abconsymbol=c

# Effective correlation and decorrelation for newforms, and weak subconvexity
for $L$-functions

Nawapan Wattanawanichkul
(with an appendix by Jesse Thorner)  1409 West Green Street, University of
Illinois Urbana-Champaign, Urbana, IL 61801, USA<EMAIL_ADDRESS>1409
West Green Street, University of Illinois Urbana-Champaign, Urbana, IL 61801,
USA<EMAIL_ADDRESS>
###### Abstract.

Let $f$ and $g$ be spectrally normalized holomorphic newforms of even weight
$k\geq 2$ on $\Gamma_{0}(q)$. If $f\neq g$, then assume that $q$ is
squarefree. For a nice test function $\psi$ supported on
$\Gamma_{0}(1)\backslash\mathbb{H}$, we establish the best known bounds
(uniform in $k$, $q$, and $\psi$) for

$\int_{\Gamma_{0}(q)\backslash\mathbb{H}}\psi(z)f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^{2}}-\mathds{1}_{f=g}\frac{3}{\pi}\int_{\Gamma_{0}(1)\backslash\mathbb{H}}\psi(z)\frac{dxdy}{y^{2}}.$

When $f=g$, our results yield an effective holomorphic variant of quantum
unique ergodicity, refining work of Holowinsky–Soundararajan and
Nelson–Pitale–Saha. When $f\neq g$, our results extend and improve the
effective decorrelation result of Huang for $q=1$. To prove our results, we
refine Soundararajan's weak subconvexity bound for Rankin–Selberg
$L$-functions.

##  1\. Introduction

Let $\Gamma_{0}(1)=\operatorname{SL}_{2}(\mathbb{Z})$, and consider the
hyperbolic surface $Y_{0}(1)=\Gamma_{0}(1)\backslash\mathbb{H}$. The volume
form is $d\mu=y^{-2}dxdy$, and $\mu(Y_{0}(1))=\pi/3$. Given bounded measurable
functions $H_{1}$ and $H_{2}$ on $Y_{0}(1)$, their Petersson inner product is
given by

$\langle H_{1},H_{2}\rangle=\int_{Y_{0}(1)}H_{1}(z)\overline{H_{2}(z)}d\mu.$

Let $\Delta=-y^{2}(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}})$ be the hyperbolic Laplacian on
$Y_{0}(1)$. Let $\\{\phi_{j}\\}_{j}$ be a basis of square-integrable
eigenfunctions of $\Delta$ given by the Hecke–Maaß cusp forms, normalized so
that $\\{\phi_{j}\\}_{j}$ is orthonormal with respect to the Petersson inner
product. We order the sequence $\\{\phi_{j}\\}_{j}$ so that the corresponding
sequence of Laplace eigenvalues $\\{\lambda_{j}\\}_{j}$ monotonically
increases.

A fundamental problem in quantum chaos asks for the limiting behavior of
$\phi_{j}$. Rudnick and Sarnak [32] conjectured that if $\psi$ is a bounded
measurable function on $Y_{0}(1)$, then

$\lim_{j\to\infty}\langle\psi,|\phi_{j}|^{2}\rangle=\frac{3}{\pi}\langle\psi,1\rangle.$
(1.1)

This is called quantum unique ergodicity (QUE). The limit (1.1) is equivalent
to the convergence in the weak-∗ topology of the probability measures
$|\phi_{j}|^{2}d\mu$ to the uniform measure $(3/\pi)d\mu$ as $j\to\infty$. The
combined work of Lindenstrauss [24] and Soundararajan [33] proves (1.1). As of
now, no rate of convergence is known unconditionally. For all $\varepsilon>0$,
a rate of convergence of
$O_{\psi,\varepsilon}(\lambda_{j}^{-1/4+\varepsilon})$ would follow from the
generalized Lindelöf hypothesis.

In this paper, we focus on a holomorphic variant of QUE. Let $z=x+iy$, and let

$f(z)=\sum_{n=1}^{\infty}\lambda_{f}(n)n^{\frac{k-1}{2}}e^{2\pi inz}=e^{2\pi
iz}+\sum_{n=2}^{\infty}\lambda_{f}(n)n^{\frac{k-1}{2}}e^{2\pi inz}$

be a normalized Hecke eigenform of even weight $k\geq 2$ and level $1$. In
other words, $f$ is a cusp form of weight $k$ and level $1$ that is also an
eigenfunction of all the Hecke operators with $\lambda_{f}(1)=1$. Consider a
sequence of such $f$ with $k\to\infty$. Write
$F_{k}(z)=\rho_{f}(1)y^{k/2}f(z)$, where $\rho_{f}(1)$ is chosen such that
$\langle 1,|F_{k}|^{2}\rangle=1$. The analogue of the QUE conjecture in this
setting is that if $\psi$ is a bounded measurable function on $Y_{0}(1)$, then

$\lim_{k\to\infty}\langle\psi,|F_{k}|^{2}\rangle=\frac{3}{\pi}\langle\psi,1\rangle.$
(1.2)

Holowinsky and Soundararajan [6, 7, 34] proved that if $\varepsilon>0$, then

$\langle\psi,|F_{k}|^{2}\rangle=\frac{3}{\pi}\langle\psi,1\rangle+O_{\psi,\varepsilon}((\log
k)^{-\frac{1}{30}+\varepsilon}),$ (1.3)

The proof of (1.3) in [7] relies on two different approaches. In [34],
Soundararajan spectrally expands the left-hand side of (1.3) in terms of
Hecke–Maaß cusp forms and unitary Eisenstein series, bounding the error term
using a weak subconvexity bound (see (1.13)). In [6], Holowinsky expands the
left-hand side of (1.3) in terms of incomplete Poincaré series and incomplete
Eisenstein series, bounding the error term by bounding shifted convolution
sums that arise from the expansion. A rate of convergence of
$O_{\psi,\varepsilon}(k^{-1/2+\varepsilon})$ follows from the generalized
Lindelöf hypothesis; up to refinements in the $k^{\varepsilon}$, this is
optimal (see [13]). Using (1.2), Rudnick [31] proved that as $k\to\infty$, the
zeros of $f$ equidistribute on $Y_{0}(1)$.

Let $C_{c}^{\infty}(Y_{0}(1))$ be the space of compactly supported, infinitely
differentiable functions on $Y_{0}(1)$. Let $\mathbb{N}$ denote the set of
positive integers. Given $M\geq 1$, let $C_{c}^{\infty}(Y_{0}(1),M)$ equal

$\Big{\\{}\psi\in C_{c}^{\infty}(Y_{0}(1))\colon\textup{ if
$a,b\in\mathbb{N}\cup\\{0\\}$, then }\sup_{z\in
Y_{0}(1)}\Big{|}y^{\max(a+b,\frac{1}{2})}\frac{\partial^{a}}{\partial
x^{a}}\frac{\partial^{b}}{\partial
y^{b}}\psi(z)\Big{|}\ll_{a,b}M^{a+b}\Big{\\}}.$ (1.4)

A close inspection of the work of Lester, Matomäki, and Radziwiłł [22, Proof
of Theorem 1.1] shows that if $M\geq 1$, $\psi\in C_{c}^{\infty}(Y_{0}(1),M)$,
and there exist absolute and effectively computable constants $A>0$ and
$\delta>0$ such that

$\langle\psi,|F_{k}|^{2}\rangle=\frac{3}{\pi}\langle\psi,1\rangle+O_{\varepsilon}(M^{A+\varepsilon}(\log
k)^{-\delta+\varepsilon}),$ (1.5)

then for any fixed $B>0$ and any hyperbolic ball
${\mathscr{B}}(z_{0},r)\subseteq\\{z\in\mathcal{F}\colon\mathrm{Im}(z)\leq
B\\}$ centered at $z_{0}$ with radius $r>0$, the following effective variant
of Rudnick's result holds:

$\frac{\\#\\{\varrho_{f}\in\mathscr{B}(z_{0},r)\colon
f(\varrho_{f})=0\\}}{\\#\\{\varrho_{f}\in\mathcal{F}\colon
f(\varrho_{f})=0\\}}=\frac{3}{\pi}\mu(\mathscr{B}(z_{0},r))+O_{B,\varepsilon}\Big{(}\frac{r}{(\log
k)^{\frac{\delta}{4+A}-\varepsilon}}\Big{)}.$ (1.6)

Following Iwaniec’s course notes [15] on holomorphic QUE, Lester, Matomäki,
and Radziwiłł [22, Theorem 1.3] proved that (1.5) holds with $A=2$ and
$\delta=31/2-4\sqrt{15}=0.008066\ldots$. Iwaniec's course notes present a
proof of holomorphic QUE that does not require Soundararajan's weak
subconvexity from [34]. However, an error was found in [15, Section 11], where
Iwaniec attempts to remove the need for weak subconvexity in the spectral
decomposition of the left-hand side of (1.3). This error propagated to the
proof [22, Theorem 1.3], rendering the proof incomplete. See Appendix A for
details. Alternatively, [22, Theorem 1.3] can be proved by refining
Soundararajan's weak subconvexity bound. We discuss this below.

Kowalski, Michel, and Vanderkam [21, Conjecture 1.4] introduced a variant of
holomorphic QUE in which the levels of holomorphic newforms are allowed to
vary. For $z=x+iy$, let

$f(z)=\sum_{n=1}^{\infty}\lambda_{f}(n)n^{\frac{k-1}{2}}e^{2\pi inz}\in
S_{k}^{\mathrm{new}}(\Gamma_{0}(q))$

be a holomorphic newform of even weight $k\geq 2$ and level $q\geq 1$ with
trivial nebentypus (see Section 5.1 for more details). We consider a sequence
of such $f$ with $kq\to\infty$. Given bounded measurable functions $H_{1}$ and
$H_{2}$ on $Y_{0}(q)=\Gamma_{0}(q)\backslash\mathbb{H}$, we define

$\langle
H_{1},H_{2}\rangle_{q}=\int_{Y_{0}(q)}H_{1}(z)\overline{H_{2}(z)}d\mu.$ (1.7)

Write

$F_{k}(z)=\rho_{f}(1)y^{k/2}f(z),$ (1.8)

where $\rho_{f}(1)$ is chosen such that
$\langle\psi,|F_{k}(z)|^{2}\rangle_{q}=1$. Kowalski, Michel, and Vanderkam
conjectured that if $\psi$ is a bounded measurable function supported on
$Y_{0}(1)$, then

$\lim_{kq\to\infty}\langle\psi,|F_{k}|^{2}\rangle_{q}=\frac{3}{\pi}\langle\psi,1\rangle_{1}.$
(1.9)

Building on the work in [6, 7, 34], Nelson, Pitale, and Saha [28, 29] proved
that there exist constants $\delta_{1},\delta_{2}>0$ such that if $q_{0}$ is
the largest squarefree divisor of $q$, then

$\langle\psi,|F_{k}|^{2}\rangle_{q}=\frac{3}{\pi}\langle\psi,1\rangle_{1}+O_{\psi}\Big{(}\Big{(}\frac{q}{q_{0}}\Big{)}^{-\delta_{1}}(\log
kq)^{-\delta_{2}}\Big{)}.$

Thus, the rate of convergence is sensitive to whether the level $q$ is
powerful.

Let $f$ and $g$ be holomorphic newforms of weight $k$ and level $q$, and let
$G_{k}$ be defined similarly to (1.8). In analogy with the variations on
holomorphic QUE mentioned above, it is natural to conjecture that if $\psi$ is
a bounded measurable function supported on $Y_{0}(1)$, then

$\lim_{kq\to\infty}|\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}|=0.$ (1.10)

Huang [6] proved that if $f$ and $g$ are normalized Hecke eigenforms of weight
$k$ and level $1$, with $f\neq g$, $M\geq 1$, and $\psi\in
C_{c}^{\infty}(Y_{0}(1),M)$, then for all $\varepsilon>0$, we have

$\langle\psi,\overline{F_{k}}G_{k}\rangle_{1}\ll_{\varepsilon}M^{\frac{5}{3}}(\log
k)^{-1.19\times 10^{-41}+\varepsilon}.$ (1.11)

Constantinescu [3] studied another variation of the decorrelation problem
considering normalized Hecke eigenforms $f$ and $g$ of level $1$ and weights
$k_{1}$ and $k_{2}$, respectively. He showed that the decorrelation values
dissipate as $\max(k_{1},k_{2})$ tends to infinity.

Now, consider holomorphic newforms $f$ and $g$ of weight $k$ and level $q$. If
$f\neq g$, then we assume that $q$ is squarefree. In this paper, we establish
the strongest known unconditional upper bound on
$|\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\mathds{1}_{f=g}\frac{3}{\pi}\langle\psi,1\rangle_{1}|$.
When $f=g$, this addresses the rate of convergence in (1.9) with effective
dependence on $k$, $q$, and $\psi$. When $f\neq g$, this addresses the rate of
convergence in (1.11) and extends such result to squarefree levels. For $f=g$
and $q=1$, our result provides a faster rate of convergence than in (1.3) and
[22, Theorem 1.3].

######  Theorem 1.1.

Let $q\geq 1$ be an integer, and let $q_{0}$ be the largest squarefree divisor
of $q$. Let $k\geq 2$ be an even integer. Let $f$ and $g$ be holomorphic
newforms of weight $k$, level $q$, and trivial nebentypus. Let $F_{k}$ and
$G_{k}$ be defined as in (1.8). Assume that if $f\neq g$, then $q$ is
squarefree. Let $\theta\in[0,7/64]$ denote the best bound towards the
generalized Ramanujan conjecture for Hecke–Maaß cusp forms. If
$\varepsilon>0$, $M\geq 1$, and $\psi\in C_{c}^{\infty}(Y_{0}(1),M)$, then

$\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\mathds{1}_{f=g}\frac{3}{\pi}\langle\psi,1\rangle_{1}\ll_{\varepsilon}M^{4}\Big{(}\frac{q}{q_{0}}\Big{)}^{-\frac{1}{12}(2\sqrt{3}-3)(1-2\theta)+\varepsilon}(\log
kq)^{-(\frac{7}{2}-2\sqrt{3})+\varepsilon}.$

If $q=1$ and $f=g$, then we can replace $M^{4}$ with $M^{(23-2\sqrt{3})/12}$.

######  Corollary 1.2.

Let $f$ be a normalized Hecke eigenform of even weight $k\geq 2$ and level
$1$. Equation (1.6) holds with $A=\frac{1}{12}(23-2\sqrt{3})=1.627991\ldots$
and $\delta=\frac{7}{2}-2\sqrt{3}=0.035898\ldots$. In particular, if $B>0$ is
fixed, then the zeros of $f$ equidistribute with respect to $d\mu$ within any
hyperbolic ball
${\mathscr{B}}(z_{0},r)\subseteq\\{z\in\mathcal{F}\colon\mathrm{Im}(z)\leq
B\\}$ of radius $r\geq(\log k)^{-1/157}$.

The main new result that enables us to prove Theorem 1.1 is a refinement of
Soundararajan's weak subconvexity bound. To describe this refinement, let
$m\geq 1$ be an integer, and let $\mathfrak{F}_{m}$ be the set of all cuspidal
automorphic representations of
$\operatorname{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$ with unitary central
character. For convenience, we normalize each $\pi\in\mathfrak{F}_{m}$ so that
the central character is trivial on the diagonally embedded copy of the
positive reals. Consider $\pi\in\mathfrak{F}_{m}$ and
$\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$. Let
$\mathfrak{C}(\pi\times\pi^{\prime})$ denote the analytic conductor of the
Rankin–Selberg $L$-function $L(s,\pi\times\pi^{\prime})$ (see Section 3.2 for
the definitions). Let $\varepsilon>0$. The convexity bound for
$L(s,\pi\times\pi^{\prime})$ is

$L(\tfrac{1}{2},\pi\times\pi^{\prime})\ll_{m,m^{\prime},\varepsilon}\mathfrak{C}(\pi\times\pi^{\prime})^{\frac{1}{4}+\varepsilon},$
(1.12)

while the generalized Riemann hypothesis implies that

$L(\tfrac{1}{2},\pi\times\pi^{\prime})\ll_{m,m^{\prime},\varepsilon}\mathfrak{C}(\pi\times\pi^{\prime})^{\varepsilon}.$

If the exponent $1/4+\varepsilon$ in (1.12) could be replaced by any fixed
exponent less than $1/4$ (such a bound is called a subconvexity bound), then
Theorem 1.1 would hold with a power saving rate of convergence in both $k$ and
$q$. Currently, no such exponent is known to exist.

The proofs in [7, 9, 28, 29] crucially rely on saving a power of $\log
C(\pi\times\pi^{\prime})$ over (1.12). Soundararajan [34] proved the following
weak subconvexity bound: If $\pi$ satisfies the generalized Ramanujan
conjecture (GRC), then for any $\varepsilon>0$,

$L(\tfrac{1}{2},\pi\times\pi^{\prime})\ll_{m,\pi^{\prime},\varepsilon}\frac{\mathfrak{C}(\pi\times\pi^{\prime})^{\frac{1}{4}}}{[\log\mathfrak{C}(\pi\times\pi^{\prime})]^{1-\varepsilon}}.$
(1.13)

In particular, if $\pi$ satisfies GRC, then (1.13) provides a logarithmic
improvement over (1.12) when $\pi^{\prime}$ is fixed. In order to explicate
the dependence on $\psi$ as in Theorem 1.1, one must first explicate the
dependence on $\pi^{\prime}$ in (1.13). Alternatively, when $\pi$ satisfies
GRC, one might appeal to the bound

$L(\tfrac{1}{2},\pi\times\pi^{\prime})\ll_{m,m^{\prime}}\frac{\mathfrak{C}(\pi\times\pi^{\prime})^{\frac{1}{4}}}{[\log\mathfrak{C}(\pi\times\pi^{\prime})]^{1/(10^{17}(mm^{\prime})^{3})}},$

proved by Soundararajan and Thorner [35]. While this bound is fully uniform in
both $\pi$ and $\pi^{\prime}$, the power of
$\log\mathfrak{C}(\pi\times\pi^{\prime})$ that it saves is quite small. In
this paper, we prove the following refinement of (1.13), which we expect to be
useful in settings beyond what we consider here.

######  Theorem 1.3.

Let $\pi\in\mathfrak{F}_{m}$ and $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$.
Assume that $L(s,\pi\times\pi^{\prime})$ is entire and $\pi$ satisfies
GRC.111Our proof shows that we only require $\pi$ to satisfy GRC at the non-
archimedean places. If $0<\varepsilon<1$, then

$L(\tfrac{1}{2},\pi\times\pi^{\prime})\ll_{m,m^{\prime},\varepsilon}\mathfrak{C}(\pi^{\prime})^{\varepsilon}\frac{\mathfrak{C}(\pi\times\pi^{\prime})^{\frac{1}{4}}}{[\log\mathfrak{C}(\pi\times\pi^{\prime})]^{1-\varepsilon}}.$

######  Remark.

With some additional work, one can show that there exists an effectively
computable constant $\Cl[abcon]{Li-thm1.3}=\Cr{Li-thm1.3}(m,m^{\prime})>0$
such that the factor $\mathfrak{C}(\pi^{\prime})^{\varepsilon}$ in Theorem 1.3
can be refined to $\exp[\Cr{Li-
thm1.3}(\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime}))/\log\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})]$.

### Acknowledgements

The author would like to thank Jesse Thorner for suggesting this project and
providing invaluable feedback that significantly improved this paper. The
author also thanks Peter Humphries, Maksym Radziwiłł, and Bingrong Huang for
very helpful discussion and comments on the manuscript.

##  2\. Outline of the paper

In Section 3, we record some properties of standard $L$-functions and
Rankin–Selberg $L$-functions, and establish useful bounds for partial sums of
Dirichlet coefficients.

In Section 4, we prove Theorem 1.3.

In Section 5, we discuss some background on $\operatorname{GL}(2)$ automorphic
forms, including holomorphic newforms, Hecke–Maaß cusp forms, Eisenstein
series, incomplete Eisenstein series, and incomplete Poincaré series. We also
recall the spectral decomposition and the Poincaré decomposition.

In Section 6, we prove Theorem 1.1 assuming Propositions 6.1 and 6.2.

In Section 7, we establish useful lower bounds for the adjoint lift of a
holomorphic newform. We also prove bounds for $L$-functions that will be
useful for proving Propositions 6.1.

In Section 8, we prove Propositions 6.1 and 6.2.

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

In this section and for the rest of the paper, if $\nu$ is a parameter, then
we write $f(z)=O_{\nu}(g(z))$, or equivalently, $f(z)\ll_{\nu}g(z)$, if there
exists an effectively computable constant $c$, depending at most on $m$,
$m^{\prime}$, $\varepsilon$, and $\nu$, such that $|f(z)|\leq c|g(z)|$ for all
$z$ under consideration. If no $\nu$ is present, then $c$ depends at most on
$m$, $m^{\prime}$, and $\varepsilon$.

###  3.1. Standard $L$-functions

Let $m\geq 1$ be an integer, let $\mathbb{A}_{\mathbb{Q}}$ denote the ring of
adèles over $\mathbb{Q}$, and let $\mathfrak{F}_{m}$ be the set of all
cuspidal automorphic representations of
$\mathrm{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$ whose central characters are
normalized to be trivial on the positive reals. If $\pi\in\mathfrak{F}_{m}$,
then for each place $v$ of $\mathbb{Q}$, there exists a smooth admissible
representation $\pi_{v}$ of $\operatorname{GL}_{m}(F_{v})$ such that $\pi$ can
be written as the restricted tensor product $\otimes_{v}\pi_{v}$. Given
$\pi\in\mathfrak{F}_{m}$, let $\widetilde{\pi}\in\mathfrak{F}_{m}$ be the
contragredient representation and $N_{\pi}$ be the conductor of $\pi$. When
$v$ is non-archimedean and corresponds with a prime $p$, then the local
$L$-function $L(s,\pi_{p})$ is defined in terms of the Satake parameters
$A_{\pi}(p)=\\{\alpha_{1,\pi}(p),\ldots,\alpha_{m,\pi}(p)\\}$ by

$L_{p}(s,\pi)=\prod_{j=1}^{m}(1-\alpha_{j,\pi}(p)p^{-s})^{-1}=\sum_{k=0}^{\infty}\frac{\lambda_{\pi}(p^{k})}{p^{ks}}.$
(3.1)

If $p\nmid N_{\pi}$, then for all $j$, we have that $\alpha_{j,\pi}(p)\neq 0$.
If $p\mid N_{\pi}$, it might be the case that there exists $j$ such that
$\alpha_{j,\pi}(p)=0$. The standard $L$-function $L(s,\pi)$ associated to
$\pi$ is

$L(s,\pi)=\prod_{p}L_{p}(s,\pi)=\sum_{n=1}^{\infty}\frac{\lambda_{\pi}(n)}{n^{s}}.$

The Euler product and Dirichlet series converge absolutely when
$\mathrm{Re}(s)>1$.

At the archimedean place of $\mathbb{Q}$, there are $m$ Langlands parameters
$\mu_{\pi}(j)\in\mathbb{C}$ from which we define

$L_{\infty}(s,\pi)=\pi^{-\frac{ms}{2}}\prod_{j=1}^{m}\Gamma\Big{(}\frac{s+\mu_{\pi}(j)}{2}\Big{)}.$

Luo, Rudnick, and Sarnak [25] and Müeller and Speh [27] proved that there
exists $\theta_{m}\in[0,\frac{1}{2}-\frac{1}{m^{2}+1}]$ such that we have the
uniform bounds

$|\alpha_{j,\pi}(p)|\leq p^{\theta_{m}}\qquad\textup{ and
}\qquad\operatorname{Re}(\mu_{j,\pi}(v))\geq-\theta_{m},$ (3.2)

and GRC asserts that in (3.2), one may take $\theta_{m}=0$. By the combined
work of Kim and Sarnak [20, Appendix 2] and Blomer and Brumley [1], we have
the bound $\theta_{2}\leq 7/64$. We have $N_{\pi}=N_{\widetilde{\pi}}$, and we
have the equalities of sets

$\\{\alpha_{j,\widetilde{\pi}}(p)\\}=\\{\overline{\alpha_{j,\pi}(p)}\\},\qquad\\{\mu_{j,\widetilde{\pi}}(v)\\}=\\{\overline{\mu_{j,\pi}(v)}\\}.$

Let $r_{\pi}$ be the order of the pole of $L(s,\pi)$ at $s=1$. The completed
$L$-function

$\Lambda(s,\pi)=(s(1-s))^{r_{\pi}}N_{\pi}^{s/2}L(s,\pi)L_{\infty}(s,\pi)$

is entire of order 1, and there exists a complex number $W(\pi)$ of modulus 1
such that for all $s\in\mathbb{C}$, we have the functional equation
$\Lambda(s,\pi)=W(\pi)\Lambda(1-s,\widetilde{\pi})$. The analytic conductor of
$\pi$ [13] is given by

$\mathfrak{C}(\pi,t)\coloneqq
N_{\pi}\prod_{j=1}^{m}(3+|it+\mu_{\pi}(j)|),\qquad\mathfrak{C}(\pi)\coloneqq\mathfrak{C}(\pi,0).$
(3.3)

We define $a_{\pi}(p^{k})$ by the Dirichlet series identity

$-\frac{L^{\prime}}{L}(s,\pi)=\sum_{p}\sum_{k=1}^{\infty}\frac{\sum_{j=1}^{m}\alpha_{j,\pi}(p)^{k}\log
p}{p^{ks}}=\sum_{p}\sum_{k=1}^{\infty}\frac{a_{\pi}(p^{k})\log
p}{p^{ks}},\quad\operatorname{Re}(s)>1.$

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

Let $\pi\in\mathfrak{F}_{m}$ and $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$.
For each $p\nmid N_{\pi}N_{\pi^{\prime}}$, define

$L_{p}(s,\pi\times\pi^{\prime})=\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}\frac{1}{1-\alpha_{j,\pi}(p)\alpha_{j^{\prime},\pi^{\prime}}(p)p^{-s}}.$

Jacquet, Piatetski-Shapiro, and Shalika [17] proved the following theorem.

######  Theorem 3.1.

If $(\pi,\pi^{\prime})\in\mathfrak{F}_{m}\times\mathfrak{F}_{m^{\prime}}$,
then there exist

  1. (1)

complex numbers
$(\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(p))_{j=1}^{m}{}_{j^{\prime}=1}^{m^{\prime}}$
for each prime $p\,|\,N_{\pi}N_{\pi^{\prime}}$, from which we define

$L_{p}(s,\pi\times\pi^{\prime})=\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}\frac{1}{1-\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(p)p^{-s}},\quad
L_{p}(s,\widetilde{\pi}\times\widetilde{\pi}^{\prime})=\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}\frac{1}{1-\overline{\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(p)}p^{-s}};$

  2. (2)

complex numbers
$(\mu_{\pi\times\pi^{\prime}}(j,j^{\prime}))_{j=1}^{m}{}_{j^{\prime}=1}^{m^{\prime}}$,
from which we define

$\displaystyle
L_{\infty}(s,\pi\times\pi^{\prime})=\pi^{-\frac{mm^{\prime}s}{2}}\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}\Gamma\Big{(}\frac{s+\mu_{\pi\times\pi^{\prime}}(j,j^{\prime})}{2}\Big{)},$
$\displaystyle
L_{\infty}(s,\widetilde{\pi}\times\widetilde{\pi}^{\prime})=\pi^{-\frac{mm^{\prime}s}{2}}\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}\Gamma\Big{(}\frac{s+\overline{\mu_{\pi\times\pi^{\prime}}(j,j^{\prime})}}{2}\Big{)};$

  3. (3)

an arithmetic conductor, denoted as $N_{\pi\times\pi^{\prime}}$; and

  4. (4)

a complex number $W(\pi\times\pi^{\prime})$ of modulus 1

such that the Rankin–Selberg $L$-function

$L(s,\pi\times\pi^{\prime})=\prod_{p}L_{p}(s,\pi\times\pi^{\prime})=\sum_{n=1}^{\infty}\frac{\lambda_{\pi\times\pi^{\prime}}(n)}{n^{s}}$

converges absolutely for $\operatorname{Re}(s)>1$ with a pole at $s=1$ of
order

$r_{\pi\times\pi^{\prime}}=\begin{cases}1&\mbox{if
$\pi^{\prime}=\widetilde{\pi}$,}\\\ 0&\mbox{otherwise,}\end{cases}$

the completed $L$-function

$\Lambda(s,\pi\times\pi^{\prime})=(s(1-s))^{r_{\pi\times\pi^{\prime}}}N_{\pi\times\pi^{\prime}}^{s/2}L(s,\pi\times\pi^{\prime})L_{\infty}(s,\pi\times\pi^{\prime})$
(3.4)

is entire of order 1, and
$\Lambda(s,\pi\times\pi^{\prime})=W(\pi\times\pi^{\prime})\Lambda(1-s,\widetilde{\pi}\times\widetilde{\pi}^{\prime})$.

We can explicitly determine the numbers
$\mu_{\pi\times\pi^{\prime}}(j,j^{\prime})$ (resp.
$\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(p)$ at primes
$p\,|\,N_{\pi}N_{\pi^{\prime}}$) using the archimedean case of the local
Langlands correspondence [27] (resp. [35, Appendix]). These descriptions and
(3.2) yield the bounds

$|\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(p)|\leq
p^{\theta_{m}+\theta_{m^{\prime}}},\qquad\operatorname{Re}(\mu_{\pi\times\pi^{\prime}}(j,j^{\prime}))\geq-(\theta_{m}+\theta_{m^{\prime}}).$
(3.5)

If $k\geq 1$ is an integer, then we define

$a_{\pi\times\pi^{\prime}}(p^{k})=\begin{cases}a_{\pi}(p^{k})a_{\pi^{\prime}}(p^{k})&\text{if
}p\nmid N_{\pi}N_{\pi^{\prime}},\\\
\sum_{j=1}^{m}\sum_{j=1}^{m^{\prime}}\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(p)^{k}&\text{otherwise.}\end{cases}$
(3.6)

We define $\Lambda_{\pi\times\pi^{\prime}}(n)$ by the Dirichlet series
identity

$-\frac{L^{\prime}}{L}(s,\pi\times\pi^{\prime})=\sum_{p}\sum_{k=1}^{\infty}\frac{a_{\pi\times\pi^{\prime}}(p^{k})\log
p}{p^{ks}}=\sum_{n=1}^{\infty}\frac{\Lambda_{\pi\times\pi^{\prime}}(n)}{n^{s}},\quad\operatorname{Re}(s)>1.$
(3.7)

As with $L(s,\pi)$, we define the analytic conductor

$\displaystyle\mathfrak{C}(\pi\times\pi^{\prime},t)\coloneqq
N_{\pi\times\pi^{\prime}}\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}(3+|it+\mu_{\pi\times\pi^{\prime}}(j,j^{\prime})|),\quad\mathfrak{C}(\pi\times\pi^{\prime})\coloneqq\mathfrak{C}(\pi\times\pi^{\prime},0).$
(3.8)

The combined work of Bushnell and Henniart [9] and Brumley [11, Appendix]
yields

$\mathfrak{C}(\pi\times\pi^{\prime},t)\ll\mathfrak{C}(\pi\times\pi^{\prime})(3+|t|)^{mm^{\prime}},\quad\mathfrak{C}(\pi\times\pi^{\prime})\ll\mathfrak{C}(\pi)^{m^{\prime}}\mathfrak{C}(\pi^{\prime})^{m}.$
(3.9)

###  3.3. Bounds for partial sums of Dirichlet coefficients

Consider $\pi\in\mathfrak{F}_{m}$ and
$\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$. Brumley [35, Appendix] and Jiang,
Lü, and Wang [18] proved useful inequalities for the coefficients
$\Lambda_{\pi\times\pi^{\prime}}(n)$ and $\lambda_{\pi\times\pi^{\prime}}(n)$,
respectively.

######  Lemma 3.2 (Lemma 2.2 in [35]).

Let $\pi\in\mathfrak{F}_{m}$ and $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$.
If $n\geq 1$ is an integer, then

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

######  Lemma 3.3 (Lemma 3.1 in [18]).

Let $\pi\in\mathfrak{F}_{m}$ and $\pi^{\prime}\in\mathfrak{F}_{m}$. If $n\geq
1$ is an integer, then

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

Lemmas 3.2 and 3.3 are useful to us because we assume that $\pi$ satisfies
GRC. To handle the contribution from $\pi^{\prime}$, we use two results, the
first of which is due to Li.

######  Theorem 3.4 (Theorem 2 in [23]).

There exists an absolute and effectively computable constant
$\Cl[abcon]{Li}>0$ such that if $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$ and
$1<\sigma\leq 3$, then

$L(\sigma,\pi^{\prime}\times\widetilde{\pi}^{\prime})\ll(\sigma-1)^{-1}\exp\Big{(}\Cr{Li}{m^{\prime}}^{2}\frac{\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}{\log\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}\Big{)}\ll(\sigma-1)^{-1}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$

The second result is a Brun–Titchmarsh type upper bound with mild dependence
on $\pi^{\prime}$.

######  Lemma 3.5.

Let $\pi\in\mathfrak{F}_{m}$ and $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$.
If $x\geq 2$ and $\pi$ satisfies GRC, then

$\sum_{n\leq x}|\Lambda_{\pi\times\pi^{\prime}}(n)|\ll
x\sqrt{\log\mathfrak{C}(\pi^{\prime})}.$

###### Proof.

By Lemma 3.2 and the Cauchy–Schwarz inequality, we have that

$\sum_{n\leq x}|\Lambda_{\pi\times\pi^{\prime}}(n)|\leq\Big{(}\sum_{n\leq
x}\Lambda_{\pi\times\widetilde{\pi}}(n)\Big{)}^{\frac{1}{2}}\Big{(}\sum_{n\leq
x}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)\Big{)}^{\frac{1}{2}}.$
(3.10)

Since $\pi$ satisfies GRC, it follows from the prime number theorem that

$\Big{(}\sum_{n\leq
x}\Lambda_{\pi\times\widetilde{\pi}}(n)\Big{)}^{\frac{1}{2}}\leq\Big{(}\sum_{n\leq
x}m^{2}\Lambda(n)\Big{)}^{\frac{1}{2}}\ll\sqrt{x}.$

We now estimate the second factor of (3.10). Fix a compactly supported,
infinitely differentiable function $\phi$ with $\phi^{(j)}(y)\ll_{j}1$ such
that $\phi$ is supported on $[1/2,5/2]$ and is at least $1$ on $[1,2]$. It
follows that

$\sum_{x\leq n\leq
2x}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)\leq\sum_{n=1}^{\infty}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)\phi(n/x).$
(3.11)

Let $\rho$ range over all zeros of
$L(s,\pi^{\prime}\times\widetilde{\pi}^{\prime})$. Applying the Mellin
inversion formula and pushing the contour all the way to the left, we find
that

$\sum_{n=1}^{\infty}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)\phi(n/x)=\frac{1}{2\pi
i}\int_{3-i\infty}^{3+i\infty}-\frac{L^{\prime}}{L}(s,\pi^{\prime}\times\widetilde{\pi}^{\prime})x^{s}\widehat{\phi}(s)\,ds=x\widehat{\phi}(1)-\sum_{\rho}x^{\rho}\widehat{\phi}(\rho).$
(3.12)

By (3.2) and (3.5), for any integer $k\geq 0$, there are $m^{\prime 2}$
trivial zeros with $\operatorname{Re}(\rho)\leq-2k+1$. Then the contribution
from trivial zeros is $\ll x$. For nontrivial zeros, we have that
$|x^{\rho}|\leq x$ trivially, and $\widehat{\phi}(1)\ll 1$. Hence, (3.12)
becomes

$\sum_{n=1}^{\infty}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)\phi(n/x)\ll
x+\sum_{\rho}x\widehat{\phi}(\rho).$ (3.13)

Now, it remains to bound $\sum_{\rho}\widehat{\phi}(\rho)$. For any real
number $t$, we have

$|\\{\rho=\beta+i\gamma:|\gamma-t|\leq
1\\}|\ll\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime},t)\ll\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})+\log(|t|+3).$

Write $\rho=\beta+i\gamma$. By integration by parts and the derivative
estimates of $\phi$, we also have that
$|\widehat{\phi}(\rho)|\ll_{\phi}(|\gamma|^{2}+1)^{-1}$. Therefore,

$\sum_{\rho}\widehat{\phi}(\rho)\ll\int_{-\infty}^{\infty}(\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})+\log(|t|+3))\frac{dt}{|t|^{2}+1}\ll\log\mathfrak{C}(\pi^{\prime}).$
(3.14)

By (3.11), (3.13), and (3.14), if $x\geq 2$, then $\sum_{x\leq n\leq
2x}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)\ll
x\log\mathfrak{C}(\pi^{\prime})$. Therefore, the lemma follows upon using
dyadic decomposition and (3.10). ∎

For future convenience, we record some helpful bounds obtained from applying
Lemma 3.3, the Cauchy–Schwarz inequality, and Theorem 3.4.

######  Lemma 3.6.

Let $\pi\in\mathfrak{F}_{m}$ and $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$.
Let $\pi$ satisfy GRC. If $x\geq 2$ and $1<\sigma\leq 2$, then

$\displaystyle\sum_{n\leq
x}\frac{|\lambda_{\pi\times\pi^{\prime}}(n)|}{n}\ll(\log
x)^{\frac{m^{2}+1}{2}}\mathfrak{C}(\pi^{\prime})^{\varepsilon},$ (3.15)
$\displaystyle|L(\sigma+it,\pi\times\pi^{\prime})|\ll(\sigma-1)^{-\frac{m^{2}+1}{2}}\mathfrak{C}(\pi^{\prime})^{\varepsilon},$
(3.16)
$\displaystyle|L^{\prime}(\sigma+it,\pi\times\pi^{\prime})|\ll(\sigma-1)^{-\frac{m^{2}+3}{2}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$
(3.17)

###### Proof.

We first prove (3.16) Appealing to Lemma 3.3 and the Cauchy–Schwarz
inequality, we have that

$\displaystyle|L(\sigma+it,\pi\times\pi^{\prime})|=\Big{|}\sum_{n=1}^{\infty}\frac{\lambda_{\pi\times\pi^{\prime}}(n)}{n^{\sigma+it}}\Big{|}$
$\displaystyle\leq\sum_{n=1}^{\infty}\frac{\sqrt{\lambda_{\pi\times\widetilde{\pi}}(n)\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)}}{n^{\sigma}}$
(3.18)
$\displaystyle\leq\Big{(}\sum_{n=1}^{\infty}\frac{\lambda_{\pi\times\widetilde{\pi}}(n)}{n^{\sigma}}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{n=1}^{\infty}\frac{\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)}{n^{\sigma}}\Big{)}^{\frac{1}{2}}.$

Let $\tau_{d}(n)$ denote the $d$-divisor function, the number of ordered
factorizations of $n$ into $d$ positive integers. Since
$\pi\in\mathfrak{F}_{m}$ satisfies GRC, we have
$\lambda_{\pi\times\widetilde{\pi}}(n)\leq\tau_{m^{2}}(n)$. It follows that

$\Big{(}\sum_{n=1}^{\infty}\frac{\lambda_{\pi\times\widetilde{\pi}}(n)}{n^{\sigma}}\Big{)}^{\frac{1}{2}}\leq\Big{(}\sum_{n=1}^{\infty}\frac{\tau_{m^{2}}(n)}{n^{\sigma}}\Big{)}^{\frac{1}{2}}=\zeta(\sigma)^{\frac{m^{2}}{2}}\ll(\sigma-1)^{-\frac{m^{2}}{2}}.$
(3.19)

By Theorem 3.4, we have that

$\Big{(}\sum_{n=1}^{\infty}\frac{\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)}{n^{\sigma}}\Big{)}^{\frac{1}{2}}=L(\sigma,\pi^{\prime}\times\widetilde{\pi}^{\prime})^{\frac{1}{2}}\ll(\sigma-1)^{-\frac{1}{2}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$
(3.20)

The assertion (3.16) then follows by inserting the bounds from (3.19) and
(3.20) into (3.18). To prove (3.17) , we combine (3.16) with Cauchy's integral
formula for $L^{\prime}(\sigma+it,\pi\times\pi^{\prime})$. Lastly, to prove
(3.15) , we first observe that

$\sum_{n\leq x}\frac{|\lambda_{\pi\times\pi^{\prime}}(n)|}{n}\leq
e\sum_{n=1}^{\infty}\frac{|\lambda_{\pi\times\pi^{\prime}}(n)|}{n^{1+1/\log(ex)}}.$

The result now follows from (3.16) and (3.18)–(3.20) (each with
$\sigma=1+1/\log(ex)$). ∎

######  Lemma 3.7.

Let $\pi\in\mathfrak{F}(m)$ and $\pi^{\prime}\in\mathfrak{F}(m^{\prime})$. Let
$\pi$ satisfy GRC. If $x\geq 2$, then

$\sum_{n\leq x}|\lambda_{\pi\times\pi^{\prime}}(n)|\ll x(\log
x)^{\frac{m^{2}+1}{2}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$ (3.21)

Moreover, if $1\leq y\leq x$, then

$\sum_{x\leq n\leq x+y}|\lambda_{\pi\times\pi^{\prime}}(n)|\ll
x^{3/4}y^{1/4}(\log
x)^{\frac{m^{4}+2}{4}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$ (3.22)

###### Proof.

The first assertion follows from (3.15) of Lemma 3.6 and Rankin's trick:

$\sum_{n\leq x}|\lambda_{\pi\times\pi^{\prime}}(n)|\ll x\sum_{n\leq
x}\frac{|\lambda_{\pi\times\pi^{\prime}}(n)|}{n}\ll x(\log
x)^{\frac{m^{2}+1}{2}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$

For the second assertion, it follows from Lemma 3.3 and the Cauchy–Schwarz
inequality that

$\sum_{x<n\leq
x+y}|\lambda_{\pi\times\pi^{\prime}}(n)|\leq\Big{(}\sum_{x<n\leq
x+y}\lambda_{\pi\times\widetilde{\pi}}(n)\Big{)}^{\frac{1}{2}}\Big{(}\sum_{x<n\leq
x+y}\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)\Big{)}^{\frac{1}{2}}.$

If $\pi$ satisfies GRC, then
$\lambda_{\pi\times\widetilde{\pi}}(n)\leq\tau_{m^{2}}(n)$. Since
$\lambda_{\pi\times\widetilde{\pi}}(n)$ is nonnegative, we use the
Cauchy–Schwarz inequality and Rankin's trick to find that

$\Big{(}\sum_{x<n\leq
x+y}\lambda_{\pi\times\widetilde{\pi}}(n)\Big{)}^{\frac{1}{2}}\leq
y^{\frac{1}{4}}\Big{(}\sum_{x<n\leq
x+y}\lambda_{\pi\times\widetilde{\pi}}(n)^{2}\Big{)}^{\frac{1}{4}}\leq(xy)^{\frac{1}{4}}\Big{(}\sum_{x<n\leq
x+y}\frac{\tau_{m^{2}}(n)^{2}}{n}\Big{)}^{\frac{1}{4}}.$

Since $\tau_{m^{2}}(n)^{2}\leq\tau_{m^{4}}(n)$, the above display is

$\displaystyle\leq(xy)^{\frac{1}{4}}\Big{(}\sum_{x<n\leq
x+y}\frac{\tau_{m^{4}}(n)}{n}\Big{)}^{\frac{1}{4}}$ $\displaystyle\leq
e(xy)^{\frac{1}{4}}\Big{(}\sum_{n=1}^{\infty}\frac{\tau_{m^{4}}(n)}{n^{1+1/\log(2ex)}}\Big{)}^{\frac{1}{4}}$
$\displaystyle=e(xy)^{\frac{1}{4}}\zeta(1+1/\log(2ex))^{\frac{m^{4}}{4}}\ll(xy)^{\frac{1}{4}}(\log
x)^{\frac{m^{4}}{4}}.$

Since $\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)$ is
nonnegative, we apply Rankin's trick and Theorem 3.4 to obtain

$\Big{(}\sum_{x<n\leq
x+y}\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)\Big{)}^{\frac{1}{2}}\leq
x^{\frac{1}{2}}\Big{(}\sum_{n\leq
2x}\frac{\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)}{n}\Big{)}^{\frac{1}{2}}\leq
x^{\frac{1}{2}}\Big{(}e\sum_{n=1}^{\infty}\frac{\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)}{n^{1+1/\log(2ex)}}\Big{)}^{\frac{1}{2}}\\\
=(ex)^{\frac{1}{2}}L(1+1/\log(2ex),\pi^{\prime}\times\widetilde{\pi}^{\prime})^{\frac{1}{2}}\ll
x^{\frac{1}{2}}(\log
x)^{\frac{1}{2}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$

The desired result follows. ∎

##  4\. Weak subconvexity bound

In this section, we establish our weak subconvexity bound as stated in Theorem
1.3. Our proof closely follows the framework introduced by Soundararajan in
[34]. Consequently, we shall exclude specific details mirroring
Soundararajan's method and instead focus on highlighting the distinctive
refinements in our approach.

###  4.1. Notation

Consider $\pi\in\mathfrak{F}_{m}$ and
$\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$. In this section, we assume that
$L(s,\pi\times\pi^{\prime})$ is entire, and $\pi$ satisfies GRC. For clarity,
we provide a list of the conventions and notation that we will use throughout
this section.

  * •

$C=\mathfrak{C}(\pi\times\pi^{\prime})$.

  * •

$x\geq 2$.

  * •

$S(x)=\sum_{n\leq x}\lambda_{\pi\times\pi^{\prime}}(n)$.

  * •

$\widetilde{S}(x)=\sum_{n\leq x}\lambda_{\pi\times\pi^{\prime}}(n)\log n$.

  * •

$\varepsilon\in(0,1]$.

  * •

$R=\lfloor 72m^{2}/\varepsilon^{2}\rfloor+1$.

  * •

$L=\lfloor 5(m^{2}+1)R\rfloor$.

  * •

$\underline{L}=(L,\ldots,L)$.

  * •

$X\geq\max(x,10)$.

  * •

$w\in[2,\exp((\log X)^{1/(3R)})]$ is an integer.

  * •

$T=\exp((\log\log X)^{2})$.

  * •

$\underline{\tau}=(\tau_{1},\tau_{2},\ldots,\tau_{R})\in\mathbb{R}^{R}$ with
$|\tau_{j}|\leq T$.

  * •

$l_{1},\ldots,l_{R},j_{1},\ldots,j_{R}\in\mathbb{Z}$ are nonnegative.

  * •

$\underline{l}=(l_{1},\ldots,l_{R})$ and $\underline{j}=(j_{1},\ldots,j_{R})$.

  * •

$\underline{j}\leq\underline{l}$ indicates that $0\leq j_{1}\leq
l_{1},\ldots,0\leq j_{R}\leq l_{R}$.

  * •

$\binom{\underline{l}}{\underline{j}}=\binom{l_{1}}{j_{1}}\cdots\binom{l_{R}}{j_{R}}$,
where $\binom{a}{b}$ are the binomial coefficients.

  * •

$\mathcal{O}_{\underline{l}}(x,w)=\mathcal{O}_{\underline{l}}(x,w;\underline{\tau})=\sum_{\underline{j}\leq\underline{l}}(-1)^{j_{1}+\cdots+j_{R}}\binom{\underline{l}}{\underline{j}}w^{j_{1}(1+i\tau_{1})+\cdots+j_{R}(1+i\tau_{R})}S(x/w^{j_{1}+\cdots+j_{R}})$.

  * •

$\widetilde{\mathcal{O}}_{\underline{l}}(x,w)=\mathcal{O}_{\underline{l}}(x,w;\underline{\tau})=\sum_{\underline{j}\leq\underline{l}}(-1)^{j_{1}+\cdots+j_{R}}\binom{\underline{l}}{\underline{j}}w^{j_{1}(1+i\tau_{1})+\cdots+j_{R}(1+i\tau_{R})}\widetilde{S}(x/w^{j_{1}+\cdots+j_{R}})$.

  * •

$\mathcal{L}_{\underline{l}}(s)=L(s,\pi\times\pi^{\prime})\prod_{j=1}^{R}(1-w^{1+i\tau_{j}-s})^{l_{j}}$.

###  4.2. Main results

As discussed in Soundararajan's work [34], the essence of his weak
subconvexity proof, and consequently our proof, is the slow variation of the
mean values of multiplicative functions. In our case, we consider the
multiplicative function $\lambda_{\pi\times\pi^{\prime}}(n)$. Writing
$C:=\mathfrak{C}(\pi\times\pi^{\prime})$, we will use the convexity bound for
$L(s,\pi\times\pi^{\prime})$ (see Lemma 4.4) to prove that there exists a
constant $B_{0}=B_{0}(m,m^{\prime})>0$ such that

$\sum_{n\leq x}\lambda_{\pi\times\pi^{\prime}}(n)\ll\frac{x}{\log
x}\mathfrak{C}(\pi^{\prime})^{\varepsilon},\qquad x\geq C^{\frac{1}{2}}(\log
C)^{B_{0}}.$

Next, we prove that for all $B>0$, similar cancellation holds even when $x\geq
C^{\frac{1}{2}}(\log C)^{-B}$. In particular, we have the following theorem.

######  Theorem 4.1.

Let $\pi$ and $\pi^{\prime}$ be as in Theorem 1.3 and
$C=\mathfrak{C}(\pi\times\pi^{\prime})$. If $\varepsilon>0$, $B>0$, and $x\geq
C^{\frac{1}{2}}(\log C)^{-B}$, then

$\sum_{n\leq x}\lambda_{\pi\times\pi^{\prime}}(n)\ll_{B}\frac{x}{(\log
x)^{1-\varepsilon}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$

Recall the notation in Section 4.1. Theorem 4.1 follows from our next theorem.

######  Theorem 4.2.

With the notation in Section 4.1, there exist real numbers
$\tau_{1},\ldots,\tau_{R}$, whose absolute values are at most $\exp((\log\log
X)^{2})$, such that if $2\leq x\leq X$, then

$|\mathcal{O}_{\underline{L}}(x,w;\tau_{1},\ldots,\tau_{R})|\ll\frac{x}{\log
x}(\log X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$

###  4.3. Deduction of the weak subconvexity bound

In this section, we prove Theorems 4.1 and 1.3. We start with some useful
lemmas. The first lemma is a result from Soundararajan and Thorner in [35].

######  Lemma 4.3.

Let $\pi\in\mathfrak{F}_{m}$ and $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$,
and let $\pi$ satisfy GRC. We have that

$L(\tfrac{1}{2}+it,\pi\times\pi^{\prime})\ll\mathfrak{C}(\pi\times\pi^{\prime})^{\frac{1}{4}}(|t|+1)^{\frac{mm^{\prime}}{4}}.$

###### Proof.

By [35, Corollary 1.3], we have

$L(\tfrac{1}{2},\pi\times\pi^{\prime})\ll|L(\tfrac{3}{2},\pi\times\pi^{\prime})|^{2}\frac{\mathfrak{C}(\pi\times\pi^{\prime})^{\frac{1}{4}}}{[\log\mathfrak{C}(\pi\times\pi^{\prime})]^{1/(10^{17}(mm^{\prime})^{3})}}.$

The above result is uniform in the full analytic conductor. Hence, by
recognizing the $L$-function $L(s+it,\pi\times\pi^{\prime})$ as
$L(s,\pi\times(\pi^{\prime}\otimes|\cdot|^{it}))$, we obtain the bound

$L(\tfrac{1}{2}+it,\pi\times\pi^{\prime})\ll|L(\tfrac{3}{2}+it,\pi\times\pi^{\prime})|^{2}\frac{\mathfrak{C}(\pi\times\pi^{\prime},t)^{\frac{1}{4}}}{[\log\mathfrak{C}(\pi\times\pi^{\prime},t)]^{1/(10^{17}(mm^{\prime})^{3})}}.$
(4.1)

We now write

$|L(\tfrac{3}{2}+it,\pi\times\pi^{\prime})|=\exp\Big{(}\operatorname{Re}\sum_{n=2}^{\infty}\frac{\Lambda_{\pi\times\pi^{\prime}}(n)}{n^{\frac{3}{2}+it}\log
n}\Big{)}.$ (4.2)

By triangle inequality, Lemma 3.2, and the Cauchy–Schwarz inequality, (4.2) is

$\leq\exp\Big{(}\Big{(}\sum_{n=2}^{\infty}\frac{\Lambda_{\pi\times\widetilde{\pi}}(n)}{n^{1+\frac{1}{m^{\prime
2}+1}}\log
n}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{n=2}^{\infty}\frac{\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)}{n^{2-\frac{1}{m^{\prime
2}+1}}\log n}\Big{)}^{\frac{1}{2}}\Big{)}.$ (4.3)

Since $\pi$ satisfies GRC, it follows from (3.2) and (3.5) that (4.3) is
$O(1)$, as desired. ∎

As discussed in Section 4.2, the next lemma gives an upper bound for the
partial summation $\sum_{n\leq x}\lambda_{\pi\times\pi^{\prime}}(n)$ in the
standard range of $x$, which can be obtained via the convexity bound.

######  Lemma 4.4.

Let $\pi\in\mathfrak{F}_{m}$ and $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$.
Assume that $L(s,\pi\times\pi^{\prime})$ is entire and $\pi$ satisfies GRC. If
$C=\mathfrak{C}(\pi\times\pi^{\prime})$, in the range $x\geq
x_{0}:=C^{\frac{1}{2}}(\log C)^{36m^{5}m^{\prime}}$, we have

$\sum_{n\leq x}\lambda_{\pi\times\pi^{\prime}}(n)\ll\frac{x}{\log
x}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$

###### Proof.

One can follow the proof of [34, Lemma 4.2] almost verbatim. Using the same
smoothing kernel, we have that, for any $c>1$, $x>0$, $\lambda>0$, and any
natural number $K$,

$\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}L(s,\pi\times\pi^{\prime})\frac{x^{s}}{s}\Big{(}\frac{e^{\lambda
s}-1}{\lambda s}\Big{)}^{K}\,ds=\sum_{n\leq
x}\lambda_{\pi\times\pi^{\prime}}(n)+O\Big{(}\sum_{x<n\leq
e^{K\lambda}x}|\lambda_{\pi\times\pi^{\prime}}(n)|\Big{)}.$ (4.4)

We shall take $K=\lfloor mm^{\prime}/4\rfloor+2$ and $\lambda=(\log
x)^{-(m^{4}+6)}$. By (3.22) of Lemma 3.7, the error term above is

$\ll(e^{K\lambda}-1)^{\frac{1}{4}}x(\log
x)^{\frac{m^{4}+2}{4}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}\ll\mathfrak{C}(\pi^{\prime})^{\varepsilon}\frac{x}{\log
x}.$

For the left-hand side of (4.4), we move the line of integration to
$c=\frac{1}{2}$ and appeal to Lemma 4.3. The integral is then

$\ll
C^{\frac{1}{4}}x^{\frac{1}{2}}\lambda^{-K}\int_{-\infty}^{\infty}(1+|t|)^{\frac{mm^{\prime}}{4}}\frac{dt}{(1+|t|)^{K+1}}\ll
C^{\frac{1}{4}}x^{\frac{1}{2}}(\log x)^{16m^{5}m^{\prime}}.$

Putting together the bound for the error term and the above bound in (4.4), we
have

$\sum_{n\leq x}\lambda_{\pi\times\pi^{\prime}}(n)\ll
C^{\frac{1}{4}}x^{\frac{1}{2}}(\log x)^{16m^{5}m^{\prime}}+\frac{x}{\log
x}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$

If $x\geq C^{\frac{1}{2}}(\log C)^{36m^{5}m^{\prime}}$, then the second term
dominates. ∎

###### Proof of Theorem 4.1, assuming Theorem 4.2.

Let $x_{0}$ be as in Lemma 4.4, and let $x_{0}\geq x\geq C^{\frac{1}{2}}(\log
C)^{-B}$ for any positive constant $B$. Take $w=x_{0}/x$ and $X=xw^{LR}$.
Applying Theorem 4.2 for $\lambda_{\pi\times\pi^{\prime}}(n)$, we obtain

$|\mathcal{O}_{\underline{L}}(X,w)|\ll\frac{X}{(\log
X)^{1-\varepsilon}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$ (4.5)

By definition, the left-hand side of (4.5) is

$w^{LR}\Big{|}\sum_{n\leq
X/w^{LR}}\lambda_{\pi\times\pi^{\prime}}(n)\Big{|}+O\Big{(}\sum_{j=0}^{LR-1}w^{j}\Big{|}\sum_{n\leq
X/w^{j}}\lambda_{\pi\times\pi^{\prime}}(n)\Big{|}\Big{)}.$ (4.6)

Since for $0\leq j\leq LR-1$ we have $X/w^{j}\geq xw=x_{0}$, Lemma 4.4 applies
for the range of the error term above, giving the bound
$O(\mathfrak{C}(\pi^{\prime})^{\varepsilon}X/\log X)$. From 4.5 and 4.6, we
conclude that

$\sum_{n\leq x}\lambda_{\pi\times\pi^{\prime}}(n)\ll w^{-LR}\frac{X}{(\log
X)^{1-\varepsilon}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}\ll\frac{x}{(\log
x)^{1-\varepsilon}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.\qed$

###### Proof of Theorem 1.3.

We use an approximation functional equation for
$L(\frac{1}{2},\pi\times\pi^{\prime})$ and partial summation. Following the
proof of [34, Theorem 1], we have that, for $c>1/2$

$L(\tfrac{1}{2},\pi\times\pi^{\prime})=\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}L(s+\tfrac{1}{2},\pi\times\pi^{\prime})\frac{L_{\infty}(s+\tfrac{1}{2},\pi\times\pi^{\prime})}{L_{\infty}(\tfrac{1}{2},\pi\times\pi^{\prime})}e^{s^{2}}\frac{ds}{s}\\\
+\frac{W(\pi\times\pi^{\prime})}{2\pi
i}\int_{c-i\infty}^{c+i\infty}L(s+\tfrac{1}{2},\widetilde{\pi}\times\widetilde{\pi}^{\prime})\frac{L_{\infty}(s+\tfrac{1}{2},\widetilde{\pi}\times\widetilde{\pi}^{\prime})}{L_{\infty}(\tfrac{1}{2},\widetilde{\pi}\times\widetilde{\pi}^{\prime})}e^{s^{2}}\frac{ds}{s}.$
(4.7)

Both integrals in (4.7) can be estimated similarly, so we only consider the
first one. By partial summation, we have

$L(s+\tfrac{1}{2},\pi\times\pi^{\prime})=(s+\tfrac{1}{2})\int_{1}^{\infty}\sum_{n\leq
x}\lambda_{\pi\times\pi^{\prime}}(n)\frac{dx}{x^{s+\frac{3}{2}}}.$

Therefore, the first integral in (4.7) equals

$\int_{1}^{\infty}\sum_{n\leq
x}\lambda_{\pi\times\pi^{\prime}}(n)\Big{(}\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}(s+\tfrac{1}{2})\frac{L_{\infty}(s+\tfrac{1}{2},\pi\times\pi^{\prime})}{L_{\infty}(\tfrac{1}{2},\pi\times\pi^{\prime})}e^{s^{2}}x^{-s}\frac{ds}{s}\Big{)}\frac{dx}{x^{\frac{3}{2}}}.$
(4.8)

To estimate the inner integral of (4.8), we move the line of integration
either to $\operatorname{Re}(s)=2$, or to
$\operatorname{Re}(s)=\tfrac{\theta_{m}+\theta_{m^{\prime}}}{2}$. Recall that
$C=\mathfrak{C}(\pi\times\pi^{\prime})$. By Stirling's formula, the inner
integral is

$\ll\min\Big{\\{}(\sqrt{C}/x)^{\tfrac{\theta_{m}+\theta_{m^{\prime}}}{2}},(\sqrt{C}/x)^{2}\Big{\\}}.$

Thus, (4.8) is

$\ll C^{(\theta_{m}+\theta_{m^{\prime}})/{4}}\int_{1}^{\sqrt{C}/(\log
C)^{(m^{2}+3)/(1-\theta_{m}-\theta_{m^{\prime}})}}\Big{|}\sum_{n\leq
x}\lambda_{\pi\times\pi^{\prime}}(n)\Big{|}\frac{dx}{x^{(3+\theta_{m}+\theta_{m^{\prime}})/{2}}}\\\
+C^{(\theta_{m}+\theta_{m^{\prime}})/{4}}\int_{\sqrt{C}/(\log
C)^{(m^{2}+3)/(1-\theta_{m}-\theta_{m^{\prime}})}}^{\sqrt{C}}\Big{|}\sum_{n\leq
x}\lambda_{\pi\times\pi^{\prime}}(n)\Big{|}\frac{dx}{x^{(3+\theta_{m}+\theta_{m^{\prime}})/{2}}}\\\
+C\int_{\sqrt{C}}^{\infty}\Big{|}\sum_{n\leq
x}\lambda_{\pi\times\pi^{\prime}}(n)\Big{|}\frac{dx}{x^{\frac{7}{2}}}.$

The first integral is
$\ll\mathfrak{C}(\pi^{\prime})^{\varepsilon}C^{\frac{1}{4}}/\log C$ by (3.21)
of Lemma 3.7. The second and third integrals are
$\ll\mathfrak{C}(\pi^{\prime})^{\varepsilon}C^{\frac{1}{4}}/(\log
C)^{1-\varepsilon}$ by Theorem 4.1. The desired result follows. ∎

###  4.4. Successive maxima

In this subsection, we define the points $\tau_{1},\ldots,\tau_{R}$ appearing
in Theorem 4.2. We define $\tau_{1}$ as the point $t$ in the compact set
$\mathcal{S}_{1}=[-T,T]$ where the maximum of $|L(1+1/\log
X+it,\pi\times\pi^{\prime})|$ is attained. We now remove the interval
$(\tau_{1}-(\log X)^{-\frac{1}{R}},\tau_{1}+(\log X)^{-\frac{1}{R}})$ from
$\mathcal{S}_{1}$, and let $\mathcal{S}_{2}$ denote the remaining compact set.
We define $\tau_{2}$ to be the point in $\mathcal{S}_{2}$ where the maximum
$|L(1+1/\log X+it,\pi\times\pi^{\prime})|$ is attained. We continue defining
$\tau_{1},\tau_{2},\ldots,\tau_{R}$ similarly and obtain the nested compact
sets
$\mathcal{S}_{1}\supset\mathcal{S}_{2}\supset\ldots\supset\mathcal{S}_{R}$. We
observe that if $j\neq k$, then $|\tau_{j}-\tau_{k}|\geq(\log
X)^{-\frac{1}{R}}$.

Incorporating the above construction, we obtain the following improved version
of (3.16) of Lemma 3.6.

######  Lemma 4.5.

Recall the notation in Section 4.1. Let $1\leq j\leq R$ and $t$ be a point in
$\mathcal{S}_{j}$. If $\varepsilon>0$, then

$|L(1+1/{\log
X}+it,\pi\times\pi^{\prime})|\ll\mathfrak{C}(\pi^{\prime})^{\varepsilon}(\log
X)^{\frac{3m}{2}\sqrt{\frac{1}{j}+\frac{j-1}{jR}}+\frac{\varepsilon}{4}}.$

In particular, if $t\in\mathcal{S}_{R}$, we have $|L(1+1/\log
X+it,\pi\times\pi^{\prime})|\ll\mathfrak{C}(\pi^{\prime})^{\varepsilon}(\log
X)^{\varepsilon/2}$.

###### Proof.

We apply the same technique as in the proof of Lemma 3.6 and follow the
outline of the proof of [34, Lemma 3.2]. Without any modification, [34, Lemma
3.2] gives

$|L(1+1/\log
X+i\tau_{j},\pi\times\pi^{\prime})|\leq\exp\Big{(}\operatorname{Re}\Big{(}\frac{1}{j}\sum_{n\geq
2}\frac{\Lambda_{\pi\times\pi^{\prime}}(n)}{n^{1+1/\log X}\log
n}\sum_{r=1}^{j}n^{-i\tau_{r}}\Big{)}\Big{)}.$ (4.9)

By Lemma 3.2 and the Cauchy–Schwarz inequality, we have that

$\displaystyle\sum_{n\geq 2}$
$\displaystyle\frac{|\Lambda_{\pi\times\pi^{\prime}}(n)|}{n^{1+1/\log X}\log
n}\Big{|}\sum_{r=1}^{j}n^{-i\tau_{r}}\Big{|}\leq\sum_{n\geq
2}\frac{\sqrt{\Lambda_{\pi\times\widetilde{\pi}}(n)\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)}}{n^{1+1/\log
X}\log n}\Big{|}\sum_{r=1}^{j}n^{-i\tau_{r}}\Big{|}$ (4.10)
$\displaystyle\leq\Big{(}\sum_{n\geq
2}\frac{\Lambda_{\pi\times\widetilde{\pi}}(n)}{n^{1+1/\log X}\log
n}\Big{|}\sum_{r=1}^{j}n^{-i\tau_{r}}\Big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{n\geq
2}\frac{\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)}{n^{1+1/\log
X}\log n}\Big{)}^{\frac{1}{2}}.$

Since $\pi$ satisfies GRC and $\Lambda_{\pi\times\widetilde{\pi}}(n)$ is
supported on prime powers, we obtain the bound
$\Lambda_{\pi\times\widetilde{\pi}}(n)\leq m^{2}\Lambda(n)$. Thus, (4.10) is

$\leq m\Big{(}\sum_{n\geq 2}\frac{\Lambda(n)}{n^{1+1/\log X}\log
n}\Big{|}\sum_{r=1}^{j}n^{-i\tau_{r}}\Big{|}^{2}\log L(1+1/{\log
X},\pi^{\prime}\times\widetilde{\pi}^{\prime})\Big{)}^{\frac{1}{2}}.$ (4.11)

By Theorem 3.4, we have that

$\displaystyle\log L(1+1/{\log X},\pi^{\prime}\times\widetilde{\pi}^{\prime})$
$\displaystyle\leq\log\log X+c_{1}m^{\prime
2}\frac{\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}{\log\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}+O(1).$
(4.12)

On the other hand, by the calculations on [34, p. 1488], the double sum in
(4.11) equals

$\displaystyle j(\log\log X+O(1))+2\sum_{1\leq r<r^{\prime}\leq j}\log|(\log
X)^{1/R}+O(1)|$ (4.13)
$\displaystyle\leq\Big{(}j+\frac{j(j-1)}{R}\Big{)}\log\log X+O(j).$

Using (4.11)–(4.13), we bound (4.9) by

$\exp\Big{[}\frac{m}{j}\Big{(}\Big{(}j+\frac{j(j-1)}{R}\Big{)}\log\log
X+O(j)\Big{)}^{\frac{1}{2}}\Big{(}\log\log X+c_{1}m^{\prime
2}\frac{\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}{\log\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}+O(1)\Big{)}^{\frac{1}{2}}\Big{]}.$

For numbers $a,b,c\geq 0$, the bound
$\sqrt{a+b+c}\leq\sqrt{a}+\sqrt{b}+\sqrt{c}$ holds. Applying this twice, we
bound (4.9) by

$\displaystyle\exp$
$\displaystyle\Bigg{[}m\sqrt{\frac{1}{j}+\frac{j-1}{jR}}\Bigg{(}\log\log
X+\sqrt{\log\log
X}\sqrt{c_{1}{m^{\prime}}^{2}\frac{\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}{\log\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}}\Bigg{)}$
(4.14)
$\displaystyle+O\Bigg{(}\Bigg{(}m\sqrt{\frac{1}{j}+\frac{j-1}{jR}}+\frac{m}{j}\Bigg{)}\sqrt{\log\log
X}+\frac{mm^{\prime}}{\sqrt{j}}\sqrt{c_{1}\frac{\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}{\log\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}}\Bigg{)}\Bigg{]}.$

The inequality of arithmetic and geometric means yields

$\sqrt{\log\log X}\sqrt{c_{1}m^{\prime
2}\frac{\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}{\log\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}}\leq\frac{1}{2}\log\log
X+\frac{c_{1}m^{\prime
2}}{2}\frac{\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}{\log\log\mathfrak{C}(\pi^{\prime}\times\widetilde{\pi}^{\prime})}.$
(4.15)

Hence, the lemma follows from (4.14) and (4.15). ∎

In the proof of Theorem 4.2, there will appear the quantity
$\mathcal{L}_{\underline{l}}(s)$, defined in Section 4.1. We record some
useful bounds for such quantity in the next two lemmas

######  Lemma 4.6.

Recall the notation in Section 4.1. For any $\sigma\geq 1+1/\log X$, we have
that

$\max_{|t|\leq T/2}|\mathcal{L}_{\underline{l}}(\sigma+it)|\leq\max_{|t|\leq
T}|\mathcal{L}_{\underline{l}}(1+1/\log{X}+it)|+O((\log
X)^{-1}\mathfrak{C}(\pi^{\prime})^{\varepsilon}).$

###### Proof.

Replacing [34, Lemma 3.2] with (3.16) of Lemma 3.6, we proceed just as in the
proof of [34, Lemma 5.2]. ∎

######  Lemma 4.7.

Recall the notation and hypotheses in Section 4.1. Let
$\underline{L-1}=(L-1,\ldots,L-1)$. If $\underline{l}\geq\underline{L-1}$,
then

$\max_{|t|\leq T}|\mathcal{L}_{\underline{l}}(1+1/\log{X}+it)|\ll(\log
X)^{\varepsilon/2}\mathfrak{C}(\pi^{\prime})^{\varepsilon/2}.$

###### Proof.

Replacing [34, Lemma 5.1] with Lemma 4.5 and [34, Lemma 3.2] with (3.16) of
Lemma 3.6, we proceed just as in the proof of [34, Lemma 5.3]. ∎

######  Proposition 4.8.

Recall the notation and hypotheses of Lemma 4.7. If $1\leq x\leq X$, then

$\mathcal{O}_{\underline{l}}(x,w)\ll x(\log
X)^{2\varepsilon/3}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$

###### Proof.

By (3.21) of Lemma 3.7, we may assume that $\log x\geq(\log
X)^{\varepsilon/(m^{2}+1)}$; otherwise, the desired bound for
$\mathcal{O}_{\underline{l}}(x,w)$ is trivial. If
$w\leq\exp((2RL)^{(m^{2}+1)/(3R\varepsilon-m^{2}-1)})$, which is a constant
depending only on $m,m^{\prime},$ and $\varepsilon$, then the desired bound is
also trivial, so we may assume that
$w\geq\exp((2RL)^{(m^{2}+1)/(3R\varepsilon-m^{2}-1)})$. Upon the second
assumption, we then have $x\geq w^{2RL}$. By (4.4) with $K=1$,
$\lambda=1/\sqrt{T}$, and $c>1$, we find that

$\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}L(s,\pi\times\pi^{\prime})z^{s}\Big{(}\frac{e^{s/\sqrt{T}-1}}{s/\sqrt{T}}\Big{)}\frac{ds}{s}=\sum_{n\leq
z}\lambda_{\pi\times\pi^{\prime}}(n)+O\Big{(}\sum_{z<n\leq
ze^{1/\sqrt{T}}}|\lambda_{\pi\times\pi^{\prime}}(n)|\Big{)}.$

By (3.22) of Lemma 3.7, the error term above is
$O_{\varepsilon}(\mathfrak{C}(\pi^{\prime})^{\varepsilon}z/\log z)$ in the
range $T\leq z\leq X$ where $T=\exp((\log\log X)^{2})$. Using the above
formula in the definition of $\mathcal{O}_{\underline{l}}(x,w)$, we obtain
that if $w^{2RL}\leq x\leq X$, then

$\mathcal{O}_{\underline{l}}(x,w)=\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}L(s,\pi\times\pi^{\prime})\prod_{j=1}^{R}(1-w^{1+i\tau_{j}-s})^{l_{j}}x^{s}\Big{(}\frac{e^{s/\sqrt{T}-1}}{s/\sqrt{T}}\Big{)}\frac{ds}{s}+O\Big{(}\mathfrak{C}(\pi^{\prime})^{\varepsilon}\frac{x}{\log
x}\Big{)}.$

Choose $c=1+1/\log X$ and split the above integral into two parts, when
$|\operatorname{Im}(s)|\leq T$ and when $|\operatorname{Im}(s)|>T$. For the
first range, we use Lemma 4.7 and apply the fact that
$(e^{s/\sqrt{T}}-1)/(s/\sqrt{T})\ll 1$ when $|\operatorname{Im}(s)|\leq T$, so
this portion contributes

$\ll(\log
X)^{\varepsilon/2}\mathfrak{C}(\pi^{\prime})^{\varepsilon/2}x^{1+1/\log
X}\int_{\begin{subarray}{c}|\operatorname{Im}(s)|\leq T\\\
\operatorname{Re}(s)=1+1/\log
X\end{subarray}}\Big{|}\frac{e^{s/\sqrt{T}}-1}{s/\sqrt{T}}\Big{|}\frac{|ds|}{|s|}\ll
x(\log X)^{2\varepsilon/3}\mathfrak{C}(\pi^{\prime})^{\varepsilon/2}.$

In the range $|\operatorname{Im}(s)|>T$, we use (3.16) of Lemma 3.6. Since
$(e^{s/\sqrt{T}}-1)/(s/\sqrt{T})\ll\sqrt{T}/|s|$ in this range, this portion
contributes

$\ll\mathfrak{C}(\pi^{\prime})^{\varepsilon}x(\log
X)^{\frac{m^{2}+1}{2}}\int_{\begin{subarray}{c}|\operatorname{Im}(s)|>T\\\
\operatorname{Re}(s)=1+1/\log
X\end{subarray}}\Big{|}\frac{e^{s/\sqrt{T}}-1}{s/\sqrt{T}}\Big{|}\frac{|ds|}{|s|}\ll
x\mathfrak{C}(\pi^{\prime})^{\varepsilon},$

Combining the two contributions gives us the proposition. ∎

###  4.5. Proof of Theorem 4.2

In this subsection, we prove Theorem 4.2 with the points
$\tau_{1},\ldots,\tau_{R}$ defined as in the previous subsection.

######  Lemma 4.9.

Recall the notation and hypotheses in Sections 3 and 4.1. If $0\leq\log
w\leq(\log X)^{\frac{1}{3R}}$ and $1\leq x\leq X$, then

$(\log x)\mathcal{O}_{\underline{L}}(x,w)=\sum_{d\leq
x}\Lambda_{\pi\times\pi^{\prime}}(d)\mathcal{O}_{\underline{L}}(x/d,w)+O(x(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{\varepsilon}).$

###### Proof.

The proof of this lemma is similar to that of [34, Lemma 6.1], where we
replace [34, Proposition 5.4] by our Proposition 4.8. Writing

$\log x=\log(x/w^{j_{1}+\cdots+j_{R}})+(j_{1}+\cdots+j_{R})\log w,$

we observe that $\mathcal{O}_{\underline{L}}(x,w)\log x$ equals

$\displaystyle\sum_{\underline{j}\leq\underline{L}}(-1)^{j_{1}+\cdots+j_{R}}\binom{\underline{L}}{\underline{j}}\log(x/w^{j_{1}+\cdots+j_{R}})S(x/w^{j_{1}+\cdots+j_{R}})w^{j_{1}(1+i\tau_{j})+\cdots+j_{R}(1+i\tau_{R})}$
(4.16) $\displaystyle+\log
w\sum_{\underline{j}\leq\underline{L}}(-1)^{j_{1}+\cdots+j_{R}}(j_{1}+\cdots+j_{R})\binom{\underline{L}}{\underline{j}}S(x/w^{j_{1}+\cdots+j_{R}})w^{j_{1}(1+i\tau_{j})+\cdots+j_{R}(1+i\tau_{R})}.$

Let $\underline{e}_{k}$ denote the vector with 1 in the $k$-th place and 0
elsewhere. The second line in (4.16) can be written as

$-\sum_{k=1}^{R}Lw^{1+i\tau_{k}}(\log
w){\mathcal{O}}_{\underline{L}-\underline{e}_{k}}(x/w,w),$ (4.17)

Since each component of $\underline{L}-\underline{e}_{k}$ is at least $L-1$,
we can apply Proposition 4.8. It follows that (4.17) is $\ll x(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{\varepsilon}$.

We observe that

$(\log x)S(x)=\sum_{n\leq
x}\lambda_{\pi\times\pi^{\prime}}(n)\log(n)+\sum_{n\leq
x}\lambda_{\pi\times\pi^{\prime}}(n)\log(x/n).$

By partial summation and the Dirichlet convolution identity

$\lambda_{\pi\times\pi^{\prime}}(n)\log(n)=\sum_{d\mid
n}\Lambda_{\pi\times\pi^{\prime}}(d)\lambda_{\pi\times\pi^{\prime}}(n/d),$

it follows that

$\displaystyle(\log x)S(x)$ $\displaystyle=\sum_{n\leq x}\sum_{d\mid
n}\Lambda_{\pi\times\pi^{\prime}}(d)\lambda_{\pi\times\pi^{\prime}}(n/d)+\int_{1}^{x}S(t)\frac{dt}{t}=\sum_{d\leq
x}\Lambda_{\pi\times\pi^{\prime}}(d)S(x/d)+\int_{1}^{x}S(t)\frac{dt}{t}.$

Hence, the first term of (4.16) equals

$\sum_{d\leq
x}\Lambda_{\pi\times\pi^{\prime}}(d)\mathcal{O}_{\underline{L}}(x/d,w)+\int_{1}^{x}\mathcal{O}_{\underline{L}}(t,w)\frac{dt}{t}.$

Again, by applying Proposition 4.8, the integral part is $\ll x(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{\varepsilon}$. The lemma follows.
∎

######  Lemma 4.10.

Recall the notation in Section 4.1. If $1\leq z\leq y$ and $y+z\leq X$, then

$||\mathcal{O}_{\underline{L}}(y,w)|^{2}-|\mathcal{O}_{\underline{L}}(y+z,w)|^{2}|\ll\mathfrak{C}(\pi^{\prime})^{\varepsilon}y(\log
X)^{\varepsilon}\sum_{j=0}^{LR}w^{j}\sum_{y/w^{j}<n\leq(y+z)/w^{j}}|\lambda_{\pi\times\pi^{\prime}}(n)|.$

###### Proof.

Replacing [34, Proposition 5.4] with Proposition 4.8 and using our definition
of $S(x)$, we proceed just as in the proof of [34, Lemma 6.2]. ∎

######  Proposition 4.11.

Recall the notation in Section 4.1. If $0\leq\log w\leq(\log
X)^{\frac{1}{3R}}$ and $1\leq x\leq X$, then

$|\mathcal{O}_{\underline{L}}(x,w)|\log x\\\ \ll x(\log\log
x)^{\frac{1}{2}}\mathfrak{C}(\pi^{\prime})^{\frac{\varepsilon}{4}}\Big{(}\int_{1}^{x}\log(ey)|\mathcal{O}_{\underline{L}}(y,w)|^{2}\,\frac{dy}{y^{3}}+\mathfrak{C}(\pi^{\prime})^{\varepsilon}\Big{)}^{\frac{1}{2}}+x(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$

###### Proof.

By Lemma 4.9, we are left to estimate the sum

$\sum_{d\leq
x}|\Lambda_{\pi\times\pi^{\prime}}(d)||\mathcal{O}_{\underline{L}}(x/d,w)|.$

We split this sum into two ranges $d\leq D:=\lfloor\exp((\log\log
X)^{6})\rfloor$ and $d>D$. For the first range of the sum, we use Proposition
4.8 and obtain

$\sum_{d\leq
D}|\Lambda_{\pi\times\pi^{\prime}}(d)||\mathcal{O}_{\underline{L}}(x/d,w)|\ll
x(\log X)^{2\varepsilon/3}\sum_{d\leq
D}\frac{|\Lambda_{\pi\times\pi^{\prime}}(d)|}{d}.$ (4.18)

By Lemma 3.5 and partial summation, the right-hand side of (4.18) is

$\ll x(\log
X)^{2\varepsilon/3}\log(eD)\mathfrak{C}(\pi^{\prime})^{\varepsilon}\ll x(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$ (4.19)

It remains to estimate the contribution of the terms with $d>D$. Define
$g(t)=t\log(ex/t)$ for $1\leq t\leq x$. By applying Lemma 3.3 and the
Cauchy–Schwarz inequality, we have

$\sum_{D<d\leq
x}|\Lambda_{\pi\times\pi^{\prime}}(d)||\mathcal{O}_{\underline{L}}(x/d,w)|\\\
\leq\Big{(}\sum_{D<d\leq
x}\Lambda_{\pi\times\widetilde{\pi}}(d)g(d)|\mathcal{O}_{\underline{L}}(x/d,w)|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{D<d\leq
x}\frac{\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(d)}{g(d)}\Big{)}^{\frac{1}{2}}.$
(4.20)

Using GRC for $\Lambda_{\pi\times\widetilde{\pi}}(d)$, and Lemma 3.5 and
partial summation for
$\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(d)$, we have that (4.20)
is

$\displaystyle\ll\Big{(}\sum_{D<d\leq
x}g(d)\Lambda(d)|\mathcal{O}_{\underline{L}}(x/d,w)|^{2}\Big{)}^{\frac{1}{2}}\Big{(}(\log\log(ex/D))\mathfrak{C}(\pi^{\prime})^{\varepsilon/2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\ll\Big{(}\sum_{D<d\leq
x}g(d)\Lambda(d)|\mathcal{O}_{\underline{L}}(x/d,w)|^{2}\Big{)}^{\frac{1}{2}}(\log\log
x)^{\frac{1}{2}}\mathfrak{C}(\pi^{\prime})^{\frac{\varepsilon}{4}}.$

The remaining of the proof then follows from the proof of [34, Lemma 6.3].
Define $\psi_{0}(x)=\sum_{n\leq x}(\Lambda(n)-1)=\psi(x)-x$ so that there
exists $\Cl[abcon]{pnt}>0$ such that $\psi_{0}(x)=O(x\exp(-\Cr{pnt}\sqrt{\log
x}))$ by the prime number theorem and

$\sum_{D<d\leq x}g(d)\Lambda(d)|\mathcal{O}_{\underline{L}}(x/d,w)|^{2}\\\
=\sum_{D<d\leq x}g(d)|\mathcal{O}_{\underline{L}}(x/d,w)|^{2}+\sum_{D<d\leq
x}(\psi_{0}(d)-\psi_{0}(d-1))g(d)|\mathcal{O}_{\underline{L}}(x/d,w)|^{2}.$
(4.21)

Moreover, we may rewrite the second term of $\eqref{eq:lem6.3-sound-3}$ as

$\sum_{D\leq d\leq
x}\psi_{0}(d)\Big{(}g(d)|\mathcal{O}_{\underline{L}}(x/d,w)|^{2}-g(d+1)|\mathcal{O}_{\underline{L}}(x/d,w)|^{2}\Big{)}\\\
-\psi_{0}(D)g(D+1)|\mathcal{O}_{\underline{L}}(x/(D+1),w)|^{2}.$ (4.22)

For $d>D$ and for any $A>0$, we have

$\psi_{0}(d)\ll d\exp(-(\log\log x)^{2})\ll_{A}\frac{d}{(\log X)^{A+2}}.$

The second term in (4.22) is then $\ll D^{2}(\log
X)^{-1}|\mathcal{O}_{\underline{L}}(x/(D+1),w)|^{2}\ll
x^{2}\mathfrak{C}(\pi^{\prime})^{\varepsilon}$ upon using Proposition 4.8. On
the other hand, the first term in (4.22) is

$\ll\sum_{D<d\leq x}\frac{d}{(\log
X)^{A+2}}\Big{(}g(d)\big{|}|\mathcal{O}_{\underline{L}}(x/d,w)|^{2}-|\mathcal{O}_{\underline{L}}(x/(d+1),w)|^{2}\big{|}\\\
+|g(d+1)-g(d)||\mathcal{O}_{\underline{L}}(x/(d+1),w)|^{2}\Big{)}.$ (4.23)

By Proposition 4.8, the contribution from the second line of (4.23) is

$\ll\sum_{D<d\leq x}\frac{d}{(\log X)^{A+2}}|g(d+1)-g(d)|(x/(d+1))^{2}(\log
X)^{4\varepsilon/3}\mathfrak{C}(\pi^{\prime})^{\varepsilon}\ll
x^{2}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$

On the other hand, by Lemma 4.10, the first line of (4.23) is

$\displaystyle\ll\sum_{D<d\leq x}\frac{d^{2}}{(\log
X)^{A}}\frac{x}{d}\sum_{j=0}^{LR}w^{j}\sum_{x/((d+1)w^{j})<n\leq
x/(dw^{j})}|\lambda_{\pi\times\pi^{\prime}}(n)|$
$\displaystyle\ll\frac{x}{(\log X)^{A}}\sum_{j=0}^{LR}w^{j}\sum_{n\leq
x/w^{j}}|\lambda_{\pi\times\pi^{\prime}}(n)|\frac{x}{nw^{j}},$

which is $\ll x^{2}\mathfrak{C}(\pi^{\prime})^{\varepsilon}$ upon using (3.15)
of Lemma 3.6. Thus, the second term in (4.21) is $\ll
x^{2}\mathfrak{C}(\pi^{\prime})^{\varepsilon}$.

It remains to estimate the first term in (4.21). For any $D<d\leq x$ and
$d-1\leq t\leq d$ we have that $g(d)=g(t)+O(\log x)$, and by Lemma 4.10 that

$|\mathcal{O}_{\underline{L}}(x/d,w)|^{2}=|\mathcal{O}_{\underline{L}}(x/t,w)|^{2}+O\Big{(}\mathfrak{C}(\pi^{\prime})^{\varepsilon}\frac{x}{d}(\log
X)^{\varepsilon}\sum_{j=0}^{LR}w^{j}\sum_{x/(dw^{j})<n\leq
x/((d-1)w^{j})}|\lambda_{\pi\times\pi^{\prime}}(n)|\Big{)}.$

We multiply $g(d)$ to both sides of the above equation and replace $g(d)$ by
$g(t)+O(\log x)$ on the right-hand side. Then integrating both sides from
$d-1$ to $d$ with respect to $t$ and using Proposition 4.8 gives

$g(d)|\mathcal{O}_{\underline{L}}(x/d,w)|^{2}=\int_{d-1}^{d}g(t)|\mathcal{O}_{\underline{L}}(x/t,w)|^{2}dt+O\Big{(}\mathfrak{C}(\pi^{\prime})^{\varepsilon}\frac{x^{2}}{d^{2}}(\log
X)^{1+2\varepsilon/3}\Big{)}\\\
+O\Big{(}\mathfrak{C}(\pi^{\prime})^{\varepsilon}x(\log
X)^{1+\varepsilon}\sum_{j=0}^{LR}w^{j}\sum_{x/(dw^{j})<n\leq
x/((d-1)w^{j})}|\lambda_{\pi\times\pi^{\prime}}(n)|\Big{)}.$ (4.24)

Summing (4.24) over all $D<d\leq x$ and changing the variable $t\mapsto x/y$,
the contribution from the main term on the right-hand side of (4.24) becomes

$\int_{D}^{x}t\log(ex/t)|\mathcal{O}_{\underline{L}}(x/t,w)|^{2}dt\leq
x^{2}\int_{1}^{x}|\mathcal{O}_{\underline{L}}(y,w)|^{2}\log(ey)\frac{dy}{y^{3}}.$

The error terms in (4.24) contribute

$\displaystyle\ll\mathfrak{C}(\pi^{\prime})^{\varepsilon}\frac{x^{2}}{D}(\log
X)^{1+2\varepsilon}+\mathfrak{C}(\pi^{\prime})^{\varepsilon}x(\log
X)^{1+\varepsilon}\sum_{j=0}^{LR}w^{j}\sum_{n\leq
x/(Dw^{j})}|\lambda_{\pi\times\pi^{\prime}}(n)|$
$\displaystyle\ll\mathfrak{C}(\pi^{\prime})^{\varepsilon}x^{2}+\mathfrak{C}(\pi^{\prime})^{\varepsilon}\frac{x^{2}}{D}(\log
X)^{1+\varepsilon}\sum_{j=0}^{LR}\sum_{n\leq
x/(Dw^{j})}\frac{|\lambda_{\pi\times\pi^{\prime}}(n)|}{n}\ll\mathfrak{C}(\pi^{\prime})^{\varepsilon}x^{2},$

where we use (3.15) of Lemma 3.6 in the last step. This finishes the proof. ∎

######  Lemma 4.12.

Recall the notation in Section 4.1. If $0\leq\log w\leq(\log
X)^{\frac{1}{3R}}$ and $1\leq x\leq X$, then

$\Big{(}\int_{1}^{x}|\mathcal{O}_{\underline{L}}(t,w)|^{2}\log(et)\frac{dt}{t^{3}}\Big{)}^{\frac{1}{2}}\ll\Big{(}\int_{1}^{x}|\widetilde{\mathcal{O}}_{\underline{L}}(t,w)|^{2}\frac{dt}{t^{3}\log(et)}\Big{)}^{\frac{1}{2}}+(\log
X)^{7\varepsilon/8}\mathfrak{C}(\pi^{\prime})^{\varepsilon}.$

###### Proof.

We start as in the proof of Lemma 4.9. First, we write

$\displaystyle(\log
t)\mathcal{O}_{\underline{L}}(t,w)=-\sum_{k=1}^{R}Lw^{1+i\tau_{k}}(\log
w)\mathcal{O}_{\underline{L}-\underline{e}_{k}}(t/w,w)$ (4.25)
$\displaystyle\qquad+\sum_{\underline{j}\leq\underline{L}}(-1)^{j_{1}+\cdots+j_{R}}\binom{\underline{L}}{\underline{j}}\log(t/w^{j_{1}+\cdots+j_{R}})S(t/w^{j_{1}+\cdots+j_{R}})w^{j_{1}(1+i\tau_{1})+\cdots+j_{R}(1+i\tau_{R})}.$

By writing $\log z=\log n+\log(z/n)$ and using partial summation, we have that

$(\log z)S(z)=\log z\sum_{n\leq
z}\lambda_{\pi\times\pi^{\prime}}(n)=\widetilde{S}(z)+\sum_{n\leq
z}\lambda_{\pi\times\pi^{\prime}}(n)\log(z/n)=\widetilde{S}(z)+\int_{1}^{z}\frac{S(y)}{y}dy.$

Then the second line of (4.25) may be written as

$\widetilde{\mathcal{O}}_{\underline{L}}(t,w)+\int_{1}^{t}\frac{\mathcal{O}_{\underline{L}}(y,w)}{y}dy.$
(4.26)

Inserting (4.26) into (4.25) and using Proposition 4.8, we conclude that

$\displaystyle(\log t)\mathcal{O}_{\underline{L}}(t,w)$
$\displaystyle=\widetilde{\mathcal{O}}_{\underline{L}}(t,w)+\int_{1}^{t}\frac{\mathcal{O}_{\underline{L}}(y,w)}{y}dy-\sum_{k=1}^{R}Lw^{1+i\tau_{k}}(\log
w)\mathcal{O}_{\underline{L}-\underline{e}_{k}}(t/w,w)$
$\displaystyle=\widetilde{\mathcal{O}}_{\underline{L}}(t,w)+O(t(1+\log w)(\log
X)^{2\varepsilon/3}\mathfrak{C}(\pi^{\prime})^{\varepsilon})$
$\displaystyle=\widetilde{\mathcal{O}}_{\underline{L}}(t,w)+O(t(\log
X)^{5\varepsilon/6}\mathfrak{C}(\pi^{\prime})^{\varepsilon}),$

and hence the lemma follows. ∎

Combining Proposition 4.11 and Lemma 4.12, we have that

$|\mathcal{O}_{\underline{L}}(x,w)|\ll\frac{x(\log\log
x)^{\frac{1}{2}}\mathfrak{C}(\pi^{\prime})^{\varepsilon/4}}{\log
x}\Big{(}\int_{1}^{x}|\widetilde{\mathcal{O}}_{\underline{L}}(t,w)|^{2}\frac{dt}{t^{3}\log(et)}\Big{)}^{\frac{1}{2}}+\frac{x(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{\varepsilon}}{\log x}.$ (4.27)

We estimate the integral in (4.27) as follows.

######  Proposition 4.13.

Recall the notation in Section 4.1. If $0\leq\log w\leq(\log
X)^{\frac{1}{3R}}$ and $1\leq x\leq X$, then

$\int_{1}^{x}|\widetilde{\mathcal{O}}_{\underline{L}}(t,w)|^{2}\frac{dt}{t^{3}\log(et)}\ll(\log
X)^{3\varepsilon/2}\mathfrak{C}(\pi^{\prime})^{3\varepsilon/2}.$

###### Proof.

We follow the proof of [34, Proposition 6.5]. First, we make substitution
$t=e^{y}$, obtaining

$\displaystyle\int_{1}^{x}|\widetilde{\mathcal{O}}_{\underline{L}}(t,w)|^{2}\frac{dt}{t^{3}\log(et)}$
$\displaystyle=\int_{0}^{\log
x}|\widetilde{\mathcal{O}}_{\underline{L}}(e^{y},w)|^{2}e^{-2y}\frac{dy}{1+y}$
(4.28) $\displaystyle\ll\int_{1/\log
X}^{\infty}e^{-2\alpha}\int_{0}^{\infty}|\widetilde{\mathcal{O}}_{\underline{L}}(e^{y},w)|^{2}e^{-2y(1+\alpha)}dyd\alpha.$

As in the proof of [34, Proposition 6.5], the inner integral of (4.28) is

$\ll\int_{-\infty}^{\infty}|L^{\prime}(1+\alpha+it,\pi\times\pi^{\prime})|\prod_{k=1}^{R}|1-w^{-\alpha-
it+i\tau_{k}}|^{2L}\frac{dt}{|1+\alpha+it|^{2}}.$ (4.29)

Recall that $T=\exp((\log\log X)^{2})$. We split the integral above into the
ranges when $|t|\leq T/2$ and when $|t|>T/2$. Note that $\alpha\geq 1/\log X$
in (4.28). For the latter range, we use (3.17) of Lemma 3.6 to bound
$|L^{\prime}(1+\alpha+it,\pi\times\pi^{\prime})|$. Thus, for any $A>0$, the
portion of the integral in (4.29) over $|t|>T/2$ contributes

$\ll(\log
X)^{\frac{m^{2}+3}{2}}\mathfrak{C}(\pi^{\prime})^{\varepsilon}\int_{|t|>T/2}\frac{dt}{|1+\alpha+it|^{2}}\ll_{A}\frac{\mathfrak{C}(\pi^{\prime})^{\varepsilon}}{(\log
X)^{A}}.$ (4.30)

When $|t|\leq T/2$, Lemmas 4.6 and 4.7 yield

$\displaystyle|L^{\prime}(1+\alpha+it,\pi\times\pi^{\prime})|\prod_{k=1}^{R}|1-w^{-\alpha-
it+i\tau_{k}}|^{L}$
$\displaystyle\ll\Big{|}\frac{L^{\prime}}{L}(1+\alpha+it,\pi\times\pi^{\prime})\Big{|}|\mathcal{L}_{\underline{L}}(1+\alpha+it)|$
$\displaystyle\ll\Big{|}\frac{L^{\prime}}{L}(1+\alpha+it,\pi\times\pi^{\prime})\Big{|}(\log
X)^{\varepsilon/2}\mathfrak{C}(\pi^{\prime})^{\varepsilon/2}.$

Therefore, the integral in (4.29) over $|t|\leq T/2$ contributes

$\displaystyle\ll(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{\varepsilon}\int_{|t|\leq
T/2}\Big{|}\frac{L^{\prime}}{L}(1+\alpha+it,\pi\times\pi^{\prime})\Big{|}^{2}\frac{dt}{|1+\alpha+it|^{2}}$
(4.31) $\displaystyle\ll(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{\varepsilon}\int_{-\infty}^{\infty}\Big{|}\frac{L^{\prime}}{L}(1+\alpha+it,\pi\times\pi^{\prime})\Big{|}^{2}\frac{dt}{|1+\alpha+it|^{2}}.$

Observe that the Fourier transform of the function
$e^{-y(1+\alpha)}\sum_{n\leq e^{y}}\Lambda_{\pi\times\pi^{\prime}}(n)$ is

$\int_{-\infty}^{\infty}\sum_{n\leq
e^{y}}\Lambda_{\pi\times\pi^{\prime}}(n)e^{-y(1+\alpha+it)}dy=\sum_{n=1}^{\infty}\frac{\Lambda_{\pi\times\pi^{\prime}(n)}}{n^{1+\alpha+it}}\frac{1}{1+\alpha+it}=-\frac{L^{\prime}}{L}(1+\alpha+it)\cdot\frac{1}{1+\alpha+it}.$

Thus, by Plancherel's theorem and Lemma 3.5, (4.31) is

$\ll(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{\varepsilon}\int_{-\infty}^{\infty}\Big{|}\sum_{n\leq
e^{y}}\Lambda_{\pi\times\pi^{\prime}}(n)\Big{|}^{2}e^{-(2+2\alpha)y}dy\ll\frac{(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{3\varepsilon/2}}{2\alpha},$ (4.32)

where we use that $y\geq 0$ in the last step. By (4.30) and (4.32), we then
conclude that

$\int_{0}^{\infty}|\widetilde{\mathcal{O}}_{\underline{L}}(e^{y},w)|^{2}e^{-y(2+2\alpha)}dy\ll\frac{\mathfrak{C}(\pi^{\prime})^{\varepsilon}}{(\log
X)^{A}}+\frac{(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{3\varepsilon/2}}{2\alpha}.$

Inserting the above bound into (4.28), we obtain the bound

$\displaystyle\int_{1}^{x}|\widetilde{\mathcal{O}}_{\underline{L}}(t,w)|^{2}\frac{dt}{t^{3}\log(et)}$
$\displaystyle\ll\frac{\mathfrak{C}(\pi^{\prime})^{\varepsilon}}{(\log
X)^{A}}+(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{3\varepsilon/2}\int_{1/\log
X}^{\infty}\frac{e^{-2\alpha}}{2\alpha}d\alpha$ $\displaystyle\ll(\log
X)^{\varepsilon}\mathfrak{C}(\pi^{\prime})^{3\varepsilon/2}\log\log(X).$

The proposition follows. ∎

###### Proof of Theorem 4.2.

Using the bound from Proposition 4.13 in (4.27), we then obtain Theorem 4.2 as
desired. ∎

##  5\. Background on $\operatorname{GL}(2)$ automorphic forms

Having established Theorem 1.3 in the previous section, we now shift our focus
to laying some foundation for proving Theorem 1.1. We recall here some
standard properties of classical automorphic forms, including holomorphic
newforms, Hecke–Maaß cusp forms, Eisenstein series, incomplete Eisenstein
series, and incomplete Poincaré series. We will also discuss the spectral
decomposition and the Poincaré decomposition. For complete definitions and
proofs, see, for example, [16], [29], and [30].

###  5.1. Holomorphic newforms.

Let $k\geq 2$ be an even integer and let
$\alpha\in\operatorname{GL}_{2}(\mathbb{R})$ with positive determinant, where
the element $\alpha$ acts on $\mathbb{H}$ by fractional linear
transformations. Given a function $f:\mathbb{H}\to\mathbb{C}$, we denote
$j((\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}),z)=cz+d$ and recall the
slash operator $f|_{k}\alpha(z)=(\det\alpha)^{k/2}j(\alpha,z)^{-k}f(\alpha
z)$.

A holomorphic cusp form on $\Gamma_{0}(q)$ of weight $k$ with trivial
nebentypus is a holomorphic function $f:\mathbb{H}\to\mathbb{C}$ such that
$f|_{k}\gamma=f$ for all $\gamma\in\Gamma_{0}(q)$ and $f$ vanishes at the
cusps of $\Gamma_{0}(q)$. We shall denote the vector space of holomorphic cusp
forms on $\Gamma_{0}(q)$ of weight $k$ with trivial nebentypus as
$S_{k}(\Gamma_{0}(q))$.

Define $S_{k}^{\operatorname{new}}(\Gamma_{0}(q))$ to be the subspace of
$S_{k}(\Gamma_{0}(q))$ that is orthogonal to the subspace of oldforms, denoted
as $S_{k}^{\textup{old}}(\Gamma_{0}(q))$, with respect to the Petersson inner
product. A holomorphic newform is a cusp form in
$S_{k}^{\operatorname{new}}(\Gamma_{0}(q))$ that is also an eigenform of all
of the Hecke operators and all of the Atkin–Lehner involutions. The space
$S_{k}^{\operatorname{new}}(\Gamma_{0}(q))$ has a basis of newforms, which we
shall denote as $\mathcal{B}_{k}(q)$ (see more discussion in [30, Chapter 2,
Section 5]).

Consider $f\in\mathcal{B}_{k}(q)$. Let $e(z)=e^{2\pi iz}$ and let
$\lambda_{f}(n)$ be the eigenvalue of the $n$-th Hecke operator $T_{n}$. Then
the Fourier expansion of $f$ can be written as

$f(z)=\sum_{n\geq 1}\lambda_{f}(n)n^{\frac{k-1}{2}}e(nz),$ (5.1)

where $\lambda_{f}(1)=1$. If we let $\tau(n)$ denote the number of positive
divisors of $n$, then $\lambda_{f}(n)$ is real, multiplicative and satisfies
Deligne's bound $|\lambda_{f}(n)|\leq\tau(n)$.

For any $\gamma\in\Gamma_{0}(q)$ and $f,g\in\mathcal{B}_{k}(q)$, if we write
$z^{\prime}=\gamma z=x^{\prime}+iy^{\prime}$, then

$y^{\prime k/2}f(z^{\prime})=(j(\gamma,z)/|j(\gamma,z)|)^{k}y^{k/2}f(z).$

This observation implies the $\Gamma_{0}(q)$-invariance for both $z\mapsto
y^{k}|f(z)|^{2}$ and $z\mapsto y^{k}f(z)\overline{g(z)}$. By the
Rankin–Selberg theory, we have

$|\rho_{f}(1)|^{2}\asymp\frac{(4\pi)^{k}}{kq\Gamma(k-1)L(1,\operatorname{Ad}f)}.$
(5.2)

###  5.2. Cusps of $\Gamma_{0}(q)$

We now recall some properties of cusps of $\Gamma_{0}(q)$ and the Fourier
expansion of holomorphic newforms at arbitrary cusps of $\Gamma_{0}(q)$ (see
more discussion in [29, Sections 3.4.1 and 3.4.2]). This will be crucial when
we apply Holowinsky's strategy in Section 8.2.

Let $\tau$ traverse a set of representatives for the double coset space
$\Gamma_{\infty}\backslash\Gamma_{0}(1)/\Gamma_{0}(q)$. Then
$\mathfrak{a}:=\tau^{-1}\infty\in\mathbb{P}^{1}(\mathbb{Q})$ traverses a set
of inequivalent cusps of $\Gamma_{0}(q)$. Denote the set of cusps of
$\Gamma_{0}(q)$ as
$\mathcal{C}=\mathcal{C}(\Gamma_{0}(q))=\\{\mathfrak{a}_{j}\\}_{j}$. The width
of $\mathfrak{a}_{j}=\tau_{j}^{-1}\infty$ is given by
$w_{j}=[\Gamma_{\infty}:\Gamma_{\infty}\cap\tau_{j}\Gamma_{0}(q)\tau_{j}^{-1}]$,
and the scaling matrix of $\mathfrak{a}_{j}$ is

$\sigma_{j}=\tau_{j}^{-1}\begin{pmatrix}w_{j}&\\\ &1\end{pmatrix}.$ (5.3)

By the discussion in [29, Section 3.4.1], if the bottom row of $\sigma_{j}$ is
$(c_{j},d_{j})$, then we define $c_{j}$ as the denominator of the cusp
$\mathfrak{a}_{j}$. Such $c_{j}$ is a positive divisor of $q$, and we have
that the width $w_{j}$ equals $q/(c_{j}^{2},q)=[q/c_{j}^{2},1]$. For each
positive divisor $c|q$, if we denote the set of cusps of denominator $c$ by

$\mathcal{C}[c]:=\\{\mathfrak{a}_{j}\in\mathcal{C}:c_{j}=c\\},$

then $\\#\mathcal{C}[c]$ equals $\varphi((c,q/c))$. If $q$ is squarefree, then
$\\#\mathcal{C}[c]=1$. In that case, the multiset of the widths of cusps of
$\Gamma_{0}(q)$ is $\\{d\in\mathbb{Z}:d|q,d\geq 1\\}$.

We now collect necessary properties of the Fourier expansion of $|f|^{2}$ at
the cusp $\mathfrak{a}_{j}$. For any positive divisor $c|q$ and
$d\in(\mathbb{Z}/(c,q/c))^{\times}$, we write
$\mathfrak{a}_{d/c}\in\mathcal{C}(\Gamma_{0}(q))$ for the corresponding cusp.
For $\mathfrak{a}_{j}=\mathfrak{a}_{d/c}$, we may take
$\tau_{j}=\begin{pmatrix}*&*\\\ c&d^{\prime}\end{pmatrix}$ for any integer
$d^{\prime}$ for which $(d^{\prime},c)=1$ and $d^{\prime}\equiv
d\pmod{(c,q/c)}$. Such $\tau_{j}$'s then form a set of representatives of
$\Gamma_{\infty}\backslash\Gamma_{0}(1)/\Gamma_{0}(q)$, and the corresponding
scaling matrix $\sigma_{j}$ is defined as in (5.3). Write $z_{j}=x_{j}+iy_{j}$
for the change of variable $z_{j}:=\sigma_{j}^{-1}z$. We observe that
$|f(z)|^{2}y^{k}=|f(\sigma_{j}z_{j})|^{2}\operatorname{Im}(\sigma_{j}z_{j})^{k}=|f|_{k}\sigma_{j}(z_{j})|^{2}y_{j}^{k}$.
For some coefficients $\lambda_{j}(n)\in\mathbb{C}$, we may write

$f|_{k}\sigma_{j}(z_{j})=\sum_{n\geq
1}\lambda_{j}(n)n^{\frac{k-1}{2}}e(nz_{j}).$ (5.4)

In the special case that $\mathfrak{a}_{j}=\infty$, we observe that
$\lambda_{j}(n)=\lambda(n).$

In general, $\lambda_{j}$ is not multiplicative. To circumvent the lack of
multiplicativity, we shall work with the root-mean-square of $\lambda_{j}$
taken over all cusps of a given denominator. For each positive divisor $c$ of
$q$, we define

$\lambda_{[c]}(n):=\Big{(}\frac{1}{\\#\mathcal{C}[c]}\sum_{\mathfrak{a}_{j}\in\mathcal{C}[c]}|\lambda_{j}(n)|^{2}\Big{)}^{1/2}.$
(5.5)

As discussed in [29, Section 3.4.2], $\lambda_{[c]}(n)$ turns out to be
multiplicative in a nonconventional sense, which allows us to deduce
Proposition 8.3. However, when applying Holowinsky's method in Section 8.2, we
will encounter shifted convolution sums in terms of $\lambda_{j}$, not
$\lambda_{[c]}$ as in Proposition 8.3. Hence, we have to apply the following
steps. By (5.6) and the Cauchy–Schwarz inequality, we have

$\displaystyle\sum_{a_{j}\in\mathcal{C}}|\lambda_{j}(n_{1})\lambda_{j}(n_{2})|$
$\displaystyle=\sum_{c\mid
q}\sum_{\mathfrak{a}_{j}\in\mathcal{C}[c]}|\lambda_{j}(n_{1})\lambda_{j}(n_{2})|$
(5.6) $\displaystyle\leq\sum_{c\mid
q}\\#\mathcal{C}[c]\Big{(}\sum_{\mathfrak{a}_{j}\in\mathcal{C}[c]}\frac{|\lambda_{j}(n_{1})|^{2}}{\\#\mathcal{C}[c]}\Big{)}^{1/2}\Big{(}\sum_{\mathfrak{a}_{j}\in\mathcal{C}[c]}\frac{|\lambda_{j}(n_{2})|^{2}}{\\#\mathcal{C}[c]}\Big{)}^{1/2}$
$\displaystyle=\sum_{c\mid
q}\\#\mathcal{C}[c]\lambda_{[c]}(n_{1})\lambda_{[c]}(n_{2}).$

The bound (5.6) then allows us to work with $\lambda_{[c]}$ as desired.

###  5.3. Hecke–Maaß cusp forms

A Maaß cusp form $\phi$ of level 1 is an $\Gamma_{0}(1)$-invariant
eigenfunction of the hyperbolic Laplacian
$\Delta:=-y^{2}(\partial_{x}^{2}+\partial_{y}^{2})$ on $\mathbb{H}$ that
decays rapidly at infinity. We denote the spectral parameter of $\phi$ by
$t_{\phi}$, which is given by $\Delta\phi=(1/4+t_{\phi}^{2})\phi$. A
Hecke–Maaß cusp form is a Maaß cusp form that is also an eigenfunction of all
of the Hecke operators and the involution
$T_{-1}:\phi\mapsto[z\mapsto\phi(-\overline{z})],$ which commute with one
another as well as with $\Delta$. A Hecke–Maaß cusp form $\phi$ admits a
Fourier expansion

$\phi(z)=\sum_{n\in\mathbb{Z}_{\neq
0}}\frac{\lambda_{\phi}(n)}{\sqrt{|n|}}\kappa_{\phi}(ny)e(nx),$

where
$\kappa_{\phi}(y)=2|y|^{1/2}K_{ir}(2\pi|y|)\mathrm{sgn}(y)^{\frac{1-\delta}{2}}$
with $K_{ir}$ the standard $K$-Bessel function, $\mathrm{sgn}(y)=1$ or $-1$
according as $y$ is positive or negative, and $\delta\in\\{\pm 1\\}$, the
$T_{-1}$-eigenvalue of $\phi$. A normalized Hecke–Maaß cusp form is a
Hecke–Maaß cusp form that satisfies $\lambda_{\phi}(1)=1$. In that case, the
coefficients $\lambda_{\phi}(n)$ are real and multiplicative. An orthonormal
basis of Maaß cusp forms can be chosen such that it consists of all normalized
Hecke–Maaß cusp forms.

Throughout this paper, we will normalize the Petersson norm of a Hecke–Maaß
cusp form $\phi$ to be 1, i.e., $\langle\phi,\phi\rangle_{1}=1$. We will also
assume that $\phi$ is an even Hecke–Maaß cusp form; otherwise,
$\langle\phi,|F_{k}|^{2}\rangle_{q}=0$. Additionally, define
$N_{\Gamma_{0}(1)}(T):=|\\{\phi:|t_{\phi}|\leq T\\}|$, the number of
Hecke–Maaß cusp forms such that $|t_{\phi}|\leq T$. By Weyl's law (see, for
example, [14, Corollary 11.2]), we have that

$N_{\Gamma_{0}(1)}(T)=\frac{T^{2}}{12}+O(T\log T).$ (5.7)

###  5.4. Eisenstein series

Let $s\in\mathbb{C}$, $z\in\mathbb{H}$, and
$\Gamma_{\infty}=\big{\\{}\pm\big{(}\begin{smallmatrix}1&n\\\
&1\end{smallmatrix}\big{)}:n\in\mathbb{Z}\big{\\}}$. The real-analytic
Eisenstein series of level 1 is defined as

$E(s,z)=\sum_{\Gamma_{\infty}\backslash\Gamma_{0}(1)}\operatorname{Im}(\gamma
z)^{s}.$

The Eisenstein series $E(s,z)$ converges normally for $\operatorname{Re}(s)>1$
and continues meromorphically to the half-plane $\operatorname{Re}(s)\geq
1/2$, where the map $s\mapsto E(s,z)$ is holomorphic with the exception of a
unique simple pole at $s=1$ of residue $\operatorname{res}_{s=1}E(s,z)=3/\pi$.
For all $\gamma\in\Gamma_{0}(1)$, the Eisenstein series satisfies the
invariance $E(s,\gamma z)=E(s,z)$.

Let $\zeta(s)$ be the Riemann zeta function and $K_{s}(y)$ be the standard
$K$-Bessel function. Let $\lambda_{s}(n)=\sum_{ab=n}(a/b)^{s}$ and
$\theta(s)=\pi^{-s}\Gamma(s)\zeta(2s)$. The Eisenstein series admits the
Fourier expansion

$E(s,z)=y^{s}+\frac{\theta(1-s)}{\theta(s)}y^{1-s}+\frac{2\sqrt{y}}{\theta(s)}\sum_{n\in\mathbb{Z}_{\neq
0}}\tau_{s-1/2}(|n|)K_{s-1/2}(2\pi|n|y)e(nx).$

For $\operatorname{Re}(s)=1/2$, we obtain $|\theta(1-s)/\theta(s)|=1$, and
such $E(s,z)$ is called a unitary Eisenstein series. Let $E_{t}$ denote
$E(1/2+it,\cdot)$. The work of Huang–Xu [10, Theorem 1] shows that for $y\geq
1/2$, if $z=x+iy$, then

$|E_{t}(z)|\ll\sqrt{y}(1+|t|)^{3/8+\varepsilon}.$ (5.8)

Throughout the paper, we will normalize the Petersson norm of $E_{t}$ to be 1,
i.e., $\langle E_{t},E_{t}\rangle_{1}=1.$

###  5.5. Spectral decomposition

Now we recall the spectral decomposition, which will be important when
applying Soundararajan's strategy in Section 8.1. Let
$Y_{0}(1)=\Gamma_{0}(1)\backslash\mathbb{H}$, and let $L^{2}(Y_{0}(1))$ denote
the space of square-integrable functions on $Y_{0}(1)$. Recall the definition
of the Petersson inner product in (1.7). Then one can spectrally decompose any
$\psi\in L^{2}(Y_{0}(1))$ into three types of spectra: a constant function,
Hecke–Maaß cusp forms, and unitary Eisenstein series. See, for example, [16,
Theorem 15.5].

######  Lemma 5.1 (Theorem 15.5 of [16]).

Let $\\{\phi_{j}\\}_{j}$ be an orthonormal basis of Hecke–Maaß cusp forms. For
any real number $t$, let $E_{t}:=E(1/2+it,\cdot)$ be a unitary Eisenstein
series. If $\psi\in L^{2}(Y_{0}(1))$, then

$\psi(z)=\frac{3}{\pi}\langle\psi,1\rangle_{1}+\sum_{j\geq
1}\langle\psi,\phi_{j}\rangle_{1}\phi_{j}(z)+\frac{1}{4\pi}\int_{\mathbb{R}}\langle\psi,E_{t}\rangle_{1}E_{t}(z)dt.$

The above decomposition gives us that

$\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}=\frac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}+\sum_{j\geq
1}\langle\psi,\phi_{j}\rangle_{1}\langle\phi_{j},\overline{F_{k}}G_{k}\rangle_{q}+\frac{1}{4\pi}\int_{\mathbb{R}}\langle\psi,E_{t}\rangle_{1}\langle
E_{t},\overline{F_{k}}G_{k}\rangle_{q}dt.$ (5.9)

This implies that we need some good estimates for
$\langle\phi_{j},\overline{F_{k}}G_{k}\rangle_{q}$ and $\langle
E_{t},\overline{F_{k}}G_{k}\rangle_{q}$, along with estimates for
$\langle\psi,\phi_{j}\rangle_{1}$ and $\langle\psi,E_{t}\rangle_{1}$, to
obtain a rate of convergence for
$\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}$. We recall the following
estimates for $\langle\psi,\phi_{j}\rangle_{1}$ and
$\langle\psi,E_{t}\rangle_{1}$.

######  Lemma 5.2 (Lemma 2.2 of [9]).

Let $M\geq 1$, $\psi\in C^{\infty}_{c}(Y_{0}(1),M)$, and $\phi$ be a
Hecke–Maaß cusp form with spectral parameter $t_{\phi}$. If $A\geq 0$, then

$|\langle\psi,\phi\rangle_{1}|\ll_{A}\Big{(}\frac{M}{1+|t_{\phi}|}\Big{)}^{A}\quad\text{and}\quad|\langle\psi,E_{t}\rangle_{1}|\ll_{A}\Big{(}\frac{M}{1+|t|}\Big{)}^{A}(1+|t|)^{3/8+\varepsilon}.$

In the next subsection, we shall discuss the Poincaré decomposition, which
will play an important role when applying Holowinsky's method in Section 8.2.

###  5.6. Incomplete Poincaré series and the Poincaré decomposition.

Recall the notation in Section 1. The discussion in this subsection closely
follows [6, Section 4]. Let $M\geq 1$ and $\psi\in
C^{\infty}_{c}(Y_{0}(1),M)$. Let $\check{\psi}$ be the function on
$\mathbb{H}$ to $\mathbb{C}$ such that $\check{\psi}(z)=\psi(z)$ if
$z\in\mathcal{F}$, and $\check{\psi}(z)=0$ otherwise. Let $\check{\Psi}(z)$ be
the extension of $\check{\psi}(z)$ to $\mathbb{H}$ by
$\Gamma_{\infty}=\big{\\{}\pm\big{(}\begin{smallmatrix}1&n\\\
&1\end{smallmatrix}\big{)}:n\in\mathbb{Z}\big{\\}}$. For any
$z=x+iy\in\mathbb{H}$, we denote the $m$-th Fourier coefficients of
$\check{\Psi}(z)$ as

$\Psi_{m}(y)=\int_{-1/2}^{1/2}\check{\Psi}(x+iy)e(-mx)\,dx.$

By integration by parts, we have that

$|\Psi_{m}(y)|\ll_{A}(M/|m|)^{A}.$ (5.10)

Then the Fourier expansion of $\check{\Psi}(z)$ can be defined as

$\check{\Psi}(z)=\sum_{m\in\mathbb{Z}}\Psi_{m}(y)e(mx).$

For any $\Psi\in C^{\infty}(\mathbb{R})$, a smooth function on $\mathbb{R}$,
we define the incomplete Poincaré series as

$P_{m}(z,\Psi)=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1)}e(m\operatorname{Re}(\gamma
z))\Psi(\operatorname{Im}(\gamma z)).$ (5.11)

When $m=0$, $P_{0}(z,\Psi)$ is called the incomplete Eisenstein series, also
denoted by $E(\cdot|\Psi)$. Observe that
$\psi(z)=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1)}\check{\Psi}(\gamma
z)$. Then we have that

$\psi(z)=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1)}\sum_{m\in\mathbb{Z}}\Psi_{m}(\operatorname{Im}(\gamma
z))e(m\operatorname{Re}(\gamma z))=\sum_{m\in\mathbb{Z}}P_{m}(z,\Psi_{m}).$
(5.12)

Therefore, by (5.12), we obtain that

$\displaystyle\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}$
$\displaystyle=\langle
E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\langle
E(\cdot|\Psi_{0}),1\rangle_{1}\mathds{1}_{f=g}$ (5.13)
$\displaystyle+\sum_{m\neq 0}\Big{(}\langle
P_{m}(\cdot,\Psi_{m}),\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\langle
P_{m}(\cdot,\Psi_{m}),1\rangle_{1}\mathds{1}_{f=g}\Big{)}.$

For any $\Psi\in C^{\infty}(\mathbb{R})$, let
$\widehat{\Psi}(s)=\int_{0}^{\infty}\Psi(y)y^{s-1}\,dy$ denote the Mellin
transform of $\Psi(y)$. By (5.11) and the $\Gamma_{0}(1)$-invariance of the
hyperbolic measure $y^{-2}dxdy$, we have that

$\displaystyle\langle E(\cdot|\Psi),1\rangle_{1}$
$\displaystyle=\int_{\Gamma_{0}(1)\backslash\mathbb{H}}\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1)}\Psi(\operatorname{Im}(\gamma
z))\frac{dxdy}{y^{2}}=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1)}\int_{\gamma\Gamma_{0}(1)\backslash\mathbb{H}}\Psi(\operatorname{Im}(z))\frac{dxdy}{y^{2}}.$

Upon unfolding and integrating over $dx$, we obtain that

$\displaystyle\langle E(\cdot|\Psi),1\rangle_{1}$
$\displaystyle=\int_{\Gamma_{\infty}\backslash\mathbb{H}}\Psi(y)\frac{dxdy}{y^{2}}=\int_{y=0}^{\infty}\int_{x=-1/2}^{1/2}\Psi(y)\frac{dxdy}{y^{2}}=\widehat{\Psi}(-1).$
(5.14)

Similarly, for the incomplete Poincaré series, we have that

$\langle
P_{m}(\cdot,\Psi),1\rangle_{1}=\int_{\Gamma_{\infty}\backslash\mathbb{H}}\Psi(y)e(mx)\frac{dxdy}{y^{2}}=\int_{y=0}^{\infty}\int_{x=-1/2}^{1/2}\Psi(y)e(mx)\frac{dxdy}{y^{2}}=0.$
(5.15)

By (5.13), (5.14), and (5.15), it follows that

$\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}\\\
=\langle
E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\widehat{\Psi}_{0}(-1)\mathds{1}_{f=g}+\sum_{m\neq
0}\langle P_{m}(\cdot,\Psi_{m}),\overline{F_{k}}G_{k}\rangle_{q}.$ (5.16)

##  6\. Outline of the proof for Theorem 1.1

In this section, we prove Theorem 1.1 assuming the following two key
propositions. The proofs of our key propositions rely on the bounds in Lemmas
7.3 and 7.4, which are proved using Theorem 1.3. Define

$Q=Q(f,g)=\begin{cases}q^{2},&\mbox{if $f\neq g$ and $q$ is squarefree,}\\\
N_{\operatorname{Ad}f},&\mbox{if $f=g$.}\end{cases}$ (6.1)

###  6.1. Key propositions

To derive Theorem 1.1, we bound
$\langle\psi,\overline{F_{k}}{G_{k}}\rangle_{q}-\tfrac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}$
in two different ways. Our first result, based on Soundararajan's approach in
[34], relies crucially on Theorem 1.3 and the variant of Watson's triple
product formula proved in [28, 29].

######  Proposition 6.1.

Keep the notation and hypotheses of Theorem 1.1. Let $\lambda_{f}(n)$ and
$\lambda_{g}(n)$ denote the $n$-th Hecke eigenvalues of $f$ and $g$,
respectively. If $\varepsilon>0$, then

$\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}\ll
M^{\delta+\varepsilon}(\log
kq)^{\varepsilon}\Big{(}\frac{q}{\sqrt{Q}}\Big{)}^{-(\frac{1}{2}-\theta)+\varepsilon}\prod_{p\leq
k\sqrt{Q}}\Big{(}1-\frac{\lambda_{f}(p)^{2}+\lambda_{g}(p)^{2}-1}{2p}\Big{)},$

where $\delta=\frac{23}{12}$ when $f=g$ and $\delta=2$ when $f\neq g$.

Our second result is based on Holowinsky's approach in [6].

######  Proposition 6.2.

Keep the notation and hypotheses of Theorem 1.1. Let $\lambda_{f}(n)$ and
$\lambda_{g}(n)$ denote the $n$-th Hecke eigenvalues of $f$ and $g$,
respectively. If $\varepsilon>0$, then

$\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}\ll
M^{4+\varepsilon}(\log
kq)^{\varepsilon}\Big{(}\frac{q}{\sqrt{Q}}\Big{)}^{\varepsilon}\prod_{p\leq
k\sqrt{Q}}\Big{(}1-\frac{(|\lambda_{f}(p)|-1)^{2}+(|\lambda_{g}(p)|-1)^{2}}{4p}\Big{)}.$

###  6.2. Proof of Theorem 1.1

We combine Propositions 6.1 and 6.2 and follow the optimization technique in
Iwaniec's course notes. First, observe that

$\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}\\\
\ll\min\Big{\\{}M^{\delta+\varepsilon}(\log
kq)^{\varepsilon}\Big{(}\frac{q}{\sqrt{Q}}\Big{)}^{-(\frac{1}{2}-\theta)+\varepsilon}\prod_{p\leq
k\sqrt{Q}}\Big{(}1-\frac{\lambda_{f}(p)^{2}+\lambda_{g}(p)^{2}-1}{2p}\Big{)},\\\
M^{4+\varepsilon}\Big{(}\frac{q}{\sqrt{Q}}\Big{)}^{\varepsilon}(\log
kq)^{\varepsilon}\prod_{p\leq
k\sqrt{Q}}\Big{(}1-\frac{(|\lambda_{f}(p)|-1)^{2}+(|\lambda_{g}(p)|-1)^{2}}{4p}\Big{)}\Big{\\}}.$

Let $\alpha\in[0,1]$. If $X,Y>0$, then $\min\\{X,Y\\}\leq
X^{\alpha}Y^{1-\alpha}$. By Deligne's bound, we have that
$|\lambda_{f}(p)|,|\lambda_{g}(p)|\leq 2$. Write
$\lambda_{f}=|\lambda_{f}(p)|$ and $\lambda_{g}=|\lambda_{g}(p)|$. Since

$\varpi=\min_{\alpha\in[0,1]}\max_{\lambda_{f}\in[0,2]}\max_{\lambda_{g}\in[0,2]}-\Big{(}\alpha\Big{(}\frac{\lambda_{f}^{2}+\lambda_{g}^{2}-1}{2}\Big{)}+(1-\alpha)\Big{(}\frac{(\lambda_{f}-1)^{2}+(\lambda_{g}-1)^{2}}{4}\Big{)}\Big{)}=-\Big{(}\frac{7}{2}-2\sqrt{3}\Big{)}$

and the minimum is achieved when $\alpha$ equals $\alpha_{0}=2/\sqrt{3}-1$, it
follows from Mertens' theorem that

$\displaystyle\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}$
$\displaystyle\ll M^{4-(4-\delta)\alpha_{0}+\varepsilon}(\log
kq)^{\varepsilon}\Big{(}\frac{q}{\sqrt{Q}}\Big{)}^{-\alpha_{0}(\frac{1}{2}-\theta)+\varepsilon}\prod_{p\leq
k\sqrt{Q}}\Big{(}1+\frac{\varpi}{p}\Big{)}$ (6.2) $\displaystyle\ll M^{4}(\log
kq)^{\varepsilon}\Big{(}\frac{q}{\sqrt{Q}}\Big{)}^{-(\frac{2}{\sqrt{3}}-1)(\frac{1}{2}-\theta)+\varepsilon}\Big{(}\frac{1}{\log(k\sqrt{Q})}\Big{)}^{\frac{7}{2}-2\sqrt{3}}.$

We then invoke the bound $(\log
k\sqrt{Q})^{-1}\ll(q/\sqrt{Q})^{\varepsilon}(\log kq)^{-1}$ and recall the
definition of $Q$ in (6.1). Theorem 1.1 then follows upon using the fact that
$(q/q_{0})^{1/2}\ll q/\sqrt{Q}$, which follows from [29, Proposition 2.5].

When $q=1$ and $f=g$, the exponent of $M$ in (6.2) can be refined to
$\tfrac{1}{12}(23-2\sqrt{3})$. To refine Proposition 6.2 when $q=1$, we
directly follow the proof of [22, Lemma 4.5] for the contribution from
incomplete Poincaré series and Lemma 8.9 for incomplete Eisenstein series.
Then we obtain

$\langle\psi,|F_{k}|^{2}\rangle_{1}-\frac{3}{\pi}\langle\psi,1\rangle_{1}\ll
M^{\frac{5}{3}+\varepsilon}(\log k)^{\varepsilon}\prod_{p\leq
k}\Big{(}1-\frac{(|\lambda_{f}(p)|-1)^{2}}{2p}\Big{)}.$ (6.3)

We now refine Proposition 6.1 when $q=1$ and $f=g$. We follow [22, Lemma 4.4]
for the Eisenstein series contribution, except now we slightly improve the
bound for $L(\frac{1}{2}+it,f\times f)$ by using that
$\zeta(\frac{1}{2}+it)\ll(1+|t|)^{13/84+\varepsilon}$. For the contribution
from Hecke–Maaß cusp forms, we improve the exponent of $|t_{\phi}|$ in Lemma
7.3 by using that $\mathfrak{C}(\operatorname{Ad}f\times\phi)\ll
k^{4}(1+|t_{\phi}|)^{2}$, which holds when $|t_{\phi}|\leq k$. In the proof of
Proposition 6.1, we can then improve the exponent of $|t_{\phi}|$ to $17/12$
in the range $|t_{\phi}|\leq M(\log k)^{\varepsilon/4}$ (see (8.14)). Hence,
we obtain

$\langle\psi,|F_{k}|^{2}\rangle_{1}-\frac{3}{\pi}\langle\psi,1\rangle_{1}\ll
M^{17/12+\varepsilon}(\log k)^{\varepsilon}\prod_{p\leq
k}\Big{(}1-\frac{2\lambda_{f}(p)^{2}-1}{2p}\Big{)}.$ (6.4)

Combining (6.3) and (6.4) as in the beginning of this proof, we obtain the
desired result.

##  7\. Bounds for $L$-functions

In this section, we prove useful lower bounds for the adjoint lift of a
holomorphic newform. We shall also establish some bounds for $L$-functions
that will be useful for proving Propositions 6.1 and 6.2.

Let $\pi\in\mathfrak{F}_{2}$ be non-dihedral with central character
$\omega_{\pi}$. The adjoint $L$-function of $\pi$ is given by

$L(s,\operatorname{Ad}\pi)=L(s,\pi\times\widetilde{\pi})\zeta(s)^{-1}.$

Since $\pi$ is non-dihedral, the work of Gelbert and Jacquet [4] shows that
$\operatorname{Ad}\pi\in\mathfrak{F}_{3}$. Thus, $L(s,\operatorname{Ad}\pi)$
admits all of the analytic properties of a standard $L$-function in Section
3.1. Define the gamma factor of $L(s,\operatorname{Ad}\pi)$ as

$L_{\infty}(s,\operatorname{Ad}\pi)=\pi^{-\frac{3s}{2}}\Gamma\Big{(}\frac{s+1}{2}\Big{)}\Gamma\Big{(}\frac{s+k-1}{2}\Big{)}\Gamma\Big{(}\frac{s+k}{2}\Big{)}$

if $\pi$ is a holomorphic newform of weight $k$ and

$L_{\infty}(s,\operatorname{Ad}\pi)=\pi^{-\frac{3s}{2}}\Gamma\Big{(}\frac{s}{2}\Big{)}\Gamma\Big{(}\frac{s}{2}+it\Big{)}\Gamma\Big{(}\frac{s}{2}+it\Big{)}$

if $\pi$ is a Hecke–Maaß form with eigenvalue $\lambda=\frac{1}{4}+t^{2}$.
Since $\pi$ is non-dihedral, $L(s,\operatorname{Ad}\pi)$ has no pole at $s=1$,
so the completed $L$-function

$\Lambda(s,\operatorname{Ad}\pi)=N_{\operatorname{Ad}\pi}^{s/2}L(s,\operatorname{Ad}\pi)L_{\infty}(s,\operatorname{Ad}\pi)$

is entire of order 1, and satisfies the functional equation
$\Lambda(s,\operatorname{Ad}\pi)=\Lambda(1-s,\operatorname{Ad}\pi)$.

Recall the definition of the analytic conductor in (3.3) and write
$\mathfrak{C}(\pi)=N_{\pi}K_{\pi}$ and
$\mathfrak{C}(\operatorname{Ad}\pi)=N_{\operatorname{Ad}\pi}K_{\operatorname{Ad}\pi}$.
In other words, we let $K_{\pi}$ and $K_{\operatorname{Ad}\pi}$ denote the
contributions from the spectral parameters. Then we have that

$N_{\operatorname{Ad}\pi}=N_{\pi\times\widetilde{\pi}}\mid
N_{\pi}^{2}\qquad\text{and}\qquad K_{\operatorname{Ad}\pi}\asymp K_{\pi}.$
(7.1)

In the next theorem, we shall use the above properties to refine the work of
Goldfeld, Hoffstein, and Lieman in [5, Appendix].

######  Theorem 7.1.

Let $\pi\in\mathfrak{F}_{2}$ be non-dihedral with central character
$\omega_{\pi}$. There exists an absolute and effectively computable constant
$\Cl[abcon]{zfr}>0$ such that $L(s,\operatorname{Ad}\pi)\neq 0$ in the region

$\mathrm{Re}(s)\geq
1-\frac{\Cr{zfr}}{\log(N_{\operatorname{Ad}\pi}K_{\pi}(|t|+3))},$ (7.2)

and

$L(1,\operatorname{Ad}\pi)\gg(\log N_{\operatorname{Ad}\pi}K_{\pi})^{-1}.$
(7.3)

###### Proof.

The zero-free region (7.2) holds when $t\neq 0$ by Theorem A.1 in [11,
Appendix]. When $t=0$, the proof is the same as in [5, Appendix] apart from
the observation that

$\varphi(s)=\zeta(s)L(s,\operatorname{Ad}\pi)^{2}L(s,\operatorname{Ad}\pi\times\operatorname{Ad}\pi)=L(s,(\mathbbm{1}\boxplus\operatorname{Ad}\pi)\times(\mathbbm{1}\boxplus\operatorname{Ad}\pi)).$

In particular, the analytic conductor of $\varphi(s)$ is
$O((N_{\operatorname{Ad}\pi}K_{\pi})^{8})$. This permits us to express the
zero-free region for $L(s,\operatorname{Ad}\pi)$ in terms of
$N_{\operatorname{Ad}\pi}$ instead of $N_{\pi}$ (as in [5, Appendix]).

Lastly, (7.3) follows from the proof of the main theorem in [5, Appendix],
where we replace their zero-free region with our zero-free region in (7.2). ∎

Consider $f\in\mathcal{B}_{k}(q)$. Recall the notation in Sections 1 and 5.1
and let $Q=N_{\operatorname{Ad}f}$ denote the conductor of the adjoint lift of
$f$ as in (6.1). Using Theorem 7.1, we establish a lower bound on
$L(1,\operatorname{Ad}f)$ that is sensitive to the distribution of Hecke
eigenvalues $\lambda_{f}(p)$ for $p\leq k\sqrt{Q}$. This is important for the
final steps in our proofs of Theorem 1.1.

######  Lemma 7.2.

If $f\in\mathcal{B}_{k}(q)$ and $Q$ is defined as in (6.1), then

$\frac{1}{L(1,\operatorname{Ad}f)}\ll(\log\log kq)^{9}\prod_{p\leq
k\sqrt{Q}}\Big{(}1-\frac{\lambda_{f}(p)^{2}-1}{p}\Big{)}.$

###### Proof.

Let $X\geq 2$. By partial summation, we obtain

$\sum_{n\geq X}\frac{\Lambda_{\operatorname{Ad}f}(n)}{n\log
n}=-\Big{(}\sum_{n\leq X}\Lambda_{\operatorname{Ad}f}(n)\Big{)}\frac{1}{X\log
X}+\int_{X}^{\infty}\Big{(}\sum_{n\leq
t}\Lambda_{\operatorname{Ad}f}(n)\Big{)}\frac{\log t+1}{t^{2}(\log
t)^{2}}\,dt.$ (7.4)

By [19, Theorem 2.4] and Theorem 7.1, there exist absolute and effectively
computable constants $\Cl[abcon]{PNT-adf-range}>0$ and $\Cl[abcon]{PNT-adf-
error}>0$ such that if $X\geq(k^{2}Q)^{\Cr{PNT-adf-range}}$, then

$\sum_{n\leq X}\Lambda_{\operatorname{Ad}f}(n)\ll X\exp\Big{(}-\Cr{PNT-adf-
error}\frac{\log X}{\log(k^{2}Q)+\sqrt{\log X}}\Big{)}.$

Applying the above bound in (7.4) and choosing $X=(k^{2}Q)^{(6/\Cr{PNT-adf-
error})\log\log(k^{2}Q)}$, we obtain

$\sum_{n\geq X}\frac{\Lambda_{\operatorname{Ad}f}(n)}{n\log n}\ll\frac{(\log
k^{2}Q)^{-1}}{(\log\log k^{2}Q)(\log k^{2}Q)}\ll 1.$

Therefore, we have

$\displaystyle\log L(1,\operatorname{Ad}f)=\sum_{n\geq
2}\frac{\Lambda_{\operatorname{Ad}f}(n)}{n\log n}=\sum_{2\leq n\leq
X}\frac{\Lambda_{\operatorname{Ad}f}(n)}{n\log n}+O(1)=\sum_{p\leq
X}\frac{\lambda_{\operatorname{Ad}f}(p)}{p}+O(1),$ (7.5)

where we used Deligne's bound to trivially bound the contribution from the
composite $n\leq X$.

By the Hecke relations, if $p\nmid q$, then
$\lambda_{\operatorname{Ad}f}(p)=\lambda_{f}(p^{2})=\lambda_{f}(p)^{2}-1$.
Applying this fact and another application of Deligne's bound in (7.5), we
obtain that

$\displaystyle\log L(1,\operatorname{Ad}f)$ $\displaystyle=\sum_{p\leq
k\sqrt{Q}}\frac{\lambda_{f}(p)^{2}-1}{p}+\sum_{\begin{subarray}{c}p\leq
k\sqrt{Q}\\\ p\mid
q\end{subarray}}\frac{\lambda_{\operatorname{Ad}f}(p)-\lambda_{f}(p)^{2}+1}{p}+\sum_{k\sqrt{Q}\leq
p\leq X}\frac{\lambda_{\operatorname{Ad}f}(p)}{p}+O(1)$
$\displaystyle\geq\sum_{p\leq
k\sqrt{Q}}\frac{\lambda_{f}(p)^{2}-1}{p}-\sum_{p\mid
q}\frac{6}{p}-\sum_{k\sqrt{Q}\leq p\leq X}\frac{3}{p}+O(1).$

Mertens' theorem in the form

$\sum_{p\leq x}\frac{1}{p}=\log\log x+0.26149\ldots+O\Big{(}\frac{1}{\log
x}\Big{)}$

implies that $\sum_{p\mid q}\frac{1}{p}\leq\log\log\log q+O(1)$. The lemma now
follows. ∎

In the next two lemmas, we establish some useful bounds for Section 8.1, when
we apply Soundararajan's strategy.

######  Lemma 7.3.

Recall the notation in Sections 5.1 and 5.3. Let $\phi$ be a Hecke–Maaß cusp
form with spectral parameter $t_{\phi}$. Let $f,g\in\mathcal{B}_{k}(q)$ and
$Q$ be defined as in (6.1). If $\varepsilon>0$, then

$\frac{\Lambda(\tfrac{1}{2},f\times
g\times\phi)}{\Lambda(1,\operatorname{Ad}\phi)\Lambda(1,\operatorname{Ad}f)\Lambda(1,\operatorname{Ad}g)}\ll\frac{Q^{\frac{1}{2}}(1+|t_{\phi}|)^{2\delta_{1}+\varepsilon}}{(\log
kQ)^{1-\varepsilon}L(1,\operatorname{Ad}f)L(1,\operatorname{Ad}f)},$

where $\delta_{1}=11/12$ when $f=g$ and $\delta_{1}=1$ otherwise.

###### Proof.

By Stirling's formula,
$\Gamma(s)\asymp|s|^{\sigma-\frac{1}{2}}\exp(-\frac{\pi}{2}|s|)$, and the fact
that $|\Gamma(k+it)|\leq\Gamma(k)$ for any real $t$, it follows that

$\frac{L_{\infty}(\tfrac{1}{2},f\times
g\times\phi)}{L_{\infty}(1,\operatorname{Ad}\phi)L_{\infty}(1,\operatorname{Ad}f)L_{\infty}(1,\operatorname{Ad}g)}\ll\frac{1}{k}.$
(7.6)

First, consider the case when $f=g$. Since $f$ has trivial nebentypus, $f$ is
self-dual and we have the following factorization
$L(\frac{1}{2},f\times{f}\times\phi)=L(\frac{1}{2},\operatorname{Ad}f\times\phi)L(\frac{1}{2},\phi)$.
Since $\mathfrak{C}(\operatorname{Ad}f\times\phi)\ll
k^{4}Q^{2}(1+|t_{\phi}|)^{6}$ and $\mathfrak{C}(\phi)\ll(1+|t_{\phi}|)^{2}$,
Theorem 1.3 implies that

$L(\tfrac{1}{2},\operatorname{Ad}f\times\phi)\ll(1+|t_{\phi}|)^{\frac{\varepsilon}{3}}\frac{(k^{4}Q^{2}(1+|t_{\phi}|)^{6})^{\frac{1}{4}}}{(\log(k^{4}Q^{2}(1+|t_{\phi}|)^{6}))^{1-\varepsilon}}\ll\frac{kQ^{\frac{1}{2}}(1+|t_{\phi}|)^{\frac{3}{2}+\frac{\varepsilon}{3}}}{(\log
kQ)^{1-\varepsilon}}.$ (7.7)

Using the bound $L(\frac{1}{2},\phi)\ll(1+|t_{\phi}|)^{1/3+\varepsilon/3}$
from [12, Corollary 2], the bound
$L(1,\operatorname{Ad}\phi)^{-1}\ll\log(1+|t_{\phi}|)$ from [5], and (7.7), we
conclude that

$\frac{L(\tfrac{1}{2},f\times
f\times\phi)}{L(1,\operatorname{Ad}\phi)L(1,\operatorname{Ad}f)^{2}}\ll\frac{kQ^{\frac{1}{2}}(1+|t_{\phi}|)^{\frac{11}{6}+\varepsilon}}{(\log
kQ)^{1-\varepsilon}L(1,\operatorname{Ad}f)^{2}}.$ (7.8)

Hence, combining (7.8) with (7.6), we obtain the lemma when $f=g$. In the case
that $f\neq g$, the proof proceeds the same way, except that
$\mathfrak{C}(f\times g\times\phi)\ll k^{4}q^{4}(1+|t_{\phi}|)^{8}$ and we
directly apply Theorem 1.3 with $L(\frac{1}{2},f\times g\times\phi)$. ∎

######  Lemma 7.4.

Recall the notation in Section 5.1. Let $f,g\in\mathcal{B}_{k}(q)$ and $Q$ be
defined as in (6.1). For any $s\in\mathbb{C}$, let $\xi(s)$ be the completed
zeta function. If $t\in\mathbb{R}$ and $\varepsilon>0$, then we have

$\frac{\Lambda(\frac{1}{2}+it,f\times
g)^{2}}{\xi(1+2it)^{2}\Lambda(1,\operatorname{Ad}f)\Lambda(1,\operatorname{Ad}g)}\ll\frac{Q^{\frac{1}{2}}(1+|t|)^{2\delta_{2}+\varepsilon}}{(\log
kQ)^{2-\varepsilon}L(1,\operatorname{Ad}f)L(1,\operatorname{Ad}g)},$

where $\delta_{2}=19/21$ if $f=g$ and $\delta_{2}=1$ otherwise.

###### Proof.

By Stirling's formula,
$\Gamma(s)\asymp|s|^{\sigma-\frac{1}{2}}\exp(-\frac{\pi}{2}|s|)$, and the fact
that $|\Gamma(k+it)|\leq\Gamma(k)$ for any real $t$, it follows that

$\displaystyle\frac{L_{\infty}(\tfrac{1}{2}+it,f\times
g)^{2}}{\Gamma(\frac{1}{2}+it)^{2}L_{\infty}(1,\operatorname{Ad}f)L_{\infty}(1,\operatorname{Ad}g)}\ll\frac{1}{k}.$
(7.9)

We first consider the case when $f=g$. Since $f$ has trivial nebentypus, $f$
is self-dual and we have the following factorization
$L(\tfrac{1}{2}+it,f\times
f)=L(\tfrac{1}{2}+it,\operatorname{Ad}f)\zeta(\tfrac{1}{2}+it)$. To estimate
$L(\tfrac{1}{2}+it,\operatorname{Ad}f)$, we invoke the weak subconvexity bound
in [34, Theorem 1]. We note that such theorem also applies for the shifted
$L$-function $L(s+it,\operatorname{Ad}f)$ as discussed in [34, Example 1].
Since $\mathfrak{C}(\operatorname{Ad}f,t)\ll k^{2}Q(1+|t|)^{3}$, we obtain
that

$L(\tfrac{1}{2}+it,\operatorname{Ad}f)\ll\frac{(k^{2}Q(1+|t|)^{3})^{\frac{1}{4}}}{(\log(k^{2}Q(1+|t|)^{3}))^{1-\frac{\varepsilon}{2}}}\ll\frac{k^{\frac{1}{2}}Q^{\frac{1}{4}}(1+|t|)^{\frac{3}{4}}}{(\log
kQ)^{1-\frac{\varepsilon}{2}}}.$ (7.10)

Using the bound $\zeta(1/2+it)\ll_{\varepsilon}(1+|t|)^{13/84+\varepsilon/3}$
from [2], the known bound $\zeta(1+2it)^{-1}\ll\log(1+|t|)$, and (7.10), we
conclude that

$\frac{L(\tfrac{1}{2}+it,f\times
f)^{2}}{\zeta(1+2it)^{2}L(1,\operatorname{Ad}f)^{2}}\ll\frac{kQ^{\frac{1}{2}}(1+|t|)^{\frac{38}{21}+\varepsilon}}{(\log
kQ)^{2-\varepsilon}L(1,\operatorname{Ad}f)^{2}}.$ (7.11)

Hence, combining (7.11) with (7.9), we obtain the lemma when $f=g$. The proof
for the case that $f\neq g$ proceeds in the same way, except that
$\mathfrak{C}(f\times g,t)\ll k^{2}q^{2}(1+|t|)^{4}$ and we directly apply
[34, Theorem 1] with $L(\frac{1}{2}+it,f\times g)$. ∎

##  8\. Effective correlation and decorrelation

In this section, we prove Propositions 6.1 and 6.2, which are based on
Soundararajan's approach in [34] and Holowinsky's approach in [6],
respectively. If $f\neq g$, then we assume, as stated in Theorem 1.1, that the
shared level of $f$ and $g$ is squarefree.

###  8.1. Soundararajan's approach

In this subsection, we prove Proposition 6.1. To do so, we first appeal to the
spectral decomposition in (5.9). The problem is then reduced to finding rates
of convergence for $\langle\phi,\overline{F_{k}}G_{k}\rangle_{q}$ and $\langle
E_{t},\overline{F_{k}}G_{k}\rangle_{q}$. Following the work of Nelson, Pitale,
and Saha in [29], we derive the next lemma.

######  Lemma 8.1.

Recall the notation and hypotheses as in Theorem 1.1 and the definition of $Q$
in (6.1). Let $\phi$ be a Hecke–Maaß cusp form with spectral parameter
$t_{\phi}$ and, for any $t\in\mathbb{R}$, let $E_{t}=E(1/2+it,\cdot)$ denote
the unitary Eisenstein series. If $\varepsilon>0$, then we have

$\langle\phi,\overline{F_{k}}G_{k}\rangle_{q}\ll\frac{(1+|t_{\phi}|)^{\delta_{1}+\varepsilon}}{L(1,\operatorname{Ad}f)^{\frac{1}{2}}L(1,\operatorname{Ad}g)^{\frac{1}{2}}}\frac{(q/\sqrt{Q})^{-\frac{1}{2}+\theta+\varepsilon}}{(\log
kQ)^{\frac{1}{2}-\varepsilon}},$ (8.1)

where $\delta_{1}=11/12$ if $f=g$ and $\delta_{1}=1$ otherwise, and

$\langle
E_{t},\overline{F_{k}}G_{k}\rangle_{q}\ll\frac{(1+|t|)^{\delta_{2}+\varepsilon}}{L(1,\operatorname{Ad}f)^{\frac{1}{2}}L(1,\operatorname{Ad}g)^{\frac{1}{2}}}\frac{(q/\sqrt{Q})^{-\frac{1}{2}+\varepsilon}}{(\log
kQ)^{1-\varepsilon}},$ (8.2)

where $\delta_{2}=19/21$ if $f=g$ and $\delta_{2}=1$ otherwise.

###### Proof.

Recall Ichino's generalization of Watson's formula and denote $I_{v}^{*}$ the
corresponding normalized local Ichino integral (see [29, Theorem 3.1] and [28,
Remark 4.2] for more details on the correlation and decorrelation cases,
respectively). We have that

$\frac{|\langle\phi,\overline{F_{k}}G_{k}\rangle_{q}|^{2}}{\langle\phi,\phi\rangle_{1}\langle
1,|F_{k}|^{2}\rangle_{q}\langle
1,|G_{k}|^{2}\rangle_{q}}=\frac{1}{8}\frac{\Lambda(\tfrac{1}{2},f\times
g\times\phi)}{\Lambda(1,\operatorname{Ad}\phi)\Lambda(1,\operatorname{Ad}f)\Lambda(1,\operatorname{Ad}g)}\prod_{v}I_{v}^{*}.$
(8.3)

For any prime $p$, let $n_{p}=\operatorname{ord}_{p}(q)$ and
$m_{p}=\operatorname{ord}_{p}(\frac{q}{\sqrt{Q}})$. The values of $I_{v}^{*}$
are defined and calculated explicitly in [29, Section 2] and [28, Section 4].
In particular, $I_{\infty}^{*}=1$, and

$I_{p}^{*}\begin{cases}=1,&\text{ if }p\nmid q,\\\
<10^{5}p^{-n_{p}}\tau(p^{m_{p}})^{2}p^{2\theta m_{p}},&\text{ if }p\mid
q\text{ and }f=g,\\\ =p^{-1},&\text{ if }p\mid q\text{ and }f\neq
g.\end{cases}$

Using the above calculation of $I_{v}^{*}$, we obtain that

$\prod_{v}I_{v}^{*}\ll
10^{5\omega({q}/{\sqrt{Q}})}q^{-1}\tau({q}/{\sqrt{Q}})^{2}({q}/{\sqrt{Q}})^{2\theta}\ll
q^{-1}({q}/{\sqrt{Q}})^{2\theta+2\varepsilon},$ (8.4)

upon using that
$10^{5\omega(q/\sqrt{Q})}\tau(q/\sqrt{Q})^{2}\ll(q/\sqrt{Q})^{2\varepsilon}$.
Recall that $\langle\phi,\phi\rangle_{1}=\langle
1,|F_{k}|^{2}\rangle_{q}=\langle 1,|G_{k}|^{2}\rangle_{q}=1$ by our
normalization. Inserting (8.4) and Lemma 7.3 into (8.3) and taking the square
root, we obtain that

$\langle\phi,\overline{F_{k}}G_{k}\rangle_{q}\ll\frac{Q^{\frac{1}{4}}(1+|t_{\phi}|)^{\delta_{1}+\varepsilon}}{(\log
kQ)^{\frac{1}{2}-\varepsilon}L(1,\operatorname{Ad}f)^{\frac{1}{2}}L(1,\operatorname{Ad}g)^{\frac{1}{2}}}q^{-\frac{1}{2}}(q/\sqrt{Q})^{\theta+\varepsilon},$
(8.5)

where $\delta_{1}=11/12$ when $f=g$ and $\delta_{1}=1$ otherwise. Hence, (8.1)
follows.

We now consider the case of unitary Eisenstein series. Let $\xi(s)$ be the
completed Riemann zeta function. Similarly to the case of Hecke–Maaß cusp
forms, we have

$|\langle
E_{t},\overline{F_{k}}G_{k}\rangle_{q}|^{2}=\frac{|\Lambda(1/2+it,f\times
g)|^{2}}{|\xi(1+2it)|^{2}\Lambda(1,\operatorname{Ad}f)\Lambda(1,\operatorname{Ad}g)}\prod_{v}J_{v}^{*},$
(8.6)

where $J_{\infty}^{*}=1$, and

$J_{p}^{*}\begin{cases}=1,&\text{ if }p\nmid q,\\\
<10^{5}p^{-n_{p}}\tau(p^{m_{p}})^{2},&\text{ if }p\mid q\text{ and }f=g,\\\
=p^{-1},&\text{ if }p\mid q\text{ and }f\neq g.\end{cases}$

The above formula follows more directly from the Rankin–Selberg method and
from the calculation in [29, Section 2]. For more details, see [29, Remark
3.2]. By the above calculation for $J_{v}^{*}$, we have $\prod_{v}J_{v}^{*}\ll
q^{-1}({q}/{\sqrt{Q}})^{2\varepsilon}$. Hence, (8.2) follows upon using Lemma
7.4. ∎

######  Remark.

We make the assumption that $q$ is squarefree when $f\neq g$ because to the
author's knowledge, an analogue of (8.3) that ensures a bound of the shape

$\langle\phi,\overline{F_{k}}G_{k}\rangle_{q}^{2}\ll\frac{\Lambda(\tfrac{1}{2},f\times
g\times\phi)}{\Lambda(1,\operatorname{Ad}\phi)\Lambda(1,\operatorname{Ad}f)\Lambda(1,\operatorname{Ad}g)}$

is only known when $q$ is squarefree. See Hu [8, Section 3] for partial
progress in this direction.

###### Proof of Proposition 6.1.

The idea of this proof follows from [6, Proposition 3.3]. Recall the spectral
decomposition in (5.9). We begin with estimating the tail sums when
$|t_{\phi}|\geq(kQ)^{\varepsilon}$ and $|t|\geq(kQ)^{\varepsilon}$. We repeat
the proof of Lemma 7.3, but replace the weak subconvexity bound with the
convexity bound. It follows from (8.3) and (7.3) that

$\displaystyle\langle\phi,\overline{F_{k}}G_{k}\rangle_{q}\ll(1+|t_{\phi}|)^{\delta_{1}+\varepsilon}(q/\sqrt{Q})^{-\frac{1}{2}+\theta+\varepsilon}\log(kQ).$
(8.7)

Similarly, we repeat the proof of Lemma 7.4, but replace the weak subconvexity
bound with the convexity bound. It follows from (8.6) and (7.3) that

$\langle
E_{t},\overline{F_{k}}G_{k}\rangle_{q}\ll(1+|t|)^{\delta_{2}+\varepsilon}(q/\sqrt{Q})^{-\frac{1}{2}+\varepsilon}\log(kQ).$
(8.8)

By Lemma 5.2, (8.7), and (8.8), for any $A>0$, we have that

$\displaystyle\sum_{|t_{\phi}|\geq(kQ)^{\varepsilon}}\langle\psi,\phi\rangle_{1}\langle\phi,\overline{F_{k}}G_{k}\rangle_{q}+\frac{1}{4\pi}\int_{|t|\geq(kQ)^{\varepsilon}}\langle\psi,E_{t}\rangle_{1}\langle
E_{t},\overline{F_{k}}G_{k}\rangle_{q}dt$
$\displaystyle\quad\ll_{A}\sum_{|t_{\phi}|\geq(kQ)^{\varepsilon}}\Big{(}\frac{M}{1+|t_{\phi}|}\Big{)}^{A}(1+|t_{\phi}|)^{\delta_{1}+\varepsilon}(q/\sqrt{Q})^{-\frac{1}{2}+\theta+\varepsilon}\log(kQ)$
$\displaystyle\qquad+\frac{1}{4\pi}\int_{|t|\geq(kQ)^{\varepsilon}}\Big{(}\frac{M}{1+|t|}\Big{)}^{A}(1+|t|)^{\frac{3}{8}+\varepsilon}(1+|t|)^{\delta_{2}+\varepsilon}(q/\sqrt{Q})^{-\frac{1}{2}+\varepsilon}\log(kQ)dt$
$\displaystyle\quad\ll_{A}M^{A}(q/\sqrt{Q})^{-\frac{1}{2}+\theta+\varepsilon}\log(kQ)\max_{T\geq(kQ)^{\varepsilon}}T^{\delta_{1}+2-A+\varepsilon},$

upon using Weyl's law in (5.7). If $M$ is less than a fixed power of
$\log(kQ)$ and $A$ is sufficiently large with respect to $\varepsilon$, then
the contribution in the range $|t_{\phi}|\geq(kQ)^{\varepsilon}$ and
$|t|\geq(kQ)^{\varepsilon}$ is

$O_{A}((q/\sqrt{Q})^{-\frac{1}{2}+\theta+\varepsilon}(kQ)^{-A}).$ (8.9)

Now, we consider when $|t|,|t_{\phi}|\in[M(\log
kQ)^{\varepsilon/4},(kQ)^{\varepsilon}]$. By Lemma 8.1 and (7.3), we have the
bounds

$\displaystyle\langle\phi,\overline{F_{k}}G_{k}\rangle_{q}$
$\displaystyle\ll(1+|t_{\phi}|)^{\delta_{1}+\varepsilon}(q/\sqrt{Q})^{-\frac{1}{2}+\theta+\varepsilon}(\log
kQ)^{\frac{1}{2}+\frac{\varepsilon}{2}},$ (8.10) $\displaystyle\langle
E_{t},\overline{F_{k}}G_{k}\rangle_{q}$
$\displaystyle\ll(1+|t|)^{\delta_{2}+\varepsilon}(q/\sqrt{Q})^{-\frac{1}{2}+\varepsilon}(\log
kQ)^{\frac{\varepsilon}{2}}.$

By Lemma 5.2 and the estimates in (8.10), we obtain the bound

$\displaystyle\sum_{|t_{\phi}|\in[M(\log
kQ)^{\varepsilon/4},(kQ)^{\varepsilon}]}\langle\psi,\phi\rangle_{1}\langle\phi,\overline{F_{k}}G_{k}\rangle_{q}+\frac{1}{4\pi}\int_{|t|\in[M(\log
kQ)^{\varepsilon/4},(kQ)^{\varepsilon}]}\langle\psi,E_{t}\rangle_{1}\langle
E_{t},\overline{F_{k}}G_{k}\rangle_{q}dt$ (8.11)
$\displaystyle\ll_{A}\sum_{|t_{\phi}|\in[M(\log
kQ)^{\varepsilon/4},(kQ)^{\varepsilon}]}M^{A}(1+|t_{\phi}|)^{\delta_{1}-A+\varepsilon}(q/\sqrt{Q})^{-\frac{1}{2}+\theta+\varepsilon}(\log
kQ)^{\frac{1}{2}+\frac{\varepsilon}{2}}$
$\displaystyle\qquad+\int_{|t|\in[M(\log
kQ)^{\varepsilon/4},(kQ)^{\varepsilon}]}M^{A}(1+|t|)^{\frac{3}{8}+\delta_{2}-A+\varepsilon}(q/\sqrt{Q})^{-\frac{1}{2}+\varepsilon}(\log
kQ)^{\frac{\varepsilon}{2}}dt.$

By Weyl's law in (5.7) and the fact that
$\log(kq)/\log(kQ)\ll_{\varepsilon}(q/\sqrt{Q})^{\varepsilon}$, (8.11) is

$\displaystyle\ll_{A}M^{A}(q/\sqrt{Q})^{-\frac{1}{2}+\theta+\varepsilon}(\log
kQ)^{\frac{1}{2}+\frac{\varepsilon}{2}}\max_{M(\log kQ)^{\varepsilon/4}\leq
T\leq(kQ)^{\varepsilon}}T^{2+\delta_{1}-A+\varepsilon}$
$\displaystyle\qquad+M^{A}(q/\sqrt{Q})^{-\frac{1}{2}+\varepsilon}(\log
kQ)^{\frac{\varepsilon}{2}}\int_{M(\log
kQ)^{\varepsilon/4}}^{(kQ)^{\varepsilon}}t^{\frac{3}{8}+\delta_{2}-A+\varepsilon}dt$
$\displaystyle\ll_{A}M^{2+\delta_{1}+\varepsilon}(q/\sqrt{Q})^{-\frac{1}{2}+\theta+\varepsilon}(\log
kQ)^{\frac{1}{2}+\frac{\varepsilon}{2}+\frac{\varepsilon}{4}(2+\delta_{1}-A+\varepsilon)}.$

By taking $A>2+\delta_{1}$ sufficiently large with respect to $\varepsilon$,
the contribution in the range $|t|,|t_{\phi}|\in[M(\log
kQ)^{\varepsilon/4},(kQ)^{\varepsilon}]$ is

$O_{A}(M^{2+\delta_{1}+\varepsilon}(q/\sqrt{Q})^{-\frac{1}{2}+\theta+\varepsilon}(\log
kQ)^{-A}).$ (8.12)

Next, we bound the contribution from the remaining ranges $|t_{\phi}|,|t|\leq
M(\log kQ)^{\varepsilon/4}$. By the Parseval formula, we have that

$\lVert\psi\rVert_{2}^{2}=\frac{3}{\pi}|\langle\psi,1\rangle_{1}|^{2}+\sum_{\phi}|\langle\psi,\phi\rangle_{1}|^{2}+\frac{1}{4\pi}\int_{\mathbb{R}}|\langle\psi,E_{t}\rangle_{1}|^{2}\,dt.$
(8.13)

By (1.4), we have that
$\lVert\psi\rVert_{2}^{2}=\int_{Y_{0}(1)}|\psi(z)|^{2}y^{-2}dxdy\ll 1$.
Applying the Cauchy–Schwarz inequality and Lemma 8.1, we obtain that

$\displaystyle\sum_{|t_{\phi}|\leq M(\log
kQ)^{\varepsilon/4}}\langle\psi,\phi\rangle_{1}\langle\phi,\overline{F_{k}}G_{k}\rangle_{q}$
(8.14) $\displaystyle\qquad\ll\Big{(}\sum_{|t_{\phi}|\leq M(\log
kQ)^{\varepsilon/4}}|\langle\psi,\phi\rangle_{1}|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{|t_{\phi}|\leq
M(\log
kQ)^{\varepsilon/4}}\left|\langle\phi,\overline{F_{k}}G_{k}\rangle_{q}\right|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\qquad\ll\lVert\psi\rVert_{2}\Big{(}\sum_{|t_{\phi}|\leq M(\log
kQ)^{\varepsilon/4}}\frac{(1+|t_{\phi}|)^{2\delta_{1}+2\varepsilon}}{L(1,\operatorname{Ad}f)L(1,\operatorname{Ad}g)}\frac{(q/\sqrt{Q})^{-1+2\theta+2\varepsilon}}{(\log
kQ)^{1-\varepsilon}}\Big{)}^{\frac{1}{2}}.$

By Lemma 7.2, Weyl's law, and the fact that $\prod_{p\leq
x}(1-\delta/{p})\asymp(\log x)^{-\delta}$, we have that (8.14) is

$\ll M^{\delta_{1}+1+\varepsilon}(\log
kq)^{\varepsilon}(q/\sqrt{Q})^{-\frac{1}{2}+\theta+\varepsilon}\prod_{p\leq
k\sqrt{Q}}\Big{(}1-\frac{\lambda_{f}(p)^{2}+\lambda_{g}(p)^{2}-1}{2p}\Big{)}.$
(8.15)

Similarly, for the case of Eisenstein series, it follows from the
Cauchy–Schwarz inequality, Lemma 8.1, and Lemma 7.2 that

$\displaystyle\int_{|t|\leq M(\log kQ)^{\varepsilon/4}}$
$\displaystyle\langle\psi,E_{t}\rangle_{1}\langle
E_{t},\overline{F_{k}}G_{k}\rangle_{q}\,dt$ (8.16) $\displaystyle\qquad\ll
M^{\delta_{2}+\frac{1}{2}+\varepsilon}(\log
kq)^{\varepsilon}({q}/{\sqrt{Q}})^{-\frac{1}{2}+\varepsilon}\prod_{p\leq
k\sqrt{Q}}\Big{(}1-\frac{\lambda_{f}(p)^{2}+\lambda_{g}(p)^{2}}{2p}\Big{)}.$

Putting together (8.9), (8.12), (8.15), and (8.16), we see that
$|\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\tfrac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}|$
starts to decay only if $M$ is smaller than some fixed power of $\log(kQ)$. In
such case, the estimates from (8.9) and (8.12) become negligible. We also
observe that the estimate from (8.15) is larger than the estimate from (8.16).
Therefore, the rate of convergence is dominated by the contribution from
Hecke–Maaß cups forms in (8.15). ∎

###  8.2. Holowinsky's approach

In this subsection, we prove Proposition 6.2. To apply Holowinsky's method, we
will need to unfold $\langle P_{m}(\cdot,\Psi_{m}),|F_{k}|^{2}\rangle_{q}$ and
those alike, for example, $\langle E(\cdot|\Psi_{0}),|F_{k}|^{2}\rangle_{q}$
and $\langle E_{t},|F_{k}|^{2}\rangle_{q}$. Since the level of
$P_{m}(\cdot,\Psi_{m})$ is 1, while the level of $f$ can be any positive
integer $q$, the application of the unfolding method becomes more complicated
than what we see in Section 5.6. For completeness, we provide a detailed
exposition on how to unfold $\langle
P_{m}(\cdot,\Psi_{m}),|F_{k}|^{2}\rangle_{q}$ in the next lemma. For more
discussion, see [8, Section 5.2].

######  Lemma 8.2.

Recall the notation in Section 5.2 and let $\mathbb{R}_{\geq 0}$ denote the
set of nonnegative real numbers. For any $m\in\mathbb{Z}$ and any $\Psi\in
C^{\infty}_{c}(\mathbb{R}_{\geq 0})$, a smooth compactly supported function on
$\mathbb{R}_{\geq 0}$, we have that

$\langle
P_{m}(\cdot,\Psi),|F_{k}|^{2}\rangle_{q}=\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\int_{y_{j}=0}^{\infty}\Psi(w_{j}y_{j})\int_{x_{j}=0}^{1}e(mw_{j}x_{j})|F_{k}(z)|^{2}\frac{dx_{j}\,dy_{j}}{y_{j}^{2}}.$

###### Proof.

Let $\Gamma_{\infty}\backslash\Gamma_{0}(1)/\Gamma_{0}(q)=\\{\tau\\}$ be a set
of double-coset representatives as in Section 5.2. Also, set
$w_{j}=[\Gamma_{\infty}:\Gamma_{\infty}\cap\tau_{j}\Gamma_{0}(q)\tau_{j}^{-1}]$.
We begin with decomposing $\Gamma_{\infty}\backslash\Gamma_{0}(1)$ into
$\Gamma_{0}(q)$-orbits, namely,

$\Gamma_{\infty}\backslash\Gamma_{0}(1)=\sqcup\Gamma_{\infty}\backslash(\Gamma_{\infty}\tau\Gamma_{0}(q))=\sqcup\tau(\tau^{-1}\Gamma_{\infty}\tau\cap\Gamma_{0}(q)\backslash\Gamma_{0}(q)).$

By the above decomposition and the definition of $P_{m}(\cdot,\Psi)$, we have

$\langle
P_{m}(\cdot,\Psi),|F_{k}|^{2}\rangle_{q}=\int_{\Gamma_{0}(q)\backslash\mathbb{H}}\sum_{\begin{subarray}{c}\tau\in\Gamma_{\infty}\backslash\Gamma_{0}(1)/\Gamma_{0}(q)\\\
\gamma\in\tau^{-1}\Gamma_{\infty}\tau\cap\Gamma_{0}(q)\backslash\Gamma_{0}(q)\end{subarray}}\Psi(\operatorname{Im}(\tau\gamma
z))e(m\operatorname{Re}(\tau\gamma z))|F_{k}(z)|^{2}\frac{dxdy}{y^{2}}$

Since $z\mapsto|F_{k}(z)|^{2}y^{-2}dxdy$ is $\Gamma_{0}(q)$-invariant, the
above integral equals

$\displaystyle\sum_{\tau\in\Gamma_{\infty}\backslash\Gamma_{0}(1)/\Gamma_{0}(q)}\int_{\tau^{-1}\Gamma_{\infty}\tau\cap\Gamma_{0}(q)\backslash\mathbb{H}}\Psi(\operatorname{Im}(\tau
z))e(m\operatorname{Re}(\tau z))|F_{k}(z)|^{2}\frac{dxdy}{y^{2}}$ (8.17)
$\displaystyle=$
$\displaystyle\sum_{\tau\in\Gamma_{\infty}\backslash\Gamma_{0}(1)/\Gamma_{0}(q)}\int_{\Gamma_{\infty}\cap\tau\Gamma_{0}(q)\tau^{-1}\backslash\mathbb{H}}\Psi(y)e(mx)|F_{k}(\tau^{-1}z)|^{2}\frac{dxdy}{y^{2}}.$

Now we index $\tau$ with $\tau_{j}$ and let
$\mathfrak{a}_{j}=\tau_{j}^{-1}\infty$ be the corresponding cusp. If we denote
$\mathcal{C}=\mathcal{C}(\Gamma_{0}(q))=\\{\mathfrak{a}_{j}\\}_{j}$, then we
have that (8.17) equals

$\displaystyle\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\int_{\Gamma_{\infty}\cap\tau_{j}\Gamma_{0}(q)\tau_{j}^{-1}\backslash\mathbb{H}}\Psi(y)e(mx)|F_{k}(\tau_{j}^{-1}z)|^{2}\frac{dxdy}{y^{2}}$
(8.18) $\displaystyle=$
$\displaystyle\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\int_{y=0}^{\infty}\int_{x=0}^{w_{j}}\Psi(y)e(mx)|F_{k}(\tau_{j}^{-1}z)|^{2}\frac{dxdy}{y^{2}},$

upon using that
$w_{j}=[\Gamma_{\infty}:\Gamma_{\infty}\cap\tau_{j}\Gamma_{0}(q)\tau_{j}^{-1}]$
in the last step. By the change of variable $z\mapsto w_{j}z$, (8.18) equals

$\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\int_{y=0}^{\infty}\int_{x=0}^{1}\Psi(\operatorname{Im}(w_{j}z))e(m\operatorname{Re}(w_{j}z))|F_{k}(\tau_{j}^{-1}\big{(}\begin{smallmatrix}w_{j}&\\\
&1\end{smallmatrix}\big{)}z)|^{2}\frac{dxdy}{y^{2}}.$

Recall that $\sigma_{j}=\tau_{j}^{-1}\big{(}\begin{smallmatrix}w_{j}&\\\
&1\end{smallmatrix}\big{)}$ is the scaling matrix of $\mathfrak{a}_{j}$. For
each $z\in\mathbb{H}$, we write $z_{j}=x_{j}+iy_{j}$ for the change of
variable $z_{j}=\sigma_{j}^{-1}z.$ Applying the change of variables $x\mapsto
x_{j}$ and $y\mapsto y_{j}$ and using that $\sigma_{j}z_{j}=z$ in the above
integral, we obtain the lemma. ∎

######  Remark.

In the decorrelation case, the assumption of $q$ being squarefree helps us
simplify the above unfolding method for several reasons. For each positive
divisor $c_{j}$ of $q$, there is exactly one cusp in $\mathcal{C}[c_{j}]$, and
the corresponding width $w=q/(c_{j}^{2},q)$ equals $q/c_{j}$. Moreover, as
discussed in [28], the scaling matrix $\sigma_{j}$ behaves as an Atkin-Lehner
operator, and hence $f|_{k}\sigma_{j}=\pm f$.

Consider $\langle P_{m}(\cdot,\Psi),\overline{F_{k}}G_{k}\rangle_{q}$ with $q$
squarefree. By the above remark, for any positive divisor $d$ of $q$, we have
that there is only one cusp with denominator $q/d$, and the width of such cusp
is $d$. By Lemma 8.2, we conclude that

$\langle
P_{m}(\cdot,\Psi),\overline{F_{k}}G_{k}\rangle_{q}=\sum_{d|q}\int_{y=0}^{\infty}\Psi(dy)\int_{x=0}^{1}e(mdx)F_{k}(z)\overline{G_{k}(z)}\frac{dxdy}{y^{2}}.$
(8.19)

With the notation in Section 5.6, we now estimate
$|\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\tfrac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}|$.
By the Poincaré decomposition in (5.16), we have

$\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}=\langle
E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\widehat{\Psi}_{0}(-1)\mathds{1}_{f=g}\\\
\qquad+\Big{(}\sum_{0<|m|<M(\log kQ)^{\varepsilon}}+\sum_{|m|\geq M(\log
kQ)^{\varepsilon}}\Big{)}\langle
P_{m}(\cdot,\Psi_{m}),\overline{F_{k}}G_{k}\rangle_{q}.$ (8.20)

We will estimate the contribution from each range of $m$ individually. In
doing so, we need the following propositions.

######  Proposition 8.3 (Proposition 3.17 of [29]).

Recall the notation in Theorem 1.1 and Section 5.2. Let $m\neq 0$ be an
integer, $x\geq 1$ be a real number, and $\varepsilon\in(0,1)$. Let
$q_{\diamond}^{2}$ denote the largest square divisor of $q$. If $c$ is a
positive divisor of $q$, then

$\sum_{\begin{subarray}{c}n_{1}\geq 1,n_{2}=n_{1}+m\geq 1\\\
\max(n_{1},n_{2})\leq
x\end{subarray}}|\lambda_{[c]}(n_{1})\lambda_{[c]}(n_{2})|\ll
q_{\diamond}^{\varepsilon}(\log\log(e^{e}q))^{O(1)}\frac{x\prod_{p\leq
x}(1+2|\lambda_{f}(p)|/p)}{(\log ex)^{2-\varepsilon}}.$

When $f\neq g$, we need the following proposition which follows from [28,
Theorem 3.10].

######  Proposition 8.4.

Recall the notation in Theorem 1.1 and in Section 5.2. Let $m\neq 0$ be an
integer and $x\geq 1$ be a real number. If $\varepsilon\in(0,1)$, then we have
that

$\sum_{\begin{subarray}{c}n_{1}\geq 1,n_{2}=n_{1}+m\geq 1\\\
\max(n_{1},n_{2})\leq
x\end{subarray}}|\lambda_{f}(n_{1})\lambda_{g}(n_{2})|\ll\frac{x\prod_{p\leq
x}(1+(|\lambda_{f}(p)|+|\lambda_{g}(p)|)/p)}{(\log ex)^{2-\varepsilon}}.$

###### Proof.

We follow closely the proof of [28, Theorem 3.10]. Let
$\lambda_{1}(n)=|\lambda_{f}(n)|$ and $\lambda_{2}(n)=|\lambda_{g}(n)|$, so
that $\lambda_{1}$ and $\lambda_{2}$ are nonnegative multiplicative function
satisfying

$\lambda_{1}(n)\leq\tau(n)\quad\text{and}\quad\lambda_{2}(n)\leq\tau(n).$
(8.21)

We may assume that $1\leq m\leq x$. The following setup is the same as in [28,
Theorem 3.10]. Fix $\alpha\in(0,1/2)$ to be chosen later. Set

$y=x^{\alpha},\qquad s=\alpha\log\log(x),\qquad z=x^{1/s}.$

For $x\gg_{\alpha}1$, we have $10\leq z\leq y\leq x$. Let $\mathbb{N}$ denote
the set of positive integers. For each $n_{1}\in\mathbb{N}$, write
$n_{2}=n_{1}+m$. Define the $z$-part of a positive integer to be the greatest
divisor of that integer supported on primes $p\leq z$. There exist unique
positive integers $a,b,c$ such that $\gcd(a,b)=1$ and $ac$ (resp. $bc$) is the
$z$-part of $n_{1}$ (resp. $n_{2}$); such triples $a,b,c$ satisfy

$p|abc\Rightarrow p\leq z,\qquad c\mid m,\quad\text{and}\quad\gcd(a,b)=1.$
(8.22)

Write $\mathbb{N}=\sqcup_{a,b,c}\mathbb{N}_{abc}$ for the fibers of
$n_{1}\mapsto(a,b,c)$. Then we have

$\sum_{n_{1}\in\mathbb{N}\cap[1,x]}\lambda_{1}(n_{1})\lambda_{2}(n_{2})=\sum_{a,b,c}\sum_{n_{1}\in\mathbb{N}_{abc}\cap[1,x]}\lambda_{1}(n_{1})\lambda_{2}(n_{2}).$

To estimate the contribution from $a,b,c$ for which $ac>y$ or $bc>y$, we
appeal to [6, Section 4.2]. Since their proof applies to the shifted
convolution sum of distinct multiplicative functions $\lambda_{1}$ and
$\lambda_{2}$, we then obtain the same estimate, which is $O(x(\log x)^{-2})$.

We now consider the contribution from $a,b,c$ for which $ac\leq y$ and $bc\leq
y$. The assumption (8.21) implies that
$\lambda_{1}(n_{1})\lambda_{2}(n_{2})\leq
4^{s}\lambda_{1}(ac)\lambda_{2}(bc)$, so that

$\underset{ac\leq y,bc\leq
y}{\sum_{a,b,c}\sum_{n_{1}\in\mathbb{N}_{abc}\cap[1,x]}}\lambda_{1}(n_{1})\lambda_{2}(n_{2})\leq
4^{s}\sum_{a,b,c}\lambda_{1}(ac)\lambda_{2}(bc)\cdot\\#(\mathbb{N}_{abc}\cap[1,x]).$

An application of the large sieve shows that if $ac\leq y$, $bc\leq y$, and
$x\gg y^{2}$, then

$\\#(\mathbb{N}_{abc}\cap[1,x])\ll\frac{x+(yz)^{2}}{(\log
z)^{2}}\frac{1}{c^{2}\varphi(abc^{-1}m)}.$ (8.23)

Since $(yz)^{2}\ll_{\alpha}x,(\log z)^{2}\asymp_{\alpha}(\log\log x)^{-2}(\log
x)^{2},4^{s}\ll_{\varepsilon}(\log x)^{\varepsilon}$ (for
$\alpha\ll_{\varepsilon}1$), and
$\varphi(abc^{-1}m)\geq\varphi(c^{-1}m)\varphi(a)\varphi(b)$, we see that
Proposition 8.4 follows from the bound

$\displaystyle\sum_{\begin{subarray}{c}c\mid m\\\ p\mid c\Rightarrow p\leq
z\end{subarray}}\frac{1}{c}\frac{m/c}{\varphi(m/c)}\underset{p\mid
ab\Rightarrow p\leq z}{\sum_{ac\leq y}\sum_{bc\leq
y}}\frac{\lambda_{1}(ac)\lambda_{2}(bc)}{\varphi(a)\varphi(b)}\ll(\log\log
x)^{O(1)}\prod_{p\leq
z}\Big{(}1+\frac{\lambda_{1}(p)+\lambda_{2}(p)}{p}\Big{)}.$ (8.24)

To establish (8.24), we first note that

$\displaystyle\underset{p\mid ab\Rightarrow p\leq z}{\sum_{ac\leq
y}\sum_{bc\leq
y}}\frac{\lambda_{1}(ac)\lambda_{2}(bc)}{\varphi(a)\varphi(b)}\leq\Big{(}\prod_{p\leq
z}\sum_{k\geq
0}\frac{\lambda_{1}(p^{k+\operatorname{ord}_{p}(c)})}{\varphi(p^{k})}\Big{)}\Big{(}\prod_{p\leq
z}\sum_{k\geq
0}\frac{\lambda_{2}(p^{k+\operatorname{ord}_{p}(c)})}{\varphi(p^{k})}\Big{)}.$
(8.25)

Exactly as in the proof of [28, Theorem 3.10], we have that if
$i\in\\{1,2\\}$, then

$\prod_{p\leq z}\sum_{k\geq
0}\frac{\lambda_{i}(p^{k+\operatorname{ord}_{p}(c)})}{\varphi(p^{k})}\ll\prod_{p\leq
z}(1+\lambda_{i}(p)/p)(3\operatorname{ord}_{p}(c)+3).$ (8.26)

The proposition follows once we insert (8.26) into (8.25) and use the fact as
in the proof of [28, Theorem 3.10] that

$\sum_{\begin{subarray}{c}c\mid m\\\ p\mid c\Rightarrow p\leq
z\end{subarray}}\frac{1}{c}\frac{m/c}{\varphi(m/c)}\prod_{p\leq
z}(3\operatorname{ord}_{p}(c)+3)^{2}\ll(\log\log x)^{10^{6}}.\qed$

The next proposition will be needed along with Propositions 8.3 and 8.4.

######  Proposition 8.5 (Proposition 3.18 of [29]).

Let $x\geq 2$ be a real number and $q$ be a positive integer. If
$\delta\in(0,1)$, then

$\sum_{c|q}\frac{[q/c^{2},1]\varphi((c,q/c))}{\log([q/c^{2},1]x)^{2-\delta}}\ll\frac{q(\log\log(e^{e}q))^{O(1)}}{(\log
qx)^{2-\delta}},$

with an absolute implied constant.

To prove Proposition 6.2, we first estimate the sum when $|m|\geq M(\log
kQ)^{\varepsilon/3}$ in (8.20).

######  Lemma 8.6.

Recall the notation in Theorem 1.1 and Section 5.6. If $A>0$ and
$\varepsilon\in(0,1)$ are real numbers, then we have

$\sum_{|m|\geq M(\log kQ)^{\varepsilon/3}}\langle
P_{m}(\cdot,\Psi_{m}),\overline{F_{k}}G_{k}\rangle_{q}\ll_{A,\varepsilon}M(\log
kQ)^{-A}.$

###### Proof.

By the definition of $P_{m}(z,\Psi_{m})$ and Lemma 8.2, we have

$\langle
P_{m}(\cdot,\Psi_{m}),\overline{F_{k}}G_{k}\rangle_{q}=\int_{\Gamma_{0}(q)\backslash\mathbb{H}}\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1)}e(m\operatorname{Re}(\gamma
z))\Psi_{m}(\operatorname{Im}(\gamma
z))F_{k}(z)\overline{G_{k}(z)}\frac{dxdy}{y^{2}}.$ (8.27)

Recall the discussion in Section 5.6. Since
$\operatorname{supp}\check{\psi}\subset\mathcal{F}$, the function
$\Psi(\operatorname{Im}(\gamma z))$ is nonzero only if
$\operatorname{Im}(\gamma z)\geq\sqrt{3}/2$. Invoking [14, Lemma 2.10], we
have
$\\#\\{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1):\operatorname{Im}(\gamma