# Correlations of multiplicative functions with automorphic $L$-functions

Yujiao Jiang  Yujiao Jiang
School of Mathematics and Statistics
Shandong University
Weihai
Shandong 264209
China<EMAIL_ADDRESS>and  Guangshi Lü  Guangshi Lü
School of Mathematics
Shandong University
Jinan
Shandong 250100
China<EMAIL_ADDRESS>
###### Abstract.

Let $\lambda_{\phi}(n)$ be the Fourier coefficients of a Hecke holomorphic or
Hecke–Maass cusp form on ${\rm SL}_{2}(\mathbb{Z})$, and $f$ be any
multiplicative function that satisfies two mild hypotheses. We establish a
non-trivial upper bound for the correlation $\sum_{n\leq
X}f(n)\lambda_{\phi}(n+h)$ uniformly in $0<|h|\ll X$. As applications, we
consider some special cases, including
$\lambda_{\pi}(n),\,\mu(n)\lambda_{\pi}(n)$ and any divisor-bounded
multiplicative function. Here $\lambda_{\pi}(n)$ denotes the $n$-th Dirichlet
coefficient of $\text{GL}_{m}$ automorphic $L$-function $L(s,\pi)$ for an
automorphic irreducible cuspidal representation $\pi$, and $\mu(n)$ denotes
the Möbius function. In particular, some savings are achieved for shifted
convolution problems on ${\rm GL}_{m}\times{\rm GL}_{2}\,(m\geq 4)$ and
Hypothesis C for the first time.

###### Key words and phrases:

multiplicative functions, Fourier coefficients, automorphic $L$-functions,
Möbius function, sieve method

###### 2010 Mathematics Subject Classification:

11N37, 11N36, 11F30, 11F66

##  1\. Introduction

Let $f(n)$ and $g(n)$ denote two multiplicative functions. It is a classical
and fundamental problem in analytic number theory to understand the
correlations

$\sum_{n\leq X}f(n)g(n+h),$ (1.1)

where $h$ is a non-zero integer. It is directly related to many celebrated
unsolved problems in number theory, such as the Chowla conjecture and the
divisor correlation conjecture. If $h=0$, this problem then degenerates to the
mean values of the multiplicative functions, which can be tackled by a
coherent set of tools. However, very little is known about correlations (1.1).
The difficulty here is that the shift parameter $h$ destroys the
multiplicativity. It is worth noting that Klurman [19] recently obtained an
asymptotic formula of correlation (1.1) for some bounded pretentious
multiplicative functions $f$ and $g$.

In this present work, we focus on the special case $g(n)=\lambda_{\phi}(n)$,
where $\lambda_{\phi}(n)$ is the Fourier coefficient of any ${\rm GL}_{2}$
Hecke cusp form. This coefficient has been studied intensively for
arithmetical and analytical reasons. Our goal is to obtain a slightly general
result on correlations of the coefficient $\lambda_{\phi}(n)$ with a class of
multiplicative functions. Let ${\mathcal{F}}$ denote the class of all
multiplicative functions $f$ with the following weak hypotheses:

(i) There exists some constant $c\geq 0$ such that

$\sum_{n\leq X}|f(n)|^{2}\ll X(\log X)^{c}.$

(ii) Let $P$ denote the product of primes $p$ which belong to the interval
$[(\log X)^{16(c+2)},X^{\delta}]$. There exists some constant $\gamma\geq 0$
such that

$\sum_{\begin{subarray}{c}n\leq X\\\
\left(n,P\right)=1\end{subarray}}|f(n)|^{2}\ll X\frac{(\log\log
X)^{\gamma}}{\log X},$

where the implied constant depends on $c,\delta$. Here the constant $c$ is the
same as in Hypothesis (i), and the parameter $\delta$ is some constant with
$0<\delta<1/22$.

######  Theorem 1.1.

Let $f\in{\mathcal{F}}$, and let $\lambda_{\phi}(n)$ be the Fourier
coefficients of a Hecke holomorphic or Hecke–Maass cusp form on ${\rm
SL}_{2}(\mathbb{Z})$. For any non-zero $h\ll X$, we have

$\sum_{n\leq X}f(n)\lambda_{\phi}(n+h)\ll X\frac{(\log\log
X)^{\frac{\gamma+1}{2}}}{\log X},$

where the implied constant depends on $\delta$, the form $\phi$ and these
implied constants in Hypotheses (i) and (ii).

Finally, we provide a brief overview of the proof of Theorem 1.1. Recall that
Bourgain, Sarnak and Ziegler [3] formulated a finite version of Vinogradov’s
bilinear technique, which also appeared in an earlier paper of Kátai. This is
the so-called Bourgain–Kátai–Sarnak–Ziegler (BKSZ for short) criterion, which
is very effective and widely applied to attack Sarnak’s disjointness
conjecture. Our work is primarily based on the generalized BKSZ criterion. For
our purpose, we have to generalize the BKSZ criterion to a greater extent,
when compared with the previous works [4, 14] of Cafferata–Perelli–Zaccagnini
and Jiang–Lü.

From the idea of BKSZ at the beginning, the sum of what we are concerned about

$\sum_{n\leq X}f(n)\lambda_{\phi}(n+h)$

can be rearranged as a sifted sum

$\sum_{\begin{subarray}{c}n\leq X\\\
\left(n,P\right)=1\end{subarray}}f(n)\lambda_{\phi}(n+h)$ (1.2)

and a combination of bilinear sums of type

$\sum_{m\leq X/N}\,\sum_{N<p\leq N+M}f(m)f(p)\lambda_{\phi}(pm+h),$ (1.3)

where $(\log X)^{A}\ll N\ll X^{\delta}$ and $M\asymp\sqrt{N}$. After applying
the Cauchy–Schwarz inequality, the bilinear sum (1.3) is generally reduced to
estimate

$\sum_{\begin{subarray}{c}N<p_{1},p_{2}\leq N+M\\\ p_{1}\neq
p_{2}\end{subarray}}f(p_{1})f(p_{2})\sum_{m\leq
X/N}\lambda_{\phi}(p_{1}m+h)\lambda_{\phi}(p_{2}m+h).$

To obtain cancellations in the innermost sum, we use the work [7] of Harcos on
the shifted convolution problem. The sifted sum (1.2) is a standard sieve
condition, and is estimated by Hypothesis (ii) and Lemma 4.6. To prove Lemma
4.6, we need to investigate the asymptotic behavior of sum

$\sum_{\begin{subarray}{c}n\leq X\\\ n\equiv a({\rm
mod}\,q)\end{subarray}}\lambda_{\phi}(dn)^{2},$

where $d$ is a square-free integer and $(ad,q)=1$.

##  2\. Applications

In this section, we will provide some examples that fit into the framework of
Theorem 1.1.

###  2.1. Coefficients of ${\rm GL}_{m}$ automorphic $L$-functions

Let $m\geq 2$, and let $\mathcal{A}(m)$ be the set of all cuspidal automorphic
representations of $\mathrm{GL}_{m}$ over $\mathbb{Q}$ with unitary central
character. For each $\pi\in\mathcal{A}(m)$, the corresponding $L$-function is
defined by absolutely convergent Dirichlet series as

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

for $\mathop{\mathrm{Re}}s>1$. We refer the reader to [16, Section 3] for a
more detailed overview. Here we are mainly concerned with the problem of
estimating correlations.

$\sum_{n\leq X}\lambda_{\pi}(n)\lambda_{\phi}(n+h),$ (2.1)

where $\phi$ is an arbitrary Hecke holomorphic or Hecke–Maass cusp form on
${\rm SL}_{2}(\mathbb{Z})$. The sum (2.1) is often called the shifted
convolution sum for ${\rm GL}_{m}\times{\rm GL}_{2}$. The shifted convolution
problem is to strengthen the trivial bound, which means to prove

$\sum_{n\leq X}\lambda_{\pi}(n)\lambda_{\phi}(n+h)=o\Big{(}\sum_{n\leq
X}|\lambda_{\pi}(n)\lambda_{\phi}(n+h)|\Big{)}.$

When $\pi\in\mathcal{A}(2)$, this sum (2.1) has been investigated extensively
by many authors since Selberg’s seminal paper. Non-trivial bounds of this sum
often have deep implications, for example, moment bounds and subconvexity for
$L$-functions, equidistribution of Heegner points and quantum unique
ergodicity. In particular, consider the case $\pi$ induced from $\phi$. The
best-known result in the literature is

$\sum_{n\leq
X}\lambda_{\phi}(n)\lambda_{\phi}(n+h)\ll_{\phi,\varepsilon}X^{\frac{2}{3}+\varepsilon},$

where $0<|h|\leq X^{2/3}$. One can refer to Jutila [17] for details. If $\pi$
comes from a Hecke–Maass cusp form on ${\rm SL}_{3}(\mathbb{Z})$, Munshi [20]
explored a smooth version of (2.1) and established a power saving estimate
over the trivial bound. These results for shifted convolution sums are usually
treated using either the circle method or the spectral method. The basic
strategy of circle method is to use the Kronecker delta symbol in order to
separate the variable $h$ from $n$. There are different expressions for
Kronecker delta symbol, such as $\delta$ method of Duke–Friedlander–Iwaniec,
Jutila’a variant of $\delta$ method, and Heath’s version of the Kloosterman
circle method. While the spectral method is to investigate the analytic
properties or spectral decomposition of corresponding Dirichlet series for the
shifted convolution sum. To the best of our knowledge, however, for any
$\pi\in\mathcal{A}(m)$ with $m\geq 4$, there is no non-trivial estimates for
(2.1) in the literature so far.

There are some stronger hypotheses in the beautiful work of Iwaniec, Luo and
Sarnak [13]. One of them is Hypothesis C, which states that for
$0<h_{1}\leqslant X$, and $0<\left|h_{2}\right|\leqslant X$, one has

$\sum_{m\leqslant\mathrm{X}}\mu(m)\lambda_{\phi}(m)\lambda_{\phi}\left(h_{1}m+h_{2}\right)\ll\mathrm{X}^{1/2+\varepsilon}.$

They remarked that Hypothesis C with the exponent $1/2$ replaced by $1-\delta$
for $\delta>0$ is only needed, that is to say, any small power saving is
enough for their application. We hope to take at least one small step forward
on this hypothesis, even though we obtain a small power saving of $\log X$.
Motivated by this discussion, another sum of interest is

$\sum_{n\leq X}\mu(n)\lambda_{\pi}(n)\lambda_{\phi}(n+h),$ (2.2)

where $\pi\in\mathcal{A}(m)$ with $m\geq 2$. It is worth mentioning a
surprising result of Pitt [23], who showed that

$\sum_{n\leq X}\mu(n)\lambda_{\phi}(n+h)\ll X^{1-\delta}$ (2.3)

unconditional holds for $h=-1$, where $\delta$ is some positive constant, and
$\phi$ is a Hecke holomorphic cusp form on ${\rm SL}_{2}(\mathbb{Z})$. In
fact, Pitt’s result may be viewed as the simplest special case of sum (2.2)
for $\pi\in\mathcal{A}(1)$.

In order to apply our theorem 1.1 to the sums (2.1) and (2.2), we need to
verify these two hypotheses. By the Rankin–Selberg theory and the inequality
$|\lambda_{\pi}(n)|^{2}\leq\lambda_{\pi\times\tilde{\pi}}(n)$ for all positive
integers $n$ (see [16, Lemma 3.1]), one has

$\sum_{n\leq X}|\lambda_{\pi}(n)|^{2}\leq\sum_{n\leq
X}\lambda_{\pi\times\tilde{\pi}}(n)\ll_{\pi}X.$

Moreover, by employing the sieve technique, we have

$\sum_{\begin{subarray}{c}n\leq X\\\
(n,P(Y,Z))=1\end{subarray}}|\lambda_{\pi}(n)|^{2}\ll_{\pi}X\frac{\log Y}{\log
Z}$

for any $N_{\pi}<Y<Z\leq X^{1/(30m^{2})}$, which has been shown in [15, Lemma
5.3]. Here $N_{\pi}$ is the arithmetic conductor of $\pi$ and $P(Y,Z)$ is the
product of these primes $p$ which belong to the interval $[Y,Z]$. For any
fixed $\pi\in\mathcal{A}(m)$, Hypotheses (i) and (ii) then hold with
$c=0,\gamma=1,Y=(\log X)^{32}$ and $Z=X^{1/(30m^{2})}$. Therefore, Theorem 1.1
applies the multiplicative functions $\lambda_{\pi}(n)$ and
$\mu(n)\lambda_{\pi}(n)$, and then yields the following result.

######  Theorem 2.1.

Let $\lambda_{\phi}(n)$ be the Fourier coefficients of a Hecke holomorphic or
Hecke–Maass cusp form on ${\rm SL}_{2}(\mathbb{Z})$, and $\lambda_{\pi}(n)$ be
the coefficients of $L(s,\pi)$ for any fixed $\pi\in\mathcal{A}(m)$ with
$m\geq 2$. For any non-zero $h\ll X$, we have

$\sum_{n\leq X}\lambda_{\pi}(n)\lambda_{\phi}(n+h)\ll X\frac{\log\log X}{\log
X}$

and

$\sum_{n\leq X}\mu(n)\lambda_{\pi}(n)\lambda_{\phi}(n+h)\ll X\frac{\log\log
X}{\log X},$

where the implied constant depends on $\phi$ and $\pi$ only.

######  Remark 2.1.

If the multiplicative functions $\lambda_{\pi}(n)$ and
$\mu(n)\lambda_{\pi}(n)$ are replaced by their absolute values
$|\lambda_{\pi}(n)|$ and $|\mu(n)\lambda_{\pi}(n)|$, the corresponding
estimates still hold with the same qualities. Especially, we get

$\sum_{n\leq X}|\lambda_{\pi}(n)|\,\lambda_{\phi}(n+h)\ll X\frac{\log\log
X}{\log X}$

uniformly holds in $0<|h|\ll X$. When $\pi\in\mathcal{A}(2)$ comes from the
cusp form $\phi$, Holowinsky [9] investigated the behavior of the sum among
two terms by taking absolute values and obtained

$\sum_{n\leq X}|\lambda_{\phi}(n)\,\lambda_{\phi}(n+h)|\ll\frac{X}{(\log
X)^{\delta}},$

where $\delta$ is some positive constant. He also emphasized that the method
presented in his article can not obtain a full integral power saving of $\log
X$, that is, $\delta=1$. In fact, this is determined by the Sate–Tate
conjecture, which produces the best value $\delta=2(1-8/(3\pi))\sim 0.30235$.
Compared Holowinsky’s result with that of ours, we obtain a full integral
power saving of $\log X$ up to the small term $\log\log X$ for any
$\pi\in\mathcal{A}(m)$ with $m\geq 2$. Of course, we also have to mention that
the absolute value of at least one term in our summation is removed.

In order to illustrate that Theorem 2.1 provides a non-trivial saving, we need
to estimate the correct order of magnitude of

$\sum_{n\leq X}|\lambda_{\pi}(n)\lambda_{\phi}(n+h)|.$

For simplicity of presentation, we fix the shifted parameter $h=1$ for
convenience and restrict the summation with $n$ and $n+h$ being square-free.
Next, we use the standard probabilistic heuristics to predict the main term of
sum

$S:=\sum_{n\leq X}|\mu(n)\lambda_{\pi}(n)\mu(n+1)\lambda_{\phi}(n+1)|.$

We refer to [22, Section 4] for a detailed explanation of the heuristics.
Define two sequences of random variables $(X_{p})$ and $(Y_{p})$ by

$X_{p}(n)=\big{|}\mu(p^{\operatorname{ord}_{p}(n)})\,\lambda_{\pi}(p^{\operatorname{ord}_{p}(n)})\big{|},\quad
Y_{p}(n)=\big{|}\mu(p^{\operatorname{ord}_{p}(n+1)})\,\lambda_{\phi}(p^{\operatorname{ord}_{p}(n+1)})\big{|},$

where $\operatorname{ord}_{p}(\cdot)$ is the $p$-adic valuation. Associated to
a random variable $Y:\mathbb{N}\rightarrow\mathbb{C}$ with image
$\operatorname{im}(Y)=\\{Y(n)\mid n\in\mathbb{N}\\}$, its expected value to be

$\mathbb{E}(Y)=\sum_{a\in\operatorname{im}(Y)}a\cdot\mathbb{P}(Y=a),$

where for $B\subset\mathbb{N}$, the $\mathbb{P}(B)$ is given by

$\mathbb{P}(B)=\lim_{X\rightarrow\infty}\frac{\\#\\{1\leq n\leq X\mid n\in
B\\}}{X}.$

Thus, we compute the expectations of $(X_{p}),(Y_{p})$ and $(X_{p}Y_{p})$ and
then get

$\displaystyle\mathbb{E}(X_{p})=1+\Big{(}\frac{1}{p}-\frac{1}{p^{2}}\Big{)}(|\lambda_{\pi}(p)|-1),$
$\displaystyle\quad\mathbb{E}(Y_{p})=1+\Big{(}\frac{1}{p}-\frac{1}{p^{2}}\Big{)}(|\lambda_{\phi}(p)|-1),$
$\displaystyle\mathbb{E}(X_{p}Y_{p})=1+\Big{(}\frac{1}{p}-\frac{1}{p^{2}}\Big{)}$
$\displaystyle(|\lambda_{\pi}(p)|+|\lambda_{\phi}(p)|-2).$

With these notation as above, it is reasonable to make the following
conjecture:

$S\sim\mathfrak{S}_{\pi,\phi}\Big{(}\sum_{n\leq
X}|\mu(n)\lambda_{\pi}(n)|\Big{)}\,\Big{(}\sum_{n\leq
X}|\mu(n)\lambda_{\phi}(n)|\Big{)}\frac{1}{X}$ (2.4)

as $X\rightarrow\infty$, where the singular series $\mathfrak{S}_{\pi,\phi}$
is equal to

$\mathfrak{S}_{\pi,\phi}=\prod_{p}\frac{\mathbb{E}(X_{p}Y_{p})}{\mathbb{E}(X_{p})\mathbb{E}(Y_{p})}.$

Assume that the Ramanujan conjecture holds for $\pi$ and $\phi$, that is
$|\lambda_{\pi}(n)|\leq\tau_{m}(n)$ and $|\lambda_{\phi}(n)|\leq\tau(n).$ It
is obvious that the singular series $\mathfrak{S}_{\pi,\phi}$ converges. We
further suppose that a cuspidal automorphic representation $\pi$ comes from
certain symmetric power lift of a holomorphic cusp form $\psi$ on ${\rm
SL}_{2}(\mathbb{Z})$. Let $\lambda_{{\rm sym}^{r}\psi}(n)$ denote the
Dirichlet coefficients of its $r$-th symmetric power $L$-function
$L_{r}(s,\psi)$, where $r\geq 1.$ Thanks to the Sato–Tate conjecture (which is
now a theorem of Barnet-Lamb, Geraghty, Harris and Taylor [2]) and Wirsing’s
mean value theorem on multiplicative functions [11, Theorem 1.1], we have

$\sum_{n\leq X}|\mu(n)\lambda_{{\rm sym}^{r}\psi}(n)|\sim
c_{r}(\psi)\frac{X}{\log^{\delta_{r}}X},$ (2.5)

where $\delta_{r}$ can be explicitly determined by

$\delta_{r}=1-\frac{2}{\pi}\int_{0}^{\pi}\frac{|\sin((r+1)\theta)|}{\sin\theta}(\sin\theta)^{2}\,\text{d}\theta.$

A straightforward calculation of Maple gives

$\delta_{r}=1-\frac{4(r+1)}{r(r+2)\pi}\cot\Big{(}\frac{\pi}{2(r+1)}\Big{)}.$

It is clear that $\delta_{r}$ is strictly increasing. Thus, for any $r\geq 1$,
we have

$0.15<1-\frac{8}{3\pi}=\delta_{1}\leq\delta_{r}\leq\lim_{r\rightarrow\infty}\delta_{r}=1-\frac{8}{\pi^{2}}<0.19.$

When this representation $\pi\in\mathcal{A}(m)$ corresponds to ${\rm
sym}^{m-1}\psi$, one has $\lambda_{\pi}(n)=\lambda_{{\rm sym}^{m-1}\psi}(n)$.
So we obtain from (2.4) and (2.5) that

$S\sim\big{(}\mathfrak{S}_{\pi,\phi}c_{1}(\psi)c_{m-1}(\psi)\big{)}\,\frac{X}{(\log
X)^{\delta_{1}+\delta_{m-1}}}.$

Finally, the probabilistic model gives

$\sum_{n\leq X}|\lambda_{\pi}(n)\lambda_{\phi}(n+1)|\gg\frac{X}{(\log
X)^{\delta_{1}+\delta_{m-1}}}\gg\frac{X}{(\log X)^{3/5}},$

where the implied constant depends on $\phi,m$. This shows that there exists a
$(\log X)^{3/5}$ saving at least in Theorem 1.1 due to the cancellations of
$\lambda_{\pi}(n)$ and $\lambda_{\phi}(n+1)$.

###  2.2. Divisor-bounded multiplicative functions

Let $f(n)$ be any multiplicative function which satisfies
$|f(n)|\leq\tau_{m}(n)$ for some $m>0$. Here $\tau_{m}(n)$ is the generalized
divisor function, which equals to the number of ways of writing $n$ as the
product of $m$ positive integers. If the function $\tau_{m}(n)$ meets
Hypotheses (i) and (ii), then Theorem 1.1 holds for any multiplicative
function $f(n)$ with $|f(n)|\leq\tau_{m}(n)$. First, one has the elementary
bound

$\sum_{n\leq X}\tau_{m}(n)^{2}\ll X(\log X)^{m^{2}-1},$

where the implied constant depends on $m$. Then it suffices to check that
$\tau_{m}(n)$ satisfies Hypothesis (ii). One may use sieve method to get this
condition similar to the proof of [15, Lemma 5.3]. However, we here provide an
alternate proof, which can be directly derived from the Brun–Titchmarsh
theorem on multiplicative functions. By [24, Theorem 1], we have

$\sum_{\begin{subarray}{c}n\leq X\\\
(n,P(Y,Z))=1\end{subarray}}\tau_{m}(n)^{2}\ll\frac{X}{\log
X}\exp\Big{(}m^{2}\sum_{\begin{subarray}{c}p\leq X\\\ p\nmid
P(Y,Z)\end{subarray}}\frac{1}{p}\Big{)},$ (2.6)

where $2\leq Y<Z\leq X$. Mertens’ theorem then yields

$\displaystyle\sum_{\begin{subarray}{c}p\leq X\\\ p\nmid
P(Y,Z)\end{subarray}}\frac{1}{p}=$ $\displaystyle\sum_{p\leq
X}\frac{1}{p}-\sum_{p\leq Z}\frac{1}{p}+\sum_{p\leq Y}\frac{1}{p}$
$\displaystyle=$ $\displaystyle\log\log X-\log\log Z+\log\log Y+O(1).$

Inserting this into (2.6), it follows

$\sum_{\begin{subarray}{c}n\leq X\\\
(n,P(Y,Z))=1\end{subarray}}\tau_{m}(n)^{2}\ll X(\log
X)^{m^{2}-1}\Big{(}\frac{\log Y}{\log Z}\Big{)}^{m^{2}}$

for any $2\leq Y<Z\leq X$. Hypotheses (i) and (ii) then hold with
$c=m^{2}-1,\gamma=m^{2},Y=(\log X)^{16(m^{2}+1)}$ and $Z=X^{1/23}$. Therefore
Theorem 1.1 applies to any multiplicative function $f(n)$ with
$|f(n)|\leq\tau_{m}(n)$ for some $m\geq 0$, and then yields the following
result.

######  Theorem 2.2.

Let $\lambda_{\phi}(n)$ be the Fourier coefficients of a Hecke holomorphic or
Hecke–Maass cusp form on ${\rm SL}_{2}(\mathbb{Z})$, and $f(n)$ be any
multiplicative function which satisfies $|f(n)|\leq\tau_{m}(n)$ for some
$m\geq 0$. For any non-zero $h\ll X$, we have

$\sum_{n\leq X}f(n)\lambda_{\phi}(n+h)\ll X\frac{(\log\log
X)^{\frac{m^{2}+1}{2}}}{\log X},$

where the implied constant depends on $\phi$ and $m$ only.

######  Remark 2.2.

This theorem can be applied very directly to some functions of particular
arithmetic interest:

  1. (1)

the Piltz divisor function $\tau_{z}(n)$, where $\tau_{z}(n)$ is the Dirichlet
coefficient of $\zeta(s)^{z}$ with $z\in\mathbb{C}$;

  2. (2)

the number $a_{n}$ of integral ideals with norm $n$ in an algebraic number
field $K$;

  3. (3)

the integral powers $\lambda_{\psi}(n)^{l}$ of Fourier coefficient of any
holomorphic cusp form on $\Gamma_{0}(N)$, where $l\in\mathbb{N}$.

######  Remark 2.3.

If the Fourier coefficient $\lambda_{\phi}(n)$ is replaced by any
multiplicative function $g$ with $|g(n)|\leq\tau_{m}(n)$, the corresponding
sum has been studied by many number theorist such as Nair–Tenenbaum [21],
Holowinsky [10], and Henriot [8]. It admits a very accurately bound

$\sum_{n\leq X}f(n)g(n+h)\ll\Delta(|h|)\frac{X}{(\log X)^{2}}\prod_{p\leq
X}\left(1+\frac{\left|f(p)\right|}{p}\right)\left(1+\frac{\left|g(p)\right|}{p}\right),$

where the function $\Delta(|h|)$ is explicitly given in [8] such that
$\sum_{|h|\leq X}\Delta(|h|)\ll 1$. We note that these three works also employ
some sieve techniques.

##  3\. Review of ${\rm GL}(2)$ automorphic forms

In this section, we collect some facts and conventions about automorphic forms
on ${\rm GL}(2)$, and present some estimates and formulae for their Fourier
coefficients which will be used later.

Let $k$ be an even integer, and Let $\mathcal{S}_{k}$ denote the set of
arithmetically normalized primitive cusp forms of weight $k$ on ${\rm
SL}_{2}(\mathbb{Z})$, which are eigenfunctions of all the Hecke operators. Any
$\phi\in\mathcal{S}_{k}$ has a Fourier expansion at infinity given by

$\phi(z)=\sum_{n=1}^{\infty}\lambda_{\phi}(n)n^{\frac{k-1}{2}}e(nz).$

Let $\mathcal{B}_{r}$ be the space of (weight zero) real-analytic Maass forms
of Laplacian eigenvalue $1/4+\mu^{2}$ on ${\rm SL}_{2}(\mathbb{Z})$, which are
also eigenfunctions of all the Hecke operators. Any $\phi\in\mathcal{B}_{r}$
has a Fourier expansion at infinity of the form

$\phi(z)=\sqrt{y}\sum_{n\neq 0}\lambda_{\phi}(n)K_{ir}(2\pi\left\lvert
n\right\rvert y)e(nx),$

where $K_{ir}$ is the $K\mbox{-}$Bessel function. The arithmetic normalization
above means that $\lambda_{\phi}(1)=1$. The eigenvalues $\lambda_{\phi}(n)$
enjoy the multiplicative property

$\lambda_{\phi}(m)\lambda_{\phi}(n)=\sum_{d|(m,n)}\lambda_{\phi}\Big{(}\frac{mn}{d^{2}}\Big{)}$

for all integers $m,n\geq 1.$ It follows from Rankin–Selberg theory that the
Fourier coefficients $\lambda_{\phi}(n)$ are bounded on average, namely

$\sum_{n\leq
X}|\lambda_{\phi}(n)|^{2}=c_{f}\,X+O\big{(}X^{\frac{3}{5}}\big{)}$ (3.1)

for some $c_{f}>0$. For individual terms, we have

$|\lambda_{\phi}(n)|\leq n^{\frac{7}{64}}\tau(n)$ (3.2)

for all $n\geq 1$ by the work [18] of Kim and Sarnak, where $\tau(n)$ is the
divisor function. Moreover, if $\phi\in\mathcal{S}_{k}$, it follows from
Deligne’s proof [5] of the Ramanujan–Petersson conjecture that

$|\lambda_{\phi}(n)|\leq\tau(n)$

hold for all $n\geq 1$. It is also well known that the Fourier coefficients
oscillate quite substantially. For example, the classical estimate of Wilton’s
type gives

$S_{\phi}(\alpha,X):=\sum_{n\leq X}\lambda_{\phi}(n)e(n\alpha)\ll
X^{\frac{1}{2}}\log X$

uniformly in $\alpha\in\mathbb{R}$, where the implied constant depends only on
the form $\phi$. The explicit dependence on $\phi$ is also studied by some
number theorists. We note that this estimate plays an important role in
solving shifted convolution problems, especially when the Jutila circle method
is used.

For any $\phi\in\mathcal{S}_{k}\cup\mathcal{B}_{r}$, the Hecke $L$-function is
defined by

$L(s,\phi)=\sum_{n=1}^{\infty}\frac{\lambda_{\phi}(n)}{n^{s}}.$

This has an Euler product $L(s,\phi)=\prod\limits_{p}L_{p}(s,\phi)$ with the
local factors

$L_{p}(s,\phi)=\big{(}1-\lambda_{\phi}(p)p^{-s}+p^{-2s}\big{)}^{-1}=\big{(}1-\alpha_{\phi}(p)p^{-s}\big{)}^{-1}\big{(}1-\beta_{\phi}(p)p^{-s}\big{)}^{-1},$

where $\alpha_{\phi}(p),\,\beta_{\phi}(p)$ are complex numbers with
$\alpha_{\phi}(p)+\beta_{\phi}(p)=\lambda_{\phi}(p)$ and
$\alpha_{\phi}(p)\beta_{\phi}(p)=1$.

Let $\chi$ be a primitive Dirichlet character modulo $q$ with $q\geq 1$. Then
we define the twisted symmetric square $L$-function by means of the Euler
product
$L(s,\text{sym}^{2}\phi\otimes\chi)=L_{p}(s,\text{sym}^{2}\phi\otimes\chi)$
with the local factors given by

$L_{p}(s,\text{sym}^{2}\phi\otimes\chi)=\left(1-\alpha_{\phi}(p)^{2}\chi(p)p^{-s}\right)^{-1}\left(1-\chi(p)p^{-s}\right)^{-1}\left(1-\chi(p)\beta_{p}^{2}p^{-s}\right)^{-1}.$

It is well-known that this $L$-function extends to an entire function and
satisfies a functional equation. Indeed, we have a completed $L$-function
defined as

$\Lambda\left(s,\text{sym}^{2}\phi\otimes\chi\right)=q^{3s/2}\gamma(s,\text{sym}^{2}\phi\otimes\chi)L\left(s,\operatorname{Sym}^{2}f\otimes\chi\right),$

where $\gamma(s,\text{sym}^{2}\phi\otimes\chi)$ is essentially a product of
three gamma functions $\Gamma\left(\frac{s+\kappa_{j}}{2}\right),j=1,2,3$,
with $\kappa_{j}$ depending on the weight or spectral parameter of $\phi$ and
the parity of the character $\chi$, such that

$\Lambda\left(s,\text{sym}^{2}\phi\otimes\chi\right)=\varepsilon(\text{sym}^{2}\phi\otimes\chi)\Lambda\left(1-s,\text{sym}^{2}\phi\otimes\bar{\chi}\right)$

Here $\mathop{\mathrm{Re}}\left(\kappa_{j}\right)>0$ and the
$\varepsilon$-factor satisfies
$|\varepsilon(\text{sym}^{2}\phi\otimes\chi)|=1$. Consider the Dirichlet
series

$L(s,{\phi}_{\chi}\otimes{\phi})=\sum_{n=1}^{\infty}\frac{\lambda_{\phi}(n)^{2}\chi(n)}{n^{s}}$

for $\mathop{\mathrm{Re}}s>1.$ This has the Euler product
$L(s,{\phi}_{\chi}\otimes{\phi}):=\prod\limits_{p}L_{p}(s,{\phi}_{\chi}\otimes{\phi})$
with the local factors

$L_{p}(s,{\phi}_{\chi}\otimes{\phi})=\big{(}1+\chi(p)p^{-s}\big{)}\big{(}1-\alpha_{\phi}(p)^{2}\chi(p)p^{-s}\big{)}^{-1}\big{(}1-\chi(p)p^{-s}\big{)}^{-1}\big{(}1-\beta_{\phi}(p)^{2}\chi(p)p^{-s}\big{)}^{-1}.$

This can be also written in terms of the Hecke eigenvalues

$L_{p}(s,{\phi}_{\chi}\otimes{\phi})=\big{(}1+\chi(p)p^{-s}\big{)}\big{(}1-\lambda_{\phi}(p^{2})\chi(p)p^{-s}+\lambda_{\phi}(p^{2})\chi(p)^{2}p^{-2s}-\chi(p)^{3}p^{-3s}\big{)}^{-1}.$

Comparing the Euler products of $L(s,\text{sym}^{2}\phi\otimes\chi)$ and
$L(s,{\phi}_{\chi}\otimes{\phi})$, one has

$L(s,{\phi}_{\chi}\otimes{\phi})=L(2s,\chi)^{-1}L(s,\chi)L(s,\text{sym}^{2}\phi\otimes\chi).$
(3.3)

This series $L(s,{\phi}_{\chi}\otimes{\phi})$ has a holomorphic continuation
to all $s\in\mathbb{C}$ except for possible simple pole at $s=1$. The latter
simple pole can only occur if $q=1$, in which case $\chi\equiv 1$. For
convenience, we denote
$L(s,{\phi}_{\chi}\otimes{\phi})=L(s,{\phi}\otimes{\phi})$ and
$L(s,\text{sym}^{2}\phi\otimes\chi)=L(s,\text{sym}^{2}\phi)$ when $\chi\equiv
1$.

##  4\. Some lemmas

###  4.1. Shifted convolution Problems

In this section, we shall discuss the uniform bounds for shifted convolution
sum

$\sum_{n=1}^{\infty}\lambda_{\phi}(m_{1}n+h)\lambda_{\phi}(m_{2}n+h)e^{-\frac{n}{X}}.$

If $m_{1},m_{2}$ are distinct square-free integers, the case of $h=-1$ has
been treated in the work [23, Theorem 1.4] of Pitt, which plays an important
role to estimate the bilinear forms in the proof of upper bound (2.3).
Recently, Assing, Blomer and Li [1, Theorem 2.6] generalized Pitt’s estimate
to the case of general $h$ and gave the explicit dependences on the form
$\phi$ and all shifts $m_{1},m_{2},h$. Note that the weight function
$e^{-\frac{n}{X}}$ is replaced by a smooth function $g$ with support in
$[X,2X]$ and $g^{j}(x)\ll_{j}(X/Z)^{-j}$ for all $j\geq 0$ in [1, Theorem
2.6]. Next, in view of Harcos’s result [7], we will prove another uniform
estimate for the above sum, which is enough in our applications.

######  Lemma 4.1.

Let $\phi\in\mathcal{S}_{k}\cup\mathcal{B}_{r}$. Let $m_{1},\,m_{2}$ be
positive integers with $(m_{1},m_{2})=1$ and $h\neq 0$. Then we have

$\sum_{n=1}^{\infty}\lambda_{\phi}(m_{1}n+h)\lambda_{\phi}(m_{2}n+h)e^{-\frac{n}{X}}\ll
e^{\frac{(m_{1}+m_{2})h}{2Xm_{1}m_{2}}}(m_{1}m_{2})^{\frac{4}{5}}X^{\frac{9}{10}+\varepsilon},$

where the implied constant depends on $\varepsilon$ and the form $\phi$.

###### Proof.

We denote by $\mathcal{C}$ the left side of the claimed inequality.
Introducing the new variables $n_{1}=m_{1}n+h$ and $n_{2}=m_{2}n+h$, the sum
$\mathcal{C}$ can be rewritten as

$\displaystyle\mathcal{C}=$ $\displaystyle
e^{\frac{(m_{1}+m_{2})h}{2Xm_{1}m_{2}}}\sum_{\begin{subarray}{c}n_{1},n_{2}=1\\\
m_{2}(n_{1}-h)=m_{1}(n_{2}-h)\end{subarray}}^{\infty}\lambda_{\phi}(n_{1})\lambda_{\phi}(n_{2})e^{-\frac{n_{1}}{2Xm_{1}}}e^{-\frac{n_{2}}{2Xm_{2}}}$
(4.1) $\displaystyle:=$ $\displaystyle
e^{\frac{(m_{1}+m_{2})h}{2Xm_{1}m_{2}}}D_{w}\big{(}m_{2},m_{1};(m_{2}-m_{1})h\big{)},$

where the notation $D_{w}\big{(}m_{2},m_{1};(m_{1}-m_{2})h\big{)}$ is defined
as in the work [7] of Harcos

$D_{w}\big{(}m_{2},m_{1};(m_{1}-m_{2})h\big{)}=\sum_{\begin{subarray}{c}n_{1},n_{2}=1\\\
m_{2}n_{1}-m_{1}n_{2}=(m_{2}-m_{1})h\end{subarray}}^{\infty}\lambda_{\phi}(n_{1})\lambda_{\phi}(n_{2})w(m_{2}n_{1},m_{1}n_{2})$

with the nice weight function

$w(x,y)=e^{-\frac{x+y}{2Xm_{1}m_{2}}}.$

Clearly, this weight function $w(x,y)$ satisfies the estimate

$x^{i}y^{j}w^{(i,j)}(x,y)\ll_{i,j}\Big{(}1+\frac{x}{Xm_{1}m_{2}}\Big{)}^{-1}\Big{(}1+\frac{y}{Xm_{1}m_{2}}\Big{)}^{-1}$

for all $i,j\geq 0$. By the nontrivial uniform estimate in [7, Theorem 1], we
obtain that

$D_{w}\big{(}m_{2},m_{1};(m_{2}-m_{1})h\big{)}\ll_{f,\varepsilon}(m_{1}m_{2})^{\frac{4}{5}}X^{\frac{9}{10}+\varepsilon}$

holds for any $h\neq 0$. On inserting this estimate into (4.1), this lemma
follows. ∎

###  4.2. Sieve conditions

Let $\lambda_{\phi}(n)^{2}$ be the coefficients of Rankin–Selberg convolution
$L$-function $L(s,{\phi}\otimes{\phi})$. We denote by $P(Y,Z)$ the product of
these primes $p$ which belong to the interval $[Y,Z]$, where $2\leq Y<Z$. We
emphasize that there is no restriction on $Y$ and $Z$ as in Hypothesis (ii).
In this section, we shall seek an upper estimate for the sifted sum

$\sum_{\begin{subarray}{c}n\leq X\\\
\left(n-a,P(Y,Z)\right)=1\end{subarray}}\lambda_{\phi}(n)^{2}$ (4.2)

with $0<|a|\ll X$. For this purpose, we need to investigate the distribution
for $\lambda_{\phi}(n)^{2}$ in arithmetic progressions. This means to give an
asymptotic formula for

$\sum_{\begin{subarray}{c}n\leq X\\\ n\equiv a({\rm
mod}\,q)\end{subarray}}\lambda_{\phi}(n)^{2},$

where $q$ increases together with $X$. In our application to the proof of
Theorem 1.1, the modulo $q\ll X^{\varepsilon}$ is sufficient. However, we hope
to get the modulo $q$ as large as possible such that the sum above admits an
asymptotic formula, which may be of independent interest. Since there is no
restriction $(a,q)=1$ in arithmetic progressions, we need to prove the
following.

######  Proposition 4.2.

Let $\phi\in\mathcal{S}_{k}\cup\mathcal{B}_{r}$, $d$ be a square-free integer
and $(ad,q)=1$. Then we have

$\sum_{\begin{subarray}{c}n\leq X\\\ n\equiv a({\rm
mod}\,q)\end{subarray}}\lambda_{\phi}(dn)^{2}=\frac{1}{\varphi(q)}H_{d}(1,{\phi}\otimes{\phi})G_{q}(1,{\phi}\otimes{\phi})\mathop{\textup{Res}}_{s=1}L(s,{\phi}\otimes{\phi})X+O\Big{(}d^{\frac{7}{32}}X^{\frac{3}{5}+\varepsilon}\Big{)}$

for any $q<X^{\frac{2}{5}-\varepsilon}$, where $H_{d}(1,{\phi}\otimes{\phi})$
and $G_{q}(1,{\phi}\otimes{\phi})$ are given by

$\displaystyle H_{d}(1,{\phi}\otimes{\phi})=$
$\displaystyle\prod_{p|d}\big{(}1+p^{-1}\big{)}^{-1}\big{(}\lambda_{\phi}(p)^{2}-\lambda_{\phi}(p^{2})p^{-1}+p^{-2}\big{)},$
$\displaystyle G_{q}(1,{\phi}\otimes{\phi})=$
$\displaystyle\prod_{p|q}\big{(}1+p^{-1}\big{)}^{-1}\big{(}1-\lambda_{\phi}(p^{2})p^{-1}+\lambda_{\phi}(p^{2})p^{-2}-p^{-3}\big{)},$

and the implied constant depends only on $\phi,\varepsilon$.

######  Remark 4.1.

Proposition 4.2 generalized the classical result (3.1) of Rankin–Selberg to
the situation of arithmetic progressions when $d=1$. Moreover, the error terms
of these two results have the same quality up to the arbitrary small
$\varepsilon.$

###### Proof.

It is more convenient to deal with the related smoothing sums, so we introduce
two smooth compactly supported functions $w_{\ell}$ inspired by the example in
[12, Appendix A]. Let the function $g(x)$ be defined by

$g(x)=\left\\{\begin{array}[]{ll}\exp(-1/x(1-x))&\text{ if }0<x<1,\\\ 0&\text{
otherwise.}\end{array}\right.$

Put

$G(y)=\frac{1}{C}\int_{-\infty}^{y}g(x)\text{d}x,$

where $C=\int_{0}^{1}g(x)\mathrm{d}x$ is an absolute constant. Using $G(y)$,
we construct the two functions

$w_{1}(u)=G\left(\frac{u+Y}{Y}\right)-G\left(\frac{u-X}{Y}\right)$

and

$w_{2}(u)=G\left(\frac{u}{Y}\right)-G\left(\frac{u-X+Y}{Y}\right),$

where $0<Y<X/2$ and $Y$ will be determined later. Then $w_{\ell}(u)$ satisfy
the following properties:

  * •

$w_{1}(u)=1$ for $u\in[Y,X-Y],w_{-}(u)=0$ for $u\geq X$ and $u\leq 0$;

  * •

$w_{2}(u)=1$ for $u\in[0,X],w_{+}(u)=0$ for $u\geq X+Y$ and $u\leq-Y$;

  * •

$w_{\ell}^{(j)}(u)\ll_{j}Y^{-j}$ for all $j\geq 0$ and $\ell=1,2$;

  * •

the Mellin transform of $w_{\ell}(x)$ is

$\displaystyle\widehat{w_{\ell}}(s)$
$\displaystyle=\int_{0}^{\infty}w_{\ell}(u)u^{s-1}\text{d}u$ (4.3)
$\displaystyle=\frac{1}{s\cdots(s+j-1)}\int_{0}^{\infty}w_{\ell}^{(j)}(u)u^{s+j-1}\text{d}u$
$\displaystyle\ll_{j}\frac{Y}{X^{1-\sigma}}\Big{(}\frac{X}{|s|Y}\Big{)}^{j}$

for any $j\geq 1$ and $\sigma=\mathop{\mathrm{Re}}s>0$.

  * •

trivially $\widehat{w_{\ell}}(s)\ll X^{\sigma}$ for any
$\sigma=\mathop{\mathrm{Re}}s>0$ and

$\widehat{w_{\ell}}(1)=X+O(Y).$ (4.4)

We shall detect the congruence $n\equiv a({\rm mod}\,q)$ via the orthogonality
of Dirichlet characters. Thus, we obtain the identity

$\displaystyle S(w_{\ell};q,a;d):=$
$\displaystyle\sum_{\begin{subarray}{c}n\\\ n\equiv a({\rm
mod}\,q)\end{subarray}}\lambda_{\phi}(dn)^{2}w_{\ell}(n)=\frac{1}{\varphi(q)}\sum_{\chi({\rm
mod}\,q)}\overline{\chi}(a)\sum_{n}\lambda_{\phi}(dn)^{2}\chi(n)w_{\ell}(n)$
(4.5) $\displaystyle=$
$\displaystyle\frac{1}{\varphi(q)}M(w_{\ell};q;d)+O\big{(}E(w_{\ell};q;d)\big{)},$

where $M(w_{\ell};q;d)$ and $E(w_{\ell};q;d)$ are given by

$\displaystyle M(w_{\ell};q;d)=$ $\displaystyle\sum_{\begin{subarray}{c}n\\\
(n,q)=1\end{subarray}}\lambda_{\phi}(dn)^{2}w_{\ell}(n),$ (4.6) $\displaystyle
E(w_{\ell};q;d)=$
$\displaystyle\frac{1}{\varphi(q)}\sum_{\begin{subarray}{c}\chi({\rm
mod}\,q)\\\
\chi\neq\chi_{0}\end{subarray}}\Big{|}\sum_{n}\lambda_{\phi}(dn)^{2}\chi(n)w_{\ell}(n)\Big{|}.$

Now it suffices to estimate $M(w_{\ell};q;d)$ and $E(w_{\ell};q;d)$,
respectively.

######  Lemma 4.3.

Let $M(w_{\ell};q;d)$ be defined as above. Then we have

$\displaystyle M(w_{\ell};q;d)=$ $\displaystyle
H_{d}(1,{\phi}\otimes{\phi})G_{q}(1,{\phi}\otimes{\phi})\mathop{\textup{Res}}_{s=1}L(s,{\phi}\otimes{\phi})X+O\big{(}d^{\frac{7}{32}}\tau(dq)Y^{-\frac{1}{4}}X^{\frac{3}{4}}(\log
X)^{3}\big{)}$ $\displaystyle+O\big{(}d^{\frac{7}{32}}\tau(dq)Y\big{)},$

where the implied constant depends only on $\phi$.

###### Proof.

By the Mellin inversion formula

$w_{\ell}(x)=\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}\widehat{w_{\ell}}(s)x^{-s}\text{d}s,$

we may write

$\sum_{\begin{subarray}{c}n\\\
(n,q)=1\end{subarray}}\lambda_{\phi}(dn)^{2}w_{\ell}(n)=\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}\widehat{w_{\ell}}(s)\sum_{\begin{subarray}{c}n=1\\\
(n,q)=1\end{subarray}}^{\infty}\frac{\lambda_{\phi}(dn)^{2}}{n^{s}}\text{d}s.$

The Dirichlet series appearing above has non-negative coefficients, converges
in the region $\mathop{\mathrm{Re}}s>1$, and matches the Rankin–Selberg
convolution $L$-function $L(s,{\phi}\otimes{\phi})$ except for the Euler
factors at primes $p$ dividing $dq$. Note that $d$ is square-free and
$(d,q)=1$. By the multiplicative property of $\lambda_{\phi}(n)$, we may
rewrite

$\displaystyle\sum_{\begin{subarray}{c}n=1\\\
(n,q)=1\end{subarray}}^{\infty}\frac{\lambda_{\phi}(dn)^{2}}{n^{s}}=$
$\displaystyle\prod_{p|d}\Big{(}\sum_{k=0}^{\infty}\lambda_{\phi}(p^{k+1})^{2}p^{-ks}\Big{)}\prod_{p\nmid
dq}L_{p}(s,{\phi}\otimes{\phi})$ $\displaystyle=$ $\displaystyle
L(s,{\phi}\otimes{\phi})\prod_{p|d}\Big{(}L_{p}(s,{\phi}\otimes{\phi})^{-1}\sum_{k=0}^{\infty}\lambda_{\phi}(p^{k+1})^{2}p^{-ks}\Big{)}\prod_{p|q}L_{p}(s,{\phi}\otimes{\phi})^{-1}$
$\displaystyle=$ $\displaystyle
L(s,{\phi}\otimes{\phi})H_{d}(s,{\phi}\otimes{\phi})G_{q}(s,{\phi}\otimes{\phi}),$

where $H_{d}(s,{\phi}\otimes{\phi})$ and $G_{q}(s,{\phi}\otimes{\phi})$ are
given by

$\displaystyle H_{d}(s,{\phi}\otimes{\phi})=$
$\displaystyle\prod_{p|d}\big{(}1+p^{-s}\big{)}^{-1}\big{(}\lambda_{\phi}(p)^{2}-\lambda_{\phi}(p^{2})p^{-s}+p^{-2s}\big{)},$
$\displaystyle G_{q}(s,{\phi}\otimes{\phi})=$
$\displaystyle\prod_{p|q}\big{(}1+p^{-s}\big{)}^{-1}\big{(}1-\lambda_{\phi}(p^{2})p^{-s}+\lambda_{\phi}(p^{2})p^{-2s}-p^{-3s}\big{)}.$

Therefore, the above integral equals

$\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}\widehat{w_{\ell}}(s)L(s,{\phi}\otimes{\phi})H_{d}(s,{\phi}\otimes{\phi})G_{q}(s,{\phi}\otimes{\phi}){\rm{d}}s.$
(4.7)

We evaluate (4.7) by moving the line of integration to
$\mathop{\mathrm{Re}}s=1/2$. We encounter a simple pole at $s=1$ and the
residue here is

$\widehat{w_{\ell}}(1)H_{d}(1,{\phi}\otimes{\phi})G_{q}(1,{\phi}\otimes{\phi})\mathop{\textup{Res}}_{s=1}L(s,{\phi}\otimes{\phi}).$

It follows from (3.2) and the definitions of
$H_{d}(s,{\phi}\otimes{\phi}),G_{q}(s,{\phi}\otimes{\phi})$ that

$\displaystyle H_{d}(s,{\phi}\otimes{\phi})\ll$
$\displaystyle\prod_{p|d}\big{(}p^{\frac{7}{32}}+p^{-\frac{9}{32}+\varepsilon}\big{)}\ll
d^{\frac{7}{32}}\tau(d),$ (4.8) $\displaystyle
G_{q}(s,{\phi}\otimes{\phi})\ll$
$\displaystyle\prod_{p|q}\big{(}1+p^{-\frac{9}{32}+\varepsilon}\big{)}\ll\tau(q)$

for any $\mathop{\mathrm{Re}}s\geq 1/2$. Combining these with (4.4), the
residue equals to

$H_{d}(1,{\phi}\otimes{\phi})G_{q}(1,{\phi}\otimes{\phi})\mathop{\textup{Res}}_{s=1}L(s,{\phi}\otimes{\phi})X+O\big{(}d^{\frac{7}{32}}\tau(dq)Y\big{)}.$
(4.9)

It remains to bound the integral on the line $\mathop{\mathrm{Re}}s=1/2$, that
is

$\displaystyle\frac{1}{2\pi
i}\int_{\frac{1}{2}-i\infty}^{\frac{1}{2}+i\infty}\widehat{w_{\ell}}(s)L(s,{\phi}\otimes{\phi})G_{q}(s,{\phi}\otimes{\phi}){\rm{d}}s$
(4.10) $\displaystyle\ll$ $\displaystyle
d^{\frac{7}{32}}\tau(dq)\int_{-\infty}^{\infty}\Big{|}\zeta\Big{(}\frac{1}{2}+it\Big{)}L\Big{(}\frac{1}{2}+it,\text{sym}^{2}\phi\Big{)}\widehat{w_{\ell}}\Big{(}\frac{1}{2}+it\Big{)}\Big{|}\log(|t|+2){\rm{d}}t,$

where we use the decomposition (3.3) and the bound
$\zeta(1+it)^{-1}\ll\log(|t|+2)$ for any $t\in\mathbb{R}$. Moreover, the
contribution from the integration over $|t|\geq T=X^{1+\varepsilon}/Y$ is
negligibly small (i.e. $O(X^{-A})$ for any $A>0$) if we choose a sufficiently
large $j$ in (4.3). Hence, the term in the second line of (4.10) is

$\ll d^{\frac{7}{32}}\tau(dq)(\log T)^{2}X^{\frac{1}{2}}\max_{0\leq T_{1}\leq
T}\frac{1}{T_{1}+1}\int_{T_{1}/2}^{T_{1}}\Big{|}\zeta\Big{(}\frac{1}{2}\pm
it\Big{)}L\Big{(}\frac{1}{2}\pm
it,\text{sym}^{2}\phi\Big{)}\Big{|}{\rm{d}}t+X^{-A}.$

Applying the Cauchy–Schwarz inequality and inserting the second moments of
$\zeta(s)$ and $L(s,\text{sym}^{2}\phi)$ with the $t$-aspect integration

$\int_{T/2}^{T}\Big{|}\zeta\Big{(}\frac{1}{2}\pm
it\Big{)}\Big{|}^{2}{\rm{d}}t\ll T\log
T,\quad\int_{T/2}^{T}\Big{|}L\Big{(}\frac{1}{2}\pm
it,\text{sym}^{2}\phi\Big{)}\Big{|}^{2}{\rm{d}}t\ll T^{\frac{3}{2}}\log T,$

we obtain that the integral on the line $\mathop{\mathrm{Re}}s=1/2$ is less
than $d^{\frac{7}{32}}\tau(dq)T^{\frac{1}{4}}(\log T)^{3}X^{\frac{1}{2}}.$
Combining this with (4.9), we get

$\displaystyle\sum_{\begin{subarray}{c}n\\\
(n,q)=1\end{subarray}}\lambda_{\phi}(n)^{2}w_{\ell}(n)=$ $\displaystyle
H_{d}(1,{\phi}\otimes{\phi})G_{q}(1,{\phi}\otimes{\phi})\mathop{\textup{Res}}_{s=1}L(s,{\phi}\otimes{\phi})X$
$\displaystyle+O\big{(}d^{\frac{7}{32}}\tau(dq)Y^{-\frac{1}{4}}X^{\frac{3}{4}}(\log
X)^{3}\big{)}+O\big{(}d^{\frac{7}{32}}\tau(dq)Y\big{)}.$

This completes the proof of Lemma 4.3. ∎

######  Lemma 4.4.

Let $E(w_{\ell};q;d)$ be defined as in (4.6). Then we have

$E(w_{\ell};q;d)\ll\Big{(}\frac{qX}{Y}\Big{)}^{\frac{1}{4}}d^{\frac{7}{32}}X^{\frac{1}{2}+\varepsilon},$

where the implied constant depends only on $\phi,\varepsilon$.

###### Proof.

As before, we have

$\sum_{n}\lambda_{\phi}(dn)^{2}\chi(n)w_{\ell}(n)=\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}\widehat{w_{\ell}}(s)\sum_{n=1}^{\infty}\frac{\lambda_{\phi}(dn)^{2}\chi(n)}{n^{s}}\text{d}s.$

Each non-principle character $\chi({\rm mod}\,q)$ can be induced by a
primitive character $\chi({\rm mod}\,q_{1})$ with $q_{1}|q$ and $q_{1}>1$.
Then we get the Dirichlet series

$\sum_{n=1}^{\infty}\frac{\lambda_{\phi}(dn)^{2}\chi(n)}{n^{s}}=L(s,f_{\chi_{1}}\otimes
f)H_{q}(s,f_{\chi_{1}}\otimes f)G_{q}(s,f_{\chi_{1}}\otimes f),$

where $H_{q}(s,f_{\chi_{1}}\otimes f)$ and $)G_{q}(s,f_{\chi_{1}}\otimes f)$
are given by

$\displaystyle H_{q}(s,f_{\chi_{1}}\otimes f)=$
$\displaystyle\prod_{p|d}\big{(}1+\chi_{1}(n)p^{-s}\big{)}^{-1}\big{(}\lambda_{\phi}(p)^{2}\chi_{1}(n)-\lambda_{\phi}(p^{2})\chi_{1}(n)p^{-s}+\chi_{1}(n)p^{-2s}\big{)},$
$\displaystyle G_{q}(s,f_{\chi_{1}}\otimes f)=$
$\displaystyle\prod_{p|q}\big{(}1+\chi_{1}(n)p^{-s}\big{)}^{-1}\big{(}1-\lambda_{\phi}(p^{2})\chi_{1}(n)p^{-s}+\lambda_{\phi}(p^{2})\chi_{1}(n)p^{-2s}-\chi_{1}(n)p^{-3s}\big{)}.$

Following the argument of Lemma 4.3, one can derive

$\displaystyle\sum_{n}\lambda_{\phi}(dn)^{2}\chi(n)w_{\ell}(n)$
$\displaystyle\ll$ $\displaystyle
d^{\frac{7}{32}}\tau(dq)X^{\frac{1}{2}+\varepsilon}\max_{0\leq T_{1}\leq
T}\frac{1}{T_{1}+1}\int_{T_{1}/2}^{T_{1}}\Big{|}L\Big{(}\frac{1}{2}\pm
it,\chi_{1}\Big{)}L\Big{(}\frac{1}{2}\pm
it,\text{sym}^{2}\phi\otimes{\chi_{1}}\Big{)}\Big{|}{\rm{d}}t,$

where $T=X^{1+\varepsilon}/Y.$ Then the Cauchy–Schwarz inequality gives

$E(w_{\ell};q;d)\ll\frac{d^{\frac{7}{32}}\tau(dq)}{\varphi(q)}X^{\frac{1}{2}+\varepsilon}\max_{0\leq
T_{1}\leq T}\frac{1}{T_{1}+1}\sum_{q_{1}|q}(I_{1}\,I_{2})^{\frac{1}{2}}.$

Here $I_{1},I_{2}$ are moments of $L(s,\chi_{1})$ and
$L(s,\text{sym}^{2}\phi\otimes{\chi_{1}})$, respectively. More preciously,

$\displaystyle I_{1}=$ $\displaystyle\sideset{}{{}^{*}}{\sum}_{\chi({\rm
mod}\,q_{1})}\int_{T_{1}/2}^{T_{1}}\Big{|}L\Big{(}\frac{1}{2}\pm
it,\chi_{1}\Big{)}\Big{|}^{2}{\rm{d}}t,$ $\displaystyle I_{2}=$
$\displaystyle\sideset{}{{}^{*}}{\sum}_{\chi({\rm
mod}\,q_{1})}\int_{T_{1}/2}^{T_{1}}\Big{|}L\Big{(}\frac{1}{2}\pm
it,\text{sym}^{2}\phi\otimes{\chi_{1}}\Big{)}\Big{|}^{2}{\rm{d}}t.$

Recall a large sieve inequality [11, Page 179]: for any complex number
$a_{n}$, one has

$\sum_{\chi({\rm mod}\,q)}\Big{|}\sum_{n\leqslant
N}a_{n}\chi(n)\Big{|}^{2}\leq(q+N)\sum_{n\leq N}|a_{n}|^{2}.$

Together this with the approximate functional equations of $L(s,\chi_{1})$ and
$L(s,\text{sym}^{2}\phi\otimes{\chi_{1}})$, we have

$I_{1}\ll rT_{1}\log q_{1}T_{1},\quad I_{2}\ll(q_{1}T_{1})^{\frac{3}{2}}\log
q_{1}T_{1}.$

Further, we obtain

$E(w_{\ell};q;d)\ll\Big{(}\frac{qX}{Y}\Big{)}^{\frac{1}{4}}d^{\frac{7}{32}}X^{\frac{1}{2}+\varepsilon}.$

∎

Now we continue to prove Proposition 4.2. Inserting Lemma 4.3 and Lemma 4.4
into (4.5) and taking $Y=qX^{\frac{3}{5}}$, we have

$S(w_{\ell};q,a;d)=\frac{1}{\varphi(q)}G_{q}(1,f\times
f)\mathop{\textup{Res}}_{s=1}L(s,f\times
f)X+O\Big{(}d^{\frac{7}{32}}X^{\frac{3}{5}+\varepsilon}\Big{)}$

for $\ell=1,2$ and any $q<X^{\frac{2}{5}-\varepsilon}$. By the definition of
$w_{\ell}(u)$, it is easy to see

$S(w_{1};q,a;d)\leq\sum_{\begin{subarray}{c}n\leq X\\\ n\equiv a({\rm
mod}\,q)\end{subarray}}\lambda_{\phi}(dn)^{2}\leq S(w_{2};q,a;d),$

which implies Proposition 4.2. ∎

In order to estimate the sifted sum (4.2) without the restriction
$\big{(}a,P(Y,Z)\big{)}=1$, it is necessary to explore the distribution of
$\lambda_{\phi}(n)^{2}$ in the general arithmetic progressions, which do not
have the condition $(a,q)=1.$ Proposition 4.2 can yield the following result.

######  Corollary 4.5.

Let $\phi\in\mathcal{S}_{k}\cup\mathcal{B}_{r}$ and $q$ be a square-free
integer. Then we have

$\sum_{\begin{subarray}{c}n\leq X\\\ n\equiv a({\rm
mod}\,q)\end{subarray}}\lambda_{\phi}(n)^{2}=\frac{1}{q_{0}\varphi(q/q_{0})}H_{q_{0}}(1,{\phi}\otimes{\phi})G_{q/q_{0}}(1,{\phi}\otimes{\phi})\mathop{\textup{Res}}_{s=1}L(s,{\phi}\otimes{\phi})X+O\Big{(}X^{\frac{3}{5}+\varepsilon}\Big{)}$

for any $q<X^{\frac{2}{5}-\varepsilon}$, where $q_{0}=(a,q)$,
$H_{q_{0}}(1,{\phi}\otimes{\phi}),G_{q/q_{0}}(1,{\phi}\otimes{\phi})$ are as
in Proposition 4.2, and the implied constant depends only on
$\phi,\varepsilon$.

###### Proof.

Assume $a=a_{0}a_{1}$ and $q=q_{0}q_{1}$ with $(q_{0}a_{1},q_{1})=1$. Then we
obtain the identical relation

$\sum_{\begin{subarray}{c}n\leq X\\\ n\equiv a({\rm
mod}\,q)\end{subarray}}\lambda_{\phi}(n)^{2}=\sum_{\begin{subarray}{c}n\leq
X/q_{0}\\\ n\equiv a_{1}({\rm
mod}\,q_{1})\end{subarray}}\lambda_{\phi}(q_{0}n)^{2},$

where $q_{0}=(a,q)$. This corollary immediately follows from Proposition 4.2.
∎

Now we are ready to handle the sifted sum (4.2) and get the following result.

######  Lemma 4.6.

Suppose that $0<Y<Z\leq X^{\frac{1}{25}-\varepsilon}$. Let $P(Y,Z)$ denote the
product of primes $p$ which belong to the interval $[Y,Z)$. Then we have

$\sum_{\begin{subarray}{c}n\leq X\\\
\left(n-a,P(Y,Z)\right)=1\end{subarray}}\lambda_{\phi}(n)^{2}\ll X\frac{\log
Y}{\log Z}$

uniformly in any integer $a$.

###### Proof.

This lemma will follow from a standard application of sieve method and
Proposition 4.2. To estimate the contribution of main term in Proposition 4.2
to the sifted sum (4.2), we have to compute the sieve dimension. Additionally,
the sieve dimension also directly determines the choice of sieve level. So we
start to deal with a product of the type:

$\prod_{w\leq p<z}(1-g(p))^{-1},$

where $0<w<z$ and $g(p)$ is the density function. By Corollary 4.5, the
density function at primes is given by

$g(p)=\begin{cases}\frac{1}{p+1}\Big{(}\lambda_{\phi}(p)^{2}-\frac{\lambda_{\phi}(p^{2})}{p}+\frac{1}{p^{2}}\Big{)}\quad&\text{if
}p\mid a,\\\
\frac{p}{p^{2}-1}\Big{(}1-\frac{\lambda_{\phi}(p^{2})}{p}+\frac{\lambda_{\phi}(p^{2})}{p^{2}}-\frac{1}{p^{3}}\Big{)}\quad&\text{if
}p\nmid a.\end{cases}$

By the estimate (3.2), one has

$g(p)=\begin{cases}\frac{\lambda_{\phi}(p)^{2}}{p}+O\big{(}p^{-\frac{39}{32}+\varepsilon}\big{)}\quad&\text{if
}p\mid a,\\\
\frac{1}{p}+O\big{(}p^{-\frac{39}{32}+\varepsilon}\big{)}\quad&\text{if
}p\nmid a.\end{cases}$

Together this with Mertens’ theorem or prime number theorem of Rankin–Selberg
$L$-function $L(s,{\phi}\otimes{\phi})$, we have

$\prod_{w\leq p<z}(1-g(p))^{-1}\sim\frac{\log z}{\log w}.$ (4.11)

This means that the sieve dimension of the function $\lambda_{\phi}(n)^{2}$ is
one.

Next, we apply the upper-bound sieve (see, for example, [11, Corollary 6.2]).
Since the sieve dimension equals one, the parameters $\beta$ and $s$ in [11,
Corollary 6.2]) can be taken $\beta=s=10$. Thus, applying Corollary 4.5 and
(4.11), we find

$\displaystyle\sum_{\begin{subarray}{c}n\leq X\\\
\left(n-a,P(Y,Z)\right)=1\end{subarray}}\lambda_{\phi}(n)^{2}\ll$
$\displaystyle X\prod_{p|P}(1-g(p))+X^{\frac{3}{5}+\varepsilon}Z^{10}$
$\displaystyle\ll$ $\displaystyle X\frac{\log Y}{\log Z}$

for any $Z\leq X^{\frac{1}{25}-\varepsilon}.$ ∎

##  5\. Proof of Theorem 1.1

For convention, we write $H=(\log X)^{8c+16}$ and $K=X^{\delta/2}$, where $c$
and $\delta>0$ are the same constants as in Hypotheses (i) and (ii). We
emphasize that $\delta$ is a sufficiently small constant. Let $\nu\in[H,K]$,
and set

${\mathcal{I}}_{\nu}=((\nu-1)^{2},\nu^{2}],\quad\
P_{\nu}=\prod_{(H-1)^{2}<p\leq\nu^{2}}p,$
${\mathcal{P}}_{\nu}=\\{p\in{\mathcal{I}}_{\nu}\\},\quad{\mathcal{M}}_{\nu}=\Big{\\{}m\in\big{[}1,X/\nu^{2}\big{]}:(m,P_{\nu})=1\Big{\\}},$
${\mathcal{P}}_{\nu}{\mathcal{M}}_{\nu}=\\{pm:p\in{\mathcal{P}}_{\nu},m\in{\mathcal{M}}_{\nu}\\},$
${\mathcal{I}}=\bigcup_{H\leq\nu\leq K}{\mathcal{P}}_{\nu}{\mathcal{M}}_{\nu}\
\ \text{and}\ \ {\mathcal{J}}=[1,X]\setminus{\mathcal{I}}.$

The intervals above are always meant as subsets of $\mathbb{N}$. Note that
${\mathcal{P}}_{\nu}{\mathcal{M}}_{\nu}\subset[1,X]$ and each
$n\in{\mathcal{P}}_{\nu}{\mathcal{M}}_{\nu}$ can be written in a unique way as
$n=pm$ with $p\in{\mathcal{P}}_{\nu}$ and $m\in{\mathcal{M}}_{\nu}$. Moreover,
the sets ${\mathcal{P}}_{\nu}{\mathcal{M}}_{\nu}$ are pairwise disjoint for
$H\leq\nu\leq K$. First of all, we split the sum into two parts according to
the variable $n\in{\mathcal{I}}$ or $n\in{\mathcal{J}}$

$\displaystyle\sum_{n\leq X}f(n)\lambda_{\phi}(n+h)=$
$\displaystyle\sum_{n\in{\mathcal{I}}}f(n)\lambda_{\phi}(n+h)+\sum_{n\in{\mathcal{J}}}f(n)\lambda_{\phi}(n+h)$
(5.1) $\displaystyle:=$ $\displaystyle S_{\mathcal{I}}(x)+S_{\mathcal{J}}(x).$

###  5.1. Evaluation of $S_{\mathcal{I}}(x)$

Since ${\mathcal{P}}_{\nu}{\mathcal{M}}_{\nu}$ are pairwise disjoint for
$H\leq\nu\leq K$, we obtain that

$S_{\mathcal{I}}(x)\ll\sum_{H\leq\nu\leq
K}\Bigg{|}\sum_{pm\in{\mathcal{P}}_{\nu}{\mathcal{M}}_{\nu}}f(pm)\lambda_{\phi}(pm+h)\Bigg{|}.$
(5.2)

For $pm\in{\mathcal{P}}_{\nu}{\mathcal{M}}_{\nu}$, we have
$p\in{\mathcal{P}}_{\nu}$ and $m\in{\mathcal{M}}_{\nu}$. The $f(pm)$ in (5.2)
can be factored as $f(p)f(m)$ by its multiplicativity. Thus, we get

$S_{\mathcal{I}}(x)\ll\sum_{H\leq\nu\leq
K}\sum_{m\in{\mathcal{M}}_{\nu}}|f(m)|\Big{|}\sum_{p\in{\mathcal{P}}_{\nu}}f(p)\lambda_{\phi}(pm+h)\Big{|}.$

The inner sum could be estimated by using the Cauchy–Schwarz inequality

$\displaystyle\sum_{m\in{\mathcal{M}}_{\nu}}|f(m)|\,\Big{|}\sum_{p\in{\mathcal{P}}_{\nu}}f(p)\lambda_{\phi}(pm+h)\Big{|}$
(5.3) $\displaystyle\leq$
$\displaystyle\Big{(}\sum_{m\in{\mathcal{M}}_{\nu}}|f(m)|^{2}\Big{)}^{\frac{1}{2}}\,\Big{(}\sum_{m\in{\mathcal{M}}_{\nu}}\Big{|}\sum_{p\in{\mathcal{P}}_{\nu}}f(p)\lambda_{\phi}(pm+h)\Big{|}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\Big{(}\sum_{m\in{\mathcal{M}}_{\nu}}|f(m)|^{2}\Big{)}^{\frac{1}{2}}\,\Big{(}\sum_{m\leq
X/\nu^{2}}\Big{|}\sum_{p\in{\mathcal{P}}_{\nu}}f(p)\lambda_{\phi}(pm+h)\Big{|}^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\ll$
$\displaystyle\Big{(}\sum_{m\in{\mathcal{M}}_{\nu}}|f(m)|^{2}\Big{)}^{\frac{1}{2}}\,\Big{(}\sum_{p_{1},p_{2}\in{\mathcal{P}}_{\nu}}|f(p)|^{2}\,\Big{|}\sum_{m=1}^{\infty}\lambda_{\phi}(p_{1}m+h)\lambda_{\phi}(p_{2}m+h)e^{-\frac{\nu^{2}m}{X}}\Big{|}\Big{)}^{\frac{1}{2}},$

where the last step inserts a redundant weight function and uses the
inequality of arithmetic and geometric means.

The diagonal contribution in (5.3), that is $p_{1}=p_{2}$ for each $\nu$
yields at most

$\begin{split}&\Big{(}\sum_{m\in{\mathcal{M}}_{\nu}}|f(m)|^{2}\Big{)}^{\frac{1}{2}}\Big{(}e^{\frac{h}{X}}\sum_{p\in{\mathcal{P}}_{\nu}}|f(p)|^{2}\sum_{\begin{subarray}{c}n=1\\\
n\equiv
h(\text{mod}\,p)\end{subarray}}^{\infty}|\lambda_{\phi}(n)|^{2}e^{-\frac{n}{X}}\Big{)}^{\frac{1}{2}}\\\
\ll&\Big{(}\max_{p\in{\mathcal{P}}_{\nu}}\sum_{\begin{subarray}{c}n=1\\\
n\equiv
h(\text{mod}\,p)\end{subarray}}^{\infty}|\lambda_{\phi}(n)|^{2}e^{-\frac{n}{X}}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{p\in{\mathcal{P}}_{\nu},m\in{\mathcal{M}}_{\nu}}|f(pm)|^{2}\Big{)}^{\frac{1}{2}},\end{split}$
(5.4)

by using the multiplicative property of $f(n)$ and the fact $h\ll X$. We shall
use the properties of $\lambda_{\phi}(n)$ and partial summations to estimate
the first term on the second line of (5.4). Corollary 4.5 can be directly
used. After inserting the bounds in (4.8), we have

$\displaystyle\sum_{\begin{subarray}{c}n=1\\\ n\equiv
h(\text{mod}\,p)\end{subarray}}^{\infty}|\lambda_{\phi}(n)|^{2}e^{-\frac{n}{X}}\ll$
$\displaystyle\sum_{n=1}^{\infty}|\lambda_{\phi}(pn)|^{2}e^{-\frac{pn}{X}}+\max_{(p,h)=1}\sum_{\begin{subarray}{c}n=1\\\
n\equiv
h(\text{mod}\,p)\end{subarray}}^{\infty}|\lambda_{\phi}(n)|^{2}e^{-\frac{n}{X}}$
(5.5) $\displaystyle\ll$
$\displaystyle|\lambda_{\phi}(p)|^{2}\sum_{n=1}^{\infty}|\lambda_{\phi}(n)|^{2}e^{-\frac{pn}{X}}+\max_{(p,h)=1}\sum_{\begin{subarray}{c}n=1\\\
n\equiv
h(\text{mod}\,p)\end{subarray}}^{\infty}|\lambda_{\phi}(n)|^{2}e^{-\frac{n}{X}}$
$\displaystyle\ll$ $\displaystyle v^{-\frac{25}{16}}X$

for any $p\in{\mathcal{P}}_{\nu}$. Hence, by summing over $\nu$ and the
Cauchy–Schwarz inequality, the diagonal contribution to $S_{\mathcal{I}}(x)$
is less than

$\begin{split}&X^{\frac{1}{2}}\Big{(}\sum_{H\leq\nu\leq
H}v^{-\frac{25}{16}}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{H\leq\nu\leq
K}\sum_{p\in{\mathcal{P}}_{\nu},m\in{\mathcal{M}}_{\nu}}|f(pm)|^{2}\Big{)}^{\frac{1}{2}}\\\
&\ll\frac{X^{\frac{1}{2}}}{H^{\frac{9}{32}}}\Big{(}\sum_{n\leq
X}|f(n)|^{2}\Big{)}^{\frac{1}{2}}\\\ &\ll\frac{X(\log
X)^{\frac{c}{2}}}{H^{\frac{9}{32}}}.\end{split}$ (5.6)

Now we turn to treat the off-diagonal terms. For $p\not=q$, the innermost sum
over $m$ is just a shifted convolution sum. Lemma 4.1 can be applied and gives

$\sum_{m=1}^{\infty}\lambda_{\phi}(p_{1}m+h)\lambda_{\phi}(p_{2}m+h)e^{-\frac{\nu^{2}m}{X}}\ll\nu^{\frac{7}{5}}X^{\frac{9}{10}+\varepsilon}$

for any $h\ll X$. Hence, by summing over $\nu$ and the Cauchy–Schwarz
inequality, the contribution to $S_{\mathcal{I}}(x)$ of the off-diagonal terms
in the last line of (5.3) is then at most

$\displaystyle X^{\frac{9}{20}+\varepsilon}\Big{(}\sum_{H\leq\nu\leq
K}\nu^{\frac{9}{5}}\Big{)}^{\frac{1}{2}}\Big{(}\sum_{H\leq\nu\leq
K}\sum_{p\in{\mathcal{P}}_{\nu},m\in{\mathcal{M}}_{\nu}}|f(pm)|^{2}\Big{)}^{\frac{1}{2}}$
(5.7) $\displaystyle\ll K^{\frac{11}{10}}X^{\frac{9}{20}+\varepsilon}.$

Putting the estimates (5.6) and (5.7) together, and inserting the values of
parameters $H,K$, we have

$S_{\mathcal{I}}(x)\ll H^{-\frac{9}{32}}X(\log
X)^{\frac{c}{2}}+K^{\frac{11}{10}}X^{\frac{9}{20}+\varepsilon}\ll\frac{X}{\log
X}.$ (5.8)

###  5.2. Evaluation of $S_{\mathcal{J}}(x)$

In order to estimate the contribution of corresponding sums over the set
${\mathcal{J}}$, we define the following subsets of $[1,X]$:

$\begin{split}{\mathcal{J}}_{1}^{(\nu)}&=\Big{\\{}n\in[1,X]:n\ \text{has
exactly one divisor in ${\mathcal{P}}_{\nu}$ and none in $\bigcup_{H\leq
h<\nu}{\mathcal{P}}_{h}$}\Big{\\}},\\\
{\mathcal{J}}_{1}&=\bigcup_{H\leq\nu\leq K}{\mathcal{J}}_{1}^{(\nu)},\\\
{\mathcal{J}}_{2}&=\Big{\\{}n\in[1,X]:n\ \text{has at least one prime factor
in $\bigcup_{H\leq\nu\leq K}{\mathcal{P}}_{\nu}$}\Big{\\}},\\\
{\mathcal{J}}_{3}&=\Big{\\{}n\in[1,X]:n\ \text{has no prime factors in
${\bigcup_{H\leq\nu\leq K}{\mathcal{P}}_{\nu}}$}\Big{\\}}.\end{split}$

By the definitions of these subsets of $[1,X]$, it is clear that
$P_{\nu}{\mathcal{M}}_{\nu}\subset{\mathcal{J}}_{1}^{(\nu)}$. Thus we have
${\mathcal{I}}\subset{\mathcal{J}}_{1}$. Moreover,
${\mathcal{J}}_{2}\cup{\mathcal{J}}_{3}=[1,X]$ and
${\mathcal{J}}_{2}\cap{\mathcal{J}}_{3}=\emptyset$. Thus, we get

${\mathcal{J}}\subset({\mathcal{J}}_{1}\setminus{\mathcal{I}})\cup({\mathcal{J}}_{2}\setminus{\mathcal{J}}_{1})\cup{\mathcal{J}}_{3}.$

As a consequence, it follows from Hypothesis (i) that

$\begin{split}|S_{\mathcal{J}}(x)|\ll&\sum_{n\in{\mathcal{J}}_{1}\setminus{\mathcal{I}}}|f(n)\lambda_{\phi}(n+h)|+\sum_{n\in{\mathcal{J}}_{2}\setminus{\mathcal{J}}_{1}}|f(n)\lambda_{\phi}(n+h)|+\sum_{n\in{\mathcal{J}}_{3}}|f(n)\lambda_{\phi}(n+h)|\\\
\ll&X^{\frac{1}{2}}(\log
X)^{\frac{c}{2}}\bigg{(}\sum_{n\in{\mathcal{J}}_{1}\setminus{\mathcal{I}}}|\lambda_{\phi}(n+h)|^{2}+\sum_{n\in{\mathcal{J}}_{2}\setminus{\mathcal{J}}_{1}}|\lambda_{\phi}(n+h)|^{2}\bigg{)}^{\frac{1}{2}}\\\
&+\bigg{(}\sum_{n\in{\mathcal{J}}_{3}}|f(n)|^{2}\Big{)}^{\frac{1}{2}}\bigg{(}\sum_{n\in{\mathcal{J}}_{3}}|\lambda_{\phi}(n+h)|^{2}\Big{)}^{\frac{1}{2}}.\end{split}$
(5.9)

We first deal with the summation over
$n\in{\mathcal{J}}_{1}\setminus{\mathcal{I}}$. It is obvious that

${\mathcal{J}}_{1}^{(\nu)}\setminus\\\
P_{\nu}{\mathcal{M}}_{\nu}\subset{\mathcal{P}}_{\nu}\Big{(}\frac{x}{\nu^{2}},\frac{x}{(\nu-1)^{2}}\Big{]}.$

Hence, we have

$\sum_{n\in{\mathcal{J}}_{1}\setminus{\mathcal{I}}}|\lambda_{\phi}(n+h)|^{2}\ll\sum_{H\leq\nu\leq
K}\sum_{p\in{\mathcal{P}}_{\nu}}\sum_{\begin{subarray}{c}|n-X-h|\ll X/\nu\\\
n\equiv h(\text{mod}\,p)\end{subarray}}|\lambda_{\phi}(n)|^{2}.$

The innermost sum is actually related to the coefficients
$|\lambda_{\phi}(n)|^{2}$ in arithmetic progression over a short interval.
Notice that the length of short interval and the modulo of arithmetic
progression satisfies the inequality $p\ll(X/\nu)^{\frac{2}{5}-\varepsilon}.$
So we deduce from Corollary 4.5 and the estimate (4.8)that

$\sum_{\begin{subarray}{c}|n-X-h|\ll X/\nu\\\ n\equiv
h(\text{mod}\,p)\end{subarray}}|\lambda_{\phi}(n)|^{2}\ll\nu^{-\frac{41}{16}}X.$

Further, we obtain

$\displaystyle\sum_{n\in{\mathcal{J}}_{1}\setminus{\mathcal{I}}}|\lambda_{\phi}(n+h)|^{2}\ll$
$\displaystyle X\sum_{H\leq\nu\leq K}\nu^{-\frac{25}{16}}$ (5.10)
$\displaystyle\ll$ $\displaystyle H^{-\frac{9}{16}}X.$

Next, we deal with the summation over
$n\in{\mathcal{J}}_{2}\setminus{\mathcal{J}}_{1}$. On account of

${\mathcal{J}}_{2}\setminus{\mathcal{J}}_{1}\subset\bigcup_{H\leq\nu\leq
K}\\{n\in[1,X]:n\ \text{has at least two prime factors in
${\mathcal{P}}_{\nu}$}\\},$

we get

$\sum_{n\in{\mathcal{J}}_{2}\setminus{\mathcal{J}}_{1}}|\lambda_{\phi}(n+h)|^{2}\ll\sum_{H\leq\nu\leq
K}\sum_{p_{1},p_{2}\in{\mathcal{P}}_{\nu}}\sum_{\begin{subarray}{c}n\leq
X+h\\\ n\equiv h(\text{mod}\,pq)\end{subarray}}|\lambda_{\phi}(n)|^{2}.$

Similar to the arguments of (5.5), we use Corollary 4.5 again and then obtain

$\displaystyle\sum_{n\in{\mathcal{J}}_{2}\setminus{\mathcal{J}}_{1}}|\lambda_{\phi}(n+h)|^{2}\ll$
$\displaystyle X\sum_{H\leq\nu\leq
K}\Big{(}\frac{|{\mathcal{P}}_{\nu}|}{\nu^{25/16}}\Big{)}^{2}$ (5.11)
$\displaystyle\ll$ $\displaystyle H^{-\frac{1}{8}}X.$

Finally, we deal with these two summations over $n\in{\mathcal{J}}_{3}$. By
Lemma 4.6 and Hypothesis (ii), we have

$\sum_{n\in{\mathcal{J}}_{3}}|\lambda_{\phi}(n+h)|^{2}=\sum_{\begin{subarray}{c}n\leq
X+h\\\ (n-h,P)=1\end{subarray}}|\lambda_{\phi}(n)|^{2}\ll X\frac{\log\log
X}{\log X}$ (5.12)

and

$\sum_{n\in{\mathcal{J}}_{3}}|f(n)|^{2}=\sum_{\begin{subarray}{c}n\leq X\\\
(n,P)=1\end{subarray}}|f(n)|^{2}\ll X\frac{(\log\log X)^{\gamma}}{\log X},$
(5.13)

where the notation $P$ is given in Hypothesis (ii).

Now we are ready to estimate the sum $S_{\mathcal{J}}(x)$. On inserting the
estimates (5.10)–(5.13) into (5.9), we obtain

$S_{\mathcal{J}}(x)\ll H^{-\frac{1}{16}}X(\log
X)^{\frac{c}{2}}+X\frac{(\log\log X)^{\frac{\gamma+1}{2}}}{\log X}\ll
X\frac{(\log\log X)^{\frac{\gamma+1}{2}}}{\log X}.$ (5.14)

Therefore, our theorem 1.1 follows from (5.1), (5.8) and (5.14).

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