# On some estimates involving Fourier coefficients of Maass cusp forms

Qingfeng Sun  School of Mathematics and Statistics, Shandong University,
Weihai
Weihai, Shandong 264209, China<EMAIL_ADDRESS>and  Hui Wang  Department of
Mathematics, Shandong University, Jinan 250100, China<EMAIL_ADDRESS>
###### Abstract.

Let $f$ be a Hecke-Maass cusp form for $\rm SL_{2}(\mathbb{Z})$ with Laplace
eigenvalue $\lambda_{f}(\Delta)=1/4+\mu^{2}$ and let $\lambda_{f}(n)$ be its
$n$-th normalized Fourier coefficient. It is proved that, uniformly in
$\alpha,\beta\in\mathbb{R}$,

$\sum_{n\leq X}\lambda_{f}(n)e\left(\alpha n^{2}+\beta n\right)\ll
X^{7/8+\varepsilon}\lambda_{f}(\Delta)^{1/2+\varepsilon},$

where the implied constant depends only on $\varepsilon$. We also consider the
summation function of $\lambda_{f}(n)$ and under the Ramanujan conjecture we
are able to prove

$\sum_{n\leq X}\lambda_{f}(n)\ll
X^{1/3+\varepsilon}\lambda_{f}(\Delta)^{4/9+\varepsilon}$

with the implied constant depending only on $\varepsilon$.

###### Key words and phrases:

Maass cusp form, exponential sums, Fourier coefficients

###### 2000 Mathematics Subject Classification:

11F30, 11F11, 11L07

Qingfeng Sun is partially supported by the National Natural Science Foundation
of China (Grant Nos. 11871306 and 12031008)

##  1\. Introduction

The Fourier coefficients of automorphic forms contain many mysterious
properties, especially its oscillation properties, which have been intensively
studied by many number theorists. Let $\lambda_{F}(n)$ be the normalized
Fourier coefficients of an automorphic form $F$ on $\rm GL_{n}$. Among the
many criteria for evaluating the nature of oscillation, the summation function
$\sum_{n\leq X}\lambda_{F}(n)$ is certainly the most basic one, and the
related exponential sum $\sum_{n\leq X}\lambda_{F}(n)e\left(g(n)\right)$,
where as usual, $e(x)=e^{2\pi ix}$ and $g(n)$ is a real-valued function, is
another important target, with obvious application indications. In this paper,
we are concerned with the oscillating behavior of the Fourier coefficients of
cusp forms on the full modular group. More precisely, let $f$ be a holomorphic
Hecke cusp form of weight $k$ or a Hecke-Maass cusp form of Laplace eigenvalue
$\lambda_{f}(\Delta)=1/4+\mu^{2}$ $(\mu>0)$ for $\rm SL_{2}(\mathbb{Z})$ with
normalized Fourier coefficients $\lambda_{f}(n)$. Then for $f$ holomorphic,
one has the Fourier expansion

$\displaystyle f(z)=\sum_{n\geq
1}\lambda_{f}(n)n^{(k-1)/2}e(nz),\quad\operatorname{Im}(z)>0,$

while for $f$ a Maass form, we can write its Fourier expansion as

$\displaystyle f(z)=\sqrt{y}\sum_{n\neq
0}\lambda_{f}(n)K_{i\mu}(2\pi|n|y)e(nx),$

where $K_{i\mu}$ is the modified Bessel function of the third kind. The famous
Ramanujan–Petersson conjectures assert that
$\lambda_{f}(n)\ll_{\varepsilon}n^{\varepsilon}$ for any $\varepsilon>0$. This
was proved by Deligne [2] for $f$ holomorphic. For $f$ a Maass cusp form, the
best result is $\lambda_{f}(n)\ll_{\varepsilon}n^{7/64+\varepsilon}$ due to
Kim and Sarnak [18]. By Rankin–Selberg theory (see [3, Proposition 19.6]), we
have the following average result

$\displaystyle\sum_{n\leq
X}|\lambda_{f}(n)|^{2}\ll_{\varepsilon}X(X|\mu|)^{\varepsilon}.$ (1.1)

The Rankin-Selberg’s estimate in (1.1) shows that the Fourier coefficients
$\lambda_{f}(n)$ behave like constants on average. However, as $n$ grows
$\lambda_{f}(n)$ in fact vary greatly in sign, which can be seen in the
estimate (see [4, 7])

$\displaystyle\sum_{n\leq X}\lambda_{f}(n)e(n\alpha)\ll_{f}X^{1/2}\log(2X),$

which holds uniformly in $\alpha\in\mathbb{R}$. It’s worth noting that the
above estimate depends on the form $f$, whereas in certain applications such
the subconvexity problem, one may need an explicit dependence on the form $f$.
To this end, Godber [6] proved that for any $\alpha\in\mathbb{R}$ and any
$\varepsilon>0$,

$\displaystyle\sum_{n\leq X}\lambda_{f}(n)e\left(n\alpha\right)\ll
X^{1/2+\varepsilon}\lambda_{f}(\Delta)^{1/4+\varepsilon},$ (1.2)

where the implied constant depends only on $\varepsilon$. This is an
improvement over the bound
$X^{1/2+\varepsilon}\lambda_{f}(\Delta)^{1/2+\varepsilon}$ in [12, Section
8.3]. For the associated quadratic exponential sums, Pitt [24] first proved
for any $\alpha,\beta\in\mathbb{R}$ and any $\varepsilon>0$,

$\displaystyle\sum_{n\leq X}\lambda_{f}(n)e\left(\alpha n^{2}+\beta
n\right)\ll X^{15/16+\varepsilon},$ (1.3)

where the implied constant depends only on the form $f$ and $\varepsilon$.
Later, Liu and Ren [19] improved the above bound to $X^{7/8+\varepsilon}$.

It is interesting and useful to prove an estimate for the exponential sum in
(1.3), which does not depend on the form $f$. So the first aim of our paper is
to prove the following.

######  Theorem 1.1.

Let $\lambda_{f}(n)$ be the normalized Fourier coefficients of a Hecke-Maass
cusp form for $\rm SL_{2}(\mathbb{Z})$ with Laplacian eigenvalue
$\lambda_{f}(\Delta)=1/4+\mu^{2}$. For any $\alpha,\beta\in\mathbb{R}$ and any
$\varepsilon>0$, we have

$\sum_{n\leq X}\lambda_{f}(n)e\left(\alpha n^{2}+\beta n\right)\ll
X^{7/8+\varepsilon}\lambda_{f}(\Delta)^{1/2+\varepsilon},$

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

######  Remark 1.

The methods of proving Theorem 1.1 can also be adopted for holomorphic forms.
Let $f$ be a holomorphic cusp form of weight $k$ for $\rm SL_{2}(\mathbb{Z})$
with normalized Fourier coefficients $\lambda_{f}(n)$. It would have,
uniformly in $f$ and $\alpha,\beta\in\mathbb{R}$,

$\sum_{n\leq X}\lambda_{f}(n)e\left(\alpha n^{2}+\beta n\right)\ll
X^{7/8+\varepsilon}k^{1+\varepsilon}.$

The proof for Theorem 1.1 is based on the ideas introduced by [24]. That is,
taking two different approaches according to the Dirichlet approximation of
$\alpha$. Let $Q\geq 1$ be any given number. By Dirichlet’s theorem, for any
$\alpha\in\mathbb{R}$, there exists a reduced rational number $\ell/q$ with
$1\leq q\leq Q$ such that

$\displaystyle\left|\alpha-\frac{\ell}{q}\right|\leq\frac{1}{qQ}.$ (1.4)

For “larger” $q$, i.e., the oscillation of the exponential function is large,
we separate the Fourier coefficients $\lambda_{f}(n)$ and the exponential
function by the $\delta$-method as in Pitt [24] and deal the resulting sum
using techniques in Liu and Ren [19]. However, with the demand that we shall
make clear the dependence on $f$ in mind, we need to adopt a different form of
the $\delta$-method, i.e., the $\delta$-method due to Duke, Friedlander and
Iwaniec (see Section 2.5). For “smaller” $q$, we deal with it as in Pitt [24],
except for applying the result of Godber in (1.2).

Another purpose of our paper is to consider the summation function of
$\lambda_{f}(n)$, i.e.,

$\displaystyle\sum_{n\leq X}\lambda_{f}(n).$

For holomorphic cusp forms, this was first studied by Hecke [9] in 1927.
Later, Walfisz [26] proved the following estimate, i.e.,

$\displaystyle\sum_{n\leq
X}\lambda_{f}(n)\ll_{f,\vartheta}X^{\frac{1+\vartheta}{3}},$

where $\vartheta$ is exponent towards the Ramanujan-Petersson conjecture,
i.e., $|\lambda_{f}(n)|\leq n^{\vartheta}$. Then by Deligne [2], one has the
estimate $O_{f,\varepsilon}(X^{1/3+\varepsilon})$ for any $\varepsilon>0$.
Subsequently, Hafner and Ivić [8] removed the factor $X^{\varepsilon}$ of
Deligne’s result and obtained the bound $O_{f}(X^{1/3})$. The current best
record is $O_{f}\left(X^{1/3}(\log X)^{-0.1185}\right)$ due to Wu [27]. For
the case of Maass cusp forms, there are fewer results. Assuming the Ramanujan-
Petersson conjecture, one also has the estimate (see for example [5])

$\displaystyle\sum_{n\leq
X}\lambda_{f}(n)\ll_{f,\varepsilon}X^{1/3+\varepsilon}.$ (1.5)

Other interesting results without assuming the Ramanujan-Petersson conjecture
but weaker than (1.5) can be found in Hafner and Ivić [8], Lü [20], Jiang and
Lü [14] and some references therein.

Notice that all the above mentioned estimates depend on the form $f$. So the
second purpose of this paper is to strengthen the estimate in (1.5) by making
the dependence on the form $f$ explicit. Our result is the following.

######  Theorem 1.2.

Let $\lambda_{f}(n)$ be the normalized Fourier coefficients of a Hecke-Maass
cusp form for $\rm SL_{2}(\mathbb{Z})$ with Laplacian eigenvalue
$\lambda_{f}(\Delta)=1/4+\mu^{2}$. Under the Ramanujan conjecture, we have,
for any $\varepsilon>0$,

$\sum_{n\leq X}\lambda_{f}(n)\ll
X^{1/3+\varepsilon}\lambda_{f}(\Delta)^{4/9+\varepsilon},$

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

Notation. Throughout the paper, $\varepsilon$ and $A$ are arbitrarily small
and arbitrarily large positive numbers, respectively, which may be different
at each occurrence. As usual, $e(x)=e^{2\pi ix}$ and the symbol $n\sim X$
means $X<n\leq 2X$.

##  2\. Preliminaries

###  2.1. Maass cusp forms for $\rm GL_{2}$

Let $f$ be a Hecke-Maass cusp form for $\rm SL_{2}(\mathbb{Z})$ with Laplace
eigenvalue $1/4+\mu^{2}$, with the normalized Fourier coefficients
$\lambda_{f}(n)$. For $\operatorname{Re}(s)>1$, the $L$-function associated to
$f$ is given by

$\displaystyle L(s,f)=\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)}{n^{s}},$ (2.1)

which satisfies the functional equation

$\displaystyle L(1-s,f)=(-1)^{\eta}\gamma(s)L(s,f),$ (2.2)

where $\eta=0$ or 1 according as $f$ is even or odd, and

$\displaystyle\gamma(s)=\pi^{1-2s}\prod_{\pm}\Gamma\left(\frac{s+\eta\pm
i\mu}{2}\right)\Gamma\left(\frac{1-s+\eta\pm i\mu}{2}\right)^{-1}.$ (2.3)

###  2.2. Summation formulas

We first recall the Poisson summation formula over an arithmetic progression.

######  Lemma 2.1.

Let $\beta\in\mathbb{Z}$ and $c\in\mathbb{Z}_{\geq 1}$. For a Schwartz
function $f:\mathbb{R}\rightarrow\mathbb{C}$, we have

$\displaystyle\underset{\begin{subarray}{c}n\in\mathbb{Z}\\\ n\equiv\beta\bmod
c\end{subarray}}{\sum}f(n)=\frac{1}{c}\sum_{n\in\mathbb{Z}}\widehat{f}\left(\frac{n}{c}\right)e\left(\frac{n\beta}{c}\right),$

where $\widehat{f}(y)=\int_{\mathbb{R}}f(x)e(-xy)\mathrm{d}x$ is the Fourier
transform of $f$.

###### Proof.

See e.g. [13, Eq.(4.24)]. ∎

We have the following Voronoi formula for $\rm SL_{2}(\mathbb{Z})$ (see [21,
Eqs. (1.12), (1.15)]).

######  Lemma 2.2.

Let $\varphi(x)$ be a smooth function compactly supported on $\mathbb{R^{+}}$.
Let $a,\overline{a},c\in\mathbb{Z}$ with $c\neq 0,(a,c)=1$ and
$a\overline{a}\equiv 1\;(\text{{\rm mod }}c)$. Then

$\displaystyle\sum_{m=1}^{\infty}\lambda_{f}(m)e\left(\frac{am}{c}\right)\varphi(m)=c\sum_{\pm}\sum_{m=1}^{\infty}\frac{\lambda_{f}(m)}{m}e\left(\pm\frac{\overline{a}m}{c}\right)\Psi_{\pm}\left(\frac{m}{c^{2}}\right),$

where for $\sigma>-1$,

$\displaystyle\Psi_{\pm}(x)=\frac{1}{4\pi^{2}i}\int_{(\sigma)}(\pi^{2}x)^{-s}\rho_{f}^{\pm}(s)\widetilde{\varphi}(-s)\mathrm{d}s,$
(2.4)

with

$\displaystyle\rho_{f}^{\pm}(s)=\prod_{\pm}\frac{\Gamma(\frac{1+s\pm
i\mu}{2})}{\Gamma(\frac{-s\pm
i\mu}{2})}\pm\prod_{\pm}\frac{\Gamma(\frac{2+s\pm
i\mu}{2})}{\Gamma(\frac{1-s\pm i\mu}{2})}.$ (2.5)

Here $\widetilde{\varphi}(s)=\int_{0}^{\infty}\varphi(u)u^{s-1}\mathrm{d}u$ is
the Mellin transform of $\varphi$.

###  2.3. Stirling’s formula

By Stirling asymptotic formula (see [23, Section 8.4, Eq. (4.03)]), for $|\arg
s|\leq\pi-\varepsilon$, $|s|\gg 1$ and any $\varepsilon>0$,

$\displaystyle\ln\Gamma(s)=\left(s-\frac{1}{2}\right)\ln
s-s+\frac{1}{2}\ln(2\pi)+\sum_{j=1}^{K_{1}}\frac{B_{2j}}{2j(2j-1)s^{2j-1}}+O_{K_{1},\varepsilon}\left(\frac{1}{|s|^{2K_{1}+1}}\right),$

where $B_{j}$ are Bernoulli numbers. Thus for $s=\sigma+i\tau$, $\sigma$ fixed
and $|t|\geq 2$,

$\displaystyle\Gamma(\sigma+i\tau)=\sqrt{2\pi}(i\tau)^{\sigma-1/2}e^{-\pi|\tau|/2}\left(\frac{|\tau|}{e}\right)^{i\tau}\left(1+\sum_{j=1}^{K_{2}}\frac{c_{j}}{\tau^{j}}+O_{\sigma,K_{2},\varepsilon}\bigg{(}\frac{1}{|\tau|^{K_{2}+1}}\bigg{)}\right),$
(2.6)

where the constants $c_{j}$ depend on $j,\sigma$ and $\varepsilon$. Thus for
$\sigma\geq-1/2$,

$\displaystyle\rho_{f}^{\pm}(\sigma+i\tau)$ $\displaystyle=$
$\displaystyle\prod_{\pm}\frac{\Gamma(\frac{1+\sigma+i(\tau\pm\mu)}{2})}{\Gamma(\frac{-\sigma-i(\tau\pm\mu)}{2})}\pm\prod_{\pm}\frac{\Gamma(\frac{2+\sigma+i(\tau\pm\mu)}{2})}{\Gamma(\frac{1-\sigma-i(\tau\pm\mu)}{2})}$
(2.7) $\displaystyle\ll$ $\displaystyle(|\tau+\mu||\tau-\mu|)^{\sigma+1/2}.$

###  2.4. Estimates for exponential integrals

Let

$I=\int_{\mathbb{R}}w(y)e^{i\varrho(y)}dy.$

We need the following evaluation for exponential integrals which are Lemma 8.1
and Proposition 8.2 of [1] in the language of inert functions (see [17, Lemma
3.1]).

Let $\mathcal{F}$ be an index set, $Y:\mathcal{F}\rightarrow\mathbb{R}_{\geq
1}$ and under this map $T\mapsto Y_{T}$ be a function of $T\in\mathcal{F}$. A
family $\\{w_{T}\\}_{T\in\mathcal{F}}$ of smooth functions supported on a
product of dyadic intervals in $\mathbb{R}_{>0}^{d}$ is called $Y$-inert if
for each $j=(j_{1},\ldots,j_{d})\in\mathbb{Z}_{\geq 0}^{d}$ we have

$\displaystyle
C(j_{1},\ldots,j_{d})=\sup_{T\in\mathcal{F}}\sup_{(y_{1},\ldots,y_{d})\in\mathbb{R}_{>0}^{d}}Y_{T}^{-j_{1}-\cdots-
j_{d}}\left|y_{1}^{j_{1}}\cdots
y_{d}^{j_{d}}w_{T}^{(j_{1},\ldots,j_{d})}(y_{1},\ldots,y_{d})\right|<\infty.$

######  Lemma 2.3.

Suppose that $w=w_{T}(y)$ is a family of $Y$-inert functions, with compact
support on $[Z,2Z]$, so that $w^{(j)}(y)\ll(Z/Y)^{-j}$. Also suppose that
$\varrho$ is smooth and satisfies $\varrho^{(j)}(y)\ll H/Z^{j}$ for some
$H/Y^{2}\geq R\geq 1$ and all $y$ in the support of $w$.

  1. (1)

If $|\varrho^{\prime}(y)|\gg H/Z$ for all $y$ in the support of $w$, then
$I\ll_{A}ZR^{-A}$ for $A$ arbitrarily large.

  2. (2)

If $\varrho^{\prime\prime}(y)\gg H/Z^{2}$ for all $y$ in the support of $w$,
and there exists $y_{0}\in\mathbb{R}$ such that $\varrho^{\prime}(y_{0})=0$
(note $y_{0}$ is necessarily unique), then

$I=\frac{e^{i\varrho(y_{0})}}{\sqrt{\varrho^{\prime\prime}(y_{0})}}F(y_{0})+O_{A}(ZR^{-A}),$

where $F(y_{0})$ is an $Y$-inert function (depending on $A$) supported on
$y_{0}\asymp Z$.

We also need the second derivative test (see [11, Lemma 5.1.3]).

######  Lemma 2.4.

Let $\varrho(x)$ be real and twice differentiable on the open interval $[a,b]$
with $\varrho^{\prime\prime}(x)\gg\lambda_{0}>0$ on $[a,b]$. Let $w(x)$ be
real on $[a,b]$ and let $V_{0}$ be its total variation on $[a,b]$ plus the
maximum modulus of $w(x)$ on $[a,b]$. Then

$\displaystyle I\ll\frac{V_{0}}{\sqrt{\lambda_{0}}}.$

###  2.5. The circle method

Let $\delta:\mathbb{Z}\rightarrow\\{0,1\\}$ be defined as $\delta(0)=1$ and
$\delta(n)=0$ for $n\neq 0$. We will use a version of the $\delta$-method by
Duke, Friedlander and Iwaniec (see [13, Chapter 20]) which states that for any
$n\in\mathbb{Z}$ and $C\in\mathbb{R}^{+}$, we have

$\displaystyle\delta(n)=\frac{1}{C}\sum_{1\leq c\leq
C}\;\frac{1}{c}\;\sideset{}{{}^{\star}}{\sum}_{a\bmod{c}}e\left(\frac{na}{c}\right)\int_{\mathbb{R}}g(c,\zeta)e\left(\frac{n\zeta}{cC}\right)\mathrm{d}\zeta,$
(2.8)

where the $\star$ on the sum indicates that the sum over $a$ is restricted to
$(a,c)=1$. The function $g$ has the following properties (see [13, (20.158),
(20.159)] and [10, Lemma 15])

$\displaystyle
g(c,\zeta)\ll|\zeta|^{-A},\;\;\;\;\;\;g(c,\zeta)=1+O\left(\frac{C}{c}\left(\frac{c}{C}+|\zeta|\right)^{A}\right)$
(2.9)

for any $A>1$ and

$\displaystyle\frac{\partial^{j}}{\partial\zeta^{j}}g(c,\zeta)\ll|\zeta|^{-j}\min\left(|\zeta|^{-1},\frac{C}{c}\right)\log
C,\quad j\geq 1.$

###  2.6. Some estimates

We quote the following results from Pitt (see [24, Section 2]), Tolev (see
[25, Section 2]) and Liu and Ren (see [19, Lemma 3.3]) respectively.

######  Lemma 2.5.

Let $c_{1}$ be the largest square-free factor of $c\in\mathbb{N}$ such that
$c=c_{1}c_{2}$, $(c_{1},c_{2})=1$. Then for any $\varepsilon>0$ and
$\beta\in\mathbb{R}$, we have

$\displaystyle\sum_{n\sim X}S(m,n,c)e(\alpha n^{2}+\beta
n)\ll(Xc)^{1/2+\varepsilon}+\mathbf{T}^{1/2},$

where

$\displaystyle\mathbf{T}=(m,c)^{1/2}\tau^{2}(c_{1})c_{1}^{1/2}c_{2}\sum_{c=c_{3}c_{4}}c_{4}^{1/2}\sum_{u\bmod
c_{3}\atop(u,c_{3})=1}\sum_{1\leq h<X}\min\left\\{X,\left(2\left\|2\alpha
h+\frac{u}{c_{3}}\right\|\right)^{-1}\right\\}.$ (2.10)

Here $\|x\|$ denote the distance from $x$ to the nearest integer.

######  Lemma 2.6.

Let $M\geq 2$. Then for any $\varepsilon>0$, there exists a smooth function
$G(M,x)$, which is periodic with period one and satisfies

$\begin{split}\min(M,\|x\|^{-1})\leq G(M,x).\end{split}$

Moreover, $G(M,x)$ has a Fourier expansion

$\begin{split}G(M,x)=\sum_{n}b(n)e(nx)\end{split}$

with coefficients satisfying $b(n)\ll\log M$ and

$\begin{split}\sum_{|n|>M^{1+\varepsilon}}|b(n)|\ll_{A,\varepsilon}M^{-A}\end{split}$

for any constant $A>0$.

######  Lemma 2.7.

Let $f(n)$ denote the largest square-free factor of $n$ such that
$(f(n),n/f(n))=1$. Then we have

$\begin{split}\sum_{n\leq X}f^{-1/4}(n)\ll X^{3/4}\end{split}$

and

$\begin{split}\sum_{n\leq X}f^{-1/2}(n)\ll X^{1/2}\log X.\end{split}$

We also need the following estimate (see Karatsuba [16, Chapter VI, §2, Lemma
5]).

######  Lemma 2.8.

Let

$\displaystyle\alpha=\frac{\ell}{q}+\frac{\theta}{q^{2}},\quad(\ell,q)=1,\quad
q\geq 1,\quad|\theta|\leq 1.$

Then for any $\beta\in\mathbb{R}$, $U>0$ and $P\geq 1$, we have

$\displaystyle\sum_{x=1}^{P}\min\left\\{U,\left\|\alpha
x+\beta\right\|^{-1}\right\\}\leq 6\left(\frac{P}{q}+1\right)(U+q\log q).$

##  3\. Proof of Theorem 1.1

We assume $\mu>X^{\varepsilon}$, otherwise Theorem 1.1 follows from Liu and
Ren [19]. By dyadic subdivision it suffices to prove the required estimate for
the sum

$\displaystyle S(X,\alpha,\beta)=\sum_{n\sim X}\lambda_{f}(n)e\left(\alpha
n^{2}+\beta n\right).$

Let $V(x)\in\mathcal{C}_{c}^{\infty}(3/4,9/4)$ be identically one on $[1,2]$
with derivatives satisfying $V^{(j)}(x)\ll_{j}1$ for any integer $j\geq 0$.
Then we can write $S(X,\alpha,\beta)$ as

$\displaystyle S(X,\alpha,\beta)=\sum_{n\sim X}e\left(\alpha n^{2}+\beta
n\right)\sum_{m=1}^{\infty}\lambda_{f}(m)V\left(\frac{m}{X}\right)\delta(n-m),$
(3.1)

where $\delta(n)=\left\\{\begin{aligned} &1\,\textit{ if }n=0,\\\ &0\,\textit{
if }n\neq 0\end{aligned}\right.$ is the Kronecker delta function.

Plugging the identity (2.8) for $\delta(n)$ into (3.1) and exchanging the
order of integration and summations, we get

$\displaystyle S(X,\alpha,\beta)$ $\displaystyle=$
$\displaystyle\frac{1}{C}\int_{\mathbb{R}}\sum_{1\leq c\leq
C}\frac{g(c,\zeta)}{c}\sum_{n\sim X}e\left(\alpha n^{2}+\beta
n+\frac{n\zeta}{cC}\right)\;$
$\displaystyle\times\sideset{}{{}^{\star}}{\sum}_{a\bmod{c}}e\left(\frac{na}{c}\right)\sum_{m=1}^{\infty}\lambda_{f}(m)e\left(-\frac{ma}{c}\right)V\left(\frac{m}{X}\right)e\left(-\frac{m\zeta}{cC}\right)\mathrm{d}\zeta,$

where $C>1$ is a parameter to be chosen later. Note that the contribution from
$|\zeta|\leq X^{-G}$ for $G>0$ sufficiently large is negligible. Moreover, by
the first property in (2.9), we can restrict $\zeta$ in the range $|\zeta|\leq
X^{\varepsilon}$ up to an negligible error. So we can insert a smooth
partition of unity for the $\zeta$-integral and write $S(X,\alpha,\beta)$ as

$\displaystyle S(X,\alpha,\beta)$ $\displaystyle=$
$\displaystyle\sum_{X^{-G}\ll\Xi\ll
X^{\varepsilon}\atop\text{dyadic}}\frac{1}{C}\int_{\mathbb{R}}\varpi\left(\frac{\zeta}{\Xi}\right)\sum_{1\leq
c\leq C}\frac{g(c,\zeta)}{c}\sum_{n\sim X}e\left(\alpha n^{2}+\beta
n+\frac{n\zeta}{cC}\right)\;$
$\displaystyle\times\sideset{}{{}^{\star}}{\sum}_{a\bmod{c}}e\left(\frac{na}{c}\right)\sum_{m=1}^{\infty}\lambda_{f}(m)e\left(-\frac{ma}{c}\right)V\left(\frac{m}{X}\right)e\left(-\frac{m\zeta}{cC}\right)\mathrm{d}\zeta+O_{A}(X^{-A}),$

where $\varpi(x)\in\mathcal{C}_{c}^{\infty}(1,2)$ satisfying
$\varpi^{(j)}(x)\ll_{j}1$ for any integer $j\geq 0$. Without loss of
generality, we only consider the contribution from $\zeta>0$ (the proof for
$\zeta<0$ is entirely similar). By abuse of notation, we still write the
contribution from $\zeta>0$ as $S(X,\alpha,\beta)$.

Next we break the $c$-sum $\sum_{1\leq c\leq C}$ into dyadic segments $c\sim
C_{0}$ with $1\ll C_{0}\ll C$ and write

$\displaystyle S(X,\alpha,\beta)=\sum_{X^{-G}\ll\Xi\ll
X^{\varepsilon}\atop\text{dyadic}}\sum_{1\ll C_{0}\ll
C\atop\text{dyadic}}S(X,\alpha,\beta,C_{0},\Xi)+O_{A}(X^{-A})$ (3.2)

with

$\displaystyle S(X,\alpha,\beta,C_{0},\Xi)$ $\displaystyle=$
$\displaystyle\frac{1}{C}\int_{\mathbb{R}}\varpi\left(\frac{\zeta}{\Xi}\right)\sum_{c\sim
C_{0}}\frac{g(c,\zeta)}{c}\sum_{n\sim X}e\left(\alpha n^{2}+\beta
n+\frac{n\zeta}{cC}\right)\;$ (3.3)
$\displaystyle\times\sideset{}{{}^{\star}}{\sum}_{a\bmod{c}}e\left(\frac{na}{c}\right)\sum_{m=1}^{\infty}\lambda_{f}(m)e\left(-\frac{ma}{c}\right)V\left(\frac{m}{X}\right)e\left(-\frac{m\zeta}{cC}\right)\mathrm{d}\zeta.$

We now proceed to estimate $S(X,\alpha,\beta,C_{0},\Xi)$ for $1\ll C_{0}\ll
C$. Applying Lemma 2.2 with
$\varphi(x)=V\left(x/X\right)e\left(-x\zeta/(cC)\right)$ to transform the sum
over $m$ we get

$\displaystyle\sum_{m=1}^{\infty}\lambda_{f}(m)e\left(-\frac{ma}{c}\right)V\left(\frac{m}{X}\right)e\left(-\frac{m\zeta}{cC}\right)=c\sum_{\pm}\sum_{m=1}^{\infty}\frac{\lambda_{f}(m)}{m}e\left(\mp\frac{\overline{a}m}{c}\right)\Psi^{\pm}\left(\frac{m}{c^{2}},c,\zeta\right),$
(3.4)

where by (2.4),

$\displaystyle\Psi^{\pm}\left(x,c,\zeta\right)=\frac{1}{4\pi^{2}}\int_{\mathbb{R}}(\pi^{2}xX)^{-\sigma-i\tau}\rho_{f}^{\pm}(\sigma+i\tau)V^{\dagger}\left(\frac{\zeta
X}{cC},-\sigma-i\tau\right)\mathrm{d}\tau$ (3.5)

with $\rho_{f}^{\pm}(s)$ defined in (2.5) and

$\displaystyle
V^{\dagger}(r,s)=\int_{0}^{\infty}V(x)e(-rx)x^{s-1}\mathrm{d}x.$

Plugging (3.4) into (3.3), we obtain

$\displaystyle S(X,\alpha,\beta,C_{0},\Xi)$ $\displaystyle=$
$\displaystyle\frac{1}{C}\sum_{\pm}\int_{\mathbb{R}}\varpi\left(\frac{\zeta}{\Xi}\right)\sum_{c\sim
C_{0}}g(c,\zeta)\sum_{m=1}^{\infty}\frac{\lambda_{f}(m)}{m}\Psi^{\pm}\left(\frac{m}{c^{2}},c,\zeta\right)$
(3.6) $\displaystyle\times\sum_{n\sim X}e\left(\alpha n^{2}+\beta
n+\frac{n\zeta}{cC}\right)S(n,\mp m;c)\mathrm{d}\zeta,$

where $S(n,m;c)$ is the classical Kloosterman sum.

The integral $\Psi^{\pm}\left(x,c,\zeta\right)$ has the following properties.

######  Lemma 3.1.

Let $r=\zeta X/cC$ and $\zeta\asymp\Xi$.

(1) Suppose $X\Xi/(cC)\gg X^{\varepsilon}$. Then for $\mu^{1-\varepsilon}\ll
r\ll\mu^{1+\varepsilon}$,
$\Psi^{\pm}\left(x,c,\zeta\right)=\mathbf{\Psi}_{1}+\mathbf{\Psi}_{2}$, where
$\mathbf{\Psi}_{1}$ is negligibly small unless $xX\ll\mu^{1+\varepsilon}$, in
which case

$\mathbf{\Psi}_{1}\ll(rXx)^{1/2},$

and $\mathbf{\Psi}_{2}$ is negligibly small unless $xX\ll
r\mu^{1+\varepsilon}$, in which case

$\mathbf{\Psi}_{2}\ll(xX)^{1/2}.$

(2) Suppose $X\Xi/(cC)\gg X^{\varepsilon}$. Then for $r\ll\mu^{1-\varepsilon}$
or $r\gg\mu^{1+\varepsilon}$, $\Psi^{\pm}\left(x,c,\zeta\right)$ is negligibly
small unless $x\asymp\max\\{r^{2},\mu^{2}\\}/X$, in which case

$\displaystyle\Psi^{\pm}(x,c,\zeta)\ll(xX)^{1/2}.$

(3) If $X\Xi/(cC)\ll X^{\varepsilon}$, then $\Psi^{\pm}\left(x,c,\zeta\right)$
is negligibly small unless $xX\ll\mu^{2+\varepsilon}$, in which case

$\displaystyle\Psi^{\pm}\left(x,c,\zeta\right)\ll(xX)^{1/2+\varepsilon}.$

###### Proof.

For the case $r\asymp X\Xi/(cC)\gg X^{\varepsilon}$, we apply the stationary
phase to the integral $V^{\dagger}\left(r,-\sigma-i\tau\right)$. Write

$\displaystyle
V^{\dagger}\left(r,-\sigma-i\tau\right)=\int_{0}^{\infty}V\left(u\right)u^{-\sigma-1}\exp\left(i\varrho(u)\right)\mathrm{d}u,$

where $\varrho(u)=-2\pi ru-\tau\log u$. Note that

$\displaystyle\varrho^{\prime}(u)$ $\displaystyle=$ $\displaystyle-2\pi
r-\tau/u,$ $\displaystyle\varrho^{(j)}(u)$ $\displaystyle=$
$\displaystyle\tau(-1)^{j}(j-1)!u^{-j}\asymp|\tau|,\qquad j=2,3,\ldots.$

By repeated integration by parts one shows that
$V^{\dagger}\left(r,-\sigma-i\tau\right)$ is negligibly small unless
$|\tau|\asymp r$ and $\tau<0$ (note that $r>0$ here). The stationary point is
$u_{0}=-\tau/(2\pi r)$. Applying Lemma 2.3 (2) with $Y=Z=1$ and $H=R=\tau\gg
X^{\varepsilon}$, we have

$\displaystyle
V^{\dagger}\left(r,-\sigma-i\tau\right)=\tau^{-1/2}V^{\natural}_{\sigma}\left(\frac{-\tau}{2\pi
r}\right)e\left(-\frac{\tau}{2\pi}\log\frac{-\tau}{2\pi
er}\right)+O_{A}\left(X^{-A}\right),$ (3.7)

where $V_{\sigma}^{\natural}(x)$ is an inert function (depending on $A$ and
$\sigma$) supported on $x\asymp 1$. Plugging (3.7) into (3.5), we obtain

$\displaystyle\Psi^{\pm}\left(x,c,\zeta\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\int_{-\infty}^{0}(\pi^{2}xX)^{-\sigma-i\tau}\rho_{f}^{\pm}(\sigma+i\tau)$
$\displaystyle\times\tau^{-1/2}V^{\natural}_{\sigma}\left(\frac{-\tau}{2\pi
r}\right)e\left(-\frac{\tau}{2\pi}\log\frac{-\tau}{2\pi
er}\right)\mathrm{d}\tau+O_{A}\left(X^{-A}\right),$

where $r=\zeta X/(cC)>0$. Making a change of variable $\tau\rightarrow-r\tau$,

$\displaystyle\Psi^{\pm}\left(x,c,\zeta\right)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-r}}{4\pi^{2}}\int_{0}^{\infty}(\pi^{2}xX)^{-\sigma+ir\tau}\rho_{f}^{\pm}\left(\sigma-
ir\tau\right)$ (3.8)
$\displaystyle\times\tau^{-1/2}V^{\natural}_{\sigma}\left(\frac{\tau}{2\pi}\right)e\left(\frac{r\tau}{2\pi}\log\frac{\tau}{2\pi
e}\right)\mathrm{d}\tau+O_{A}\left(X^{-A}\right),$

where by (2.5),

$\displaystyle\rho_{f}^{\pm}\left(\sigma-
ir\tau\right)=\prod_{\pm}\frac{\Gamma\left(\frac{1+\sigma-i(r\tau\pm\mu)}{2}\right)}{\Gamma\left(\frac{-\sigma+i(r\tau\pm\mu)}{2}\right)}\pm\prod_{\pm}\frac{\Gamma\left(\frac{2+\sigma-i(r\tau\pm\mu)}{2}\right)}{\Gamma\left(\frac{1-\sigma+i(r\tau\pm\mu)}{2}\right)}.$
(3.9)

(1) For $\mu^{1-\varepsilon}\ll r\ll\mu^{1+\varepsilon}$, we divide the range
of $\tau$ into two pieces:

$(0,\infty)=\\{\tau||r\tau-\mu|\leq\mu^{\varepsilon}\\}\cup\\{\tau||r\tau-\mu|>\mu^{\varepsilon}\\}:=\mathbf{I}_{1}+\mathbf{I}_{2}$

and correspondingly denote by the integral over $\mathbf{I}_{j}$ by
$\mathbf{\Psi}_{j}$, $j=1,2$. Then by (2.7),

$\displaystyle\mathbf{\Psi}_{1}$ $\displaystyle\ll$ $\displaystyle
r^{1/2}(xX)^{-\sigma}\int_{\mathbf{I}_{1}}(|r\tau-\mu||r\tau+\mu|)^{\sigma+1/2}\left|V^{\natural}_{\sigma}\left(\frac{\tau}{2\pi}\right)\right|\mathrm{d}\tau$
$\displaystyle\ll$ $\displaystyle
r^{1/2}\mu^{1/2+\varepsilon}(xX/\mu^{1+\varepsilon})^{-\sigma}.$

By taking $\sigma$ sufficiently large, one sees that $\mathbf{\Psi}_{1}$ is
negligibly small unless $xX\ll\mu^{1+\varepsilon}$, in which case by taking
$\sigma=-1/2$ we have the estimate

$\displaystyle\mathbf{\Psi}_{1}\ll(rxX)^{1/2}.$ (3.10)

For $\tau\in\mathbf{I}_{2}$, by (3.9) and Stirling’s approximation in (2.6),
we have

$\displaystyle\rho_{f}^{\pm}\left(\sigma-
ir\tau\right)=\left(\prod_{\pm}\left(\frac{|r\tau\pm\mu|}{2e}\right)^{-i(r\tau\pm\mu)}|r\tau\pm\mu|^{\sigma+1/2}\right)$
$\displaystyle\times\big{(}h_{\sigma,1}(r\tau+\mu)h_{\sigma,1}(r\tau-\mu)\pm
h_{\sigma,2}(r\tau+\mu)h_{\sigma,2}(r\tau-\mu)\big{)}+O_{\sigma,A}\big{(}X^{-A}\big{)},$
(3.11)

where $h_{\sigma,j}(x)$, $j=1,2$, satisfy $h_{\sigma,j}(x)\ll_{\sigma,j,A}1$
and $x^{\ell}h_{\sigma,j}^{(\ell)}(x)\ll_{\sigma,j,\ell,A}x^{-1}$ for any
integer $\ell\geq 1$. Then by (3.8) and (3),

$\displaystyle\mathbf{\Psi}_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-r}}{4\pi^{2}}\int_{0}^{\infty}(\pi^{2}xX)^{-\sigma+ir\tau}\left(\prod_{\pm}\left(\frac{|r\tau\pm\mu|}{2e}\right)^{-i(r\tau\pm\mu)}|r\tau\pm\mu|^{\sigma+1/2}\right)$
(3.12)
$\displaystyle\times\big{(}h_{\sigma,1}(r\tau+\mu)h_{\sigma,1}(r\tau-\mu)\pm
h_{\sigma,2}(r\tau+\mu)h_{\sigma,2}(r\tau-\mu)\big{)}$
$\displaystyle\times\tau^{-1/2}V^{\natural}_{\sigma}\left(\frac{\tau}{2\pi}\right)e\left(\frac{r\tau}{2\pi}\log\frac{\tau}{2\pi
e}\right)\mathrm{d}\tau+\mathbf{\Psi}_{3},$

where

$\displaystyle\mathbf{\Psi}_{3}\ll
r^{1/2}(xX)^{-\sigma}\int_{\mathbf{I}_{1}}(|r\tau-\mu||r\tau+\mu|)^{\sigma+1/2}\left|V^{\natural}_{\sigma}\left(\frac{\tau}{2\pi}\right)\right|\mathrm{d}\tau\ll
r^{1/2}\mu^{1/2+\varepsilon}(xX/\mu^{1+\varepsilon})^{-\sigma},$

which can be negligibly small unless $xX\ll\mu^{1+\varepsilon}$, in which case
by taking $\sigma=-1/2$ we have

$\displaystyle\mathbf{\Psi}_{3}\ll(rxX)^{1/2}.$ (3.13)

Denote the first term in (3.12) by $\mathbf{\Psi}_{2}^{0}$. Then

$\displaystyle\mathbf{\Psi}_{2}^{0}\ll r^{1/2}(xX)^{-\sigma}\int_{\tau\asymp
1}(|r\tau-\mu||r\tau+\mu|)^{\sigma+1/2}\mathrm{d}\tau\ll
r\mu^{1/2+\varepsilon}\left(\frac{xX}{r\mu^{1+\varepsilon}}\right)^{-\sigma},$

which can be negligibly small unless $xX\ll r\mu^{1+\varepsilon}$, in which
case by taking $\sigma=-1/2$,

$\displaystyle\mathbf{\Psi}_{2}^{0}=(-r)^{1/2}(xX)^{1/2}\int_{0}^{\infty}G(\tau)\exp\left(i\eta(\tau)\right)\mathrm{d}\tau,$

where, temporarily,

$\displaystyle
G(\tau)=\frac{1}{4\pi\sqrt{\tau}}V^{\natural}_{\sigma}\left(\frac{\tau}{2\pi}\right)\big{(}h_{\sigma,1}(r\tau+\mu)h_{\sigma,1}(r\tau-\mu)\pm
h_{\sigma,2}(r\tau+\mu)h_{\sigma,2}(r\tau-\mu)\big{)}$

with $\sigma=-1/2$, and

$\displaystyle\eta(\tau)=r\tau\log\frac{\pi
xX}{2e}-(r\tau+\mu)\log\frac{|r\tau+\mu|}{2e}-(r\tau-\mu)\log\frac{|r\tau-\mu|}{2e}+r\tau\log\tau.$

Note that

$\displaystyle\eta^{\prime}(\tau)$ $\displaystyle=$
$\displaystyle-r\log\frac{|r\tau-\mu||r\tau+\mu|}{2\pi xX\tau},$
$\displaystyle\eta^{\prime\prime}(\tau)$ $\displaystyle=$
$\displaystyle-r\left(\frac{1}{\tau-\mu/r}+\frac{1}{\tau+\mu/r}-\frac{1}{\tau}\right)$

and

$\displaystyle\int_{|r\tau-\mu|>\sqrt{r}}\left|\frac{\mathrm{d}G(\tau)}{\mathrm{d}\tau}\right|\mathrm{d}\tau\ll\max_{\tau\asymp
1}\left\\{\frac{r}{|r\tau-\mu|^{2}},\frac{r}{|r\tau+\mu|^{2}},1\right\\}\ll
1.$

Moreover, for $|r\tau-\mu|>\sqrt{r}$ and $\mu^{1-\varepsilon}\ll
r\ll\mu^{1+\varepsilon}$,

$\displaystyle\eta^{\prime\prime}(\tau)\asymp r\max\limits_{\tau\asymp
1}|\tau-\mu/r|^{-1}.$

Then by Lemma 2.4,

$\displaystyle(-r)^{1/2}(xX)^{1/2}\int_{|r\tau-\mu|>\sqrt{r}}G(\tau)\exp\left(i\eta(\tau)\right)\mathrm{d}\tau$
$\displaystyle\ll$
$\displaystyle(xX)^{1/2}\min\limits_{|r\tau-\mu|>\sqrt{r},\tau\asymp
1}|\tau-\mu/r|^{1/2}$ $\displaystyle\ll$ $\displaystyle(xX)^{1/2}.$

Trivially, we have

$\displaystyle(-r)^{1/2}(xX)^{1/2}\int_{|r\tau-\mu|\leq\sqrt{r}}G(\tau)\exp\left(i\eta(\tau)\right)\mathrm{d}\tau\ll(xX)^{1/2}.$

Assembling the above results, we conclude that

$\displaystyle\mathbf{\Psi}_{2}^{0}\ll(xX)^{1/2}.$ (3.14)

Then the first statement follows from (3.10) and (3.12)–(3.14).

(2) For $r\ll\mu^{1-\varepsilon}$, we take $\sigma=-1/2$ in (3.8) to get

$\displaystyle\Psi^{\pm}\left(x,c,\zeta\right)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-r}}{4\pi^{2}}\int_{0}^{\infty}(\pi^{2}xX)^{1/2+ir\tau}\rho_{f}^{\pm}\left(-\frac{1}{2}-ir\tau\right)$
(3.15)
$\displaystyle\times\tau^{-1/2}V^{\natural}\left(\frac{\tau}{2\pi}\right)e\left(\frac{r\tau}{2\pi}\log\frac{\tau}{2\pi
e}\right)\mathrm{d}\tau+O_{A}\left(X^{-A}\right),$

where $V^{\natural}(x)=V^{\natural}_{-1/2}(x)$ and by (3.9),

$\displaystyle\rho_{f}^{\pm}\left(-\frac{1}{2}-ir\tau\right)=\prod_{\pm}\frac{\Gamma\left(\frac{1/2-i(r\tau\pm\mu)}{2}\right)}{\Gamma\left(\frac{1/2+i(r\tau\pm\mu)}{2}\right)}\pm\prod_{\pm}\frac{\Gamma\left(\frac{3/2-i(r\tau\pm\mu)}{2}\right)}{\Gamma\left(\frac{3/2+i(r\tau\pm\mu)}{2}\right)}.$

Since $r\ll\mu^{1-\varepsilon}$, using Stirling’s approximation in (2.6), we
derive

$\displaystyle\rho_{f}^{\pm}\left(-\frac{1}{2}-ir\tau\right)=\left(\frac{\mu-r\tau}{2e}\right)^{-i(r\tau-\mu)}\left(\frac{r\tau+\mu}{2e}\right)^{-i(r\tau+\mu)}$
$\displaystyle\qquad\times\big{(}h_{1}(r\tau-\mu)h_{1}(r\tau+\mu)\pm
h_{2}(r\tau-\mu)h_{2}(r\tau+\mu)\big{)}+O_{A}\big{(}X^{-A}\big{)},$ (3.16)

where $h_{j}(x)$, $j=1,2$, satisfy $h_{j}(x)\ll_{j}1$ and
$x^{\ell}h_{j}^{(\ell)}(x)\ll_{j,\ell,A}x^{-1}$ for any integer $\ell\geq 1$.
Plugging (3) into (3.15), one has

$\displaystyle\Psi^{\pm}\left(x,c,\zeta\right)=\frac{(-rxX)^{1/2}}{4\pi}\int_{0}^{\infty}V_{0}^{\pm}(\tau)\exp\left(i\varrho_{0}(\tau)\right)\mathrm{d}\tau+O_{A}\left(X^{-A}\right),$
(3.17)

where

$\displaystyle
V_{0}^{\pm}(\tau)=\frac{1}{\sqrt{\tau}}V^{\natural}\left(\frac{\tau}{2\pi}\right)\big{(}h_{1}(r\tau-\mu)h_{1}(r\tau+\mu)\pm
h_{2}(r\tau-\mu)h_{2}(r\tau+\mu)\big{)}$

satisfying
$\mathrm{d}^{\ell}V_{0}^{\pm}(\tau)/\mathrm{d}\tau^{\ell}\ll_{\ell}1$ for any
integer $\ell\geq 0$, and

$\displaystyle\varrho_{0}(\tau)=r\tau\log\frac{\pi
xX}{2e}-(r\tau-\mu)\log\frac{\mu-r\tau}{2e}-(r\tau+\mu)\log\frac{r\tau+\mu}{2e}+r\tau\log\tau.$

We compute

$\displaystyle\varrho^{\prime}_{0}(\tau)$ $\displaystyle=$
$\displaystyle-r\log\frac{\mu^{2}-r^{2}\tau^{2}}{2\pi xX\tau},$
$\displaystyle\varrho^{\prime\prime}_{0}(\tau)$ $\displaystyle=$
$\displaystyle-r\left(\frac{1}{\tau-\mu/r}+\frac{1}{\tau+\mu/r}-\frac{1}{\tau}\right)\asymp
r,\quad j=2,3,\ldots.$

By repeated integration by parts one shows that
$\Psi^{\pm}\left(x,c,\zeta\right)$ is negligibly small unless
$xX\asymp\mu^{2}$. By the second derivative test in Lemma 2.4, we have

$\displaystyle\Psi^{\pm}\left(x,c,\zeta\right)\ll(xX)^{1/2}.$

For $r\gg\mu^{1+\varepsilon}$, the proof is similar as that for the case
$r\ll\mu^{1-\varepsilon}$ and we will be brief. In this case, the formula
(3.17) still holds. Thus repeated integration by parts shows that
$\Psi^{\pm}\left(x,c,\zeta\right)$ is negligibly small unless $xX\asymp
r^{2}$. Note that the total variation of $V_{0}^{\pm}(\tau)$ is bounded by 1
and the second derivative of the phase function is of size $r$. By the second
derivative test in Lemma 2.4, we have

$\displaystyle\Psi^{\pm}\left(x,c,\zeta\right)\ll(xX)^{1/2}.$

This proves the second statement of the lemma.

(3) For $X\Xi/(cC)\ll X^{\varepsilon}$, we have $r\asymp X\Xi/(cC)\ll
X^{\varepsilon}$. By repeated integration by parts, one has (see [22, Lemma
5])

$\displaystyle
V^{\dagger}(r,\sigma+i\tau)\ll_{\sigma}\min\left\\{1,\left(\frac{1+|r|}{|\tau|}\right)^{j}\right\\}.$

Thus

$V^{\dagger}\left(\frac{\zeta
X}{cC},-\sigma-i\tau\right)\ll\left(\frac{X^{\varepsilon}}{|\tau|}\right)^{j},$

which implies that the contribution from $|\tau|\geq X^{\varepsilon}$ can be
arbitrarily small by taking $j$ sufficiently large. Using (3.5) and the
trivial estimate $V^{\dagger}(r,\sigma+i\tau)\ll 1$, we have

$\displaystyle\Psi^{\pm}\left(x,c,\zeta\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\int_{|\tau|\leq
X^{\varepsilon}}(\pi^{2}xX)^{-\sigma-i\tau}\rho_{f}^{\pm}(\sigma+i\tau)V^{\dagger}\left(\frac{\zeta
X}{cC},-\sigma-i\tau\right)\mathrm{d}\tau+O_{A}(X^{-A})$ $\displaystyle\ll$
$\displaystyle(xX)^{-\sigma}\int_{|\tau|\leq
X^{\varepsilon}}(|\tau+\mu||\tau-\mu|)^{\sigma+1/2}\mathrm{d}\tau$
$\displaystyle\ll$
$\displaystyle\mu^{1+\varepsilon}\left(xX/\mu^{2}\right)^{-\sigma},$

which implies that the contribution from $xX\gg\mu^{2+\varepsilon}$ is
negligible. For $xX\ll\mu^{2+\varepsilon}$, we shift the line of integration
in (3.5) to $\sigma=-1/2$ to get

$\displaystyle\Psi^{\pm}\left(x,c,\zeta\right)\ll(xX)^{1/2+\varepsilon}.$

This finishes the proof of the lemma. ∎

Now we return to the evaluation of $S(X,\alpha,\beta,C_{0},\Xi)$ in (3.6).
Applying Lemma 2.5, we have

$\displaystyle
S(X,\alpha,\beta,C_{0},\Xi)\ll\sup_{\zeta\asymp\Xi}\frac{X^{\varepsilon}}{C}\sum_{\pm}\sum_{c\sim
C_{0}}\sum_{m}\frac{|\lambda_{f}(m)|}{m}\left|\Psi^{\pm}\left(\frac{m}{c^{2}},c,\zeta\right)\right|\left((Xc)^{1/2+\varepsilon}+\textbf{T}^{1/2}\right),$

where T is given by (2.10). By Lemma 3.1, we obtain

$\displaystyle S(X,\alpha,\beta,C_{0},\Xi)$ $\displaystyle\ll$
$\displaystyle\textbf{1}_{\frac{X\Xi}{C\mu^{1+\varepsilon}}\ll
C_{0}\ll\frac{X\Xi}{C\mu^{1-\varepsilon}}}\sum_{i=1}^{2}S_{i}+\textbf{1}_{\frac{X\Xi}{C\mu^{1-\varepsilon}}\ll
C_{0}\ll\frac{X^{1-\varepsilon}\Xi}{C}}S_{3}$ (3.18)
$\displaystyle+\textbf{1}_{1\leq
C_{0}\ll\frac{X\Xi}{C\mu^{1+\varepsilon}}}S_{4}+\textbf{1}_{\frac{X^{1-\varepsilon}\Xi}{C}\ll
C_{0}\ll C}S_{5},$

where $\textbf{1}_{\mathbf{A}}=1$ is $\mathbf{A}$ is true and equals 0
otherwise,

$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{X^{\varepsilon}}{C}\sum_{c\sim C_{0}}\sum_{m\ll
c^{2}\mu^{1+\varepsilon}/X}\frac{|\lambda_{f}(m)|}{m}\left(\frac{mX^{2}\Xi}{c^{3}C}\right)^{1/2}\left((Xc)^{1/2+\varepsilon}+\textbf{T}^{1/2}\right),$
$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\frac{X^{\varepsilon}}{C}\sum_{c\sim C_{0}}\sum_{m\ll
c\mu^{1+\varepsilon}\Xi/C}\frac{|\lambda_{f}(m)|}{m}\left(\frac{mX}{c^{2}}\right)^{1/2}\left((Xc)^{1/2+\varepsilon}+\textbf{T}^{1/2}\right),$
$\displaystyle S_{3}$ $\displaystyle=$
$\displaystyle\frac{X^{\varepsilon}}{C}\sum_{c\sim C_{0}}\sum_{m\asymp
c^{2}\mu^{2}/X}\frac{|\lambda_{f}(m)|}{m}\left(\frac{mX}{c^{2}}\right)^{1/2}\left((Xc)^{1/2+\varepsilon}+\textbf{T}^{1/2}\right),$
$\displaystyle S_{4}$ $\displaystyle=$
$\displaystyle\frac{X^{\varepsilon}}{C}\sum_{c\sim C_{0}}\sum_{m\asymp
X\Xi^{2}/C^{2}}\frac{|\lambda_{f}(m)|}{m}\left(\frac{mX}{c^{2}}\right)^{1/2}\left((Xc)^{1/2+\varepsilon}+\textbf{T}^{1/2}\right)$

and

$\displaystyle S_{5}$ $\displaystyle=$
$\displaystyle\frac{X^{\varepsilon}}{C}\sum_{c\sim C_{0}}\sum_{m\ll
c^{2}\mu^{2+\varepsilon}/X}\frac{|\lambda_{f}(m)|}{m}\left(\frac{mX}{c^{2}}\right)^{1/2+\varepsilon}\left((Xc)^{1/2+\varepsilon}+\textbf{T}^{1/2}\right).$

Obviously, $S_{3}$ can be dominated by $S_{5}$. We use the strategy of Liu and
Ren [19] to deal with $S_{1}$, and $S_{2}$–$S_{5}$ can be estimated similarly.
Firstly, by (2.10) and Lemma 2.6 we have,

$\begin{split}\textbf{T}&\ll(m,c)^{1/2}c_{1}^{1/2+\varepsilon}c_{2}\sum_{c=c_{3}c_{4}}c_{4}^{1/2}\sum_{1\leq
h<X}\sum_{u\bmod c_{3}}G\left(X,2\alpha h+\frac{u}{c_{3}}\right)\\\
&=(m,c)^{1/2}cc_{1}^{-1/2+\varepsilon}\sum_{c=c_{3}c_{4}}c_{4}^{1/2}\sum_{1\leq
h<X}\sum_{u\bmod c_{3}}\left\\{b(0)+\sum_{|n|\geq 1}b(n)e\left(2n\alpha
h+\frac{nu}{c_{3}}\right)\right\\}\\\
&\ll|T_{0}|+|T_{1}|+|T_{2}|+X^{-A},\end{split}$

say, where $c_{1},c_{2}$ are defined as in Lemma 2.5. Trivially, we have

$\displaystyle T_{0}$ $\displaystyle=$
$\displaystyle(m,c)^{1/2}cc_{1}^{-1/2+\varepsilon}\sum_{c=c_{3}c_{4}}c_{4}^{1/2}\sum_{1\leq
h<X}\sum_{u\bmod c_{3}}b(0)$ (3.19) $\displaystyle\ll$ $\displaystyle
X(m,c)^{1/2}c^{2+\varepsilon}c_{1}^{-1/2}.$

Moreover,

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle(m,c)^{1/2}cc_{1}^{-1/2+\varepsilon}\sum_{c=c_{3}c_{4}}c_{4}^{1/2}\sum_{1\leq
h<X}\sum_{u\bmod c_{3}}\sum_{1\leq n\leq X^{1+\varepsilon}}b(n)e\left(2n\alpha
h+\frac{nu}{c_{3}}\right)$ (3.20) $\displaystyle\ll$ $\displaystyle(\log
X)(m,c)^{1/2}c^{2}c_{1}^{-1/2+\varepsilon}\sum_{c=c_{3}c_{4}}c_{4}^{-1/2}\sum_{1\leq
k\leq
X^{1+\varepsilon}/c_{3}}\min\left\\{X,\left\|2c_{3}k\alpha\right\|^{-1}\right\\},$

which is based on the orthogonality of additive characters and the elementary
estimate $\sum_{h\leq X}e(\xi h)\ll\min(X,\|\xi\|^{-1})$. Finally,

$\begin{split}T_{2}&=(m,c)^{1/2}cc_{1}^{-1/2+\varepsilon}\sum_{c=c_{3}c_{4}}c_{4}^{1/2}\sum_{1\leq
h<X}\sum_{u\bmod c_{3}}\sum_{1\leq n\leq
X^{1+\varepsilon}}b(-n)e\left(-2n\alpha h-\frac{nu}{c_{3}}\right),\end{split}$

which can be estimated similarly as $T_{1}$. Hence by Cauchy–Schwartz
inequality, the Rankin–Selberg estimate in (1.1) and the above estimates, we
have

$\begin{split}S_{1}&\ll\frac{X^{1+\varepsilon}\Xi^{1/2}}{C^{3/2}}\sum_{c\sim
C_{0}}\frac{1}{c^{3/2}}\sum_{m\ll
c^{2}\mu^{1+\varepsilon}/X}\frac{|\lambda_{f}(m)|}{m^{1/2}}(Xc)^{1/2+\varepsilon}+\frac{X^{1+\varepsilon}\Xi^{1/2}}{C^{3/2}}\sum_{c\sim
C_{0}}\frac{1}{c^{3/2}}\sum_{m\ll
c^{2}\mu^{1+\varepsilon}/X}\frac{|\lambda_{f}(m)|}{m^{1/2}}\textbf{T}^{1/2}\\\
&\ll\frac{X^{3/2+\varepsilon}\Xi^{1/2}}{C^{3/2}}\sum_{c\sim
C_{0}}\frac{1}{c^{1-\varepsilon}}\left(\frac{c^{2}\mu^{1+\varepsilon}}{X}\right)^{1/2+\varepsilon}+\frac{X^{1+\varepsilon}\Xi^{1/2}}{C^{3/2}}\sum_{i=0}^{2}\sum_{c\sim
C_{0}}\frac{1}{c^{3/2}}\sum_{m\ll
c^{2}\mu^{1+\varepsilon}/X}\frac{|\lambda_{f}(m)|}{m^{1/2}}T_{i}^{1/2}\\\
&\ll\frac{X^{1+\varepsilon}\mu^{1/2+\varepsilon}}{C^{3/2}}C_{0}^{1+\varepsilon}+\sum_{i=0}^{2}S_{1i},\end{split}$

recalling $\Xi\ll X^{\varepsilon}$, where

$\begin{split}S_{1i}&=\frac{X^{1+\varepsilon}}{C^{3/2}}\sum_{c\sim
C_{0}}\frac{1}{c^{3/2}}\sum_{m\ll
c^{2}\mu^{1+\varepsilon}/X}\frac{|\lambda_{f}(m)|}{m^{1/2}}T_{i}^{1/2}.\end{split}$

By (3.19), Cauchy–Schwartz inequality, the Rankin–Selberg estimate in (1.1)
and Lemma 2.7, we have

$\displaystyle S_{10}$ $\displaystyle\ll$
$\displaystyle\frac{X^{3/2+\varepsilon}}{C^{3/2}}\sum_{c\sim
C_{0}}c^{-1/2+\varepsilon}c_{1}^{-1/4}\sum_{m\ll
c^{2}\mu^{1+\varepsilon}/X}\frac{|\lambda_{f}(m)|}{m^{1/2}}(m,c)^{1/4}$ (3.21)
$\displaystyle\ll$
$\displaystyle\frac{X^{3/2+\varepsilon}}{C^{3/2}}\sum_{c\sim
C_{0}}c^{-1/2+\varepsilon}c_{1}^{-1/4}\left(\sum_{m\ll
c^{2}\mu^{1+\varepsilon}/X}|\lambda_{f}(m)|^{2}\right)^{1/2}\left(\sum_{m\ll
c^{2}\mu^{1+\varepsilon}/X}\frac{(m,c)^{1/2}}{m}\right)^{1/2}$
$\displaystyle\ll$
$\displaystyle\frac{X^{1+\varepsilon}\mu^{1/2+\varepsilon}}{C^{3/2}}C_{0}^{5/4+\varepsilon}.$

Similarly, by (3.20), Cauchy–Schwartz inequality, the Rankin–Selberg estimate
in (1.1) and Lemmas 2.7-2.8, we have

$\displaystyle S_{11}$ $\displaystyle\ll$
$\displaystyle\frac{X^{1+\varepsilon}}{C^{3/2}}\sum_{c\sim
C_{0}}c^{-\frac{1}{2}}c_{1}^{-\frac{1}{4}+\varepsilon}\sum_{m\ll
c^{2}\mu^{1+\varepsilon}/X}\frac{|\lambda_{f}(m)|}{m^{1/2}}(m,c)^{\frac{1}{4}}\left(\sum_{c=c_{3}c_{4}}c_{4}^{-\frac{1}{2}}\sum_{1\leq
k\leq\frac{X^{1+\varepsilon}}{c_{3}}}\min\left\\{X,\left\|2c_{3}k\alpha\right\|^{-1}\right\\}\right)^{1/2}$
(3.22) $\displaystyle\ll$
$\displaystyle\frac{X^{1/2+\varepsilon}\mu^{1/2+\varepsilon}}{C^{3/2}}\left(\sum_{c\sim
C_{0}}c^{1+\varepsilon}c_{1}^{-1/2+\varepsilon}\right)^{1/2}\left(\sum_{c_{4}\ll
C_{0}}c_{4}^{-1/2}\sum_{c_{3}\sim C_{0}/c_{4}}\sum_{1\leq
k\leq\frac{X^{1+\varepsilon}}{c_{3}}}\min\left\\{X,\left\|2c_{3}k\alpha\right\|^{-1}\right\\}\right)^{1/2}$
$\displaystyle\ll$
$\displaystyle\frac{X^{1/2+\varepsilon}\mu^{1/2+\varepsilon}}{C^{3/2}}C_{0}^{3/4+\varepsilon}\left(\sum_{c_{4}\ll
C_{0}}c_{4}^{-1/2}\sum_{k^{\prime}\ll
X^{1+\varepsilon}}\tau(k^{\prime})\min\left\\{X,\left\|2\alpha
k^{\prime}\right\|^{-1}\right\\}\right)^{1/2}$ $\displaystyle\ll$
$\displaystyle\frac{X^{1/2+\varepsilon}\mu^{1/2+\varepsilon}}{C^{3/2}}C_{0}^{3/4+\varepsilon}\left(C_{0}^{1/2+\varepsilon}\left(\frac{X^{1+\varepsilon}}{q}+1\right)(X+q\log
q)\right)^{1/2}$ $\displaystyle\ll$
$\displaystyle\frac{X^{1+\varepsilon}\mu^{1/2+\varepsilon}}{C^{3/2}}C_{0}^{1+\varepsilon}\left(\frac{X}{q}+\frac{q}{X}\right)^{1/2},$

and the estimate for $S_{12}$ is entirely similar as that for $S_{11}$.
Therefore, by (3.21) and (3.22), we have

$\displaystyle S_{1}$ $\displaystyle\ll$
$\displaystyle\frac{X^{1+\varepsilon}\mu^{1/2+\varepsilon}}{C^{3/2}}C_{0}^{5/4+\varepsilon}+\frac{X^{1+\varepsilon}\mu^{1/2+\varepsilon}}{C^{3/2}}C_{0}^{1+\varepsilon}\left(\frac{X}{q}+\frac{q}{X}\right)^{1/2}.$
(3.23)

Similarly, we get

$\displaystyle S_{2}$ $\displaystyle\ll$
$\displaystyle\frac{X^{1+\varepsilon}\mu^{1/2+\varepsilon}}{C^{3/2}}C_{0}^{5/4+\varepsilon}+\frac{X^{1+\varepsilon}\mu^{1/2+\varepsilon}}{C^{3/2}}C_{0}^{1+\varepsilon}\left(\frac{X}{q}+\frac{q}{X}\right)^{1/2},$
(3.24) $\displaystyle S_{4}$ $\displaystyle\ll$
$\displaystyle\frac{X^{3/2+\varepsilon}}{C^{2}}C_{0}^{3/4+\varepsilon}+\frac{X^{3/2+\varepsilon}}{C^{2}}C_{0}^{1/2+\varepsilon}\left(\frac{X}{q}+\frac{q}{X}\right)^{1/2}$
(3.25)

and

$\displaystyle
S_{3},S_{5}\ll\frac{X^{1/2+\varepsilon}\mu^{1+\varepsilon}}{C}C_{0}^{7/4+\varepsilon}+\frac{X^{1/2+\varepsilon}\mu^{1+\varepsilon}}{C}C_{0}^{3/2+\varepsilon}\left(\frac{X}{q}+\frac{q}{X}\right)^{1/2}.$
(3.26)

Plugging (3.23)–(3.26) into (3.18) and then inserting the resulting upper
bounds into (3.2), we have

$\begin{split}S(X,\alpha,\beta)\ll&\frac{X^{9/4+\varepsilon}\mu^{1+\varepsilon}}{C^{11/4}}+\frac{X^{2+\varepsilon}\mu^{1+\varepsilon}}{C^{5/2}}\left(\frac{X}{q}+\frac{q}{X}\right)^{1/2}\\\
&+X^{1/2+\varepsilon}\mu^{1+\varepsilon}C^{3/4+\varepsilon}+X^{1/2+\varepsilon}\mu^{1+\varepsilon}C^{1/2+\varepsilon}\left(\frac{X}{q}+\frac{q}{X}\right)^{1/2}.\end{split}$

We take $C=X^{1/2}$ to balance the contribution and obtain

$\begin{split}&S(X,\alpha,\beta)\ll
X^{7/8+\varepsilon}\mu^{1+\varepsilon}+X^{3/4+\varepsilon}\mu^{1+\varepsilon}\left(\frac{X}{q}+\frac{q}{X}\right)^{1/2}.\\\
\end{split}$ (3.27)

We will apply (3.27) for larger $q$. For smaller $q$, we follow closely Pitt
[24] and combine the results of Godber [6] to give an estimate for the aimed
exponential sums in Theorem 1.1, which does not depend on $f$.

######  Lemma 3.2.

Let $\lambda_{f}(n)$ be the normalized Fourier coefficients of a Maass cusp
form for $\rm SL_{2}(\mathbb{Z})$ with Laplacian eigenvalue
$\lambda_{f}(\Delta)=1/4+\mu^{2}$. Let $\alpha$ satisfy (1.4). Then for any
$\varepsilon>0$, we have

$\displaystyle\sum_{n\leq X}\lambda_{f}(n)e\left(\alpha n^{2}+\beta
n\right)\ll
X^{1/2+\varepsilon}q^{1/2}\lambda_{f}(\Delta)^{1/4+\varepsilon}+X^{3/2+\varepsilon}Q^{-1/2}\lambda_{f}(\Delta)^{1/4+\varepsilon},$

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

###### Proof.

By Godber [6, Theorem 1.2], for any $\gamma\in\mathbb{R}$,

$\displaystyle\sum_{n\leq X}\lambda_{f}(n)e(\gamma n)\ll
X^{1/2+\varepsilon}\lambda_{f}(\Delta)^{1/4+\varepsilon},$ (3.28)

where the impled constant depends only on $\varepsilon$. Then Lemma 3.2 holds
by following closely [24, Section 5] and replacing the estimate $\sum_{n\leq
X}\lambda_{f}(n)e(\gamma n)\ll X^{1/2}\log X$ by (3.28). ∎

Completion of the proof of Theorem 1.1. Take $Q=X^{5/4}$. If the Dirichlet
approximant to $\alpha$ has $q\leq X^{3/4}\lambda_{f}(\Delta)^{1/2}$, we apply
Lemma 3.2, whereas if $X^{3/4}\lambda_{f}(\Delta)^{1/2}<q\leq Q$ then we apply
$\eqref{S(X,alpha,beta)}$. In either case we obtain the estimate
$O(X^{7/8+\varepsilon}\lambda_{f}(\Delta)^{1/2+\varepsilon})$ as in the
statement of Theorem 1.1.

##  4\. Proof of Theorem 1.2

By dyadic subdivision, we only need to estimate the sum

$\displaystyle\sum_{n\sim X}\lambda_{f}(n).$

Let $h$ be a nonnegative smooth function supported in $[1,2]$ such that
$h(x)=1$ for $x\in[1+\eta,\,2-\eta]$ ($\eta>0$) and
$h^{(j)}(x)\ll_{j}\eta^{-j}$ for any integer $j\geq 0$. Assume the
Ramanujan–Petersson conjecture $\lambda_{f}(n)\ll n^{\varepsilon}$. Then we
have

$\displaystyle\sum_{n\sim X}\lambda_{f}(n)=\sum_{n\geq
1}\lambda_{f}(n)h\left(\frac{n}{X}\right)+O\big{(}\eta
X^{1+\varepsilon}\big{)}.$

By Mellin inversion, one has

$h(x)=\frac{1}{2\pi i}\int_{(2)}\widetilde{h}(s)x^{-s}\mathrm{d}s,$

where $\widetilde{h}(s)=\int_{0}^{\infty}h(x)x^{s-1}\mathrm{d}x$. It follows
that

$\displaystyle\sum_{n\sim X}\lambda_{f}(n)=\frac{1}{2\pi
i}\int_{(2)}\widetilde{h}(s)L(s,f)X^{s}\mathrm{d}s+O\big{(}\eta
X^{1+\varepsilon}\big{)}.$ (4.1)

By shifting the line of integration in (4.1) to Re$(s)=-\varepsilon$, we get

$\displaystyle\sum_{n\sim
X}\lambda_{f}(n)=\frac{X^{-\varepsilon}}{2\pi}\int_{\mathbb{R}}X^{it}\widetilde{h}(-\varepsilon+it)L(-\varepsilon+it,f)\mathrm{d}t+O\big{(}\eta
X^{1+\varepsilon}\big{)}.$ (4.2)

Repeated integration by parts shows that for any integer $j\geq 1$,

$\displaystyle\widetilde{h}(s)=\frac{(-1)^{j}}{s(s+1)\cdots(s+j-1)}\int_{0}^{\infty}h^{(j)}(x)x^{s+j-1}\mathrm{d}x\ll\eta(\eta|s|)^{-j}.$
(4.3)

Thus for $T\geq 1$ and $j>2$,

$\displaystyle\int_{|t|\geq
T}X^{it}\widetilde{h}(-\varepsilon+it)L(-\varepsilon+it,f)\mathrm{d}t$
$\displaystyle\ll$ $\displaystyle\int_{|t|\geq
T}\eta(\eta|t|)^{-j}\lambda_{f}(\Delta)^{1/2+\varepsilon}|t|^{1+\varepsilon}\mathrm{d}t$
$\displaystyle\ll$ $\displaystyle\lambda_{f}(\Delta)^{1/2+\varepsilon}\eta
T^{2}(\eta T)^{-j},$

where we have used the convexity bound (see [13, (5.20)])

$\displaystyle
L(\sigma+it,f)\ll_{\sigma,\varepsilon}(\left(1+|t+\mu|\right)\left(1+|t-\mu|\right))^{(1-\sigma)/2+\varepsilon}.$

Consequently, by taking $j$ sufficiently large, the contribution from
$T\geq(\lambda_{f}(\Delta)X)^{\varepsilon}\eta^{-1}$ is negligible. Moreover,
by taking $j=1$ in (4.3), for $T_{1}=X^{1/3}\lambda_{f}(\Delta)^{-\theta}$,
where $\theta$ is a parameter to be optimized later,

$\displaystyle\int_{|t|\leq
T_{1}}X^{it}\widetilde{h}(-\varepsilon+it)L(-\varepsilon+it,f)\mathrm{d}t$
$\displaystyle\ll$ $\displaystyle\int_{|t|\leq
T_{1}}(1+|t|)^{-1}\lambda_{f}(\Delta)^{1/2+\varepsilon}(1+|t|)^{1+\varepsilon}\mathrm{d}t$
$\displaystyle\ll$
$\displaystyle\lambda_{f}(\Delta)^{1/2-\theta+\varepsilon}X^{1/3}.$

Therefore, by inserting a dyadic smooth partition of unit to the $t$-integral
in (4.2), one has

$\displaystyle\sum_{n\sim
X}\lambda_{f}(n)=\frac{X^{-\varepsilon}}{2\pi}\sum_{T_{1}\ll
T\ll(\lambda_{f}(\Delta)X)^{\varepsilon}\eta^{-1}\atop
T\,\text{dyadic}}\mathcal{I}(T)+O\left(\eta
X^{1+\varepsilon}+\lambda_{f}(\Delta)^{1/2-\theta+\varepsilon}X^{1/3}\right),$
(4.4)

where

$\displaystyle\mathcal{I}(T):=\int_{\mathbb{R}}X^{it}\widetilde{h}(-\varepsilon+it)L(-\varepsilon+it,f)\omega\left(\frac{|t|}{T}\right)\mathrm{d}t$
(4.5)

with $\omega(x)\in C_{c}^{\infty}(1,2)$ satisfying $\omega^{(j)}(x)\ll_{j}1$
for any integer $j\geq 0$.

For $T>|\mu|^{1+\varepsilon}$, we apply the functional equation (2.2) for
$L(-\varepsilon+it,f)$, introduce the series (2.1) and then integrate termwise
to obtain

$\displaystyle\mathcal{I}(T)$ $\displaystyle\ll$
$\displaystyle\sup_{x\in[1,\,2]}\bigg{|}\int_{\mathbb{R}}\frac{(Xx)^{it}}{\varepsilon-
it}\gamma(1+\varepsilon-it)L(1+\varepsilon-
it,f)\omega\left(\frac{|t|}{T}\right)\mathrm{d}t\bigg{|}$ $\displaystyle\ll$
$\displaystyle\sup_{x\in[1,\,2]}\bigg{|}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)}{n^{1+\varepsilon}}\int_{\mathbb{R}}\frac{(Xxn)^{it}}{\varepsilon-
it}\gamma(1+\varepsilon-
it)\omega\left(\frac{|t|}{T}\right)\mathrm{d}t\bigg{|},$

where $\gamma(s)$ is defined in (2.3). By Stirling’s formula in (2.6), for
$|t|\asymp T>|\mu|^{1+\varepsilon}$,

$\displaystyle\gamma(\sigma+it)$ $\displaystyle=$
$\displaystyle\pi^{1-2\sigma-2it}\prod_{\pm}\frac{\Gamma\left(\frac{\sigma+\eta+i(t\pm\mu)}{2}\right)}{\Gamma\left(\frac{1-\sigma+\eta-i(t\pm\mu)}{2}\right)}$
(4.6) $\displaystyle=$
$\displaystyle\pi^{1-2\sigma-2it}\prod_{\pm}\bigg{(}(t\pm\mu)^{\sigma-\frac{1}{2}}g_{\sigma}(t\pm\mu)\bigg{)}\exp\bigg{(}i\sum_{\pm}(t\pm\mu)\log\frac{|t\pm\mu|}{2e}\bigg{)},$

where $g_{\sigma}(t)$ satisfies
$t^{\ell}g_{\sigma}^{(\ell)}(t)\ll_{\ell,\sigma}1.$ It follows that

$\displaystyle\gamma(\sigma+it)\ll\big{(}t^{2}+\mu^{2}\big{)}^{\sigma-\frac{1}{2}}\ll
t^{2\sigma-1}.$ (4.7)

Now we use (4.7) together with $(\varepsilon-it)^{-1}=it^{-1}+O(t^{-2})$ to
derive

$\displaystyle\mathcal{I}(T)\ll\sup_{x\in[1,\,2]}\bigg{|}\sum_{n=1}^{\infty}\frac{\lambda_{f}(n)}{n^{1+\varepsilon}}\mathfrak{J}(Xxn,T)\bigg{|}+T^{\varepsilon},$
(4.8)

where

$\displaystyle\mathfrak{J}(P,T)=\int_{\mathbb{R}}P^{it}\gamma(1+\varepsilon-
it)\omega\left(\frac{|t|}{T}\right)\frac{\mathrm{d}t}{t}.$

Making a change of variable $t\rightarrow Tt$, one has

$\displaystyle\mathfrak{J}(P,T)=\int_{\mathbb{R}}P^{iTt}\gamma(1+\varepsilon-
iTt)\omega\left(|t|\right)\frac{\mathrm{d}t}{t}.$

By (4.6) we have

$\displaystyle\mathfrak{J}(P,T)=\int_{\mathbb{R}}u(t)e(\varsigma(t))\mathrm{d}t,$

where
$u(t)=\pi^{-1-2\varepsilon}t^{-1}(t^{2}T^{2}-\mu^{2})^{1/2+\varepsilon}\omega\left(|t|\right)\prod_{\pm}g_{1+\varepsilon}(-tT\pm\mu)$
and

$\displaystyle\varsigma(t)=\frac{tT}{2\pi}\log(\pi^{2}P)-\frac{1}{2\pi}\sum_{\pm}(tT\pm\mu)\log\frac{|tT\pm\mu|}{2e}.$

We can apply the stationary phase analysis to $\mathfrak{J}(P,T)$. Without
loss of generality, we consider the case $t>0$, since the case $t<0$ can be
treated similarly. Note that

$\displaystyle\varsigma^{\prime}(t)=-\frac{T}{2\pi}\log\frac{t^{2}T^{2}-\mu^{2}}{4\pi^{2}P}$

and

$\displaystyle\varsigma^{\prime\prime}(t)=-\frac{T}{2\pi}\frac{2tT^{2}}{t^{2}T^{2}-\mu^{2}}\asymp
T.$

We also have $u^{(j_{1})}(t)\ll_{j_{1}}T^{1+\varepsilon}$ and
$\varsigma^{(j_{2})}(t)\ll_{j_{2}}T$ for any integer $j_{1}\geq 0$ and
$j_{2}\geq 2$. Then integration by parts shows that the integral is negligibly
small unless $T\asymp\sqrt{P}$. The stationary point which is the solution to
the equation $\varsigma^{\prime}(t)=0$ is
$t_{0}=T^{-1}(\mu^{2}+4\pi^{2}P)^{1/2}$. By Lemma 2.3, we have

$\displaystyle\mathfrak{J}(P,T)=\int_{\mathbb{R}}u(t)e(\varsigma(t))\mathrm{d}t=T^{1/2+2\varepsilon}F_{\natural}(t_{0})e(\varsigma(t_{0}))+O_{A}\Big{(}T^{-A}\Big{)},$
(4.9)

where $F_{\natural}(x)$ is an $1$-inert function (depending on $A$) supported
on $x\asymp 1$. Plugging (4.9) into (4.8) and estimating the resulting sum
over $n$ trivially, we obtain

$\displaystyle\mathcal{I}(T)\ll T^{1/2+\varepsilon}.$ (4.10)

If $T_{1}>|\mu|^{1+\varepsilon}$, then $T\gg T_{1}>|\mu|^{1+\varepsilon}$. If
$T_{1}\leq|\mu|^{1+\varepsilon}$, i.e.,
$X^{1/3}\lambda_{f}(\Delta)^{-\theta}\leq|\mu|^{1+\varepsilon}$, then

$\displaystyle X\leq\lambda_{f}(\Delta)^{3(1/2+\theta)}.$ (4.11)

In this case, for $T\leq|\mu|^{1+\varepsilon}$, we move the line of
integration in (4.5) to $\operatorname{Re}(s)=1/2$ and apply (4.3) with $j=1$
to get

$\displaystyle\mathcal{I}(T)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}X^{\frac{1}{2}+it+\varepsilon}\widetilde{h}\left(\frac{1}{2}+it\right)L\left(\frac{1}{2}+it,f\right)\omega\left(\frac{|\frac{1}{2}+it+\varepsilon|}{T}\right)\mathrm{d}t$
$\displaystyle=$
$\displaystyle-X^{\frac{1}{2}+\varepsilon}\int_{1}^{2}h^{\prime}(x)x^{\frac{1}{2}}\int_{\mathbb{R}}\frac{(xX)^{it}}{1/2+it}L\left(\frac{1}{2}+it,f\right)\omega\left(\frac{|\frac{1}{2}+it+\varepsilon|}{T}\right)\mathrm{d}t\mathrm{d}x.$

Applying the subconvexity bound due to Jutila and Motohashi [15],

$L\left(\frac{1}{2}+it,f\right)\ll(|t|+\mu)^{1/3+\varepsilon},$

we obtain

$\displaystyle\mathcal{I}(T)\ll X^{1/2+\varepsilon}\int_{|t|\asymp
T}\frac{1}{t}(|t|+\mu)^{1/3+\varepsilon}\mathrm{d}t\ll
X^{1/2+\varepsilon}\lambda_{f}(\Delta)^{1/6+\varepsilon},$

which by (4.11) is bounded by
$X^{1/3}\lambda_{f}(\Delta)^{\theta/2+5/12+\varepsilon}$. This estimate
combined with (4.10) when plugged into (4.4) implies

$\displaystyle\sum_{n\sim
X}\lambda_{f}(n)\ll(\lambda_{f}(\Delta)X)^{\varepsilon}\eta^{-1/2}+\eta
X^{1+\varepsilon}+\lambda_{f}(\Delta)^{1/2-\theta+\varepsilon}X^{1/3}+\lambda_{f}(\Delta)^{\theta/2+5/12+\varepsilon}X^{1/3}.$

Take $\eta=X^{-2/3}$ and $\theta=\frac{1}{18}$. We conclude that

$\displaystyle\sum_{n\sim X}\lambda_{f}(n)\ll
X^{1/3+\varepsilon}\lambda_{f}(\Delta)^{4/9+\varepsilon}.$

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