# Tuning of magnetic frustration in $S=1/2$ Kagomé lattices
{[Cu3(CO3)2$(bpe)_{3}$](ClO4)2}n and {[Cu3(CO3)2$(bpy)_{3}$](ClO4)2}n through
rigid and flexible ligands

M. O. Ajeesh  A. Yogi  M. Padmanabhan  R. Nath<EMAIL_ADDRESS>School of
Physics, Indian Institute of Science Education and Research,
Thiruvananthapuram-695016, India  School of Chemistry, Indian Institute of
Science Education and Research, Thiruvananthapuram-695016, India

###### Abstract

Single crystalline and polycrystalline samples of $S=1/2$ Kagomé lattices
{[Cu3(CO3)2$(bpe)_{3}$](ClO4)2}n and {[Cu3(CO3)2$(bpy)_{3}$](ClO4)2}n,
respectively were synthesized. Their structural and magnetic properties were
characterized by means of x-ray diffraction and magnetization measurements.
Both compounds crystalize in a hexagonal structure (space group $P-6$)
consisting of CuO4 Kagomé layers in the $ab$-plane but linked along $c$
direction through either rigid $bpy$ or flexible $bpe$ ligands to form 3D
frame works. Magnetic measurements reveal that both the compounds undergo
ferromagnetic ordering ($T_{\rm C}$) at low temperatures and the $T_{\rm C}$
and the extent of frustration could be tuned by changing the nature of the
pillar ligands. {[Cu3(CO3)2$(bpe)_{3}$](ClO4)2}n which is made up of flexible
$bpe$ ligands has a $T_{\rm C}$ of 5.7 K and a Curie-Weiss temperature
($\theta_{\rm CW}$) of -39.7 K giving rise to a frustration parameter of
$\frac{|\theta_{\rm CW}|}{T_{\rm C}}$ $\simeq$ 6.96. But the replacement of
$bpe$ by a more rigid and electronically delocalized $bpy$ ligand leads to an
enhanced $T_{\rm C}$ $\simeq$ 9.3 K and a reduced frustration parameter of
$\frac{|\theta_{\rm CW}|}{T_{\rm C}}$ $\simeq$ 3.54.

###### keywords:

A. Magnetically ordered materials, D. Exchange and superexchange

##  1 Introduction

Frustrated magnetism is one of the most fascinating topics in condensed matter
physics which attracts enormous attention from both experimental and
theoretical researchers [1, 2]. The $S=1/2$ Kagomé lattices consisting of
vertex-sharing triangular plaquettes of antiferromagnetically (AF) interacting
$S=1/2$ ions in two-dimension (2D) are known to be highly frustrated in
nature. In real materials, the combination of geometric frustration and strong
quantum effects results in the persistence of spin fluctuations down to low
temperatures. This leads to various unusual ground state properties including
quantum spin liquid (QSL) [3]. The exact ground state of $S=1/2$ AF Kagomé
lattice is still controversial since theory predicts different possibilities
of the excitation spectra. It could be either gapped or gapless QSL ground
state [2, 4, 5, 6, 7, 8]. With these variety of magnetic ground states, the
Kagomé system becomes an interesting candidate with possibilities of novel
low-energy and low-temperature magnetic properties.

Cu3V2O7(OH)2.2H2O is the first material realization of $S=1/2$ AF Kagomé
lattice in which the interaction was found to be anisotropic and it under goes
a weak magnetic ordering at low temperature [9, 10]. Later, ZnCu3(OH)6Cl2 was
realized to be a perfect $S=1/2$ Kagomé lattice with isotropic AF Heisenberg
interaction [11]. Neutron scattering and $\mu$SR experiments suggest that the
material does not undergo magnetic ordering down to 50 mK [12, 13]. Heat
capacity follows a power law behaviour over a wide intermediate $T$-range [12]
and NMR shift which is a measure of intrinsic susceptibility follows a power
law. All these experimental features point to a possible gapless QSL ground
state in ZnCu3(OH)6Cl2. A gapless QSL ground state is also reported in
vanadium oxyfluoride [NH4]2[C7H14N][V7O6F18] [14]. Subsequently, a series of
Kagomé lattice compounds have been synthesized where BaCu3V2O8(OH)2 was found
to order at 9 K [15], SrCr9pGa12-9pO19 shows a spin-glass transition at
$T_{\rm g}$ $\simeq$ 4 K, but retains fluctuations down to low temperature
[16], and also some of the compounds show different unusual ground state
properties [17, 18, 19].

Recently, {[Cu3(CO3)2$(bpe)_{3}$](ClO4)2}n (abbreviated as Cu-$bpe$), with a
planar S = 1/2 Kagomé layer pillared by a flexible organic ligand $bpe$ [$bpe$
= 1,2-di(4-pyridyl)ethane] was reported along with its structural and magnetic
characterizations [20]. It has a hexagonal structure with $P-6$ space group
and lattice constants $a$ = 9.3139(6) Å, $b$ = 9.3139(6) Å, and $c$ =
13.3133(9) Å. As seen from the crystal structure [Fig. 1(a)], the Cu2+ ions
are chelated to CO32- anions to form a perfectly 2D Kagomé layer in the
$ab$-plane. The adjacent Kagomé layers are further interconnected via $bpe$
pillar ligands along the $c$-axis forming a three dimensional (3D) structure
[Fig. 1(b)] [20]. In the layer the Cu - O bond lengths vary from 1.949(4) Å to
2.763(5) Å while both of the out of the plane Cu - N bond lengths are 1.987(3)
Å suggesting that the CuO4N2 octahdra takes a highly distorted structure.
Magnetic susceptibility $\chi$($T$) shows a ferromagnetic transition ($T_{\rm
C}$) at low temperature. The magnetization ($M$) vs. applied field ($H$) at
$T$=2 K is reported to show a small hysteresis loop with a coercive field of
8.5 Oe. Heat capacity data do not show any well defined peak at the $T_{\rm
C}$ [20]. On the other hand, 1H NMR spin-lattice relaxation rate ($1/T_{1}$)
and spectral measurements by Kikuchi et al.,[21] clearly suggest the
ferromagnetic ordering at the $T_{\rm C}$.

In this paper we report synthesis, structural, and magnetic characterization
of a new Kagomé system {[Cu3(CO3)2$(bpy)_{3}$](ClO4)2}n [abbreviated as
Cu-$bpy$] which contains a very rigid and electronically delocalized ligand
$bpy$ [$bpy$ = 1,2-di(4-pyridyl)ethylene] interconnecting the adjacent layers.
To understand the role of pillar ligands on the magnetic properties of the
Kagomé systems we have also reinvestigated the recently reported Cu-$bpe$
Kagomé system with flexible $bpe$ pillar ligand having no electronic
delocalization between the ends.

Figure 1:  (a) A section of the 2D Kagomé layer in Cu-$bpe$ formed by Cu2+ and
CO32- in the $ab$-plane. (b) 3D framework of Cu-$bpe$ showing the Kagomé
layers pillared by the $bpe$ ligand along the $c$-axis.

##  2 Experimental Details

The Kagomé compound Cu-$bpe$ was synthesized following the procedure reported
in Ref. [20]. Hexagonal plate-like violet crystals formed (90% yield) were
filtered and washed with distilled water and then air-dried. A good quality
single crystal with dimension [$0.4\times 0.4~{}\times 0.05$] mm3 was used for
x-ray diffraction (XRD) (Bruker APEX-II machine with Mo Kα1 radiation of
wavelength $\lambda$ = 0.71073 Å) study. The phase purity of the sample was
checked by powder XRD (PANalytical machine with Cu Kα radiation of $\lambda$ =
1.54060 Å at room temperature) and matching with the simulated pattern.

Since the above procedure failed to produce the expected Kagomé compound with
the $bpy$ ligand, we made use of a modified method for the synthesis of new
compound Cu-$bpy$. To a solution of Cu(ClO4)2.6H2O (1.853 g, 5 mM, in 100 mL
distilled water and 50 mL aqueous 20% ammonia), a $bpy$ solution (0.912 g, 5
mM, in 50 mL aqueous 20% ammonia and 20 mL methanol) was added and stirred for
1 hour. After 3 days, blue-violet polycrystalline powder of Cu-$bpy$ was
formed in high yield (92%) by fixing atmospheric CO2 as CO32- anion. The solid
compound was filtered, washed with water and methanol and then air-dried. Our
repeated attempts to get single crystal form of Cu-$bpy$ by varying
experimental conditions turned out to be unsuccessful. However, the
polycrystalline sample obtained was phase-pure form of the Kagomé structure as
confirmed by powder XRD studies along with FTIR and CHN analysis.

DC magnetic susceptibility $\chi$ (= $M/H$, where $M$ is magnetization and $H$
is applied magnetic field) was measured as a function of $T$ (2 K $\leq$ $T$
$\leq$ 300 K) and at different applied fields. A magnetization isotherm ($M$
vs. $H$) was measured at $T$ = 4 K by varying the magnetic field up to 7 T.
Both $\chi$ and magnetization isotherm were measured using a 7 T SQUID - VSM
(Quantum Design). AC magnetic susceptibility ($\chi_{ac}$ = $\chi^{\prime}$ \+
i$\chi^{\prime\prime}$, where $\chi^{\prime}$ is the real part and
$\chi^{\prime\prime}$ is the imaginary part of the susceptibility)
measurements were performed using a separate AC susceptibility setup (Cryobind
AC susceptometer) as a function of $T$ and at different frequencies.

##  3 Results

###  3.1 X-ray diffraction

Single crystal XRD was performed on a good quality single crystal of Cu-$bpe$.
It is found to be crystallizing in the hexagonal structure with space group
$P-6$ and lattice constants $a$ = 9.4102(6) Å, $b$ = 9.4102(6) Å, $c$ =
13.3253(8) Å, and Vcell = 1021.89(11) Å3 which are consistent with the
previous report [20]. Powder XRD data of Cu-$bpe$ is shown in Fig. 2(a) where
all the peaks are indexed based on the above structural data. As shown in Fig.
2(b), very similar peaks appear in the powder XRD pattern of Cu-$bpy$
suggesting that Cu-$bpy$ is isostructural to Cu-$bpe$. Rietveld refinement was
performed using the FullProf software with the structural parameters reported
in Ref. [20] as the starting parameters [22]. Since $bpy$ is restricted in the
$trans$-configuration because of the -HC=CH- double bond between the pyridyl
moieties, C and H atoms corresponding to the orientational disorder (C9 and
H9) seen in Cu-$bpe$ were removed and the occupancy of C8 and H8 atoms were
corrected to one. The room-temperature powder XRD pattern along with the
Rietveld refinement fit is shown in Fig. 2(b). The Rietveld refinements
confirmed the phase purity of Cu-$bpy$ and also the hexagonal structure (space
group: $P-6$), same as that of Cu-$bpe$. The best fit was obtained with
$\chi^{2}$ $\simeq$ 1.86 and the lattice parameters obtained from refinement
are $a$ = 9.3024(5) Å, $b$ = 9.3024(5) Å, $c$ = 13.3724(7) Å, and Vcell =
1157.17 Å3.

Figure 2: (a) Powder XRD pattern of Cu-$bpe$ recorded at room temperature.
($hkl$) values of the corresponding peaks are indicated. (b) Powder XRD
pattern (open circles) recorded at room temperature for Cu-$bpy$. The solid
line through the experimental points are the Rietveld refinement profile
calculated using structural parameters of Cu-$bpe$. The short vertical bars
mark the fitted Bragg peak positions. The lowermost curve represents the
difference between the experimental and calculated intensities.

###  3.2 DC Magnetization

The DC magnetic susceptibility ($\chi$) data of Cu-$bpe$ and Cu-$bpy$ as a
function of $T$ at an applied magnetic field of 1 T are shown in Fig. 3(a) and
3(b), respectively. At high $T$s, $\chi$($T$) increases in a Curie-Weiss
manner with decreasing $T$ and then shows a rapid increase at low temperature.
This rapid increase which occurs below about 20 K for both the compounds is an
indication of ferromagnetic (FM) correlation.

In order to extract the magnetic parameters, the $\chi$($T$) data at high $T$
region are fitted by the expression

$\chi=\chi_{0}+\frac{C}{T+\theta_{\rm CW}}\,,$ (1)

where $\chi_{0}$ is the temperature independent contribution consisting of
core diamagnetism of the core shells and Van-Vleck paramagnetism of the open
shells of Cu2+ ions. The second term is the Curie-Weiss (CW) law with $C$ (=
$N_{A}\mu_{\rm eff}^{2}/3k_{B}$, where $N_{A}$ is Avogadro’s number, $\mu$eff
is the effective magnetic moment, and $k_{B}$ is Boltzmann’s constant) being
the Curie constant and $\theta_{\rm CW}$ the Curie-Weiss temperature. The
extracted parameters for both the compounds are mentioned in Table-1. The
calculated values of $\mu$eff from $C$ for both the compounds are in good
agreement with the theoretical value [$\mu$eff = $g\sqrt{S(S+1)}$$\mu$B] of
1.73 $\mu$B expected for $S=1/2$ systems assuming $g$-factor $g$ = 2. The core
diamagnetic susceptibility ($\chi_{\rm{core}}$) was calculated by adding the
contributions from individual ions to be $-5.2534\times 10^{-4}$ cm3/mol and
$-4.9126\times 10^{-4}$ cm3/mol for Cu-bpe and Cu-bpy, respectively [23]. The
Van-Vleck paramagnetic susceptibility ($\chi_{\rm VV}$) was estimated by
subtracting $\chi_{\rm{core}}$ from $\chi_{0}$ to be $\sim 4.974\times
10^{-4}$ cm3/mol and $\sim 3.012\times 10^{-4}$ cm3/mol for Cu-bpe and Cu-bpy,
respectively.

Figure 3: DC Magnetic susceptibility $\chi$ (left $y$-axis) and $\chi^{-1}$ (right $y$-axis) of (a) Cu-$bpe$ and (b) Cu-$bpy$ vs. temperature $T$ at an applied magnetic field $H$ = 1 T. The solid red lines are the CW fits to the data for the temperature range 189 - 300 K and 154 - 185 K for Cu-$bpe$ and Cu-$bpy$, respectively. Insets: ZFC and FC susceptibilities of Cu-$bpe$ and Cu-$bpy$ vs. $T$ at low fields. Table 1: Magnetic parameters obtained by fitting Eq. 1 to $\chi$($T$) data in the $T$-range (189 - 300 K) and (154 - 185 K) for Cu-$bpe$ and Cu-$bpy$, respectively. $\mu$eff was calculated using the experimental $C$ value. | $\chi_{0}$ | $C$ | $\theta_{\rm CW}$ |  $\mu$eff
---|---|---|---|---
|  (cm3/mol.Cu) |  (cm3.K/mol.Cu) | (K) |  ($\mu$B)
Cu-$bpe$ |  -0.279(5)$\times$10-4 | 0.378(4) | -39.7(9) | 1.738(2)
Cu-$bpy$ |  -1.904(4)$\times$10-4 | 0.397(6) | -33.0(2) | 1.781(3)

We also measured the low temperature Zero Field Cooled(ZFC) and Field
Cooled(FC) susceptibilities at a small applied field [see insets of Fig. 3(a)
and 3(b)]. No significant splitting between ZFC and FC susceptibilities was
observed for both the compounds. This confirms the absence of any glassy
component and spin reorientation effect in the ordered state. To further
confirm the FM correlation and also to check whether there is any field
induced effect at low temperature, we measured $M$ vs. $H$ at $T$ = 4 K for
both the compounds (Fig. 4). A typical hysteresis loop was observed with a
coercive field $H_{\rm c}$ $\simeq$ 600 Oe and 60 Oe for Cu-$bpe$ and
Cu-$bpy$, respectively suggesting a weak ferromagnetism in both the compounds
below $T_{\rm C}$. Moreover, the value of $H_{\rm c}$ reported in Ref. [20]
for Cu-$bpe$ is two orders of magnitude less than our observed value. The
magnetization was found to reach a saturation value of 0.9 $\mu$B/Cu at about
1.6 T for both the compounds which is close to the expected spin only value of
(= $gS$$\mu$B) 1 $\mu$B, assuming $g$ = 2. No signature of any field induced
transition was obtained in the $M$ vs. $H$ curve.

Figure 4:  $M$ vs. $H$ for (a) Cu-$bpe$ and (b) Cu-$bpy$ at $T$ = 4 K. Figure
5:  Real part of AC magnetic susceptibility ($\chi_{ac}^{\prime}$) of (a)
Cu-$bpe$ and (b) Cu-$bpy$ as a function of $T$ at an applied AC field of
$H_{\rm ac}$ = 0.177 Oe for different frequencies.

###  3.3 AC Susceptibility

Since AC susceptibility ($\chi_{ac}$) is very sensitive to magnetic ordering,
one can gain information about the nature of the magnetic ordering by
measuring the frequency dependent $\chi_{ac}$. Figure 5(a) and 5(b) present
the real part of AC susceptibility ($\chi_{ac}^{\prime}$) of Cu-$bpe$ and
Cu-$bpy$, respectively as a function of $T$ measured at an applied AC field of
$H_{\rm ac}$ = 0.177 Oe for three different frequencies 15 Hz, 420 Hz, and
1000 Hz. The $\chi_{ac}^{\prime}$ for Cu-$bpe$ shows a broad maximum at
$T_{\rm C}$ $\simeq$ 5.7 K while for Cu-$bpy$ the maximum is more pronounced
at $T_{\rm C}$ $\simeq$ 9.3 K, reflecting their $T_{\rm C}$s. The broad maxima
for both the compounds remain frequency independent, indicating the absence of
spin-glass behaviour below $T_{\rm C}$ [24]. This is also a characteristic
feature of bulk ferromagnet.

##  4 Discussion and Conclusion

In the crystal structure, the angle $\angle$Cu-O-Cu is 1700 in the Kagomé
plane. According to Goodenough-Kanamori rule, the superexchange interaction
through a non-magnetic ion will favour ferromagnetism if the angle between
them is close to 900 and antiferromagnetism if the angle is close to 1800 [25,
26]. This suggests that the intra-layer exchange interaction ($J_{1}$) in both
the compounds should be antiferromagnetic. According to the mean field theory
$\theta_{\rm CW}$ is the sum of all the exchange couplings [27]. But the
negative value of $\theta_{\rm CW}$ indicates that the dominant interaction is
FM. Since $J_{1}$ is expected to be AFM, this dominant FM interaction is most
likely originating from the inter-layer superexchange ($J_{2}$) through $bpe$
and $bpy$ ligands between Cu2+ ions in the adjacent layers. In the present
case, estimation of individual exchange couplings from the values of
$\theta_{\rm CW}$ and $T_{C}$ is not trivial since it may involve another
exchange coupling $J_{3}$ along the hexagon. An estimation of individual
exchange couplings through electronic band structure calculations would be
useful to understand the exchange mechanism in such compounds [28]. FM inter-
layer interactions are also reported in Kagomé systems Cu(1,3-bdc) and
KFe3(OH)6(SO4)2 with similar geometries [29, 30]. There are also other copper
based metal-organic complexes having similar ligands which are reported to
show dominant FM interactions [31].

As discussed before, the distorted CuO4N2 octahedra gives rise to an
anisotropic CuO4 arrangement in the plane. In such a geometry, the
d${}_{x^{2}-y^{2}}$ orbital of Cu2+ ion and 2p orbital of O2- ion, which are
responsible for super-exchange are aligned orthogonal to each other [32]. Due
to this kind of orbital arrangements, the super-exchange between Cu2+ ions are
very weak in the plane and the interaction between the planes via ligands
dominates.

With the estimated values of $\theta_{\rm CW}$ and $T_{\rm C}$, the
frustration in the spin system can be quantified. The frustration parameter
$f$, which is an empirical measure of frustration [33] is given by,

$f=\frac{|\theta_{\rm CW}|}{T_{\rm C}}\,.$ (2)

Typically spin systems with $f$ $>$ 5-10 are considered to be moderately
frustrated. The frustration parameter $f$ for Cu-$bpe$ and Cu-$bpy$ was found
to be 6.96 and 3.54, respectively suggesting that Cu-$bpe$ is more frustrated
than Cu-$bpy$. This can be explained by the structural differences seen in
$bpe$ and $bpy$ ligands which is evident from Fig. 6. The difference between
the $bpe$ and $bpy$ ligands is that, the spacer between the pyridyl rings is
having a single bond ( -CH2-CH2\- ) in $bpe$ compared to a double bond (
-HC=CH- ) in $bpy$. The presence of double bond in Cu-$bpy$ is responsible for
facilitating strong exchange interaction between the Kagomé layers compared to
Cu-$bpe$. Therefore, replacement of the flexible $bpe$ by a more rigid and
electronically delocalized $bpy$ ligand leads to an increase of $T_{\rm C}$
from 5.7 K to 9.3 K and decrease of $f$ from 6.96 to 3.54.

Figure 6:  Structure of (a) $bpe$ and (b) $bpy$ ligands. The portion enclosed
by the circle in (a) is pyridyl ring and in (b) is the spacer.

Ground state properties of such complexes can further be tuned by modifying
the nature of the pillar ligand and increasing the inter-planar distance.
Owing to the perfectly planar Kagomé structure and the tunability of inter-
layer exchange coupling through appropriate linker molecules, this kind of
Kagomé systems would serve as promising candidates for exploring exotic ground
state properties.

## Acknowledgments

We would like to thank Dr. B. R. Sekhar (IOP Bhubaneswar) for SQUID-VSM and
Dr. P. S. Anil Kumar (IISc Bangalore) for AC susceptibility measurements. DST
India is acknowledged for financial support.

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z)\geq\sqrt{3}/2\\}\ll 1.$ Thus, by the rapid decay of $\Psi(y)$ in (5.10) and
the inequality of arithmetic and geometric means, we obtain that

$\displaystyle\langle P_{m}(\cdot,\Psi_{m}),\overline{F_{k}}G_{k}\rangle_{q}$
$\displaystyle\ll_{A}\int_{\Gamma_{0}(q)\backslash\mathbb{H}}\\#\\{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1):\operatorname{Im}(\gamma
z)\geq\tfrac{\sqrt{3}}{2}\\}(M/|m|)^{A}|F_{k}(z)\overline{G_{k}(z)}|\frac{dxdy}{y^{2}}$
$\displaystyle\ll_{A}(M/|m|)^{A}(\langle F_{k},F_{k}\rangle_{q}+\langle
G_{k},G_{k}\rangle_{q})\ll_{A}(M/|m|)^{A}.$

Hence, the lemma follows upon summing over $m\geq M(\log kQ)^{\varepsilon/3}$
and taking $A$ sufficiently large with respect to $\varepsilon$. ∎

The next lemma estimates the contribution from the $m^{\text{th}}$ incomplete
Poincaré series, where we shall sum up over the range $1\leq|m|\leq M(\log
kQ)^{\varepsilon/3}$. With the notation in Section 5.6, we know that
$\check{\psi}(z)=\psi(z)|_{\mathcal{F}}$ and $\check{\Psi}$ is the extension
of $\check{\psi}$ to $\mathbb{H}$ by $\Gamma_{\infty}$. Since $\psi\in
C_{c}^{\infty}(Y_{0}(1),M)$, we have that for $M\geq 1$

$y^{\max(j,1/2)}\Psi_{m}^{(j)}(y)=\int_{-1/2}^{1/2}y^{\max(j,1/2)}\frac{\partial^{j}}{\partial
y^{j}}\check{\Psi}(x+iy)e(-mx)\,dx\ll_{j}M^{j}.$ (8.28)

######  Lemma 8.7.

Recall the notation in Section 5.6 and the estimate in (8.28). Define

$M_{f}(x):=\frac{\prod_{p\leq
x}(1+2|\lambda_{f}(p)|/p)}{L(1,\operatorname{Ad}f)(\log ex)^{2}}.$

and let $q_{\diamond}^{2}$ denote the largest square divisor of $q$. If
$\varepsilon\in(0,1)$, then we have

$\sum_{1\leq|m|\leq M(\log kQ)^{\varepsilon/3}}\langle
P_{m}(\cdot,\Psi_{m}),\overline{F_{k}}G_{k}\rangle_{q}\ll_{\varepsilon}M^{4+\varepsilon}(\log
kq)^{\varepsilon}q_{\diamond}^{\varepsilon}M_{f}(kq)^{\frac{1}{2}}M_{g}(kq)^{\frac{1}{2}}.$

###### Proof.

We first prove the case when $f=g$. Recall that
$F_{k}(z)=\rho_{f}(1)y^{k/2}f(z)$ and
$|f(z)|^{2}y^{k}=|f|_{k}\sigma_{j}|^{2}(z_{j})y_{j}^{k}$ as discussed in
Sections 5.1 and 5.2. By Lemma 8.2 and the Fourier expansion of $f$ in (5.4),
we have

$\langle
P_{m}(\cdot,\Psi_{m}),|F_{k}|^{2}\rangle_{q}=|\rho_{f}(1)|^{2}\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\sum_{\begin{subarray}{c}n_{1}\geq
1\\\ n_{2}=n_{1}+mw_{j}\geq
1\end{subarray}}\lambda_{j}(n_{1})\lambda_{j}(n_{2})(n_{1}n_{2})^{\frac{k-1}{2}}\\\
\cdot\int_{0}^{\infty}\exp(-2\pi(n_{1}+n_{2})y_{j})\Psi_{m}(w_{j}y_{j})y_{j}^{k-2}\,dy_{j},$
(8.29)

upon integrating over $x_{j}$.

We then use the method in Luo–Sarnak [26, Section 2] to deal with the integral
in (8.29). Define the Mellin tranform of $\Psi_{m}$ as
$\widehat{\Psi}_{m}(s)=\int_{0}^{\infty}\Psi_{m}(y)y^{s-1}\,dy$. Since
$\Psi_{m}\in\mathcal{C}_{c}^{\infty}(\mathbb{R}_{\geq 0})$,
$\widehat{\Psi}_{m}$ is entire. For any integer $j\geq 0$ and
$\operatorname{Re}(s)<0$, integrating by parts $j$ times and applying the
bound in (8.28), we have that

$\widehat{\Psi}_{m}(s)\ll_{j}\frac{M^{j}}{(1+|s|)^{j}},$ (8.30)

and hence (8.30) also holds for any real $j\geq 0$. We also have the Mellin
inversion of $\widehat{\Psi}_{m}(s)$, namely, for any $A>0$ and $x>0$

$\Psi_{m}(x)=\frac{1}{2\pi
i}\int_{(-A)}\widehat{\Psi}_{m}(s)x^{-s}ds=\frac{1}{2\pi
i}\int_{(A)}\widehat{\Psi}_{m}(-s)x^{s}ds.$ (8.31)

Applying the Mellin inversion, we obtain that the integral in (8.29) becomes

$\displaystyle\int_{(A)}\widehat{\Psi}_{m}(-s)\int_{0}^{\infty}y_{j}^{s+k-2}w_{j}^{s}\exp(-2\pi(n_{1}+n_{2})y_{j})\,dy_{j}\,\frac{ds}{2\pi
i}$ (8.32)
$\displaystyle=\int_{(A)}\frac{\widehat{\Psi}_{m}(-s)w_{j}^{s}}{(2\pi(n_{1}+n_{2}))^{s+k-1}}\int_{0}^{\infty}(2\pi(n_{1}+n_{2})y_{j})^{s+k-1}\exp(-2\pi(n_{1}+n_{2})y_{j})\,\frac{dy_{j}}{y_{j}}\,\frac{ds}{2\pi
i}$
$\displaystyle=\frac{\Gamma(k-1)}{(4\pi)^{k-1}}\Big{(}\frac{1}{\frac{n_{1}+n_{2}}{2}}\Big{)}^{k-1}\int_{(A)}\frac{\widehat{\Psi}_{m}(-s)w_{j}^{s}}{(4\pi(\frac{n_{1}+n_{2}}{2}))^{s}}\frac{\Gamma(s+k-1)}{\Gamma(k-1)}\,\frac{ds}{2\pi
i},$

by using the definition of $\Gamma$-function in the last step. By Stirling's
formula, if $\operatorname{Re}(s)=A>0$, then, as in [26, Eq. (2.3)], we have
that

$\frac{\Gamma(s+k-1)}{\Gamma(k-1)}=(k-1)^{A}\Big{(}1+O_{A}\Big{(}\frac{(1+|s|)^{2}}{k}\Big{)}\Big{)}\ll_{A}k^{A}(1+|s|)^{2}.$

Using (8.30) with $j=3+\varepsilon$, we have that the integral in (8.32) is

$\ll_{A,\varepsilon}\int_{(A)}\frac{M^{3+\varepsilon}}{(1+|s|)^{3+\varepsilon}}\frac{w_{j}^{A}k^{A}}{(4\pi(\frac{n_{1}+n_{2}}{2}))^{A}}(1+|s|)^{2}\,|ds|\ll_{A}M^{3+\varepsilon}\max\Big{(}1,\frac{\max(n_{1},n_{2})}{w_{j}k}\Big{)}^{-A}.$
(8.33)

Hence, using (8.32) and (8.33) in (8.29), we conclude that

$\langle
P_{m}(\cdot,\Psi_{m}),|F_{k}|^{2}\rangle_{q}\ll_{A}\frac{M^{3+\varepsilon}|\rho_{f}(1)|^{2}\Gamma(k-1)}{(4\pi)^{k-1}}\\\
\cdot\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\sum_{\begin{subarray}{c}n_{1}\geq
1\\\ n_{2}=n_{1}+mw_{j}\geq
1\end{subarray}}\lambda_{j}(n_{1})\lambda_{j}(n_{2})\Big{(}\frac{\sqrt{n_{1}n_{2}}}{\frac{n_{1}+n_{2}}{2}}\Big{)}^{k-1}\max\Big{(}1,\frac{\max(n_{1},n_{2})}{w_{j}k}\Big{)}^{-A}.$
(8.34)

Upon using (5.2), (5.6), and applying the inequality of arithmetic and
geometric means to the factor $\sqrt{n_{1}n_{2}}/((n_{1}+n_{2})/2)$, we have
that (8.34) is

$\ll_{A}\frac{M^{3+\varepsilon}}{L(1,\operatorname{Ad}f)kq}\sum_{c\mid
q}\\#\mathcal{C}[c]\sum_{\begin{subarray}{c}n_{1}\geq 1\\\
n_{2}=n_{1}+mw_{c}\geq
1\end{subarray}}\lambda_{[c]}(n_{1})\lambda_{[c]}(n_{2})\max\Big{(}1,\frac{\max(n_{1},n_{2})}{w_{c}k}\Big{)}^{-A}.$
(8.35)

We decompose the inner sum of (8.35) dyadically and applying Proposition 8.3.
Then the inner sum of (8.35) is

$\displaystyle\ll_{A}\sum_{N=0}^{\infty}\sum_{\begin{subarray}{c}n_{1}\geq
1,n_{2}=n_{1}+mw_{c}\geq 1,\\\ \lfloor 2^{N-1}\rfloor
w_{c}k\leq\max(n_{1},n_{2})\leq
2^{N}w_{c}k\end{subarray}}|\lambda_{[c]}(n_{1})\lambda_{[c]}(n_{2})|\max\Big{(}1,\frac{\max(n_{1},n_{2})}{w_{c}k}\Big{)}^{-A}$
(8.36)
$\displaystyle\ll_{A}\sum_{N=0}^{\infty}q_{\diamond}^{\varepsilon}(\log\log(e^{e}q))^{O(1)}\frac{2^{N}w_{c}k\prod_{p\leq
2^{N}w_{c}k}(1+2|\lambda_{f}(p)|/p)}{\log(e2^{N}w_{c}k)^{2-\frac{\varepsilon}{3}}}\cdot
2^{-NA}$
$\displaystyle\ll_{A}q_{\diamond}^{\varepsilon}(\log\log(e^{e}q))^{O(1)}\frac{w_{c}k\prod_{p\leq
w_{c}k}(1+2|\lambda_{f}(p)|/p)}{\log(ew_{c}k)^{2-\frac{\varepsilon}{3}}}$
$\displaystyle\qquad\cdot\Big{(}\sum_{N=0}^{\infty}2^{N(1-A)}\Big{(}\frac{\log(ew_{c}k)}{\log(e2^{N}w_{c}k)}\Big{)}^{2-\frac{\varepsilon}{3}}\prod_{w_{c}k\leq
p\leq 2^{N}w_{c}k}(1+2|\lambda_{f}(p)|/p)\Big{)}.$

Using Deligne's bound $|\lambda_{f}(p)|\leq 2$ and choosing $A=2$, we have
that the summation in the last line above is $O(1)$. Hence, by inserting the
bound from (8.36) into (8.35) and using the facts that $w_{c}=[q/c^{2},1]$ and
$\\#\mathcal{C}[c]=\varphi((c,q/c))$, we obtain that

$\displaystyle\langle P_{m}(\cdot,\Psi_{m}),|F_{k}|^{2}\rangle_{q}\ll
M^{3+\varepsilon}\frac{q_{\diamond}^{\varepsilon}(\log\log(e^{e}q))^{O(1)}}{L(1,\operatorname{Ad}f)q}\sum_{c\mid
q}\frac{\varphi((c,q/c))[q/c^{2},1]}{(\log([q/c^{2},1]k))^{2-\frac{\varepsilon}{3}}}\prod_{p\leq
kq}(1+2|\lambda_{f}(p)|/p).$

Applying Proposition 8.5 and the definition of $M_{f}(x)$, we conclude that

$\displaystyle\langle P_{m}(\cdot,\Psi_{m}),|F_{k}|^{2}\rangle_{q}$
$\displaystyle\ll
M^{3+\varepsilon}\frac{q_{\diamond}^{\varepsilon}(\log\log(e^{e}q))^{O(1)}}{L(1,\operatorname{Ad}f)(\log
kq)^{2-\frac{\varepsilon}{3}}}\prod_{p\leq kq}(1+2|\lambda_{f}(p)|/p)$
$\displaystyle\ll M^{3+\varepsilon}(\log
kq)^{\frac{2\varepsilon}{3}}q_{\diamond}^{\varepsilon}M_{f}(kq).$

The lemma then follows by summing the above estimate over $m$ in $[1,M(\log
kQ)^{\varepsilon/3}]$.

In the case that $f\neq g$, we can proceed in the same manner, except now we
replace Lemma 8.2 by (8.19) and Proposition 8.3 by Proposition 8.4 in the
above proof. ∎

It remains for us to estimate the contribution from the incomplete Eisenstein
series. We start with writing the Fourier expansion

$E(z|\Psi_{0})=a_{\Psi_{0},0}(y)+\sum_{|l|\geq 1}a_{\Psi_{0},l}(y)e(lx)$

and recalling the following lemma.

######  Lemma 8.8 (Lemma 4.2 of [9]).

Let $M\geq 1$, and let $\Psi\in C^{\infty}_{c}(\mathbb{R}_{\geq 0})$ be a
function such that for all integer $j\geq 0$,
$y^{\max(j,1/2)}\Psi^{(j)}(y)\ll_{j}M^{j}$. If
$\theta(s)=\pi^{-s}\Gamma(s)\zeta(2s)$, then for all $y>0$

$a_{\Psi,0}(y)=\frac{3}{\pi}\widehat{\Psi}(-1)+\frac{1}{2\pi}\int_{\mathbb{R}}\widehat{\Psi}(-1/2-it)\Big{(}y^{\frac{1}{2}+it}+\frac{\theta(1/2-it)}{\theta(1/2+it)}y^{\frac{1}{2}-it}\Big{)}\,dt.$
(8.37)

For any integer $l\neq 0$, if $\varepsilon>0$ and $A\geq 0$, then for all
$y>0$

$a_{\Psi,l}(y)\ll_{A}\tau(|l|)\sqrt{y}M^{\frac{2}{3}+\varepsilon}\Big{(}\frac{M}{|ly|}\Big{)}^{A}\Big{(}1+\frac{1}{|ly|}\Big{)}^{\varepsilon}.$
(8.38)

In the next lemma, we state the estimate for the incomplete Eisenstein series.
We will see that the expected main term
$\tfrac{3}{\pi}\widehat{\Psi}_{0}(-1)\mathds{1}_{f=g}$ is obtained in this
step.

######  Lemma 8.9.

Recall the definitions of $M_{f}(x)$ and $q_{\diamond}$ in Lemma 8.7, the
notation in Section 5.6, and the bound in (8.28). If $\varepsilon\in(0,1)$,
then we have that

$\langle
E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\widehat{\Psi}_{0}(-1)\mathds{1}_{f=g}\ll
M^{\frac{5}{3}+\varepsilon}q_{\diamond}^{\varepsilon}(\log
kq)^{\varepsilon}M_{f}(kq)^{\frac{1}{4}}M_{g}(kq)^{\frac{1}{4}}.$

To prove Lemma 8.9, we shall begin by using the ideas from [6] and [22]. Let
$1\leq Y\leq(\log kq)^{5}$ be chosen later. Let $h$ be a fixed nonnegative
smooth compactly supported function on $\mathbb{R}$. Assume
$\operatorname{supp}h\subset[1,2],h^{(j)}(y)\ll_{j}1$, and
$\widehat{h}(-1)=\int_{0}^{\infty}h(y)y^{-2}\,dy=\pi/3$. By integration by
parts, $\widehat{h}(-s)\ll(1+|s|)^{-A}$ for any $s>0$ and $A\geq 0$. Define
the smoothed incomplete Eisenstein series

$E^{Y}(z|h)=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1)}h(Y\operatorname{Im}(\gamma
z)).$

By Mellin inversion, we have

$E^{Y}(z|h)=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1)}\int_{(2)}\widehat{h}(-s)Y^{s}\operatorname{Im}(\gamma
z)^{s}\,\frac{ds}{2\pi i}=\int_{(2)}\widehat{h}(-s)Y^{s}E(s,z)\,\frac{ds}{2\pi
i}.$

Moving the contour to $\operatorname{Re}(s)=1/2$ and using that
$\operatorname{res}_{s=1}E(s,z)=3/\pi$, we obtain that

$E^{Y}(z|h)=Y+\int_{(1/2)}\widehat{h}(-s)Y^{s}E(s,z)\,\frac{ds}{2\pi i}.$
(8.39)

Define

$I(Y):=\langle
E^{Y}(\cdot|h)E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}=\int_{(2)}\widehat{h}(-s)Y^{s}\langle
E(s,\cdot)E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}\,\frac{ds}{2\pi
i}.$

By (8.39), we obtain that

$I(Y)=Y\langle
E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}+\int_{(1/2)}\widehat{h}(-s)Y^{s}\langle
E(s,\cdot)E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}\,\frac{ds}{2\pi
i}.$

Hence, we have that

$I(Y)Y^{-1}-\langle E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}\\\
\qquad=Y^{-1}\int_{-\infty}^{\infty}\widehat{h}(-1/2-it)Y^{1/2+it}\langle
E(1/2+it,\cdot)E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}\,\frac{dt}{2\pi}.$
(8.40)

We now estimate the term $\langle
E(1/2+it,\cdot)E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}$ in the
right-hand side of (8.40). We know from (5.8) that $|E(1/2+it,z)|\ll
y^{1/2}(1+|t|)^{3/8}$ for any $y\geq 1/2$, and know that
$y^{1/2}\Psi_{0}(y)\ll 1$ due to (8.28). Since $E(1/2+it,z)$ is
$\Gamma_{0}(1)$-invariant and $\Psi_{0}(y)$ is nonzero only if
$y\geq\sqrt{3}/{2}$, it follows that

$\displaystyle|E(z|\Psi_{0})E(1/2+it,z)|$
$\displaystyle\leq\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1)}|\Psi_{0}(\operatorname{Im}\gamma
z)E(1/2+it,\gamma z)|$ (8.41)
$\displaystyle\ll\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1)}|\Psi_{0}(\operatorname{Im}\gamma
z)|\sqrt{\operatorname{Im}\gamma z}(1+|t|)^{\frac{3}{8}}$
$\displaystyle\ll(1+|t|)^{\frac{3}{8}}\\#\\{\gamma\in\Gamma_{\infty}\backslash\Gamma_{0}(1):\operatorname{Im}\gamma
z\geq\sqrt{3}/2\\},$

which is $O((1+|t|)^{3/8})$ upon using [14, Lemma 2.10]. Using (8.41) and the
inequality of arithmetic and geometric means, we have that

$\langle
E(1/2+it,\cdot)E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}\ll(1+|t|)^{\frac{3}{8}}(\langle
1,|F_{k}|^{2}\rangle_{q}+\langle
1,|G_{k}|^{2}\rangle_{q})\ll(1+|t|)^{\frac{3}{8}}.$ (8.42)

Thus, inserting (8.42) into (8.40) and using the rapid decay of
$\widehat{h}(-1/2-it)$, we have that

$I(Y)Y^{-1}-\langle E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}\ll
Y^{-\frac{1}{2}},$ (8.43)

which implies that to estimate $\langle
E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}$, we need to estimate
$I(Y)$.

Similarly to Lemma 8.2, we can apply the unfolding method with $I(Y)$ as
follows.

$I(Y)=\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\int_{y_{j}=0}^{\infty}h(Yw_{j}y_{j})\int_{x_{j}=0}^{1}E(w_{j}z_{j}|\Psi_{0})F_{k}(z)\overline{G_{k}(z)}\frac{dx_{j}dy_{j}}{y_{j}^{2}}.$
(8.44)

We first consider $I(Y)$ in the case that $f=g$. By the fact that
$|F_{k}(z)|^{2}=|\rho_{f}(1)|^{2}|f|_{k}\sigma_{j}(z_{j})|^{2}y_{j}^{k}$ as in
Section 5.1, we have that

$\displaystyle
I(Y)=|\rho_{f}(1)|^{2}\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\int_{y_{j}=0}^{\infty}h(Yw_{j}y_{j})\int_{x_{j}=0}^{1}E(w_{j}z_{j}|\Psi_{0})|f|_{k}\sigma_{j}(z_{j})|^{2}y_{j}^{k}\frac{dx_{j}dy_{j}}{y_{j}^{2}}.$
(8.45)

In (8.45), we write
$E(w_{j}z_{j}|\Psi_{0})=\sum_{l\in\mathbb{Z}}a_{\Psi_{0},l}(w_{j}y_{j})e(lw_{j}x_{j})$,
expand $f|_{k}\sigma_{j}(z_{j})$ as in (5.4), and integrate (8.45) over
$x_{j}$. Then it follows that

$I(Y)=|\rho_{f}(1)|^{2}\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\sum_{l\in\mathbb{Z}}\sum_{\begin{subarray}{c}n_{1}\geq
1\\\ n_{2}=n_{1}+w_{j}l\geq
1\end{subarray}}\lambda_{j}(n_{1})\lambda_{j}(n_{2})(n_{1}n_{2})^{\frac{k-1}{2}}\\\
\cdot\int_{0}^{\infty}h(Yw_{j}y_{j})a_{\Psi_{0},l}(w_{j}y_{j})\exp(-2\pi(n_{1}+n_{2})y_{j})y_{j}^{k}\frac{dy_{j}}{y_{j}^{2}}.$
(8.46)

If we let

$I_{0}(Y)=|\rho_{f}(1)|^{2}\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\sum_{n\geq
1}\lambda_{j}(n)^{2}n^{k-1}\int_{0}^{\infty}h(Yw_{j}y_{j})a_{\Psi_{0},0}(w_{j}y_{j})\exp(-4\pi
ny_{j})y_{j}^{k}\frac{dy_{j}}{y_{j}^{2}},$ (8.47)

and, for $l\neq 0$, let

$I_{l}(Y)=|\rho_{f}(1)|^{2}\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\sum_{\begin{subarray}{c}n_{1}\geq
1\\\ n_{2}=n_{1}+lw_{j}\geq
1\end{subarray}}\lambda_{j}(n_{1})\lambda_{j}(n_{2})(n_{1}n_{2})^{\frac{k-1}{2}}\\\
\cdot\int_{0}^{\infty}h(Yw_{j}y_{j})a_{\Psi_{0},l}(w_{j}y_{j})\exp(-2\pi(n_{1}+n_{2})y_{j})y_{j}^{k}\frac{dy_{j}}{y_{j}^{2}},$
(8.48)

then $I(Y)=\sum_{l\in\mathbb{Z}}I_{l}(Y)$.

On the other hand, in the case that $f\neq g$, by the remark of Lemma Remark,
we obtain the following simpler expressions for $I_{0}(Y)$ and $I_{l}(Y)$ with
$l\neq 0$:

$I_{0}(Y)=\rho_{f}(1)\overline{\rho_{g}(1)}\sum_{d\mid q}\sum_{n\geq
1}\lambda_{f}(n)\lambda_{g}(n)n^{k-1}\int_{0}^{\infty}h(Ydy)a_{\Psi_{0},0}(dy)\exp(-4\pi
ny)y^{k}\frac{dy}{y^{2}},$ (8.49)
$I_{l}(Y)=\rho_{f}(1)\overline{\rho_{g}(1)}\sum_{d\mid
q}\sum_{\begin{subarray}{c}n_{1}\geq 1\\\ n_{2}=n_{1}+dl\geq
1\end{subarray}}\lambda_{f}(n_{1})\lambda_{g}(n_{2})(n_{1}n_{2})^{\frac{k-1}{2}}\\\
\cdot\int_{0}^{\infty}h(Ydy)a_{\Psi_{0},l}(dy)\exp(-2\pi(n_{1}+n_{2})y)y^{k}\frac{dy}{y^{2}}.$
(8.50)

Recall that to prove Lemma 8.9, we need to estimate $I(Y)$. We first estimate
the contribution from $I_{l}(Y)$ when $l\neq 0$ in the next lemma.

######  Lemma 8.10.

Recall the notation in Lemma 8.9 and the definition of $I_{l}(Y)$ in (8.48).
If $\varepsilon\in(0,1)$, then

$\sum_{l\in\mathbb{Z}\backslash\\{0\\}}I_{l}(Y)Y^{-1}\ll
M^{\frac{5}{3}+\varepsilon}Y^{\frac{1}{2}+\varepsilon}q_{\diamond}^{\varepsilon}(\log
kq)^{\varepsilon}M_{f}(kq)^{\frac{1}{2}}M_{g}(kq)^{\frac{1}{2}}.$

###### Proof.

We first prove the case when $f=g$. The proof uses similar ideas as in Lemma
8.7. By applying Mellin inversion for $h(Yw_{j}y_{j})$, Stirling's formula,
and the rapid decay of $\widehat{h}(-s)$, we obtain that

$\int_{0}^{\infty}h(Yw_{j}y_{j})\exp(-2\pi(n_{1}+n_{2})y_{j})y_{j}^{k}\frac{dy_{j}}{y_{j}^{2}}\ll_{A}\frac{\Gamma(k-1)}{(4\pi)^{k-1}}\Big{(}\frac{1}{\frac{n_{1}+n_{2}}{2}}\Big{)}^{k-1}\max\Big{(}1,\frac{\max(n_{1},n_{2})}{Yw_{j}k}\Big{)}^{-A}.$

By using (8.38), the above bound, and the fact that
$\operatorname{supp}h\subset[1,2]$, we have that the integral in (8.48) is

$\ll_{A}M^{\frac{2}{3}+{\frac{\varepsilon}{2}}}Y^{-\frac{1}{2}+{\frac{\varepsilon}{2}}}\tau(|l|)\Big{(}\frac{MY}{|l|}\Big{)}^{A}\frac{\Gamma(k-1)}{(4\pi)^{k-1}}\Big{(}\frac{1}{\frac{n_{1}+n_{2}}{2}}\Big{)}^{k-1}\max\Big{(}1,\frac{\max(n_{1},n_{2})}{Yw_{j}k}\Big{)}^{-A}.$
(8.51)

Applying (8.51) in (8.48), we then obtain that

$I_{l}(Y)\ll_{A}|\rho_{f}(1)|^{2}\frac{\Gamma(k-1)}{(4\pi)^{k-1}}M^{\frac{2}{3}+{\frac{\varepsilon}{2}}}Y^{-\frac{1}{2}+{\frac{\varepsilon}{2}}}\tau(|l|)\Big{(}\frac{MY}{|l|}\Big{)}^{A}\\\
\cdot\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\sum_{\begin{subarray}{c}n_{1}\geq
1\\\ n_{2}=n_{1}+lw_{j}\geq
1\end{subarray}}\lambda_{j}(n_{1})\lambda_{j}(n_{2})\Big{(}\frac{\sqrt{n_{1}n_{2}}}{\frac{n_{1}+n_{2}}{2}}\Big{)}^{k-1}\max\Big{(}1,\frac{\max(n_{1},n_{2})}{Yw_{j}k}\Big{)}^{-A}$
(8.52)

Invoking (5.2) and (5.6), and applying the inequality of arithmetic and
geometric means for $\sqrt{n_{1}n_{2}}/((n_{1}+n_{2})/2)$ in (8.52), we have
that

$I_{l}(Y)\ll_{A}\frac{M^{\frac{2}{3}+{\frac{\varepsilon}{2}}}Y^{-\frac{1}{2}+{\frac{\varepsilon}{2}}}\tau(|l|)\big{(}\frac{MY}{|l|}\big{)}^{A}}{kqL(1,\operatorname{Ad}f)}\\\
\cdot\sum_{c|q}\\#\mathcal{C}[c]\sum_{\begin{subarray}{c}n_{1}\geq 1\\\
n_{2}=n_{1}+lw_{c}\geq
1\end{subarray}}\lambda_{[c]}(n_{1})\lambda_{[c]}(n_{2})\max\Big{(}1,\frac{\max(n_{1},n_{2})}{Yw_{c}k}\Big{)}^{-A}.$
(8.53)

Similarly to the proof of Lemma 8.7, applying Proposition 8.3 dyacdically in
(8.53) and using the facts that $w_{c}=[q/c^{2},1]$ and
$\\#\mathcal{C}[c]=\varphi((c,q/c))$, we obtain that

$I_{l}(Y)\ll_{A}\frac{M^{\frac{2}{3}+{\frac{\varepsilon}{2}}}Y^{-\frac{1}{2}+{\frac{\varepsilon}{2}}}\tau(|l|)\big{(}\frac{MY}{|l|}\big{)}^{A}}{kqL(1,\operatorname{Ad}f)}q_{\diamond}^{\varepsilon}(\log\log(e^{e}q))^{O(1)}\\\
\cdot\sum_{c|q}\frac{\varphi((c,q/c))[q/c^{2},1]}{\log([q/c^{2},1]k)^{2-{\frac{\varepsilon}{2}}}}Yk\prod_{p\leq
Ykq}(1+2|\lambda_{f}(p)|/p).$

By Proposition 8.5 and observing that the contribution from the product when
$kq\leq p\leq Ykq$ is trivial since we assume $1\leq Y\leq(\log kq)^{5}$, we
then have

$\displaystyle I_{l}(Y)$
$\displaystyle\ll_{A}\frac{M^{\frac{2}{3}+{\frac{\varepsilon}{2}}}Y^{-\frac{1}{2}+{\frac{\varepsilon}{2}}}\tau(|l|)\big{(}\frac{MY}{|l|}\big{)}^{A}}{kqL(1,\operatorname{Ad}f)}\frac{qq_{\diamond}^{\varepsilon}(\log\log(e^{e}q))^{O(1)}}{(\log
kq)^{2-{\frac{\varepsilon}{2}}}}Yk\prod_{p\leq kq}(1+2|\lambda_{f}(p)|/p)$
$\displaystyle\ll_{A}M^{\frac{2}{3}+{\frac{\varepsilon}{2}}}Y^{\frac{1}{2}+{\frac{\varepsilon}{2}}}\tau(|l|)\big{(}\tfrac{MY}{|l|}\big{)}^{A}q_{\diamond}^{\varepsilon}(\log
kq)^{\varepsilon}M_{f}(kq),$

by the definition of $M_{f}(x)$ in Lemma 8.7. Therefore, we obtain that

$\displaystyle\sum_{l\in\mathbb{Z}\backslash\\{0\\}}I_{l}(Y)Y^{-1}$
$\displaystyle\ll_{A}M^{\frac{2}{3}+{\frac{\varepsilon}{2}}}Y^{-\frac{1}{2}+{\frac{\varepsilon}{2}}}q_{\diamond}^{\varepsilon}(\log
kq)^{\varepsilon}M_{f}(kq)\sum_{l\in\mathbb{Z}_{\neq
0}}\tau(|l|)\big{(}\tfrac{MY}{|l|}\big{)}^{A}$ $\displaystyle\ll
M^{\frac{5}{3}+\varepsilon}Y^{\frac{1}{2}+\varepsilon}q_{\diamond}^{\varepsilon}(\log
kq)^{\varepsilon}M_{f}(kq),$

when taking $A=1+\varepsilon/2$ in the last step.

In the case that $f\neq g$, we proceed similarly, except that $I_{l}(Y)$ is
now simpler (see (8.50)) and we need to replace Proposition 8.3 by Proposition
8.4 in the above proof. ∎

######  Lemma 8.11.

Recall the notation in Lemma 8.9 and the definition of $I_{0}(Y)$ in (8.47).
If $\varepsilon\in(0,1)$, then we have that

$I_{0}(Y)Y^{-1}-\frac{3}{\pi}\widehat{\Psi}_{0}(-1)\mathds{1}_{f=g}\ll
M^{1+\varepsilon}Y^{-\frac{1}{2}}(\log kQ)^{\varepsilon}.$

###### Proof.

By (8.44) and the expression of $a_{\Psi_{0},0}(y)$ in (8.37), we have that

$I_{0}(Y)=\frac{3}{\pi}\widehat{\Psi}_{0}(-1)\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\int_{y_{j}=0}^{\infty}h(Yw_{j}y_{j})\int_{x_{j}=0}^{1}F_{k}(z)\overline{G_{k}(z)}\frac{dx_{j}dy_{j}}{y_{j}^{2}}+\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\int_{y_{j}=0}^{\infty}h(Yw_{j}y_{j})\\\
\cdot\int_{x_{j}=0}^{1}\Big{(}\frac{1}{2\pi}\int_{\mathbb{R}}\widehat{\Psi}_{0}(-\tfrac{1}{2}-it)\Big{(}(w_{j}y_{j})^{\frac{1}{2}+it}+\frac{\theta(\frac{1}{2}-it)}{\theta(\frac{1}{2}+it)}(w_{j}y_{j})^{\frac{1}{2}-it}\Big{)}\,dt\Big{)}F_{k}(z)\overline{G_{k}(z)}\frac{dx_{j}dy_{j}}{y_{j}^{2}}.$
(8.54)

Invoking (8.39), we obtain that the first summation on the right-hand side of
(8.54) equals

$\langle E^{Y}(\cdot|h),\overline{F_{k}}G_{k}\rangle_{q}=Y\langle
1,\overline{F_{k}}G_{k}\rangle_{q}+\int_{(1/2)}\widehat{h}(-s)Y^{s}\langle
E(s,\cdot),\overline{F_{k}}G_{k}\rangle_{q}\,\frac{ds}{2\pi i}.$ (8.55)

Applying (8.2), the rapid decay of $\widehat{h}(-s)$, and (7.3), we have that

$\displaystyle\langle E^{Y}(\cdot|h),\overline{F_{k}}G_{k}\rangle_{q}$
$\displaystyle=Y\mathds{1}_{f=g}+O\Big{(}(q/\sqrt{Q})^{-\frac{1}{2}+\varepsilon}Y^{\frac{1}{2}}(\log
kQ)^{\varepsilon}\Big{)}.$ (8.56)

Since $\operatorname{supp}h\subset[1,2]$, so $h(Yw_{j}y_{j})$ is nonzero only
if $w_{j}y_{j}\asymp Y^{-1}$. Then, by the rapid decay of $\widehat{\Psi}(s)$
as in (8.30), the second sum on the right-hand side of (8.54) is

$\displaystyle\ll(M^{1+\varepsilon}Y^{-\frac{1}{2}})\sum_{\mathfrak{a}_{j}\in\mathcal{C}}\int_{y_{j}=0}^{\infty}h(Yw_{j}y_{j})\int_{x_{j}=0}^{1}|F_{k}(z)||G_{k}(z)|\frac{dx_{j}dy_{j}}{y_{j}^{2}}.$
(8.57)

By the inequality of arithmetic and geometric means, we have that (8.57) is

$\displaystyle\ll M^{1+\varepsilon}Y^{-\frac{1}{2}}(\langle
E^{Y}(\cdot|h),|F_{k}|^{2}\rangle_{q}+\langle
E^{Y}(\cdot|h),|G_{k}|^{2}\rangle_{q})$ (8.58) $\displaystyle\ll
M^{1+\varepsilon}Y^{-\frac{1}{2}}(Y+(q/\sqrt{Q})^{-\frac{1}{2}+\varepsilon}Y^{\frac{1}{2}}(\log
kQ)^{\varepsilon}),$

where we apply (8.56) in the last step.

Inserting (8.56) and (8.58) in (8.54), we obtain that

$I_{0}(Y)=\frac{3}{\pi}\widehat{\Psi}_{0}(-1)\Big{(}Y\mathds{1}_{f=g}+O\big{(}(q/\sqrt{Q})^{-\frac{1}{2}+\varepsilon}Y^{\frac{1}{2}}(\log
kQ)^{\varepsilon}\big{)}\Big{)}\\\
+O\Big{(}M^{1+\varepsilon}Y^{-\frac{1}{2}}\Big{(}Y+(q/\sqrt{Q})^{-\frac{1}{2}+\varepsilon}Y^{\frac{1}{2}}(\log
kQ)^{\varepsilon}\Big{)}\Big{)}.$

Hence, the lemma follows from the facts that
$\frac{3}{\pi}\widehat{\Psi}_{0}(-1)\ll 1$ and $Y\geq 1$. ∎

###### Proof of Lemma 8.9.

Applying Lemmas 8.10 and 8.11 in (8.43), we obtain that

$\displaystyle\langle E(\cdot|\Psi_{0}),\overline{F_{k}}G_{k}\rangle_{q}-$
$\displaystyle\frac{3}{\pi}\widehat{\Psi}_{0}(-1)\mathds{1}_{f=g}$
$\displaystyle\ll M^{1+\varepsilon}Y^{-\frac{1}{2}}(\log
kQ)^{\varepsilon}+M^{\frac{5}{3}+\varepsilon}Y^{\frac{1}{2}+\varepsilon}q_{\diamond}^{\varepsilon}(\log
kq)^{\varepsilon}M_{f}(kq)^{\frac{1}{2}}M_{g}(kq)^{\frac{1}{2}}+Y^{-\frac{1}{2}}$
$\displaystyle\ll M^{\frac{5}{3}+\varepsilon}q_{\diamond}^{\varepsilon}(\log
kq)^{\varepsilon}M_{f}(kq)^{\frac{1}{4}}M_{g}(kq)^{\frac{1}{4}},$

upon choosing $Y=M_{f}(kq)^{-\frac{1}{2}}M_{f}(kq)^{-\frac{1}{2}}$ and noting
that $Q\leq q^{2}$. ∎

###### Proof of Proposition 6.2.

Consider the factor $M_{f}(kq)$. When applying Lemma 7.2 and the facts that
$\prod_{p\leq x}(1-\delta/p)\asymp(\log x)^{-\delta}$ and
$\log(kq)/\log(kQ)\ll(q/\sqrt{Q})^{\varepsilon}$, we have that

$M_{f}(kq)\ll\prod_{p\leq
k\sqrt{Q}}\Big{(}1-\frac{(|\lambda_{f}(p)|-1)^{2}}{p}\Big{)}\Big{(}\frac{q}{\sqrt{Q}}\Big{)}^{\varepsilon}(\log
kq)^{\varepsilon}.$ (8.59)

Using Lemmas 8.6, 8.7, and 8.9 in (5.16), we obtain that

$\langle\psi,\overline{F_{k}}G_{k}\rangle_{q}-\frac{3}{\pi}\langle\psi,1\rangle_{1}\mathds{1}_{f=g}\ll_{A}M^{\frac{5}{3}+\varepsilon}q_{\diamond}^{{\varepsilon}}(\log
kq)^{{\varepsilon}}M_{f}(kq)^{\frac{1}{4}}M_{g}(kq)^{\frac{1}{4}}\\\
+M^{4+\varepsilon}q_{\diamond}^{\varepsilon}(\log
kq)^{{\varepsilon}}M_{f}(kq)^{\frac{1}{2}}M_{g}(kq)^{\frac{1}{2}}+M(\log
kQ)^{-A}.$

Therefore, by (8.59) and the fact that
$q_{\diamond}^{\varepsilon}\ll(q/\sqrt{Q})^{\varepsilon}$, we conclude that

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

##  Appendix A The $\mathrm{GL}_{2}\times\mathrm{GL}_{3}$ approximate
functional equation and QUE

by Jesse Thorner

Let $f$ be a holomorphic cuspidal Hecke eigenform of even weight $k\geq 2$ and
$\phi$ be a Hecke–Maaß cusp form, both of level 1. Each corresponds with a
certain unitary cuspidal automorphic representation of
$\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$. Under the stated hypotheses, the
Hecke eigenvalues $\lambda_{f}(n)$ and $\lambda_{\phi}(n)$ of $f$ and $\phi$
(respectively) are all real and multiplicative.

Let $F_{k}(x+iy)=\rho_{f}(1)y^{k/2}f(x+iy)$, where $\rho_{f}(1)$ is a
normalizing constant so that if $\langle\cdot,\cdot\rangle$ is the Petersson
inner product for $Y_{0}(1)$, then $\langle 1,|F_{k}|^{2}\rangle=1$. Iwaniec
[15, Section 11] and Lester, Matomäki, and Radziwiłł [22, Section 4] describe
a method for showing that $\lim_{k\to\infty}\langle\phi,|F_{k}|^{2}\rangle=0$
without a weak subconvexity bound for
$L(\tfrac{1}{2},\mathrm{Ad}\,f\times\phi)$. This appendix shows that their
proof is incomplete without assuming bounds for $L(s,\mathrm{Ad}\,f)$ near
$\mathrm{Re}(s)=1$ that are not yet known. While the dependence on $\phi$ is
kept effective in [22], we suppress it here for simplicity.

It is proved in [15, (3.2) and (11.1)] and [22, (4.2)] that

$|\langle\phi,|F_{k}|^{2}\rangle|\ll_{A,\phi,\varepsilon}\frac{(\log\log
k)^{3}}{\sqrt{k}}|L(\tfrac{1}{2},\mathrm{Ad}\,f\times\phi)|^{\frac{1}{2}}\prod_{p\leq
k}\Big{(}1-\frac{\lambda_{f}(p^{2})}{p}\Big{)}+\frac{1}{(\log k)^{A}}.$ (A.1)

A bound for $L(\frac{1}{2},\mathrm{Ad}\,f\times\phi)$ essentially of the form

$|L(\tfrac{1}{2},\mathrm{Ad}\,f\times\phi)|\ll_{A,\phi,\varepsilon}\sum_{n\leq
k^{2}(\log
k)^{\varepsilon}}\frac{|\lambda_{f}(n^{2})\lambda_{\phi}(n)|}{\sqrt{n}}+\frac{k}{(\log
k)^{A}}$ (A.2)

(a weak form of the approximate functional equation) was asserted without
proof in [15, Section 11.1] and [22, (4.3)]. These appear to neglect the
expression for $L(s,\mathrm{Ad}\,f\times\phi)$ below.

######  Theorem A.1.

Let $f$ be a holomorphic cuspidal Hecke eigenform and $\phi$ be a Hecke–Maaß
newform, each of level $1$. There exists an absolute constant
$0<\omega\leq\frac{1}{10}$ and a function $H(s)$ such that in the region
$\mathrm{Re}(s)\geq\frac{1}{2}-\omega$, $H(s)$ is holomorphic, $H(s)\asymp 1$,
and

$L(s,\mathrm{Ad}\,f\times\phi)=H(s)L(2s,\mathrm{Ad}\,f)L(2s,\mathrm{Ad}\,\phi)\sum_{p\mid
n\implies p\geq 19}\frac{\lambda_{f}(n^{2})\lambda_{\phi}(n)}{n^{s}}.$

###### Proof.

By multiplicativity of the Hecke eigenvalues, it suffices to work locally. Let
$p$ be prime. For any $\pi\in\mathfrak{F}_{2}$ of conductor $1$, we have that

$\displaystyle L_{p}(s,\pi)$
$\displaystyle=\prod_{j\in\\{-1,1\\}}\frac{1}{1-\alpha_{\pi}(p)^{j}p^{-s}}=\sum_{j=0}^{\infty}\frac{\lambda_{\pi}(p^{j})}{p^{js}},$
$\displaystyle L_{p}(s,\mathrm{Ad}\,\pi)$
$\displaystyle=\prod_{j\in\\{-1,0,1\\}}\frac{1}{1-\alpha_{\pi}(p)^{2j}p^{-s}}=\sum_{j=0}^{\infty}\frac{\lambda_{\mathrm{Ad}\,\pi}(p^{j})}{p^{js}}.$

We write $\alpha_{p}=\alpha_{f}(p)$, $\beta_{p}=\alpha_{\phi}(p)$, and

$L_{p}(s,\mathrm{Ad}\,f\times\phi)=\prod_{j\in\\{-1,0,1\\}}\,\prod_{\ell\in\\{-1,1\\}}\frac{1}{1-\alpha_{p}^{2j}\beta_{p}^{\ell}p^{-s}}=\sum_{j=0}^{\infty}\frac{\lambda_{\mathrm{Ad}\,f\times\phi}(p^{j})}{p^{js}}.$

Note that

$\lambda_{f}(p^{j})=\sum_{r=0}^{j}\alpha_{p}^{2r-j},\qquad\lambda_{\phi}(p^{j})=\lambda_{\rho}(p^{j})=\sum_{r=0}^{j}\beta_{p}^{2r-j}.$

By (A.3) and the Ramanujan conjecture for $\pi$, we have the bounds

$|\alpha_{p}|=1,\qquad\max\\{|\beta_{p}|,|\beta_{p}|^{-1}\\}\leq p^{7/64}.$
(A.3)

Recall the well-known Ramanujan-type identity

$\sum_{j=0}^{\infty}\frac{\lambda_{f}(p^{j})\lambda_{\phi}(p^{j})}{p^{js}}=\sum_{j=0}^{\infty}\frac{(\sum_{r=0}^{j}\alpha_{p}^{2r-j})(\sum_{r=0}^{j}\beta_{p}^{2r-j})}{p^{js}}=(1-p^{-2s})L_{p}(s,f\times\phi).$
(A.4)

We will now establish a similar identity for
$L_{p}(s,\mathrm{Ad}\,f\times\phi)$. Consider the rational function

$\frac{1+(\beta_{p}+\beta_{p}^{-1})p^{-s}-(\alpha_{p}^{2}+1+\alpha_{p}^{-2})p^{-2s}}{(1-\alpha_{p}^{2}\beta_{p}p^{-s})(1-\alpha_{p}^{2}\beta_{p}^{-1}p^{-s})(1-\alpha_{p}^{-2}\beta_{p}p^{-s})(1-\alpha_{p}^{-2}\beta_{p}^{-1}p^{-s})}.$
(A.5)

When $\mathrm{Re}(s)>7/64$, we expand (A.5) as a polynomial in $p^{-s}$ times
a product of four geometric sums and conclude that (A.5) equals

$\sum_{j=0}^{\infty}\frac{(\sum_{r=0}^{2j}\alpha_{p}^{2r-2j})(\sum_{r=0}^{j}\beta_{p}^{2r-j})}{p^{js}}=\sum_{j=0}^{\infty}\frac{\lambda_{f}(p^{2j})\lambda_{\phi}(p^{j})}{p^{js}}.$

On the other hand, we identify (A.5) as a product of local $L$-functions,
namely

$L_{p}(s,\mathrm{Ad}\,f\times\phi)\frac{1+\lambda_{\phi}(p)p^{-s}-\lambda_{f}(p^{2})p^{-2s}}{L_{p}(s,\phi)}.$

Therefore, if $\mathrm{Re}(s)>7/64$, then

$L_{p}(s,\mathrm{Ad}\,f\times\phi)=\frac{L_{p}(s,\phi)}{1+\lambda_{\phi}(p)-\lambda_{f}(p^{2})p^{-2s}}\sum_{j=0}^{\infty}\frac{\lambda_{f}(p^{2j})\lambda_{\phi}(p^{j})}{p^{js}},$
(A.6)

provided that the denominator for the leading factor on the right-hand side is
nonzero. We verify that if $p\geq 19$ and $\mathrm{Re}(s)\geq\frac{2}{5}$,
then $|1+\lambda_{\phi}(p)p^{-s}-\lambda_{f}(p^{2})p^{-2s}|>0.06$ using (A.3)
and Mathematica 12. For such $p$ and $s$, we find that

$\frac{L_{p}(s,\phi)}{1+\lambda_{\phi}(p)p^{-s}-\lambda_{f}(p^{2})p^{-2s}}=1+(\lambda_{f}(p^{2})+\lambda_{\phi}(p^{2}))p^{-2s}+\cdots,$
(A.7)

which agrees with the beginning of the expansion

$L_{p}(2s,\mathrm{Ad}\,f)L(2s,\mathrm{Ad}\,\phi)=1+(\lambda_{f}(p^{2})+\lambda_{\phi}(p^{2}))p^{-2s}+\cdots.$

This leads us to define

$H_{p}(s)=\frac{L_{p}(s,\phi)}{(1+\lambda_{\phi}(p)p^{-s}-\lambda_{f}(p^{2})p^{-2s})L_{p}(2s,\mathrm{Ad}\,f)L_{p}(2s,\mathrm{Ad}\,\phi)}.$

By our above numerical calculation and (A.3), we know that if $p\geq 19$, then
in the region $\mathrm{Re}(s)\geq\frac{2}{5}$, $H_{p}(s)$ is holomorphic and

$\displaystyle L_{p}(s,\mathrm{Ad}\,f\times\phi)$
$\displaystyle=H_{p}(s)L_{p}(2s,\mathrm{Ad}\,f)L_{p}(2s,\mathrm{Ad}\,\phi)\sum_{j=0}^{\infty}\frac{\lambda_{f}(p^{2j})\lambda_{\phi}(p^{j})}{p^{js}},$
$\displaystyle H_{p}(s)$
$\displaystyle=1-\lambda_{f}(p)^{2}\lambda_{\phi}(p)p^{-3s}+(\lambda_{f}(p)^{2}(\lambda_{\phi}(p)^{2}+1)-2)p^{-4s}+\cdots.$

Since the coefficients of $p^{-s}$ and $p^{-2s}$ in the expansion of
$H_{p}(s)$ equal zero, (A.3) implies that there exists an absolute constant
$0<\omega\leq\frac{1}{10}$ such that in the region
$\mathrm{Re}(s)\geq\frac{1}{2}-\omega$, the functions

$\mathcal{H}_{1}(s)=\prod_{p\geq
19}H_{p}(s),\qquad\mathcal{H}_{2}(s)=\prod_{p<19}\frac{L_{p}(s,\mathrm{Ad}\,f\times\phi)}{L_{p}(2s,\mathrm{Ad}\,f)L_{p}(2s,\mathrm{Ad}\,\phi)}$

are holomorphic and $\asymp 1$ in the region. To finish, we choose
$H(s)=\mathcal{H}_{1}(s)\mathcal{H}_{2}(s)$. ∎

Using Theorem A.1 and the ideas in [16, Section 5.2], we prove an approximate
functional equation for $L(s,\mathrm{Ad}\,f\times\phi)$. Let $\kappa$ be the
root number of $L(s,\mathrm{Ad}\,f\times\phi)$,
$L_{\infty}(s,\mathrm{Ad}\,f\times\phi)$ be the gamma factor, and

$A>2,\qquad 0<\varepsilon<1,\qquad B=\frac{2A+6}{\varepsilon}+\frac{1}{2}>3.$
(A.8)

Using the functional equation for $L(s,\mathrm{Ad}\,f\times\phi)$ and Theorem
A.1, we derive the identity

$\displaystyle L(\tfrac{1}{2},\mathrm{Ad}\,f\times\phi)=(1+\kappa)\sum_{p\mid
n\implies p\geq 19}\frac{\lambda_{f}(n^{2})\lambda_{\phi}(n)}{\sqrt{n}}W(n),$
(A.9) $\displaystyle W(n)=\frac{1}{2\pi
i}\int_{B-i\infty}^{B+i\infty}H(\tfrac{1}{2}+s)L(1+2s,\mathrm{Ad}\,f)L(1+2s,\mathrm{Ad}\,\phi)\frac{L_{\infty}(\tfrac{1}{2}+s,\mathrm{Ad}\,f\times\phi)}{L_{\infty}(\tfrac{1}{2},\mathrm{Ad}\,f\times\phi)}\frac{e^{s^{2}}}{n^{s}}\frac{ds}{s}.$

We proceed to estimate $W(n)$.

######  Lemma A.2.

With the notation and hypotheses in Theorem A.1, (A.8), and (A.9), there holds

$W(n)\ll_{A,\phi,\varepsilon}\begin{cases}(k^{2}/n)^{B}&\mbox{if $n>k^{2}(\log
k)^{\varepsilon}$,}\\\ (\log k)^{3}\log\log k&\mbox{if $n\leq k^{2}(\log
k)^{\varepsilon}$.}\end{cases}$

###### Proof.

Recall (A.8) and (A.9). Write $s=\sigma+it$. For $-\omega<\sigma\leq B$,
define

$Y(s)=Y_{\mathrm{Ad}\,f\times\phi}(s)=H(\tfrac{1}{2}+s)L(1+2s,\mathrm{Ad}\,f)L(1+2s,\mathrm{Ad}\,\phi)\frac{L_{\infty}(\tfrac{1}{2}+s,\mathrm{Ad}\,f\times\phi)}{L_{\infty}(\tfrac{1}{2},\mathrm{Ad}\,f\times\phi)}\frac{e^{s^{2}}}{n^{s}},$

so that

$W(n)=\frac{1}{2\pi i}\int_{B-i\infty}^{B+i\infty}Y(s)\frac{ds}{s}.$

Theorem A.1 implies that $H(\tfrac{1}{2}+s)\ll 1$. Using GRC for $f$ and
proceeding as in [23, Theorem 2] for $\phi$, we find that

$|L(1+2s,\operatorname{Ad}\,f)L(1+2s,\operatorname{Ad}\,\phi)|\ll_{\phi}\begin{cases}1&\mbox{if
$\sigma=B$,}\\\ (\log k)^{3}(|t|+1)&\mbox{if $\sigma=\frac{1}{2\log
k}$.}\end{cases}$

Also, Stirling's formula yields

$\frac{L_{\infty}(\tfrac{1}{2}+s,\mathrm{Ad}\,f\times\phi)}{L_{\infty}(\tfrac{1}{2},\mathrm{Ad}\,f\times\phi)}\frac{e^{s^{2}}}{n^{s}}\ll_{A,\phi,\varepsilon}\Big{(}\frac{k^{2}}{n}\Big{)}^{\sigma}(|t|+1)^{-200}.$

It follows that

$Y(\sigma+it)\ll_{A,\phi,\varepsilon}\begin{cases}(k^{2}/n)^{B}(|t|+1)^{-100}&\mbox{if
$\sigma=B$,}\\\ n^{-1/(2\log k)}(\log k)^{3}(|t|+1)^{-100}&\mbox{if
$\sigma=\tfrac{1}{2\log k}$.}\end{cases}$ (A.10)

When $n>k^{2}(\log k)^{\varepsilon}$, we estimate the integral in (A.9) on the
line $\sigma=B$. Otherwise, we push the contour to $\sigma=1/(2\log k)$. The
lemma follows from (A.10) in both cases. ∎

######  Remark.

In our application of the Lemma A.2, we will sum
$\lambda_{f}(n^{2})\lambda_{\phi}(n)$ over $n\leq k^{2}(\log k)^{\varepsilon}$
instead of $n\leq k^{2+\varepsilon}$ because our result is sensitive to powers
of $\log k$. Consequently, our bounds for $W(n)$ would not improve in a manner
useful to our application if we push the contour for the integral that defines
$W(n)$ further to the left. In order to push further to the left while
obtaining something useful for our application, we would need bounds for
$L(1+2s,\mathrm{Ad}\,f)L(1+2s,\mathrm{Ad}\,\phi)$ near $\mathrm{Re}(s)=1$ than
are stronger than what is currently known.

Theorem A.1 and Lemma A.2 yield the following bound for
$L(\frac{1}{2},\mathrm{Ad}\,f\times\phi)$.

######  Theorem A.3.

If $f$ is a holomorphic cuspidal Hecke eigenform of even weight $k\geq 2$ and
level $1$, $\phi$ is a Hecke–Maaß cusp form of level $1$, $A>2$, and
$\varepsilon>0$, then

$\displaystyle L(\tfrac{1}{2},\mathrm{Ad}\,f\times\phi)$
$\displaystyle=(1+\kappa)\sum_{\begin{subarray}{c}n\leq k^{2}(\log
k)^{\varepsilon}\\\ p\mid n\implies p\geq
19\end{subarray}}\frac{\lambda_{f}(n^{2})\lambda_{\phi}(n)}{\sqrt{n}}W(n)+O_{A,\phi,\varepsilon}\Big{(}\frac{k}{(\log
k)^{2A+2}}\Big{)}$ $\displaystyle\ll_{A,\phi,\varepsilon}(\log k)^{3}(\log\log
k)\sum_{n\leq k^{2}(\log
k)^{\varepsilon}}\frac{|\lambda_{f}(n^{2})\lambda_{\varphi}(n)|}{\sqrt{n}}+\frac{k}{(\log
k)^{2A+2}}.$

###### Proof.

We estimate the tail of the sum in (A.9) using Lemma A.2, the Cauchy–Schwarz
inequality, Rankin–Selberg theory, and (A.8). We have that

$\displaystyle\sum_{\begin{subarray}{c}n>k^{2}(\log k)^{\varepsilon}\\\ p\mid
n\implies p\geq
19\end{subarray}}\frac{|\lambda_{f}(n^{2})\lambda_{\phi}(n)|}{\sqrt{n}}W(n)$
$\displaystyle\ll_{A,\phi,\varepsilon}k^{2B}\sum_{n>k^{2}(\log
k)^{\varepsilon}}\frac{|\lambda_{f}(n^{2})\lambda_{\phi}(n)|}{n^{B+\frac{1}{2}}}$
(A.11) $\displaystyle\ll_{A,\phi,\varepsilon}k^{2B}\Big{(}\sum_{n>k^{2}(\log
k)^{\varepsilon}}\frac{|\lambda_{f}(n^{2})|^{2}}{n^{B+\frac{1}{2}}}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{n>k^{2}(\log
k)^{\varepsilon}}\frac{|\lambda_{\phi}(n)|^{2}}{n^{B+\frac{1}{2}}}\Big{)}^{\frac{1}{2}}.$

Since

$\sum_{n\leq x}|\lambda_{\phi}(n)|^{2}\leq\sum_{n\leq
x}\lambda_{\phi\times\phi}(n)\ll_{\phi}x$ (A.12)

by Lemma 3.3, Theorem 3.1, and contour integration, it follows from partial
summation that

$\sum_{n>k^{2}(\log
k)^{\varepsilon}}\frac{|\lambda_{\phi}(n)|^{2}}{n^{B+\frac{1}{2}}}\ll_{B,\phi,\varepsilon}k^{1-2B}(\log
k)^{(\frac{1}{2}-B)\varepsilon}.$

Using (A.3), we find that $\sum_{n\leq
x}|\lambda_{f}(n^{2})|^{2}\leq\sum_{n\leq x}\tau_{9}(n)\ll x(\log x)^{8}$ and

$\sum_{n>k^{2}(\log
k)^{\varepsilon}}\frac{|\lambda_{f}(n^{2})|^{2}}{n^{B+\frac{1}{2}}}\ll_{B,\phi,\varepsilon}k^{1-2B}(\log
k)^{8+(\frac{1}{2}-B)\varepsilon}.$

We conclude that (A.11) is $O_{A,\phi,\varepsilon}k(\log k)^{-2A-2}$. The
condition $p\mid n\implies p\geq 19$ is dropped by nonnegativity, so the
theorem follows. ∎

We now apply Theorem A.3 to (A.1).

######  Theorem A.4.

If $f$ is a holomorphic cuspidal Hecke eigenform of even weight $k\geq 2$ and
level $1$, $\phi$ is a Hecke–Maaß cusp form of level $1$, $A>2$, and
$\varepsilon>0$, then

$\langle\phi,|F_{k}|^{2}\rangle\ll_{A,\phi,\varepsilon}(\log
k)^{\frac{3}{2}+\varepsilon}\prod_{p\leq
k}\Big{(}1-\frac{\lambda_{f}(p^{2})+\frac{1}{4}(1-\lambda_{f}(p^{2}))^{2}}{p}\Big{)}+\frac{1}{(\log
k)^{A}}.$

###### Proof.

By Theorem A.3 and the Cauchy–Schwarz inequality,
$L(\frac{1}{2},\mathrm{Ad}\,f\times\phi)^{1/2}$ is

$\ll_{A,\phi,\varepsilon}(\log k)^{\frac{3}{2}}(\log\log
k)^{\frac{1}{2}}\Big{(}\sum_{n\leq k^{2}(\log
k)^{\varepsilon}}\frac{\lambda_{f}(n^{2})^{2}}{\sqrt{n}}\Big{)}^{\frac{1}{4}}\Big{(}\sum_{n\leq
k^{2}(\log
k)^{\varepsilon}}\frac{\lambda_{\phi}(n)^{2}}{\sqrt{n}}\Big{)}^{\frac{1}{4}}+\frac{1}{(\log
k)^{A+1}}.$ (A.13)

It follows from (A.12) and partial summation that (A.13) is

$\ll_{A,\phi,\varepsilon}k^{\frac{1}{4}}(\log
k)^{\frac{3}{2}+\frac{\varepsilon}{8}}(\log\log
k)^{\frac{1}{2}}\Big{(}\sum_{n\leq k^{2}(\log
k)^{\varepsilon}}\frac{\lambda_{f}(n^{2})^{2}}{\sqrt{n}}\Big{)}^{\frac{1}{4}}+\frac{1}{(\log
k)^{A+1}}.$ (A.14)

We apply [22, (4.4)] to conclude that (A.14) is

$\displaystyle\ll_{A,\phi,\varepsilon}\sqrt{k}(\log
k)^{\frac{3}{2}+\varepsilon}\prod_{p\leq k^{2}(\log
k)^{\varepsilon}}\Big{(}1-\frac{\lambda_{f}(p^{2})+\frac{1}{4}(1-\lambda_{f}(p^{2})^{2})}{p}\Big{)}+\frac{1}{(\log
k)^{A+1}}$ $\displaystyle\ll_{A,\phi,\varepsilon}\sqrt{k}(\log
k)^{\frac{3}{2}+\varepsilon}\prod_{p\leq
k}\Big{(}1-\frac{\lambda_{f}(p^{2})+\frac{1}{4}(1-\lambda_{f}(p^{2})^{2})}{p}\Big{)}+\frac{1}{(\log
k)^{A+1}}.$

The desired result follows from (7.2), (7.3), and (A.1). ∎

The quality of the available bounds on $\langle\phi,|F_{k}|^{2}\rangle$
affects the rate of convergence in holomorphic QUE. By the Hecke relations,
$\lambda_{f}(p^{2})=\lambda_{f}(p)^{2}-1$. Assume that there exists an
absolute constant $\theta\geq 0$ such that

$\displaystyle\langle\phi,|F_{k}|^{2}\rangle$
$\displaystyle\ll_{A,\phi,\varepsilon}(\log
k)^{\theta+\varepsilon}\prod_{p\leq
k}\Big{(}1-\frac{\lambda_{f}(p^{2})+\frac{1}{4}(1-\lambda_{f}(p^{2})^{2})}{p}\Big{)}+\frac{1}{(\log
k)^{A}}$ (A.15) $\displaystyle\asymp_{A,\phi,\varepsilon}(\log
k)^{\varepsilon}\prod_{p\leq
k}\Big{(}1-\frac{\lambda_{f}(p)^{2}-1+\frac{1}{4}(1-(\lambda_{f}(p)^{2}-1)^{2})-\theta}{p}\Big{)}+\frac{1}{(\log
k)^{A}}.$

By a different method, Holowinsky (see [7, (1.2)] or [15, (1.14)]) proved that

$\langle\phi,|F_{k}|^{2}\rangle\ll_{\phi,\epsilon}(\log
k)^{\varepsilon}\prod_{p\leq
k}\Big{(}1-\frac{(|\lambda_{f}(p)|-1)^{2}}{2p}\Big{)}.$

We synthesize this bound with (A.13) using the fact that if $\xi\in(0,1)$ and
$a,b>0$, then $\min\\{a,b\\}\leq a^{\xi}b^{1-\xi}$. Let $\xi\in(0,1)$, and let
$A$ be sufficiently large with respect to $\xi$.

$\langle\phi,|F_{k}|^{2}\rangle\ll_{A,\phi,\varepsilon}(\log
k)^{-\delta+\varepsilon},\quad\delta=\max_{0\leq\xi\leq
1}~{}\min_{0\leq\lambda\leq
2}\Big{(}\frac{\xi}{2}(\lambda-1)^{2}+(1-\xi)\Big{(}\lambda^{2}-1+\frac{1}{4}(1-(\lambda^{2}-1)^{2})-\theta\Big{)}\Big{)}.$

(See [15, §12] and [22, §4.3].) One calculates $\delta=0$ when
$\theta\geq\frac{1}{4}$, and $\delta>0$ when $\theta<\frac{1}{4}$. Since
Theorem A.4 establishes (A.15) with $\theta=\frac{3}{2}$, the method in [15,
22] apparently fails to produce any rate of convergence with which
$\langle\phi,|F_{k}|^{2}\rangle\to 0$ as $k\to\infty$ without assuming much
better bounds on $L(s,\mathrm{Ad}\,f)$ near $\mathrm{Re}(s)=1$ than we
currently know (see the remark following the proof of Lemma A.2).

If we replace $\lambda_{f}(n^{2})$ with $\lambda_{\mathrm{Ad}\,f}(n)$, then
there exist absolute constants $z\geq 19$ and
$0<\omega^{\prime}\leq\frac{1}{10}$ such that if
$\mathrm{Re}(s)\geq\frac{1}{2}-\omega^{\prime}$, then

$\mathcal{H}_{1}(s)=\prod_{p>z}\frac{1}{L_{p}(2s,\mathrm{Ad}\,f)(1-\lambda_{f}(p^{2})p^{-2s}+\lambda_{\phi}(p)p^{-3s})},\qquad\mathcal{H}_{2}(s)=\prod_{p\leq
z}\frac{L_{p}(s,\mathrm{Ad}\,f\times\phi)}{L_{p}(2s,\mathrm{Ad}\,f)}$

converge absolutely and are each $\asymp 1$. If
$H(s)=\mathcal{H}_{1}(s)\mathcal{H}_{2}(s)$, then

$L(s,\mathrm{Ad}\,f\times\phi)=H(s)L(2s,\mathrm{Ad}\,f)\sum_{p\mid n\implies
p>z^{\prime}}\frac{\lambda_{\mathrm{Ad}\,f}(n)\lambda_{\phi}(n)}{n^{s}},\qquad\mathrm{Re}(s)\geq\frac{1}{2}-\omega^{\prime}.$

We still arrive at Theorem A.4 because of the contribution of
$L(2s,\mathrm{Ad}\,f)$ in the ensuing version of Lemma A.2.

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