abconsymbol=c

# A new zero-free region for Rankin–Selberg $L$-functions

Gergely Harcos  Alfréd Rényi Institute of Mathematics, POB 127, Budapest
H-1364, Hungary<EMAIL_ADDRESS>and  Jesse Thorner  Department of
Mathematics, University of Illinois, Urbana, IL 61801, USA
<EMAIL_ADDRESS>
###### Abstract.

Let $\pi$ and $\pi^{\prime}$ be cuspidal automorphic representations of
$\mathrm{GL}(n)$ and $\mathrm{GL}(n^{\prime})$ with unitary central
characters. We establish a new zero-free region for all
$\mathrm{GL}(1)$-twists of the Rankin–Selberg $L$-function
$L(s,\pi\times\pi^{\prime})$, generalizing Siegel’s celebrated work on
Dirichlet $L$-functions. As an application, we prove a Siegel–Walfisz bound
for the Dirichlet coefficients of
$-L^{\prime}(s,\pi\times\pi^{\prime})/L(s,\pi\times\pi^{\prime})$.

The first author was supported by the Rényi Intézet Lendület Automorphic
Research Group and NKFIH (National Research, Development and Innovation
Office) grant K 143876.

##  1\. Introduction and statement of results

In 1896, Hadamard and de la Vallée Poussin independently proved that the
Riemann zeta function $\zeta(s)$ does not vanish in the half-plane
$\mathrm{Re}(s)\geq 1$. This statement is equivalent to the asymptotic form of
the prime number theorem. In 1899, de la Vallée Poussin established the
classical zero-free region for $\zeta(s)$, which was quickly extended to
Dirichlet $L$-functions. In particular, let $\chi\,(\mathrm{mod}\,\,q)$ be a
Dirichlet character. There exists an absolute and effectively computable
constant $\Cl[abcon]{ZetaZFR}>0$ such that $L(s,\chi)$ has at most one zero
$\beta$ (necessarily real and simple) in the region

$\mathrm{Re}(s)\geq 1-\Cr{ZetaZFR}/\log(q(|\mathrm{Im}(s)|+3)).$

If $\beta$ exists, then $\chi$ is quadratic. In this case, Siegel’s lower
bound on $L(1,\chi)$ implies that for any $\varepsilon>0$, there exists an
ineffective constant
$\Cl[abcon]{SiegelDirichlet}=\Cr{SiegelDirichlet}(\varepsilon)>0$ such that

(1.1) $L(\sigma,\chi)\neq 0,\qquad\sigma\geq
1-\Cr{SiegelDirichlet}q^{-\varepsilon}.$

See [25, 27] for the original references.

The method of de la Vallée Poussin can be modified to establish a zero-free
region for many $L$-functions. Specifically, let $F$ be a number field, let
$\mathbb{A}_{F}$ be the ring of adeles over $F$, and let $\mathfrak{F}_{n}$ be
the set of cuspidal automorphic representations $\pi$ of
$\mathrm{GL}_{n}(\mathbb{A}_{F})$ whose central character $\omega_{\pi}$ is
unitary. Given $\pi\in\mathfrak{F}_{n}$, let
$\widetilde{\pi}\in\mathfrak{F}_{n}$ be the contragredient and $L(s,\pi)$ be
the $L$-function of $\pi$. Let $C(\pi)\geq 3$ denote the analytic conductor,
which measures the arithmetic and spectral complexity of $\pi$. For
convenience, we introduce the subset $\mathfrak{F}_{n}^{*}$ consisting of
$\pi\in\mathfrak{F}_{n}$ for which $\omega_{\pi}$ is trivial on the diagonally
embedded positive reals. For each $\pi\in\mathfrak{F}_{n}$, there exist unique
$\pi^{*}\in\mathfrak{F}_{n}^{*}$ and $t_{\pi}\in\mathbb{R}$ such that

$\pi=\pi^{*}\otimes|\det|^{it_{\pi}},\qquad L(s,\pi)=L(s+it_{\pi},\pi^{*}).$

Given $(\pi,\pi^{\prime})\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}$,
let $L(s,\pi\times\pi^{\prime})$ be the associated Rankin–Selberg
$L$-function, whose basic properties were established by Jacquet, Piatetski-
Shapiro, and Shalika [11, 12, 13]. If
$(\pi,\pi^{\prime})\in\mathfrak{F}_{n}^{*}\times\mathfrak{F}_{n^{\prime}}^{*}$,
then $L(s,\pi\times\pi^{\prime})$ is holomorphic away from a possible pole at
$s=1$, which occurs if and only if $\pi^{\prime}=\widetilde{\pi}$. If
$\pi^{\prime}\in\mathfrak{F}_{1}^{*}$ is trivial, then
$L(s,\pi\times\pi^{\prime})=L(s,\pi)$.

In 1981, Shahidi [24] proved that $L(s,\pi\times\pi^{\prime})$ does not vanish
in the half-plane $\mathrm{Re}(s)\geq 1$. In some cases, the method of de la
Vallée Poussin can extend this zero-free region. Brumley [7, Theorem A.1] and
Humphries and Thorner [9, Theorem 2.1] (see also Moreno [20]) proved that
there exists an effectively computable constant
$\Cl[abcon]{ZFR2_{s}td}=\Cr{ZFR2_{s}td}(n,n^{\prime},[F:\mathbb{Q}])>0$ with
the following property. If
$(\pi,\pi^{\prime})\in\mathfrak{F}_{n}^{*}\times\mathfrak{F}_{n^{\prime}}^{*}$,
and

(1.2)
$\pi=\widetilde{\pi}\qquad\textup{or}\qquad\pi^{\prime}=\widetilde{\pi}^{\prime}\qquad\textup{or}\qquad\pi^{\prime}=\widetilde{\pi},$

then $L(s,\pi\times\pi^{\prime})$ has at most one zero $\beta$ (necessarily
real and simple) in the region

$\mathrm{Re}(s)\geq
1-\Cr{ZFR2_{s}td}/\log(C(\pi)C(\pi^{\prime})(|\mathrm{Im}(s)|+3)).$

If the exceptional zero $\beta$ exists, then

(1.3)
$\pi=\widetilde{\pi}\quad\textup{and}\quad\pi^{\prime}=\widetilde{\pi}^{\prime}\qquad\textup{or}\qquad\pi^{\prime}=\widetilde{\pi}.$

If $\pi^{\prime}\in\mathfrak{F}_{1}^{*}$ is trivial, then we recover the
standard zero-free region for $L(s,\pi)$.

There are limited (though important) cases where (1.3) holds and exceptional
zeros have been precluded altogether. For example, if
$\pi^{\prime}\in\mathfrak{F}_{1}^{*}$ is trivial and
$\pi\in\mathfrak{F}_{2}^{*}\cup\mathfrak{F}_{3}^{*}$, then the exceptional
zero does not exist [1, 5, 6]. See [18, 22] for examples when
$\pi^{\prime}\in\mathfrak{F}_{2}^{*}$ and
$\pi\in\mathfrak{F}_{2}^{*}\cup\mathfrak{F}_{3}^{*}$. Generalizing (1.1),
Jiang–Lü–Thorner–Wang [14, Section 4] and Humphries–Thorner [8, Theorem 2.4]
proved that if $\chi\in\mathfrak{F}_{1}^{*}$ is quadratic, then for all
$\varepsilon>0$, there exists an ineffective constant
$\Cl[abcon]{Siegel_{p}re}=\Cr{Siegel_{p}re}(\pi,\varepsilon)>0$ such that

$L(\sigma,\pi\otimes\chi)\neq 0,\qquad
L(\sigma,\pi\otimes(\widetilde{\pi}\otimes\chi))\neq 0,\qquad\sigma\geq
1-\Cr{Siegel_{p}re}C(\chi)^{-\varepsilon}.$

It is unclear how to execute the method of de la Vallée Poussin for all
$(\pi,\pi^{\prime})\in\mathfrak{F}_{n}^{*}\times\mathfrak{F}_{n^{\prime}}^{*}$
that do not satisfy (1.2). In this setting, Brumley ([2], [16, Theorem A.1])
made the first uniform improvement over [24], proving that for all
$\varepsilon>0$, there exists an effectively computable constant
$\Cl[abcon]{ZFR_{B}rumley}=\Cr{ZFR_{B}rumley}(n,n^{\prime},F,\varepsilon)>0$
such that $L(s,\pi\times\pi^{\prime})\neq 0$ when

(1.4) $\mathrm{Re}(s)\geq
1-\Cr{ZFR_{B}rumley}/((C(\pi)C(\pi^{\prime}))^{n+n^{\prime}-1+\varepsilon}(|\mathrm{Im}(s)|+1)^{n^{\prime}n[F:\mathbb{Q}](1-\frac{1}{n+n^{\prime}})+\varepsilon}).$

See also Zhang [28], who halved the exponent of $|\mathrm{Im}(s)|+1$ in (1.4),
and Sarnak [23] and Gelbart–Lapid [3], who established zero-free regions using
Eisenstein series and the Maaß–Selberg relations.

In this paper, we develop a new method for establishing zero-free regions for
$L$-functions. We apply our method to extend Siegel’s celebrated result (1.1)
to every $\mathrm{GL}_{1}$-twist of $L(s,\pi\times\pi^{\prime})$, where
$(\pi,\pi^{\prime})\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}$. Our
proof relies crucially on the group structure of $\mathfrak{F}_{1}$, not just
$\mathfrak{F}_{1}^{*}$. We substantially improve (1.4) in the
$\mathrm{GL}_{1}$-twist aspect, but the dependence on $\pi$ and $\pi^{\prime}$
is no longer effective.

######  Theorem 1.1.

Let $\pi\in\mathfrak{F}_{n}$ and $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$.
For all $\varepsilon>0$, there exists an ineffective constant
$\Cl[abcon]{ZFR2}=\Cr{ZFR2}(\pi,\pi^{\prime},\varepsilon)>0$ such that for any
$\chi\in\mathfrak{F}_{1}$ with analytic conductor $C(\chi)$, we have

(1.5)
$|L(\sigma,\pi\times(\pi^{\prime}\otimes\chi))|\geq\Cr{ZFR2}C(\chi)^{-\varepsilon},\qquad\sigma\geq
1-\Cr{ZFR2}C(\chi)^{-\varepsilon}.$

######  Remark.

By replacing $\chi$ with $\chi|\det|^{it}$, the bound (1.5) becomes

(1.6)
$|L(\sigma+it,\pi\times(\pi^{\prime}\otimes\chi))|\geq\Cr{ZFR2}C(it,\chi)^{-\varepsilon},\qquad\sigma\geq
1-\Cr{ZFR2}C(it,\chi)^{-\varepsilon},$

where $C(it,\chi)\ll C(\chi)(|t|+1)^{[F:\mathbb{Q}]}$. In particular, there
exists an ineffective constant
$\Cl[abcon]{C777}=\Cr{C777}(\pi,\pi^{\prime},\varepsilon)>0$ such that

$|L(\sigma+it,\pi\times\pi^{\prime})|\geq\Cr{C777}(|t|+1)^{-\varepsilon},\qquad\sigma\geq
1-\Cr{C777}(|t|+1)^{-\varepsilon}.$

We provide an application of Theorem 1.1. Let $\mathcal{O}_{F}$ be the ring of
integers of $F$, and define the norm $\mathrm{N}=\mathrm{N}_{F/\mathbb{Q}}$ on
the nonzero ideals $\mathfrak{a}$ of $\mathcal{O}_{F}$ by
$\mathrm{N}\mathfrak{a}=|\mathcal{O}_{F}/\mathfrak{a}|$. Let $\mathfrak{q}$ be
a nonzero ideal of $\mathcal{O}_{F}$, and let $I(\mathfrak{q})$ be the group
of fractional ideals that are coprime to $\mathfrak{q}$. Let $P(\mathfrak{q})$
be the subgroup of $I(\mathfrak{q})$ consisting of principal fractional ideals
$(\alpha)$ such that $\alpha$ is totally positive and $\alpha\equiv
1\,(\mathrm{mod}\,\,\mathfrak{q})$. The narrow ray class group
$\mathrm{Cl}(\mathfrak{q})$ is the finite abelian quotient
$I(\mathfrak{q})/P(\mathfrak{q})$. Let $\mathscr{P}(\mathfrak{q})$ be the set
of primitive ray class characters that induce the characters of
$\mathrm{Cl}(\mathfrak{q})$.

If there exists $u\in\mathbb{R}$ such that
$(\pi,\pi^{\prime})\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}$
satisfies $\pi^{\prime}=\widetilde{\pi}\otimes|\det|^{iu}$, we define
$\mathcal{M}_{\pi\times\pi^{\prime}}(x)=x^{1-iu}/(1-iu)$. For all other pairs
$(\pi,\pi^{\prime})\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}$, we
define $\mathcal{M}_{\pi\times\pi^{\prime}}(x)=0$. We also define
$\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})$ by the Dirichlet series
identity

(1.7)
$\sum_{\mathfrak{a}}\frac{\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})}{\mathrm{N}\mathfrak{a}^{s}}=-\frac{L^{\prime}}{L}(s,\pi\times\pi^{\prime}),\qquad\mathrm{Re}(s)>1.$

For a class $\mathcal{C}\in\mathrm{Cl}(\mathfrak{q})$ and $x\geq 2$, the bound

(1.8)
$\mathcal{E}_{\pi\times\pi^{\prime}}(x;\mathfrak{q},\mathcal{C}):=\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{a}\leq
x\\\
\mathfrak{a}\in\mathcal{C}\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})-\frac{1}{|\mathrm{Cl}(\mathfrak{q})|}\sum_{\chi\in\mathscr{P}(\mathfrak{q})}\mathcal{M}_{\pi\times(\pi^{\prime}\otimes\chi)}(x)=o(x)$

is equivalent to the fact that for all $\chi\in\mathscr{P}(\mathfrak{q})$, the
$L$-functions $L(s,\pi\times(\pi^{\prime}\otimes\chi))$ do not vanish on the
line $\mathrm{Re}(s)\geq 1$.

Unconditionally, Brumley’s narrow zero-free region (1.4) suffices to prove
that there exist effectively computable constants
$\Cl[abcon]{Brumley_{P}NTAP1},\Cl[abcon]{Brumley_{P}NTAP2}\in(0,1)$ depending
at most on $n$, $n^{\prime}$, and $[F:\mathbb{Q}]$ such that
$\mathcal{E}_{\pi\times\pi^{\prime}}(x;\mathfrak{q},\mathcal{C})\ll_{\pi,\pi^{\prime}}x(\log
x)^{-\Cr{Brumley_{P}NTAP1}}$ as long as $\mathrm{N}\mathfrak{q}\leq(\log
x)^{\Cr{Brumley_{P}NTAP2}}$. The bound
$\mathcal{E}_{\pi\times\pi^{\prime}}(x;\mathfrak{q},\mathcal{C})\ll_{\pi,\pi^{\prime},A}x(\log
x)^{-A}$ for all $A>0$ holds in very limited cases, depending on the zero-free
regions proved in [7, 8, 9, 14]. Theorem 1.1 permits a significant
improvement.

######  Theorem 1.2.

Let $\pi\in\mathfrak{F}_{n}$, $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$,
$x\geq 2$, and $A>0$. For a nonzero ideal $\mathfrak{q}$ of $\mathcal{O}_{F}$,
let $\mathrm{Cl}(\mathfrak{q})$ be the associated ray class group, and let
$\mathcal{C}\in\mathrm{Cl}(\mathfrak{q})$. If
$\mathcal{E}_{\pi\times\pi^{\prime}}(x;\mathfrak{q},\mathcal{C})$ is given by
(1.8) and $\mathrm{N}\mathfrak{q}\leq(\log x)^{A}$, then

$\mathcal{E}_{\pi\times\pi^{\prime}}(x;\mathfrak{q},\mathcal{C})\ll_{\pi,\pi^{\prime},A}x(\log
x)^{-A}.$

The implied constant is ineffective.

######  Remark.

Let $q\geq 1$ be an integer. If $F=\mathbb{Q}$, $\mathfrak{q}=(q)$, and $q$ is
coprime to the conductors of $\pi$ and $\pi^{\prime}$, then Theorem 1.2 states
that if $a\in\mathbb{Z}$ and $\gcd(a,q)=1$, then

$\sum_{\begin{subarray}{c}m\leq x\\\ m\equiv
a\,(\mathrm{mod}\,\,q)\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(m)=\frac{\mathcal{M}_{\pi\times\pi^{\prime}}(x)}{\varphi(q)}+O_{\pi,\pi^{\prime},A}\left(\frac{x}{(\log
x)^{A}}\right),\qquad q\leq(\log x)^{A},$

where $\varphi(q)$ is Euler’s totient function. This generalizes the
Siegel–Walfisz theorem [25, 27].

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

Let $F$ be a number field with adele ring $\mathbb{A}_{F}$. Let $D_{F}$ be the
absolute discriminant of $F$, $\mathcal{O}_{F}$ the ring of integers of $F$,
and $\mathrm{N}=\mathrm{N}_{F/\mathbb{Q}}$ the norm defined on nonzero ideals
$\mathfrak{a}$ of $\mathcal{O}_{F}$ by
$\mathrm{N}\mathfrak{a}=|\mathcal{O}_{F}/\mathfrak{a}|$. For a place $v$ of
$F$, let $v\mid\infty$ (resp. $v\nmid\infty$) denote that $v$ is archimedean
(resp. non-archimedean), and let $F_{v}$ be the corresponding completion of
$F$. Each $v\nmid\infty$ corresponds with a prime ideal $\mathfrak{p}$ of
$\mathcal{O}_{F}$.

###  2.1. Standard $L$-functions

Let $\mathfrak{F}_{n}$ be the set of cuspidal automorphic representations
$\pi$ of $\mathrm{GL}_{n}(\mathbb{A}_{F})$ whose central character
$\omega_{\pi}$ is unitary, and let $\mathfrak{F}_{n}^{*}$ be the subset of
those $\pi$’s for which $\omega_{\pi}$ is trivial on the diagonally embedded
positive reals. Every $\pi\in\mathfrak{F}_{n}$ can be written uniquely as

(2.1)
$\pi=\pi^{*}\otimes|\det|^{it_{\pi}},\qquad\pi^{*}\in\mathfrak{F}_{n}^{*},\quad
t_{\pi}\in\mathbb{R}.$

If $\pi\in\mathfrak{F}_{n}^{*}$, then for each place $v$, there exists an
irreducible admissible representation $\pi_{v}$ of $\mathrm{GL}_{n}(F_{v})$,
with $\pi_{v}$ ramified for at most finitely many $v$, such that $\pi$ is a
restricted tensor product $\otimes_{v}\pi_{v}$. When $v\nmid\infty$ and
$\mathfrak{p}$ corresponds with $v$, then we write $\pi_{v}$ and
$\pi_{\mathfrak{p}}$ interchangeably. For each prime ideal $\mathfrak{p}$,
there exist $n$ Satake parameters $(\alpha_{j,\pi}(\mathfrak{p}))_{j=1}^{n}$
such that the standard $L$-function $L(s,\pi)$ of $\pi$ is the absolutely
convergent Euler product

$L(s,\pi)=\prod_{\mathfrak{p}}L(s,\pi_{\mathfrak{p}})=\prod_{\mathfrak{p}}\prod_{j=1}^{n}\frac{1}{1-\alpha_{j,\pi}(\mathfrak{p})\mathrm{N}\mathfrak{p}^{-s}},\qquad\mathrm{Re}(s)>1.$

For $v\mid\infty$, define

$\Gamma_{v}(s)=\begin{cases}\pi^{-s/2}\Gamma(s/2)&\mbox{if
$F_{v}=\mathbb{R}$,}\\\ 2(2\pi)^{-s}\Gamma(s)&\mbox{if
$F_{v}=\mathbb{C}$.}\end{cases}$

There exist $n$ Langlands parameters $(\mu_{j,\pi}(v))_{j=1}^{n}$ such that

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

Let $\mathfrak{q}_{\pi}$ denote the conductor of $\pi$, and let
$\mathbbm{1}\in\mathfrak{F}_{1}^{*}$ denote the trivial representation (whose
$L$-function is the Dedekind zeta function $\zeta_{F}(s)$). Let
$\delta_{\pi}=1$ if $\pi=\mathbbm{1}$, and $\delta_{\pi}=0$ otherwise. If

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

then the completed $L$-function

$\Lambda(s,\pi)=(s(1-s))^{\delta_{\pi}}(D_{F}^{n}\mathrm{N}\mathfrak{q}_{\pi})^{s/2}L_{\infty}(s,\pi)L(s,\pi)$

is entire of order $1$. There exists $W(\pi)\in\mathbb{C}$ of modulus $1$ such
that $\Lambda(s,\pi)$ satisfies the functional equation

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

The nontrivial zeros of $L(s,\pi)$ are the zeros of $\Lambda(s,\pi)$, and they
lie in the critical strip $0<\mathrm{Re}(s)<1$. It is conjectured (GRH) that
they actually lie on the critical line $\mathrm{Re}(s)=1/2$. Finally, the
analytic conductor is $C(\pi)=C(0,\pi)$, where

$C(it,\pi)=D_{F}^{n}\mathrm{N}\mathfrak{q}_{\pi}\prod_{v\mid\infty}\prod_{j=1}^{n}(|\mu_{j,\pi}(v)+it|+3)^{[F_{v}:\mathbb{R}]}.$

More generally, these quantities are defined for all $\pi\in\mathfrak{F}_{n}$,
and we have in particular

$L(s,\pi)=L(s+it_{\pi},\pi^{*}),\qquad C(it,\pi)=C(it+it_{\pi},\pi^{*}).$

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

Let $\pi\in\mathfrak{F}_{n}^{*}$ and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}^{*}$. For each
$\mathfrak{p}\mid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$, there exist
complex numbers $\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})$
with $1\leq j\leq n$ and $1\leq j^{\prime}\leq n$ such that if

$L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}^{\prime})=\begin{cases}\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}(1-\alpha_{j,\pi}(\mathfrak{p})\alpha_{j^{\prime},\pi^{\prime}}(\mathfrak{p})\mathrm{N}\mathfrak{p}^{-s})^{-1}&\mbox{if
$\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$,}\\\
\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}&\mbox{if
$\mathfrak{p}\mid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$,}\end{cases}$

then the Rankin–Selberg $L$-function $L(s,\pi\times\pi^{\prime})$ equals the
absolutely convergent product

$L(s,\pi\times\pi^{\prime})=\prod_{\mathfrak{p}}L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}^{\prime})=\sum_{\mathfrak{a}}\frac{\lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})}{\mathrm{N}\mathfrak{a}^{s}},\qquad\mathrm{Re}(s)>1.$

The absolute convergence for $\mathrm{Re}(s)>1$ ensures that

(2.2)
$|\alpha_{j,\pi}(\mathfrak{p})\alpha_{j^{\prime},\pi^{\prime}}(\mathfrak{p})|\leq\mathrm{N}\mathfrak{p},\qquad|\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})|\leq\mathrm{N}\mathfrak{p}.$

By [15, Lemma 3.1], we have the bound

(2.3)
$|\lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})|\leq\sqrt{\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{a})\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{a})}.$

Let $\mathfrak{q}_{\pi\times\pi^{\prime}}$ be the conductor of
$L(s,\pi\times\pi^{\prime})$. For each $v\mid\infty$, $1\leq j\leq n$, and
$1\leq j^{\prime}\leq n^{\prime}$, there exists a Langlands parameter
$\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)$ such that if

(2.4)
$L_{\infty}(s,\pi\times\pi^{\prime})=\prod_{v\mid\infty}\prod_{j=1}^{n}\prod_{j=1}^{n^{\prime}}\Gamma_{v}(s+\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)),\qquad\delta_{\pi\times\pi^{\prime}}=\begin{cases}1&\mbox{if
$\pi^{\prime}=\widetilde{\pi}$,}\\\ 0&\mbox{otherwise,}\end{cases}$

then the completed $L$-function

(2.5)
$\Lambda(s,\pi\times\pi^{\prime})=(s(1-s))^{\delta_{\pi\times\pi^{\prime}}}(D_{F}^{n^{\prime}n}\mathrm{N}\mathfrak{q}_{\pi\times\pi^{\prime}})^{s/2}L_{\infty}(s,\pi\times\pi^{\prime})L(s,\pi\times\pi^{\prime})$

is an entire function of order 1. There exists
$W(\pi\times\pi^{\prime})\in\mathbb{C}$ of modulus $1$ such that
$\Lambda(s,\pi\times\pi^{\prime})$ satisfies the functional equation

(2.6)
$\Lambda(s,\pi\times\pi^{\prime})=W(\pi\times\pi^{\prime})\Lambda(1-s,\widetilde{\pi}\times\widetilde{\pi}^{\prime})=W(\pi\times\pi^{\prime})\overline{\Lambda(1-\overline{s},\pi\times\pi^{\prime})}.$

The absolute convergence of the Euler product representation of
$L(s,\pi\times\pi^{\prime})$ ensures that
$\mathrm{Re}(\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v))\geq-1$. The
nontrivial zeros of $L(s,\pi\times\pi^{\prime})$ are the zeros of
$\Lambda(s,\pi\times\pi^{\prime})$, and they lie in the critical strip
$0<\mathrm{Re}(s)<1$. It is conjectured (GRH) that they actually lie on the
critical line $\mathrm{Re}(s)=1/2$. Finally, the analytic conductor is
$C(\pi\times\pi^{\prime})=C(0,\pi\times\pi^{\prime})$, where

(2.7)
$C(it,\pi\times\pi^{\prime})=D_{F}^{n^{\prime}n}\mathrm{N}\mathfrak{q}_{\pi\times\pi^{\prime}}\prod_{v\mid\infty}\prod_{j=1}^{n}\prod_{j=1}^{n^{\prime}}(|\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)+it|+3)^{[F_{v}:\mathbb{R}]}.$

We have the bounds [7, Lemma A.2]

(2.8)
$C(it,\pi\times\pi^{\prime})\ll_{n,n^{\prime}}C(\pi\times\pi^{\prime})(|t|+1)^{n^{\prime}n[F:\mathbb{Q}]}\ll_{n,n^{\prime}}C(\pi)^{n^{\prime}}C(\pi^{\prime})^{n}(|t|+1)^{n^{\prime}n[F:\mathbb{Q}]}.$

More generally, for $\pi\in\mathfrak{F}_{n}$ and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$, we have

$L(s,\pi\times\pi^{\prime})=L(s+it_{\pi}+it_{\pi^{\prime}},\pi^{*}\times\pi^{\prime*}),\qquad
C(it,\pi\times\pi^{\prime})=C(it+it_{\pi}+it_{\pi^{\prime}},\pi^{*}\times\pi^{\prime*}).$

###  2.3. Local bounds

Let $\pi\in\mathfrak{F}_{n}$. At each prime ideal $\mathfrak{p}$ of
$\mathcal{O}_{F}$ and each $v\mid\infty$, there exists
$\theta_{n}\in[0,1/2-1/(n^{2}+1)]$ such that [19, 21]

(2.9)
$|\alpha_{j,\pi}(\mathfrak{p})|\leq\mathrm{N}\mathfrak{p}^{\theta_{n}},\qquad\mathrm{Re}(\mu_{j,\pi}(v))\geq-\theta_{n}.$

Using the explicit descriptions of
$\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})$ in [26, Appendix]
and $\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})$ in [21, Section
3], we find that if $\pi\in\mathfrak{F}_{n}$ and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$, then

(2.10)
$|\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})|\leq\mathrm{N}\mathfrak{p}^{\theta_{n}+\theta_{n^{\prime}}},\qquad\mathrm{Re}(\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v))\geq-\theta_{n}-\theta_{n^{\prime}}.$

###  2.4. Isobaric sums

The Langlands theory of Eisenstein series associates to any $r$-tuple
$(\pi_{1},\dotsc,\pi_{r})\in\mathfrak{F}_{n_{1}}\times\dotsb\times\mathfrak{F}_{n_{r}}$
an automorphic representation of
$\mathrm{GL}_{n_{1}+\dotsb+n_{r}}(\mathbb{A}_{F})$, the isobaric sum, denoted
$\Pi=\pi_{1}\boxplus\dotsb\boxplus\pi_{r}$. The contragredient is
$\widetilde{\Pi}=\widetilde{\pi}_{1}\boxplus\dotsb\boxplus\widetilde{\pi}_{r}$,
and $L(s,\Pi)=\prod_{j=1}^{r}L(s,\pi_{j})$.

######  Lemma 2.1 ([6, Lemma a]).

If $\Pi$ is an isobaric sum, then the Rankin–Selberg $L$-function

$L(s,\Pi\times\widetilde{\Pi})=\prod_{j=1}^{r}\prod_{k=1}^{r}L(s,\pi_{j}\times\widetilde{\pi}_{k})=\sum_{\mathfrak{a}}\frac{\lambda_{\Pi\times\widetilde{\Pi}}(\mathfrak{a})}{\mathrm{N}\mathfrak{a}^{s}},\qquad\mathrm{Re}(s)>1$

has nonnegative Dirichlet coefficients, as does its logarithm.

###  2.5. Convexity

Our proofs require strong bounds for Rankin–Selberg $L$-functions and their
derivatives. We state our next lemma for $\pi\in\mathfrak{F}_{n}^{*}$ and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}^{*}$, and this suffices in light of
the discussion in Section 2.2.

######  Lemma 2.2.

For $\pi\in\mathfrak{F}_{n}^{*}$ and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}^{*}$, consider the holomorphic
function

$\mathcal{L}(s,\pi\times\pi^{\prime})=\left(\frac{s-1}{s+1}\right)^{\delta_{\pi\times\pi^{\prime}}}L(s,\pi\times\pi^{\prime}),\qquad\mathrm{Re}(s)>-1.$

If $j\geq 0$, $\sigma\geq 0$, $t\in\mathbb{R}$, and $\varepsilon>0$, then

$\mathcal{L}^{(j)}(\sigma+it,\pi\times\pi^{\prime})\ll_{n,n^{\prime},[F:\mathbb{Q}],j,\varepsilon}C(it,\pi\times\pi^{\prime})^{\max(1-\sigma,0)/2+\varepsilon}.$

In particular, if $\delta_{\pi\times\pi^{\prime}}=0$ or $|t|>1$, then

$L^{(j)}(\sigma+it,\pi\times\pi^{\prime})\ll_{n,n^{\prime},[F:\mathbb{Q}],j,\varepsilon}C(it,\pi\times\pi^{\prime})^{\max(1-\sigma,0)/2+\varepsilon}.$

###### Proof.

Without loss of generality, $0<\varepsilon<1$. It suffices to show for all
$\sigma\geq-\varepsilon/2$ that

(2.11)
$\mathcal{L}(\sigma+it,\pi\times\pi^{\prime})\ll_{n,n^{\prime},[F:\mathbb{Q}],\varepsilon}C(it,\pi\times\pi^{\prime})^{\max(1-\sigma,0)/2+3\varepsilon/4}.$

Indeed, assuming this statement and using the bound

$|\mathcal{L}^{(j)}(s,\pi\times\pi^{\prime})|\leq
j!(2/\varepsilon)^{j}\sup_{|z-s|=\varepsilon/2}|\mathcal{L}(z,\pi\times\pi^{\prime})|$

that follows from Cauchy’s formula and the triangle inequality for complex
line integrals, we readily obtain the required conclusion (for all $j\geq 0$,
$\sigma\geq 0$, and $t\in\mathbb{R}$).

It remains to prove (2.11) for all $\sigma\geq-\varepsilon/2$. Assume first
that $\sigma\geq 1+\varepsilon/2$. It follows from (2.3) and the
Cauchy–Schwarz inequality that

$|L(\sigma+it,\pi\times\pi^{\prime})|\leq(L(\sigma,\pi\times\widetilde{\pi})L(\sigma,\pi^{\prime}\times\widetilde{\pi}^{\prime}))^{1/2}.$

The right-hand side is decreasing in $\sigma$, hence by [17, Theorem 2] we
infer that

$\log|L(\sigma+it,\pi\times\pi^{\prime})|\ll_{n,n^{\prime},[F:\mathbb{Q}],\varepsilon}\frac{\log
C(\pi\times\widetilde{\pi})}{\log\log C(\pi\times\widetilde{\pi})}+\frac{\log
C(\pi^{\prime}\times\widetilde{\pi}^{\prime})}{\log\log
C(\pi^{\prime}\times\widetilde{\pi}^{\prime})}.$

Then, using [26, Lemma 2.1], we conclude that

(2.12)
$L(\sigma+it,\pi\times\pi^{\prime})\ll_{n,n^{\prime},[F:\mathbb{Q}],\varepsilon}C(it,\pi\times\pi^{\prime})^{\varepsilon/4}.$

Concerning the application of [26, Lemma 2.1], three remarks are in order.
First, this result is meant for $F=\mathbb{Q}$, but a generalization to an
arbitrary number field $F$ is straightforward (with extra constants depending
on $[F:\mathbb{Q}]$ in the exponents). Second, [26] tacitly assumes that each
central character is trivial on the diagonally embedded positive reals, but
this assumption is not used in the proof of [26, Lemma 2.1]. Third, we rely on
the fact that $C(it,\pi\times\pi^{\prime})$ in (2.12) equals
$C(\pi\times(\pi^{\prime}\otimes|\det|^{it}))$.

Finally, we verify (2.11) in the strip
$-\varepsilon/2\leq\sigma<1+\varepsilon/2$. We express the boundary value
$L(-\varepsilon/2+it,\pi\times\widetilde{\pi})$ from
$L(1+\varepsilon/2-it,\pi\times\widetilde{\pi})$ using the functional equation
(2.6) and the definitions (2.4)–(2.5). Then, applying [4, Lemma 3.2] and
recalling (2.7), we obtain from (2.12) the bound

(2.13)
$L(-\varepsilon/2+it,\pi\times\pi^{\prime})\ll_{n,n^{\prime},[F:\mathbb{Q}],\varepsilon}C(it,\pi\times\pi^{\prime})^{1/2+3\varepsilon/4}.$

We interpolate between (2.12) and (2.13) using the Phragmén–Lindelöf
principle, and the lemma follows. ∎

###  2.6. A weak Brun–Titchmarsh bound

Let $\pi\in\mathfrak{F}_{n}$ and $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$.
Recall the Dirichlet series definition of
$\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})$ in (1.7).

######  Lemma 2.3.

Let $(\pi,\pi^{\prime})\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}$.
There exists an effectively computable constant
$\Cl[abcon]{BTrange}=\Cr{BTrange}(n,n^{\prime},[F:\mathbb{Q}])>0$ such that if

$X\geq\max(C(\pi\times\widetilde{\pi}),C(\pi^{\prime}\times\widetilde{\pi}^{\prime}))^{\Cr{BTrange}}\qquad\text{and}\qquad
1\leq T\leq X^{1/(16\max(n,n^{\prime})^{2}[F:\mathbb{Q}])},$

then

(2.14) $\sum_{X<\mathrm{N}\mathfrak{a}\leq
Xe^{1/T}}|\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})|\ll_{n,n^{\prime},[F:\mathbb{Q}]}\frac{X}{T}.$

###### Proof.

By [26, Proposition A.1], we have the bound

(2.15)
$|\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})|\leq\sqrt{\Lambda_{\pi\times\widetilde{\pi}}(\mathfrak{a})\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{a})}.$

Combined with the Cauchy–Schwarz inequality, (2.15) yields

(2.16) $\sum_{X<\mathrm{N}\mathfrak{a}\leq
Xe^{1/T}}|\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})|\leq\left(\sum_{X<\mathrm{N}\mathfrak{a}\leq
Xe^{1/T}}\Lambda_{\pi\times\widetilde{\pi}}(\mathfrak{a})\right)^{1/2}\left(\sum_{X<\mathrm{N}\mathfrak{a}\leq
Xe^{1/T}}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{a})\right)^{1/2}.$

By [9, Proposition 4.1], there exist absolute and effectively computable
constants $\Cl[abcon]{BT1}>0$ and $\Cl[abcon]{BT2}>0$ such that if $\log\log
C(\pi\times\widetilde{\pi})\geq\Cr{BT1}n^{4}[F:\mathbb{Q}]^{2}$, $X\geq
e^{\Cr{BT2}n^{4}[F:\mathbb{Q}]^{2}}C(\pi\times\widetilde{\pi})^{32n^{2}[F:\mathbb{Q}]}$,
and $1\leq T\leq X^{1/(16n^{2}[F:\mathbb{Q}])}$, then

$\sum_{X<\mathrm{N}\mathfrak{a}\leq
Xe^{1/T}}\Lambda_{\pi\times\widetilde{\pi}}(\mathfrak{a})\ll
n^{2}[F:\mathbb{Q}]\frac{X}{T}.$

A careful inspection of the proof of [9, Proposition 4.1] shows if the lower
bound on $C(\pi\times\widetilde{\pi})$ is removed, then the dependency on $n$
and $[F:\mathbb{Q}]$ in the implied constant and the range of $X$ become
worse, which is not a concern here. We conclude that there exists an
effectively computable constant
$\Cr{BTrange}=\Cr{BTrange}(n,[F:\mathbb{Q}])>0$ such that if $X\geq
C(\pi\times\widetilde{\pi})^{\Cr{BTrange}}$ and $1\leq T\leq
X^{1/(16n^{2}[F:\mathbb{Q}])}$, then

$\sum_{X<\mathrm{N}\mathfrak{a}\leq
Xe^{1/T}}\Lambda_{\pi\times\widetilde{\pi}}(\mathfrak{a})\ll_{n,[F:\mathbb{Q}]}\frac{X}{T}.$

We similarly bound the second factor on the right-hand side of (2.16), and
(2.14) follows. ∎

######  Corollary 2.4.

Let
$(\pi,\pi^{\prime},\chi)\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}\times\mathfrak{F}_{1}$.
If

$X\geq 1\qquad\text{and}\qquad X\geq Y\geq
X^{1-1/(16\max(n,n^{\prime})^{2}[F:\mathbb{Q}])},$

then

(2.17) $\sum_{X<\mathrm{N}\mathfrak{a}\leq
X+Y}|\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(\mathfrak{a})|\ll_{\pi,\pi^{\prime}}Y.$

###### Proof.

We define
$X_{0}=\max(C(\pi\times\widetilde{\pi}),C(\pi^{\prime}\times\widetilde{\pi}^{\prime}))^{\Cr{BTrange}}$.
If $X\geq X_{0}$, then (2.17) follows from Lemma 2.3 upon setting $T=X/Y$. If
$X<X_{0}$, then (2.17) follows from (2.15) and the Cauchy–Schwarz inequality.
∎

##  3\. Preliminaries

We begin with a finiteness statement on $\mathrm{GL}_{1}$-twists.

######  Lemma 3.1.

Let $\pi\in\mathfrak{F}_{n}$ and $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$.
There are finitely many $\chi\in\mathfrak{F}_{1}^{*}$ such that
$L(s,\pi\times(\pi^{\prime}\otimes\chi))$ has a pole.

###### Proof.

Without loss of generality, $\pi\in\mathfrak{F}_{n}^{*}$ and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}^{*}$. Let
$\chi\in\mathfrak{F}_{1}^{*}$ be such that
$L(s,\pi\times(\pi^{\prime}\otimes\chi))$ has a pole. Then
$\pi^{\prime}\otimes\chi=\widetilde{\pi}$, hence $n=n^{\prime}$, and $\chi$
can only ramify where $\pi$ or $\pi^{\prime}$ ramifies. Moreover, $\chi^{n}$
is uniquely determined by the equation
$\omega_{\pi^{\prime}}\chi^{n}=\overline{\omega_{\pi}}$. Since
$F_{v}^{\times}/{F_{v}^{\times}}^{n}$ is finite for every place $v$ of $F$,
the result follows. ∎

We proceed with two lemmata on residues.

######  Lemma 3.2.

Let $f(s)$, $g(s)$, $h(s)$ be holomorphic in an open neighborhood of
$s_{0}\in\mathbb{C}$. Assume that $|f(s_{0})|=|g(s_{0})|$. Then, for any
integer $k\geq 1$, the residue

$\mathop{\mathrm{Res}}_{s=s_{0}}\frac{(f(s)g(s))^{k}}{(s-s_{0})^{2k}}h(s)$

equals $|f(s_{0})|=|g(s_{0})|$ times a $\mathbb{C}$-linear combination of
monomials of the derivative values $f^{(\ell)}(s_{0})$, $g^{(\ell)}(s_{0})$,
$h^{(\ell)}(s_{0})$ for $\ell\in\\{0,\dotsc,2k-1\\}$. The monomials in the
linear combination, and the modulus of each coefficient in the linear
combination depend at most on $k$.

###### Proof.

Expanding $f(s)$, $g(s)$, $h(s)$ into Taylor series around $s_{0}$, we obtain

$\mathop{\mathrm{Res}}_{s=s_{0}}\frac{(f(s)g(s))^{k}}{(s-s_{0})^{2k}}h(s)=\sum_{\begin{subarray}{c}\ell_{1},\dotsc,\ell_{2k+1}\geq
0\\\
\ell_{1}+\dotsb+\ell_{2k+1}=2k-1\end{subarray}}\frac{h^{(\ell_{2k+1})}(s_{0})}{\ell_{1}!\dotsb\ell_{2k+1}!}\prod_{j=1}^{k}f^{(\ell_{j})}(s_{0})g^{(\ell_{j+k})}(s_{0}).$

Since $\ell_{1}+\dotsb+\ell_{2k+1}=2k-1$, at least one of
$\ell_{1},\ldots,\ell_{2k}$ equals zero. Consequently, the $j$-product above
has a factor of $f(s_{0})$ or $g(s_{0})$, and the result follows. ∎

We apply Lemma 3.2 to study the residues of an auxiliary $L$-function. For
$(\pi,\pi^{\prime},\chi)\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}\times\mathfrak{F}_{1}$,
we define

$\Pi=\pi\boxplus\pi\otimes\chi\boxplus\widetilde{\pi}^{\prime}\boxplus\widetilde{\pi}^{\prime}\otimes\overline{\chi}\qquad\text{and}\qquad
D(s)=L(s,\Pi\times\widetilde{\Pi}).$

We have the factorization

(3.1) $\displaystyle D(s)=$
$\displaystyle~{}L(s,\pi\times\widetilde{\pi})^{2}L(s,\pi^{\prime}\times\widetilde{\pi}^{\prime})^{2}L(s,\pi\times(\pi^{\prime}\otimes\chi))^{2}L(s,\widetilde{\pi}\times(\widetilde{\pi}^{\prime}\otimes\overline{\chi}))^{2}$
$\displaystyle\cdot
L(s,\pi\times(\widetilde{\pi}\otimes\chi))L(s,\pi^{\prime}\times(\widetilde{\pi}^{\prime}\otimes\chi))L(s,\widetilde{\pi}\times\widetilde{\pi}^{\prime})L(s,\pi\times(\pi^{\prime}\otimes\chi^{2}))$
$\displaystyle\cdot
L(s,\pi\times(\widetilde{\pi}\otimes\overline{\chi}))L(s,\pi^{\prime}\times(\widetilde{\pi}^{\prime}\otimes\overline{\chi}))L(s,\pi\times\pi^{\prime})L(s,\widetilde{\pi}\times(\widetilde{\pi}^{\prime}\otimes\overline{\chi}^{2})).$

Let

(3.2)
$Q=(C(\pi)C(\pi^{\prime}))^{2(n+n^{\prime})}C(\chi)^{(n+n^{\prime})^{2}}.$

######  Lemma 3.3.

Let
$(\pi,\pi^{\prime},\chi)\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}\times\mathfrak{F}_{1}$,
$x>1$, $\varepsilon\in(0,1)$, and $\beta\in(1-\varepsilon/2,1)$. Recall the
notations (2.1), (3.1), (3.2), and let $\mathcal{S}$ be the set of poles of
$D(s)$. Assume that the following $L$-functions are entire:

(3.3) $L(s,\pi\times\pi^{\prime}),\qquad
L(s,\pi\times(\pi^{\prime}\otimes\chi)),\qquad
L(s,\pi\times(\pi^{\prime}\otimes\chi^{2})).$

If $\chi^{*}$ is trivial, then assume also that $|t_{\chi}|>1$. We have that

$\sum_{s_{0}\in\mathcal{S}}\mathop{\mathrm{Res}}_{s=s_{0}}D(s)x^{s-\beta}\Gamma(s-\beta)\ll_{n,n^{\prime},[F:\mathbb{Q}],\beta,\varepsilon}|L(1,\pi\times(\pi^{\prime}\otimes\chi))|(Qx)^{\varepsilon}.$

###### Proof.

First, assume that $\chi^{*}$ is trivial, in which case
$\chi=|\det|^{it_{\chi}}$ with $|t_{\chi}|>1$. The initial assumptions imply
that $\mathcal{S}=\\{1,1-it_{\chi},1+it_{\chi}\\}$. For each choice of
$s_{0}\in\mathcal{S}$, the order of the pole of $D(s)$ at $s=s_{0}$ is even,
say $2k$. Specifically, $k=2$ for $s_{0}=1$, and $k=1$ for $s_{0}=1\pm
it_{\chi}$. We consider the decomposition

$(s-s_{0})^{2k}D(s)x^{s-\beta}\Gamma(s-\beta)=(f(s)g(s))^{k}h(s),$

where

  * •

$f(s)=L(s+it_{\chi},\pi\times\pi^{\prime})$ and
$g(s)=L(s-it_{\chi},\widetilde{\pi}\times\widetilde{\pi}^{\prime})$ for
$s_{0}=1$;

  * •

$f(s)=L(s,\widetilde{\pi}\times\widetilde{\pi}^{\prime})$ and
$g(s)=L(s+2it_{\chi},\pi\times\pi^{\prime})$ for $s_{0}=1-it_{\chi}$;

  * •

$f(s)=L(s,\pi\times\pi^{\prime})$ and
$g(s)=L(s-2it_{\chi},\widetilde{\pi}\times\widetilde{\pi}^{\prime})$ for
$s_{0}=1+it_{\chi}$.

These three cases correspond to the three lines in (3.1), with $f(s)^{k}$ and
$g(s)^{k}$ occurring on the relevant line.

Now we apply Lemma 3.2 in conjunction with Lemma 2.2 and (2.8). The functions
$f(s)$, $g(s)$, $h(s)$ defined above are holomorphic in the open disk
$|s-s_{0}|<1-\beta$. Moreover,

$|f(s_{0})|=|g(s_{0})|=|L(1+it_{\chi},\pi\times\pi^{\prime})|.$

We are finished upon noting that, at the point $s_{0}$, the $\ell$-th
derivative of $s\mapsto x^{s-\beta}$ is bounded by $x^{\varepsilon/2}(\log
x)^{\ell}$, while

$\left|\Gamma^{(\ell)}(s_{0}-\beta)\right|\leq\int_{0}^{\infty}r^{-\beta}|\log
r|^{\ell}e^{-r}\,dr.$

Second, assume that $\chi^{*}$ is nontrivial. The initial assumptions imply
that $\mathcal{S}=\\{1\\}$, and the order of the pole at $s=1$ is 4. We
consider the decomposition

$(s-1)^{4}D(s)x^{s-\beta}\Gamma(s-\beta)=(f(s)g(s))^{2}h(s),$

where

$f(s)=L(s,\pi\times(\pi^{\prime}\otimes\chi))\quad\textup{and}\quad
g(s)=L(s,\widetilde{\pi}\times(\widetilde{\pi}^{\prime}\otimes\overline{\chi})).$

Now we apply Lemma 3.2 in conjunction with Lemma 2.2 and (2.8), as before. ∎

We use Lemmata 2.1, 2.2, and 3.3 to prove the following conditional lower
bound on $L(1,\pi\times(\pi^{\prime}\otimes\chi))$ that will play a central
role in our proof of Theorem 1.1.

######  Proposition 3.4.

Let
$(\pi,\pi^{\prime},\chi)\in\mathfrak{F}_{n}\times\mathfrak{F}_{n^{\prime}}\times\mathfrak{F}_{1}$,
$\varepsilon\in(0,1/2)$, and $\beta\in(1-\varepsilon/8,1)$. Assume that the
$L$-functions under (3.3) are entire. If $L(\beta,\pi\times\pi^{\prime})=0$,
then

(3.4)
$|L(1,\pi\times(\pi^{\prime}\otimes\chi))|\gg_{\pi,\pi^{\prime},\beta,\varepsilon}C(\chi)^{-(n+n^{\prime})^{2}\varepsilon}.$

###### Proof.

Recall the notations (2.1) and (3.2). If $\chi^{*}$ is trivial and
$|t_{\chi}|\leq 1$, then (3.4) holds because both sides are positive
continuous functions of $t_{\chi}\in\mathbb{R}$. Henceforth, we may assume
that $\chi^{*}$ is nontrivial or $|t_{\chi}|>1$. Subject to these constraints,
we let $D(s)$ be as in (3.1) and $\mathcal{S}$ be the set of its poles. Let
$x>1$ be a parameter to be chosen later. By the initial assumptions,
$D(\beta)=0$. Hence $\mathcal{S}$ is also the set of poles of
$D(s)x^{s-\beta}\Gamma(s-\beta)$ in the half-plane $\mathrm{Re}(s)>0$, and we
shall use this below. If $\lambda_{D}(\mathfrak{a})$ is the $\mathfrak{a}$-th
Dirichlet coefficient of $D(s)$, then $\lambda_{D}(\mathfrak{a})\geq 0$ by
Lemma 2.1. Since $\lambda_{D}(\mathcal{O}_{F})=1$, we have by the residue
theorem

$\displaystyle\frac{1}{e}\leq\sum_{\mathfrak{a}}\frac{\lambda_{D}(\mathfrak{a})}{\mathrm{N}\mathfrak{a}^{\beta}}e^{-\frac{\mathrm{N}\mathfrak{a}}{x}}$
$\displaystyle=\frac{1}{2\pi
i}\int_{1-i\infty}^{1+i\infty}D(s+\beta)x^{s}\Gamma(s)\,ds$
$\displaystyle=\sum_{s_{0}\in\mathcal{S}}\mathop{\mathrm{Res}}_{s=s_{0}}D(s)x^{s-\beta}\Gamma(s-\beta)+\frac{1}{2\pi
i}\int_{1/2-i\infty}^{1/2+i\infty}D(s)x^{s-\beta}\Gamma(s-\beta)\,ds.$

We estimate the sum over $\mathcal{S}$ by Lemma 3.3 (with $\varepsilon$
replaced by $\varepsilon/4$), and the last integral by Lemma 2.2 (with
$\varepsilon$ replaced by $\varepsilon/16$) combined with (2.8) and Stirling’s
formula. We conclude that

$1\ll_{n,n^{\prime},[F:\mathbb{Q}],\beta,\varepsilon}\left(|L(1,\pi\times(\pi^{\prime}\otimes\chi))|+Qx^{-1/2}\right)(Qx)^{\varepsilon/4}.$

At this point we choose
$x=\max\left(1,Q^{2}|L(1,\pi\times(\pi^{\prime}\otimes\chi))|^{-2}\right)$. If
$x=1$, then (3.4) is trivial. Otherwise,
$x=Q^{2}|L(1,\pi\times(\pi^{\prime}\otimes\chi))|^{-2}>1$, and the previous
bound yields (3.4) after solving for
$|L(1,\pi\times(\pi^{\prime}\otimes\chi))|$:

$|L(1,\pi\times(\pi^{\prime}\otimes\chi))|\gg_{n,n^{\prime},[F:\mathbb{Q}],\beta,\varepsilon}Q^{-3\varepsilon/(4-2\varepsilon)}>Q^{-\varepsilon}.\qed$

##  4\. Proof of Theorem 1.1 for $\sigma=1$.

In this section, we prove that there exists an ineffective constant
$\Cl[abcon]{ZFR3}=\Cr{ZFR3}(\pi,\pi^{\prime},\varepsilon)>0$ such that for any
$\chi\in\mathfrak{F}_{1}$ we have

(4.1)
$|L(1,\pi\times(\pi^{\prime}\otimes\chi))|\geq\Cr{ZFR3}C(\chi)^{-\varepsilon}.$

Without loss of generality, we may assume that $\varepsilon\in(0,1/2)$. We
define

$\varepsilon^{\prime}=\frac{\varepsilon}{8(n+n^{\prime})^{2}}.$

###  4.1. An initial reduction

We shall assume that $L(s,\pi\times(\pi^{\prime}\otimes\chi))$ is holomorphic
in the open disk $|s-1|<1$, for otherwise (4.1) is clear. Moreover, if
$L(s,\pi\times(\pi^{\prime}\otimes\chi))$ has no zero in the half-plane
$\mathrm{Re}(s)>1-\varepsilon^{\prime}$, then standard methods produce a bound
much stronger than (4.1). Indeed, assume this zero-free region, and let
$\sigma\in[1,2]$. Let us work with a parameter $x\geq 2$ (to be chosen later)
and the Mellin transform pair

$\phi(r)=\max(1-r,0),\qquad\widehat{\phi}(w)=1/(w^{2}+w).$

Proceeding as in the proof of [10, Proposition 5.16], but shifting the contour
only to $\mathrm{Re}(w)=-\varepsilon^{\prime}/2$, we infer that

$-\frac{L^{\prime}}{L}(\sigma,\pi\times(\pi^{\prime}\otimes\chi))=\sum_{\mathfrak{a}}\frac{\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(\mathfrak{a})}{\mathrm{N}\mathfrak{a}^{\sigma}}\phi\left(\frac{\mathrm{N}\mathfrak{a}}{x}\right)\\\
+O_{\pi,\pi^{\prime},\varepsilon}\left(x^{-\varepsilon^{\prime}/2}\log
C(\chi)\right).$

Since $C(\chi)\geq 3$ and $0<\varepsilon^{\prime}<1/64$, the choice $x=(\log
C(\chi))^{2/\varepsilon^{\prime}}$ satisfies $x\geq 2$ and
$x^{-\varepsilon^{\prime}/2}\log C(\chi)=1$. Integrating the resulting
approximation from $\sigma=1$ to $\sigma=2$, and applying the triangle
inequality, we infer

$\displaystyle|\log L(1,\pi\times(\pi^{\prime}\otimes\chi))|$
$\displaystyle\leq\sum_{2\leq\mathrm{N}\mathfrak{a}\leq
x}\frac{|\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(\mathfrak{a})|}{\mathrm{N}\mathfrak{a}\log\mathrm{N}\mathfrak{a}}+O_{\pi,\pi^{\prime},\varepsilon}(1)$
$\displaystyle\ll_{\pi,\pi^{\prime},\varepsilon}\sum_{k=1}^{\lfloor\log
x\rfloor}\frac{1}{ke^{k}}\sum_{e^{k-1}<\mathrm{N}\mathfrak{a}\leq
e^{k}}|\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(\mathfrak{a})|+1.$

By Corollary 2.4, we conclude the bound

$|\log
L(1,\pi\times(\pi^{\prime}\otimes\chi))|\ll_{\pi,\pi^{\prime},\varepsilon}\log\log
x\ll_{\pi,\pi^{\prime},\varepsilon}\log\log\log C(\chi).$

This is much stronger than (4.1).

###  4.2. Overall strategy.

By the work in Section 4.1, we only need to prove (4.1) for those characters
$\chi\in\mathfrak{F}_{1}$ for which there exists
$(\beta,\gamma)\in(1-\varepsilon^{\prime},1)\times\mathbb{R}$ such that

$L(\beta+i\gamma,\pi\times(\pi^{\prime}\otimes\chi))=0.$

Keeping this in mind, we shall prove (4.1) in three steps. Each step relies on
an application of Proposition 3.4 and allows us to verify (4.1) for a larger
_subgroup_ of characters $\chi\in\mathfrak{F}_{1}$ than before. Let us
introduce the notation

$\mathfrak{F}_{1}^{(j)}=\\{\chi\in\mathfrak{F}_{1}:{\chi^{*}}^{j}=1\\},$

and note the chain of subgroups
$\mathfrak{F}_{1}^{(1)}\leq\mathfrak{F}_{1}^{(2)}\leq\mathfrak{F}_{1}$.

In the first step, we prove (4.1) for $\chi\in\mathfrak{F}_{1}^{(1)}$. In the
second step, we extend the validity of (4.1) to
$\chi\in\mathfrak{F}_{1}^{(2)}$. In the third step, we establish (4.1) in full
generality. For the discussion below, let us assume that
$L(s,\pi\times(\pi^{\prime}\otimes\chi))$ is entire. At each application of
Proposition 3.4, we rely on the “existence of bad zero” hypothesis that we
paraphrase as follows. For the character $\chi\in\mathfrak{F}_{1}$ under
consideration, there exists
$(\beta,\eta)\in(1-\varepsilon^{\prime},1)\times\mathfrak{F}_{1}^{(1)}$ such
that

$L(\beta,\pi\times(\pi^{\prime}\otimes\chi\eta))=0.$

However, we cannot use the pair $(\beta,\eta)$ directly because the implied
constants are not allowed to depend on $\chi$. Instead, we use that each step
operates with a given subgroup
$G\in\\{\mathfrak{F}_{1}^{(1)},\mathfrak{F}_{1}^{(2)},\mathfrak{F}_{1}\\}$,
and by the above hypothesis there exists
$(\beta,\psi)\in(1-\varepsilon^{\prime},1)\times G$ such that

(4.2) $L(s,\pi\times(\pi^{\prime}\otimes\psi))$ is entire and vanishes at
$s=\beta$.

This _weaker_ hypothesis follows from the previous one (upon setting
$\psi=\chi\eta$). Its advantage lies in the observation that since $G$ is
fixed, we can select an admissible pair $(\beta,\psi)$ solely in terms of the
triple $(\pi,\pi^{\prime},\varepsilon)$. Such selection is possible via the
axiom of choice or a more constructive approach based on some well-ordering of
the admissible pairs.

###  4.3. The case of $\chi^{*}$ trivial.

In this subsection, we prove (4.1) for all $\chi$ lying in the subgroup
$G=\mathfrak{F}_{1}^{(1)}$. Let us write $\chi=|\det|^{it}\in G$ with
$t\in\mathbb{R}$. If $L(s,\pi\times\pi^{\prime})$ has a pole, then there
exists $u\in\mathbb{R}$ such that

$\pi=\widetilde{\pi}\otimes|\det|^{iu}\qquad\textup{or}\qquad\pi^{\prime}=\widetilde{\pi}^{\prime}\otimes|\det|^{iu}\qquad\textup{or}\qquad\pi^{\prime}=\widetilde{\pi}\otimes|\det|^{iu},$

hence (4.1) holds in the stronger form

$L(1+it,\pi\times\pi^{\prime})\gg_{\pi,\pi^{\prime}}1/\log(|t|+3)$

by appealing to the zero-free region in [9, Theorem 2.1]. Therefore, we shall
assume that $L(s,\pi\times\pi^{\prime})$ is entire. Furthermore, as explained
in Section 4.2, we can assume that (4.2) holds for some
$(\beta,\psi)\in(1-\varepsilon^{\prime},1)\times G$ depending only on
$(\pi,\pi^{\prime},\varepsilon)$.

Consider the automorphic representations

(4.3)
$\pi^{\prime\prime}=\pi^{\prime}\otimes\psi\in\mathfrak{F}_{n^{\prime}}\qquad\text{and}\qquad\chi^{\prime}=\overline{\psi}\chi\in
G.$

It follows that

$L(s,\pi\times(\pi^{\prime\prime}\otimes\chi^{\prime}))=L(s,\pi\times(\pi^{\prime}\otimes\chi))\qquad\text{and}\qquad
C(\chi^{\prime})\asymp_{\pi,\pi^{\prime},\varepsilon}C(\chi),$

and (4.1) is equivalent to

(4.4)
$L(1,\pi\times(\pi^{\prime\prime}\otimes\chi^{\prime}))\gg_{\pi,\pi^{\prime\prime},\varepsilon}C(\chi^{\prime})^{-\varepsilon}.$

But (4.4) follows readily from Proposition 3.4, upon noting that
$L(\beta,\pi\times\pi^{\prime\prime})=0$ and the following $L$-functions are
entire:

(4.5) $L(s,\pi\times\pi^{\prime\prime}),\qquad
L(s,\pi\times(\pi^{\prime\prime}\otimes\chi^{\prime})),\qquad
L(s,\pi\times(\pi^{\prime\prime}\otimes\chi^{\prime 2})).$

Indeed, these $L$-functions are shifts of $L(s,\pi\times\pi^{\prime})$, which
is entire by assumption.

We have shown that (4.1) holds for all $\chi\in\mathfrak{F}_{1}^{(1)}$.
Consequently, (4.1) also holds for the elements of any _fixed coset_ of
$\mathfrak{F}_{1}^{(1)}$. We shall use this principle in the next subsection.

###  4.4. The case of $\chi^{*}$ quadratic.

In this subsection, we prove (4.1) for all $\chi$ lying in the subgroup
$G=\mathfrak{F}_{1}^{(2)}$. If $L(s,\pi\times(\pi^{\prime}\otimes\chi))$ has a
pole, then by Lemma 3.1, $\chi$ lies in finitely many cosets of
$\mathfrak{F}_{1}^{(1)}$ (depending only on $(\pi,\pi^{\prime})$), hence (4.1)
holds by the concluding remark of the previous subsection. Therefore, we shall
assume that $L(s,\pi\times(\pi^{\prime}\otimes\chi))$ is entire.

As before, we also assume that (4.2) holds for some
$(\beta,\psi)\in(1-\varepsilon^{\prime},1)\times G$ depending only on
$(\pi,\pi^{\prime},\varepsilon)$, and we need to prove (4.4) with the notation
(4.3). But (4.4) follows readily from Proposition 3.4, upon noting that
$L(\beta,\pi\times\pi^{\prime\prime})=0$ and the $L$-functions under (4.5) are
entire. Indeed, the first two $L$-functions under (4.5) are entire by
assumption, while the third $L$-function is a shift of the first one due to
$\chi\in G$.

We have shown that (4.1) holds for all $\chi\in\mathfrak{F}_{1}^{(2)}$.
Consequently, (4.1) also holds for the elements of any _fixed coset_ of
$\mathfrak{F}_{1}^{(2)}$. We shall use this principle in the next subsection.

###  4.5. The general case.

In this subsection, we prove (4.1) in general. As before, we assume that
$L(s,\pi\times(\pi^{\prime}\otimes\chi))$ is entire, and (4.2) holds for some
$(\beta,\psi)\in(1-\varepsilon^{\prime},1)\times\mathfrak{F}_{1}$ depending
only on $(\pi,\pi^{\prime},\varepsilon)$. We need to prove (4.4) with the
notation (4.3).

If $L(s,\pi\times(\pi^{\prime\prime}\otimes\chi^{\prime 2}))$ has a pole, then
by Lemma 3.1, $\chi^{\prime}$ lies in finitely many cosets of
$\mathfrak{F}_{1}^{(2)}$ depending only on $(\pi,\pi^{\prime\prime})$, hence
(4.4) holds by the concluding remark of the previous subsection. Therefore, we
shall assume that $L(s,\pi\times(\pi^{\prime\prime}\otimes\chi^{\prime 2}))$
is entire. But then (4.4) follows readily from Proposition 3.4, because
$L(\beta,\pi\times\pi^{\prime\prime})=0$ and the $L$-functions under (4.5) are
entire by assumption.

##  5\. Finishing the proof of Theorem 1.1

Now that we have unconditionally proved (4.1) for all $\pi$, $\pi^{\prime}$,
$\chi$, and $\varepsilon>0$, we will use it to finish the proof of Theorem
1.1. As before, by continuity and nonvanishing arguments, we can assume that
$L(s,\pi\times(\pi^{\prime}\otimes\chi))$ is holomorphic in the open disk
$|s-1|<1$.

###  5.1. The case of $\sigma<1$.

We compare $L(\sigma,\pi\times(\pi^{\prime}\otimes\chi))$ with
$L(1,\pi\times(\pi^{\prime}\otimes\chi))$ via the bound

$|L(1,\pi\times(\pi^{\prime}\otimes\chi))-L(\sigma,\pi\times(\pi^{\prime}\otimes\chi))|\leq(1-\sigma)\sup_{\kappa\in[\sigma,1]}|L^{\prime}(\kappa,\pi\times(\pi^{\prime}\otimes\chi))|.$

For $|L(1,\pi\times(\pi^{\prime}\otimes\chi))|$, we have the lower bound
(4.1). On the other hand, by Lemma 2.2 and (2.8), there exists a constant
$\Cl[abcon]{ZFRN2}=\Cr{ZFRN2}(\pi,\pi^{\prime},\varepsilon)>0$ such that

$|L^{\prime}(\kappa,\pi\times(\pi^{\prime}\times\chi))|\leq\Cr{ZFRN2}C(\chi)^{\varepsilon/2},\qquad\kappa\geq
1-\frac{\varepsilon}{2nn^{\prime}}.$

Define

$\Cl[abcon]{ZFRN3}=\Cr{ZFRN3}(\pi,\pi^{\prime},\varepsilon)=\min\left(\frac{\Cr{ZFR3}}{1+\Cr{ZFRN2}},\frac{\varepsilon}{2nn^{\prime}}\right).$

If $1-\Cr{ZFRN3}C(\chi)^{-\varepsilon}<\sigma<1$, then

$\displaystyle|L(\sigma,\pi\times(\pi^{\prime}\otimes\chi))|$
$\displaystyle\geq|L(1,\pi\times(\pi^{\prime}\otimes\chi))|-|L(1,\pi\times(\pi^{\prime}\otimes\chi))-L(\sigma,\pi\times(\pi^{\prime}\otimes\chi))|$
$\displaystyle\geq|L(1,\pi\times(\pi^{\prime}\otimes\chi))|-\Cr{ZFRN3}C(\chi)^{-\varepsilon}\cdot\Cr{ZFRN2}C(\chi)^{\varepsilon/2}$
$\displaystyle\geq(\Cr{ZFR3}-\Cr{ZFRN2}\Cr{ZFRN3})C(\chi)^{-\varepsilon/2}$
$\displaystyle\geq\Cr{ZFRN3}C(\chi)^{-\varepsilon/2}.$

###  5.2. The case of $\sigma>1$.

It follows from (2.2) that

(5.1)
$L(\sigma,\pi\times(\pi^{\prime}\otimes\chi))\asymp_{n,n^{\prime}}1,\qquad\sigma\geq
3.$

To prove (1.5) in the strip $1<\sigma<3$, we interpolate between (4.1) and
(5.1) by applying Phragmén–Lindelöf principle to
$1/L(s,\pi\times\pi^{\prime})$.

##  6\. Proof of Theorem 1.2

Let $x\geq 2$, $A>0$, and $y=x(\log x)^{-A}$. Consider the Mellin transform
pair

(6.1) $\displaystyle\phi(r)$
$\displaystyle=\mathbf{1}_{(0,x]}(r)+\mathbf{1}_{(x,x+y]}(r)\frac{x+y-r}{y},\qquad
r>0;$ $\displaystyle\widehat{\phi}(s)$
$\displaystyle=\int_{0}^{\infty}\phi(r)r^{s-1}\,dr=\frac{(x+y)^{s+1}-x^{s+1}}{y(s^{2}+s)},\qquad\mathrm{Re}(s)>0.$

Clearly,

$\displaystyle\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{a}\leq x\\\
\mathfrak{a}\in\mathcal{C}\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})=\sum_{\begin{subarray}{c}\mathfrak{a}\in\mathcal{C}\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})\phi(\mathrm{N}\mathfrak{a})+O\left(\sum_{x<\mathrm{N}\mathfrak{a}\leq
x+y}|\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})|\right).$

By Corollary 2.4, the error term above is $O_{\pi,\pi^{\prime}}(y)$.

Let $\widehat{\mathrm{Cl}(\mathfrak{q})}$ be the group of characters of
$\mathrm{Cl}(\mathfrak{q})$. By character orthogonality, we have that

$\sum_{\begin{subarray}{c}\mathfrak{a}\in\mathcal{C}\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})\phi(\mathrm{N}\mathfrak{a})=\frac{1}{|\mathrm{Cl}(\mathfrak{q})|}\sum_{\psi\in\widehat{\mathrm{Cl}(\mathfrak{q})}}\sum_{\mathfrak{a}}\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})\psi(\mathfrak{a})\phi(\mathrm{N}\mathfrak{a}).$

Using (2.9) and (2.10), we find that if $\chi\in\mathscr{P}(\mathfrak{q})$ is
the primitive ray class character that induces $\psi$, then

$\displaystyle\sum_{\mathfrak{a}}\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})\psi(\mathfrak{a})\phi(\mathrm{N}\mathfrak{a})-\sum_{\mathfrak{a}}\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(\mathfrak{a})\phi(\mathrm{N}\mathfrak{a})$
$\displaystyle\ll
n^{\prime}n\sum_{\mathfrak{p}\mid\mathfrak{q}\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}}\sum_{\ell\leq\frac{\log(2x)}{\log\mathrm{N}\mathfrak{p}}}\mathrm{N}\mathfrak{p}^{\ell(\theta_{n}+\theta_{n^{\prime}})}\log\mathrm{N}\mathfrak{p}$
$\displaystyle\ll_{\pi,\pi^{\prime}}x^{1-\frac{1}{n^{2}+1}-\frac{1}{(n^{\prime})^{2}+1}}(\log
x)\log\mathrm{N}\mathfrak{q}.$

In light of the hypothesis $\mathrm{N}\mathfrak{q}\leq(\log x)^{A}$ and our
choice of $y$, it follows that

$\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{a}\leq x\\\
\mathfrak{a}\in\mathcal{C}\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(\mathfrak{a})=\frac{1}{|\mathrm{Cl}(\mathfrak{q})|}\sum_{\chi\in\mathscr{P}(\mathfrak{q})}\sum_{\mathfrak{a}}\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(\mathfrak{a})\phi(\mathrm{N}\mathfrak{a})+O_{\pi,\pi^{\prime},A}(y).$

Recall the definition of
$\mathcal{E}_{\pi\times\pi^{\prime}}(x;\mathfrak{q},\mathcal{C})$ from (1.8).
Applying Mellin inversion, we obtain

$\mathcal{E}_{\pi\times\pi^{\prime}}(x;\mathfrak{q},\mathcal{C})=\frac{1}{|\mathrm{Cl}(\mathfrak{q})|}\sum_{\chi\in\mathscr{P}(\mathfrak{q})}\frac{1}{2\pi
i}\int_{3-i\infty}^{3+i\infty}-\frac{L^{\prime}}{L}(s,\pi\times(\pi^{\prime}\otimes\chi))\,\widehat{\phi}(s)\,ds\\\
-\frac{1}{|\mathrm{Cl}(\mathfrak{q})|}\sum_{\chi\in\mathscr{P}(\mathfrak{q})}\mathcal{M}_{\pi\times(\pi^{\prime}\otimes\chi)}(x)+O_{\pi,\pi^{\prime},A}(y).$

Define $\varepsilon=1/(3A+2)$. By (1.6), Lemma 2.2, (2.8), and the bound
$C(\chi)\ll_{[F:\mathbb{Q}]}\mathrm{N}\mathfrak{q}$, there exists an
ineffective constant $\Cl[abcon]{C7777}=\Cr{C7777}(\pi,\pi,\varepsilon)>0$
such that on the piecewise smooth parametric curve

$\mathscr{C}(t)=1-\Cr{C7777}(\mathrm{N}\mathfrak{q}(|t|+1))^{-\varepsilon}+it,\qquad
t\in\mathbb{R},$

there holds

(6.2)
$\frac{L^{\prime}}{L}(\mathscr{C}(t),\pi\times(\pi^{\prime}\otimes\chi))\ll_{\pi,\pi^{\prime},\varepsilon}(\mathrm{N}\mathfrak{q}(|t|+1))^{2\varepsilon}.$

Note that $\mathscr{C}^{\prime}(t)\ll_{\pi,\pi^{\prime},\varepsilon}1$ for
$t\neq 0$. We deform the line of integration to $\mathscr{C}$.

If there exists $u\in\mathbb{R}$ such that
$\pi^{\prime}\otimes\chi=\widetilde{\pi}\otimes|\det|^{iu}$, then the
integrand has a pole at $s=1-iu$ with residue

$\widehat{\phi}(1-iu)=\int_{0}^{x}r^{-iu}\,dr+\int_{x}^{x+y}\phi(r)r^{-iu}\,dr=\mathcal{M}_{\pi\times(\pi^{\prime}\otimes\chi)}(x)+O(y).$

If no such $u$ exists, then $L(s,\pi\times(\pi^{\prime}\otimes\chi))$ is
entire, and $\mathcal{M}_{\pi\times(\pi^{\prime}\otimes\chi)}(x)=0$.
Therefore, by the residue theorem,

$\displaystyle\mathcal{E}_{\pi\times\pi^{\prime}}(x;\mathfrak{q},\mathcal{C})$
$\displaystyle=\frac{1}{|\mathrm{Cl}(\mathfrak{q})|}\sum_{\chi\in\mathscr{P}(\mathfrak{q})}\frac{1}{2\pi
i}\int_{\mathscr{C}}-\frac{L^{\prime}}{L}(s,\pi\times(\pi^{\prime}\otimes\chi))\,\widehat{\phi}(s)\,ds+O_{\pi,\pi^{\prime},A}(y)$
$\displaystyle\ll_{\pi,\pi^{\prime},A}\max_{\chi\in\mathscr{P}(\mathfrak{q})}\left|\int_{\mathscr{C}}-\frac{L^{\prime}}{L}(s,\pi\times(\pi^{\prime}\otimes\chi))\,\widehat{\phi}(s)\,ds\right|+y.$

Therefore, we bound the integrand via (6.1) and (6.2):

$\mathcal{E}_{\pi\times\pi^{\prime}}(x;\mathfrak{q},\mathcal{C})\ll_{\pi,\pi^{\prime},A}\frac{x^{2}}{y}\mathrm{N}\mathfrak{q}^{2\varepsilon}\int_{-\infty}^{\infty}x^{-\Cr{C7777}(\mathrm{N}\mathfrak{q}(|t|+1))^{-\varepsilon}}(|t|+1)^{2\varepsilon-2}\,dt+y.$

Finally, we express the last integral in terms of the new variable
$r=(\mathrm{N}\mathfrak{q}(|t|+1))^{-\varepsilon}\log x$, and estimate it in a
straightforward fashion, using that $\varepsilon\in(0,1/2$):

$\mathcal{E}_{\pi\times\pi^{\prime}}(x;\mathfrak{q},\mathcal{C})\ll_{\pi,\pi^{\prime},A}\frac{x^{2}}{y}(\log
x)^{2-1/\varepsilon}\mathrm{N}\mathfrak{q}\int_{0}^{\mathrm{N}\mathfrak{q}^{-\varepsilon}\log
x}e^{-\Cr{C7777}r}r^{-3+1/\varepsilon}\,dr+y.$

Our choices of $y$ and $\varepsilon$ along with our range of
$\mathrm{N}\mathfrak{q}$ ensures that
$\mathcal{E}_{\pi\times\pi^{\prime}}(x;\mathfrak{q},\mathcal{C})\ll_{\pi,\pi^{\prime},A}y$,
as desired.

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