{l})b(\mathfrak{n})\Big{|}$
$\displaystyle\ll\Big{(}Q+\sqrt{L+N}+\frac{\sqrt{LN}}{(\log
N)^{A}}\Big{)}(\log
Q)^{2}\mathcal{A}(L)^{\frac{1}{2}}\mathcal{B}(N)^{\frac{1}{2}},$

where

$\mathcal{A}(L)=\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}|a(\mathfrak{l})|^{2},\quad\mathcal{B}(N)=\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|b(\mathfrak{n})|^{2},$

and the implied constant depends only on $A$.

###### Proof.

Using the orthogonality of multiplicative characters, we see that the left-
hand side in (8.2) is bounded by

(8.3) $\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant Q}$
$\displaystyle\frac{1}{\varphi_{F}(\mathfrak{m})}\sum_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{m})}}\Big{|}\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}a(\mathfrak{l})\chi(\mathfrak{l})\Big{|}\Big{|}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}b(\mathfrak{n})\chi(\mathfrak{n})\Big{|}$
$\displaystyle\leqslant\sum_{\mathrm{N}\mathfrak{cr}\leqslant
Q}\frac{1}{\varphi_{F}(\mathfrak{c})\varphi_{F}(\mathfrak{r})}\sideset{}{{}^{*}}{\sum}_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{r})}}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{l}\leqslant
L\\\
(\mathfrak{l},\mathfrak{c})=\mathcal{O}_{F}\end{subarray}}a(\mathfrak{l})\chi(\mathfrak{l})\Big{|}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
N\\\
(\mathfrak{n},\mathfrak{c})=\mathcal{O}_{F}\end{subarray}}b(\mathfrak{n})\chi(\mathfrak{n})\Big{|}.$

Fix $\mathfrak{c}$ and split the sum over $\mathfrak{r}$ into dyadic segments
$\mathrm{N}\mathfrak{r}\sim R_{1}$. Then apply the Cauchy–Schwarz inequality
to the dyadic segment to get

$\leqslant\Big{(}\sum_{\mathrm{N}\mathfrak{r}\sim
R_{1}}\frac{1}{\varphi_{F}(\mathfrak{r})}\sideset{}{{}^{*}}{\sum}_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{r})}}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{l}\leqslant
L\\\
(\mathfrak{l},\mathfrak{c})=\mathcal{O}_{F}\end{subarray}}a(\mathfrak{l})\chi(\mathfrak{l})\Big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{\mathrm{N}\mathfrak{r}\sim
R_{1}}\frac{1}{\varphi_{F}(\mathfrak{r})}\sideset{}{{}^{*}}{\sum}_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{r})}}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
N\\\
(\mathfrak{n},\mathfrak{c})=\mathcal{O}_{F}\end{subarray}}b(\mathfrak{n})\chi(\mathfrak{n})\Big{|}^{2}\Big{)}^{\frac{1}{2}}$

By Lemma 6.1, the above is

$\displaystyle\ll\Big{(}R_{1}+\frac{L}{R_{1}}\Big{)}^{\frac{1}{2}}\Big{(}R_{1}+\frac{N}{R_{1}}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}|a(\mathfrak{l})|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|b(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\ll\Big{(}R_{1}+\sqrt{L+N}+\frac{\sqrt{LN}}{R_{1}}\Big{)}\Big{(}\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}|a(\mathfrak{l})|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|b(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}}.$

Summing this over $R\leqslant R_{1}\leqslant Q/\mathrm{N}\mathfrak{c}$, we get

(8.4) $\Big{(}\frac{Q}{\mathrm{N}\mathfrak{c}}+\sqrt{L+N}\log
Q+\frac{\sqrt{LN}}{R}\Big{)}\Big{(}\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}|a(\mathfrak{l})|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|b(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}}.$

It remains to estimate the contribution of the primitive characters
$\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{r})}$ with
$1\leqslant\mathrm{N}\mathfrak{r}\leqslant R.$ For each of these we appeal to
the condition (8.1).

First, we define Möbius function $\mu_{F}(\mathfrak{a})$ for $F$ by

$\mu_{F}(\mathfrak{a})=\left\\{\begin{array}[]{ll}1&\mathfrak{a}=\mathcal{O}_{F},\\\
(-1)^{r}&\mathfrak{a}\text{ is a product of }r\text{ distinct prime ideals,
}\\\ 0&\mathfrak{a}\text{ is divided by square of a prime ideal.
}\end{array}\right.$

It is easy to see that $\mu_{F}(\mathfrak{a})$ is a multiplicative function
over the ideals which satisfies

$\sum_{\mathfrak{o}|\mathfrak{a}}\mu_{F}(\mathfrak{o})=\left\\{\begin{array}[]{ll}1&\mathfrak{a}=\mathcal{O}_{F},\\\
0&\text{otherwise.}\end{array}\right.$

We detect the coprimality condition
$(\mathfrak{n},\mathfrak{c})=\mathcal{O}_{F}$ by this formula and get

$\displaystyle\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant N\\\
(\mathfrak{n},\mathfrak{c})=\mathcal{O}_{F}\end{subarray}}b(\mathfrak{n})\chi(\mathfrak{n})$
$\displaystyle=\sum_{\mathfrak{o}|\mathfrak{c}}\mu_{F}(\mathfrak{o})\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
N\\\
\mathfrak{o}|\mathfrak{n}\end{subarray}}b(\mathfrak{n})\chi(\mathfrak{n})$
$\displaystyle=\sum_{\begin{subarray}{c}\mathfrak{o}|\mathfrak{c}\\\
\mathrm{N}\mathfrak{o}\leqslant
K\end{subarray}}\mu_{F}(\mathfrak{o})\sum_{\mathfrak{l}|\mathfrak{o}}\mu_{F}(\mathfrak{l})\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
N\\\
(\mathfrak{n},\mathfrak{l})=\mathcal{O}_{F}\end{subarray}}b(\mathfrak{n})\chi(\mathfrak{n})+\sum_{\begin{subarray}{c}\mathfrak{o}|\mathfrak{c}\\\
\mathrm{N}\mathfrak{o}>K\end{subarray}}\mu_{F}(\mathfrak{o})\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
N\\\
\mathfrak{o}|\mathfrak{n}\end{subarray}}b(\mathfrak{n})\chi(\mathfrak{n})$
$\displaystyle:=W_{1}+W_{2}.$

where $K$ will be chosen later. Next we estimate the innermost sum in $W_{1}$
by splitting into classes in $\mathrm{Cl}^{+}(\mathfrak{m})$, and for each
class we apply the Siegel–Walfisz hypothesis (8.1). Consequently, we have that

$\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant N\\\
(\mathfrak{n},\mathfrak{l})=\mathcal{O}_{F}\end{subarray}}b(\mathfrak{n})\chi(\mathfrak{n})=\sum_{\mathfrak{h}\in\mathrm{Cl}^{+}(\mathfrak{rl})}\chi(\mathfrak{h})\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
N\\\ \mathfrak{n}\equiv\mathfrak{h}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{rl})\end{subarray}}b(\mathfrak{n})\ll\frac{h(\mathfrak{rl})N^{\frac{1}{2}}}{(\log
N)^{9A}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|b(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}},$

where we choose $\mathfrak{h}$ to be integral representatives. This further
yields

$W_{1}\ll\frac{N\mathfrak{r}N^{\frac{1}{2}}}{(\log
N)^{A}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|b(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}}\sum_{\begin{subarray}{c}\mathfrak{o}|\mathfrak{c}\\\
\mathrm{N}\mathfrak{o}\leqslant
K\end{subarray}}\sum_{\mathfrak{l}|\mathfrak{o}}|\mu_{F}(\mathfrak{l})|N\mathfrak{l}\ll\frac{RKN^{\frac{1}{2}}d_{F}(\mathfrak{c})}{(\log
N)^{9A}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|b(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}}.$

Then we treat the sum $W_{2}$ and apply the Cauchy–Schwarz inequality getting

$W_{2}\ll\Big{(}\frac{N}{K}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|b(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}}d_{F}(\mathfrak{c}).$

Adding these two estimates for $W_{1}$ and $W_{2}$ and then choosing $K=(\log
N)^{6A}$, we obtain

(8.5) $\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant N\\\
(\mathfrak{n},\mathfrak{c})=\mathcal{O}_{F}\end{subarray}}b(\mathfrak{n})\chi(\mathfrak{n})\ll\frac{RN^{\frac{1}{2}}d_{F}(\mathfrak{c})}{(\log
N)^{3A}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|b(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}}.$

Moreover, we use the trivial bound

$\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{l}\leqslant L\\\
(\mathfrak{l},\mathfrak{c})=\mathcal{O}_{F}\end{subarray}}a(\mathfrak{l})\chi(\mathfrak{l})\ll
L^{\frac{1}{2}}\Big{(}\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}|a(\mathfrak{l})|^{2}\Big{)}^{\frac{1}{2}}.$

Combining this with the estimate (8.5) together, we obtain

(8.6) $\displaystyle\sum_{\mathrm{N}\mathfrak{r}\leqslant
R}\frac{1}{\varphi_{F}(\mathfrak{r})}\sideset{}{{}^{*}}{\sum}_{\chi\in\widehat{\mathrm{Cl}^{+}(\mathfrak{r})}}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{l}\leqslant
L\\\
(\mathfrak{l},\mathfrak{c})=\mathcal{O}_{F}\end{subarray}}a(\mathfrak{l})\chi(\mathfrak{l})\Big{|}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leqslant
N\\\
(\mathfrak{n},\mathfrak{c})=\mathcal{O}_{F}\end{subarray}}b(\mathfrak{n})\chi(\mathfrak{n})\Big{|}$
$\displaystyle\ll\frac{R^{2}\sqrt{LN}d_{F}(\mathfrak{c})}{(\log
N)^{3A}}\Big{(}\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}|a(\mathfrak{l})|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|b(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}}.$

Summing (8.6) and (8.4) over $\mathfrak{c}$, we infer the following bound for
our original sum in (8.3):

$\Big{(}Q+\sqrt{L+N}+\frac{\sqrt{LN}}{R}+\frac{R^{2}\sqrt{LN}}{(\log
N)^{3A}}\Big{)}(\log Q)^{2}\Big{(}\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}|a(\mathfrak{l})|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}|b(\mathfrak{n})|^{2}\Big{)}^{\frac{1}{2}}$

Choosing $R=(\log N)^{A}$, we complete the proof of the theorem. ∎

######  Lemma 8.2.

With the notation and conditions as in Lemma 8.1, we have

$\displaystyle\sum_{\mathrm{N}\mathfrak{m}\leqslant
Q}\frac{h(\mathfrak{m})}{\varphi_{F}(\mathfrak{m})}\max_{(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{F}}\max_{y\leqslant
x}\Big{|}{\underset{\mathrm{N}\mathfrak{ln}\leqslant
y\atop\mathfrak{ln}\equiv\mathfrak{a}\text{ in
}\mathrm{Cl}^{+}(\mathfrak{m})}{\sum_{\mathrm{N}\mathfrak{l}\leqslant
L}\sum_{\mathrm{N}\mathfrak{n}\leqslant
N}}}a(\mathfrak{l})b(\mathfrak{n})\Big{|}$
$\displaystyle\ll\Big{(}Q+\sqrt{L+N}+\frac{\sqrt{LN}}{(\log
N)^{A}}\Big{)}(\log xLN)(\log
Q)^{2}\mathcal{A}(L)^{\frac{1}{2}}\mathcal{B}(N)^{\frac{1}{2}}.$

###### Proof.

We follow the trick of Vaughan [33, Lemma 2]. If $\gamma>0$, then

$\frac{1}{\pi}\int_{-\infty}^{\infty}e^{i\beta\alpha}\frac{\sin\gamma\alpha}{\alpha}d\alpha=\begin{cases}1&\mbox{if
$0\leqslant\beta<\gamma$,}\\\ 0&\mbox{if
$\beta>\gamma$}\end{cases}=:\delta(\beta)$

via the product-to-sum identities of trigonometric functions and the identity
$e^{i\beta\alpha}=\cos\beta\alpha+i\sin\beta\alpha$. By integration by parts,
we get

$\int_{A}^{\infty}\frac{\sin Y\alpha}{\alpha}d\alpha\ll\frac{1}{YA}$

for any $Y>0$ and $A>0.$ Thus, using the product-to-sum identities of
trigonometric functions again, we have

(8.7)
$\delta(\beta)=\frac{1}{\pi}\int_{-A}^{A}e^{i\beta\alpha}\frac{\sin\gamma\alpha}{\alpha}d\alpha+O\Big{(}\frac{1}{A|\gamma-\beta|}\Big{)}.$

Taking $\gamma=\log(\lfloor y\rfloor+\frac{1}{2})$ and
$\beta=\log\mathrm{N}\mathfrak{ln}$, we obtain the equality

$\delta(\log\mathrm{N}\mathfrak{ln})=\left\\{\begin{array}[]{ll}1&\mathrm{N}\mathfrak{ln}\leqslant