Uniform bounds for Kloosterman sums of half-integral weight, same-sign case

Qihang Sun Department of Mathematics, University of Illinois, Urbana, IL 61801 [email protected]

Abstract.

In the previous paper [Sun23], the author proved a uniform bound for sums of half-integral weight Kloosterman sums. This bound was applied to prove an exact formula for partitions of rank modulo 3. That uniform estimate provides a more precise bound for a certain class of multipliers compared to the 1983 result by Goldfeld and Sarnak and generalizes the 2009 result from Sarnak and Tsimerman to the half-integral weight case. However, the author only considered the case when the parameters satisfied $\tilde{m}\tilde{n}<0$ . In this paper, we prove the same uniform bound when $\tilde{m}\tilde{n}>0$ for further applications.

Key words and phrases:

Kloosterman sum; Maass form; Kuznetsov trace formula

1. Introduction

For positive integers $N$ , $c$ and $m,n\in{\mathbb{Z}}$ , we define the generalized Kloosterman sums with a multiplier system $\nu$ as

S(m,n,c,\nu)=\sum_{\begin{subarray}{c}0\leq a,d<c\\ \gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma\end{subarray}}\overline{\nu}(\gamma)\left(\frac{\tilde{m}a+\tilde{n}d}{c}\right)

where $\Gamma=\Gamma_{0}(N)$ is a congruence subgroup of ${\rm SL}_{2}({\mathbb{Z}})$ , $\nu$ is a weight $k\in{\mathbb{R}}$ multiplier system on $\Gamma$ , and $\tilde{n}=n-\alpha_{\nu,\infty}$ as defined in (2.6). These Kloosterman sums have been studied by Goldfeld and Sarnak [GS83] and Pribitkin [Pri00]. In a previous paper [Sun23], the author proved a uniform bound for the sums of such Kloosterman sums and applied the bound to estimate the partial sums of Rademacher-type exact formulas. For example, the Fourier coefficient $G(n)$ of $\gamma(q)$ , which is a sixth order mock theta function defined in [AH91, (0.17)], can be written as [Sun23, Theorem 2.2]

G(n)=A\left(\frac{1}{3};n\right)=\frac{2\pi\,e(-\frac{1}{8})}{(24n-1)^{\frac{1}{4}}}\sum_{3|c>0}\frac{S(0,n,c,\,(\frac{\cdot}{3})\overline{\nu_{\eta}})}{c}I_{\frac{1}{2}}\left(\frac{\pi\sqrt{24n-1}}{6c}\right),

(1.1)

where $\nu_{\eta}$ is the multiplier system of weight $\frac{1}{2}$ for Dedekind’s eta-function (see (2.4)) and $I_{\kappa}$ is the $I$ -Bessel function. The author bounded the error

R_{3}(n,x)=\frac{2\pi\,e(-\frac{1}{8})}{(24n-1)^{\frac{1}{4}}}\sum_{3|c>x}\frac{S(0,n,c,\,(\frac{\cdot}{3})\overline{\nu_{\eta}})}{c}I_{\frac{1}{2}}\left(\frac{\pi\sqrt{24n-1}}{6c}\right)

(1.2)

with [Sun23, Theorem 1.6] to get

R_{3}(n,\alpha\sqrt{n})\ll_{\alpha,\varepsilon}n^{-\frac{1}{147}+\varepsilon}.

Our estimate can be applied to sums of Kloosterman sum with a class of multiplier systems. However, in the former paper [Sun23] we only focus on the case $\tilde{m}\tilde{n}<0$ . In this paper we add the complementary case $\tilde{m}\tilde{n}>0$ for further applications like Corollary 1.6.

To state the results, we need to classify our half-integral weight multipliers first:

Definition 1.1.

Let $k=\pm\frac{1}{2}$ and $\nu^{\prime}=(\frac{|D|}{\cdot})\nu_{\theta}^{2k}$ where $D$ is some even fundamental discriminant and $\nu_{\theta}$ is the multiplier for the theta function. We say a weight $k$ multiplier $\nu$ on $\Gamma=\Gamma_{0}(N)$ is admissible if it satisfies the following two conditions:

(1)
Level lifting: there exist positive integers $B$ and $M$ such that the map $\mathscr{L}:(\mathscr{L}f)(z)=f(Bz)$ gives:
- (i)
  
  an injection from weight $k$ automorphic eigenforms of the hyperbolic Laplacian $\Delta_{k}$ on $(\Gamma_{0}(N),\nu)$ to those on $(\Gamma_{0}(M),\nu^{\prime})$ and keeps the eigenvalue;
- (ii)
  
  an injection from weight $k$ holomorphic cusp forms on $(\Gamma_{0}(N),\nu)$ to weight $k$ holomorphic cusp forms on $(\Gamma_{0}(M),\nu^{\prime})$ .
Here $M$ is a multiple of $4$ and $M$ depends on $B$ .
(2)

Average Weil bound: for $x>y>0$ and $x-y\gg x^{\frac{2}{3}}$ , we have

$\sum_{N|c\,\in[x,y]}\frac{|S(m,n,c,\nu)|}{c}\ll_{N,\nu,\varepsilon}(\sqrt{x}-\sqrt{y})(\tilde{m}\tilde{n}x)^{\varepsilon}.$

The author proved [Sun23, Proposition 5.1] that the following class of multipliers satisfy both the conditions:

Lemma 1.2.

Let $\nu=(\frac{|D|}{\cdot})\nu_{\theta}$ or $\nu=(\frac{|D|}{\cdot})\nu_{\eta}$ where $D$ is a fundamental discriminant and $\nu_{\theta}$ and $\nu_{\eta}$ are the multiplier system for the standard theta function and Dedekind’s eta function, respectively (see (2.1) and (2.4)). Then both $\nu$ and its conjugate are admissible.

Let $\rho_{j}(n)$ denote the $n$ -th Fourier coefficient of an orthonormal basis $\{v_{j}(\cdot)\}_{j}$ of ${\tilde{\mathcal{L}}}_{k}(N,\nu)$ (the space of square-integrable eigenforms of the weight $k$ Laplacian with respect to $(\Gamma_{0}(N),\nu)$ , see Section 2 for details). Here are our theorems:

Theorem 1.3.

Suppose $\tilde{m}>0$ , $\tilde{n}>0$ and $\nu$ is a weight $k=\pm\frac{1}{2}$ admissible multiplier on $\Gamma_{0}(N)$ . We have

\sum_{N|c\leq X}\frac{S(m,n,c,\nu)}{c}=\!\!\!\sum_{r_{j}\in i(0,\frac{1}{4}]}\!\!\!\tau_{j}(m,n)\frac{X^{2s_{j}-1}}{2s_{j}-1}+O_{\nu,\varepsilon}\left(\left(A_{u}(m,n)+X^{\frac{1}{6}}\right)(\tilde{m}\tilde{n}X)^{\varepsilon}\right),

(1.3)

where for $B$ and $M$ in Definition 1.1, we factor $B\tilde{\ell}=t_{\ell}u_{\ell}^{2}w_{\ell}^{2}$ with $t_{\ell}$ square-free, $u_{\ell}|M^{\infty}$ positive and $(w_{\ell},M)=1$ for $\ell\in\{m,n\}$ . Here $s_{j}=\operatorname{Im}r_{j}+\frac{1}{2}$ ,

\tau_{j}(m,n)=2i^{k}\overline{\rho_{j}(m)}\rho_{j}(n)\pi^{1-2s_{j}}(4\tilde{m}\tilde{n})^{1-s_{j}}\frac{\Gamma(s_{j}+\frac{k}{2})\Gamma(2s_{j}-1)}{\Gamma(s_{j}-\frac{k}{2})}

are the coefficients in [GS83] (as corrected by [AA16, Proposition 7]), and

	$\displaystyle A_{u}(m,n)$	$\displaystyle\vcentcolon=\left(\tilde{m}^{\frac{131}{294}}+u_{m}\right)^{\frac{1}{8}}\left(\tilde{n}^{\frac{131}{294}}+u_{n}\right)^{\frac{1}{8}}(\tilde{m}\tilde{n})^{\frac{3}{16}}$
		$\displaystyle\ll(\tilde{m}\tilde{n})^{\frac{143}{588}}+\tilde{m}^{\frac{143}{588}}\tilde{n}^{\frac{3}{16}}\,u_{n}^{\frac{1}{8}}+\tilde{m}^{\frac{3}{16}}\tilde{n}^{\frac{143}{588}}\,u_{m}^{\frac{1}{8}}+(\tilde{m}\tilde{n})^{\frac{3}{16}}(u_{m}u_{n})^{\frac{1}{8}}.$

Remark.

We have the following notes for this theorem:

•

The notation $u|M^{\infty}$ means $u|M^{C}$ for some positive integer $C$ .
•

When $u_{m}$ and $u_{n}$ are both $O_{N,\nu}(1)$ , $A_{u}(m,n)\ll_{N,\nu}(\tilde{m}\tilde{n})^{\frac{143}{588}}$ .
•

In general, $A_{u}(m,n)\ll_{N,\nu}(\tilde{m}\tilde{n})^{\frac{1}{4}}$ .
•

When $k=-\frac{1}{2}$ and $r_{j}=\frac{i}{4}$ , we have $\tau_{j}(m,n)=0$ (see (2.21) and (2.22)).
•

The theorem also applies to the case $\tilde{m}<0$ and $\tilde{n}<0$ because of (2.8) by conjugation.

Based on Theorem 1.3 and [Sun23, Theorem 1.4], we are able to reformulate the Linnik-Selberg conjecture in the half-integral weight case:

Conjecture 1.4.

Suppose $k=\pm\frac{1}{2}$ and $\nu$ is a weight $k$ multiplier system on $\Gamma_{0}(N)$ . If there is no eigenvalue for the hyperbolic Laplacian $\Delta_{k}$ in $(\frac{3}{16},\frac{1}{4})$ , then

\sum_{N|c\leq X}\frac{S(m,n,c,\nu)}{c}-2X^{\frac{1}{2}}\sum_{r_{j}=\frac{i}{4}}\tau_{j}(m,n)\ll_{N,\nu,\varepsilon}|\tilde{m}\tilde{n}x|^{\varepsilon},

(1.4)

where the second sum runs over normalized eigenforms of $\Delta_{k}$ with eigenvalue $\frac{1}{4}+r_{j}^{2}=\frac{3}{16}$ .

We also get the following bound by the properties of Bessel functions.

Theorem 1.5.

With the same setting as Theorem 1.3, for $\beta=\frac{1}{2}$ or $\frac{3}{2}$ , we have

\sum_{N|c>\alpha\sqrt{\tilde{m}\tilde{n}}}\frac{S(m,n,c,\nu)}{c}\mathscr{B}_{\beta}\left(\frac{4\pi\sqrt{\tilde{m}\tilde{n}}}{c}\right)\ll_{\alpha,\nu,\varepsilon}A_{u}(m,n)(\tilde{m}\tilde{n})^{\varepsilon}

(1.5)

except the case when $\tilde{m}$ , $\tilde{n}$ and $k=\pm\frac{1}{2}$ are all of the same sign such that $\sum_{r_{j}=\frac{i}{4}}\tau_{j}(m,n)\neq 0$ . Here $\mathscr{B}_{\beta}$ is the Bessel function $I_{\beta}$ or $J_{\beta}$ . Note that $A_{u}(m,n)\ll_{N,\nu}(\tilde{m}\tilde{n})^{\frac{143}{588}}$ when $u_{m}$ and $u_{n}$ are $O_{N,\nu}(1)$ .

1.1. Application

In [BO12], Bringmann and Ono constructed Maass-Poincaré series to prove that the Fourier coefficients of weight $k\in{\mathbb{Z}}+\frac{1}{2}$ , $k\leq\frac{1}{2}$ harmonic Maass forms have exact formulas, i.e. they equal infinite sums of Kloosterman sums (up to Fourier coefficients from holomorphic theta functions when the weight is $\frac{1}{2}$ ). The specific case, weight $\frac{1}{2}$ , is crucial as it is related to the rank of partitions. Readers may also refer to [BO06] or [Sun23, Section 4] for further introduction.

We can denote the weight $\frac{1}{2}$ Maass-Poincaré series as $P_{\mathfrak{a}}(z,m,\Gamma_{0}(N),\frac{1}{2},s,\nu)$ for cusp $\mathfrak{a}$ and multiplier $\nu$ of $\Gamma_{0}(N)$ as [BO12, (3.4)]. The exact formulas are derived from $P_{\mathfrak{a}}$ at $s=\frac{3}{4}$ , while we only know the convergence at $\operatorname{Re}s>1$ (see [BO12, Lemma 3.1, (3.7)]). Bringmann and Ono called a weight $\frac{1}{2}$ harmonic Maass form on $(\Gamma_{0}(N),\nu)$ is good if those Maass-Poincaré series corresponding to nontrivial terms in the principal parts of $f$ are individually convergent. They conclude the exact formula for its Fourier coefficients only when $f$ is good.

With Theorem 1.5 and [BO12, Remark (1) after Theorem 3.2], we have

Corollary 1.6.

Suppose $f$ is a weight $\frac{1}{2}$ harmonic Maass form on $(\Gamma_{0}(N),\nu)$ and $f$ only has non-trivial principal part at cusp $\infty$ . Then $f$ is good if $\nu$ is admissible (Definition 1.1).

The paper is organized as follows. In Section 2 we introduce notations on Kloosterman sums and Maass forms. Section 3 revisits the trace formula by Proskurin [Pro79] and discusses certain properties crucial for the subsequent proof. Section 4 is about bounds on translations of the test function. The proof of Theorem 1.3 is presented in Section 5 and is divided into two cases: the weight $k=\frac{1}{2}$ and $k=-\frac{1}{2}$ . Readers are advised to take particular note of the Remark following Proposition 5.2 and be mindful of the weight context in Section 5.

2. Background

In this section we recall some basic notions on Maass forms with general weight and multiplier. Let $\Gamma=\Gamma_{0}(N)$ for some $N\geq 1$ denote our congruence subgroup and ${\mathbb{H}}$ denote the upper half complex plane. Fixing the argument in $(-\pi,\pi]$ , for any $\gamma\in{\rm SL}_{2}({\mathbb{R}})$ and $\tau=x+iy\in{\mathbb{H}}$ , we define the automorphic factor

j(\gamma,\tau)\vcentcolon=\frac{c\tau+d}{|c\tau+d|}=e^{i\arg(c\tau+d)}

and the weight $k$ slash operator

f|_{k}\gamma\vcentcolon=j(\gamma,\tau)^{-k}f(\gamma\tau)

for $k\in{\mathbb{R}}$ . We say that $\nu:\Gamma\to{\mathbb{C}}^{\times}$ is a multiplier system of weight $k$ if

(i)

$|\nu|=1$ ,
(ii)

$\nu(-I)=e^{-\pi ik}$ , and

(iii)

$\nu(\gamma_{1}\gamma_{2})=w_{k}(\gamma_{1},\gamma_{2})\nu(\gamma_{1})\nu(\gamma_{2})$ for all $\gamma_{1},\gamma_{2}\in\Gamma$ , where

w_{k}(\gamma_{1},\gamma_{2})\vcentcolon=j(\gamma_{2},\tau)^{k}j(\gamma_{1},\gamma_{2}\tau)^{k}j(\gamma_{1}\gamma_{2},\tau)^{-k}.

If $\nu$ is a multiplier system of weight $k$ , then it is also a multiplier system of weight $k\pm 2$ and its conjugate $\overline{\nu}$ is a multiplier system of weight $-k$ .

We are interested in the following multiplier systems of weight $\frac{1}{2}$ and their conjugates of weight $-\frac{1}{2}$ . The theta-multiplier $\nu_{\theta}$ on $\Gamma_{0}(4)$ is given by

\theta(\gamma z)=\nu_{\theta}(\gamma)\sqrt{cz+d}\;\theta(z),\quad\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma_{0}(4)

(2.1)

where

\theta(z)\vcentcolon=\sum_{n\in{\mathbb{Z}}}e(n^{2}z),\quad\nu_{\theta}(\gamma)=\left(\frac{c}{d}\right)\varepsilon_{d}^{-1},\quad\varepsilon_{d}=\left\{\begin{array}[]{ll}1&d\equiv 1\ (\mathrm{mod}\ 4),\\ i&d\equiv 3\ (\mathrm{mod}\ 4),\end{array}\right.

and $\left(\frac{\cdot}{\cdot}\right)$ is the extended Kronecker symbol such that $\left(\tfrac{-1}{d}\right)=(-1)^{\frac{d-1}{2}}=\varepsilon_{d}^{2}$ for odd $d\in{\mathbb{Z}}$ . The eta-multiplier $\nu_{\eta}$ on ${\rm SL}_{2}({\mathbb{Z}})$ is given by

\eta(\gamma z)=\nu_{\eta}(\gamma)\sqrt{cz+d}\;\eta(z),\quad\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in{\rm SL}_{2}({\mathbb{Z}})

(2.2)

where

\eta(z)\vcentcolon=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})\quad\text{and}\quad q=e(z)\vcentcolon=e^{2\pi iz}.

(2.3)

Explicit formulas for $\nu_{\eta}$ were given by Rademacher [Rad73, (74.11), (74.12)]:

\nu_{\eta}(\gamma)=e\left(-\frac{1}{8}\right)e^{-\pi is(d,c)}e\left(\frac{a+d}{24c}\right),\quad s(d,c)\vcentcolon=\sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}\right\rfloor-1\right),

(2.4)

for all $c\in{\mathbb{Z}}$ and also given by Knopp [Kno70]:

\nu_{\eta}(\gamma)=\left\{\begin{array}[]{ll}\left(\dfrac{d}{c}\right)e\left(\dfrac{1}{24}\Big{(}(a+d)c-bd(c^{2}-1)-3c\Big{)}\right)&\text{if }c\text{ is odd,}\\ \left(\dfrac{c}{d}\right)e\left(\dfrac{1}{24}\Big{(}(a+d)c-bd(c^{2}-1)+3d-3-3cd\Big{)}\right)&\text{if }c\text{ is even,}\end{array}\right.

(2.5)

for $c>0$ . The properties $\nu_{\eta}\left(\pm\begin{pmatrix}1&b\\ 0&1\end{pmatrix}\right)=e\left(\tfrac{b}{24}\right)$ and $\nu_{\eta}(-\gamma)=i\nu_{\eta}(\gamma)$ for $c>0$ are convenient in computations.

2.1. Kloosterman sums

For any cusp $\mathfrak{a}$ of $\Gamma=\Gamma_{0}(N)$ , let $\Gamma_{\mathfrak{a}}$ denote its stabilizer in $\Gamma$ . For example, $\Gamma_{\infty}=\{\pm\begin{pmatrix}1&b\\ 0&1\end{pmatrix}:b\in{\mathbb{Z}}\}$ . Let $\sigma_{\mathfrak{a}}\in{\rm SL}_{2}({\mathbb{R}})$ denote its scaling matrix satisfying $\sigma_{\mathfrak{a}}\infty=\mathfrak{a}$ and $\sigma_{\mathfrak{a}}^{-1}\Gamma_{\mathfrak{a}}\sigma_{\mathfrak{a}}=\Gamma_{\infty}$ . We define $\alpha_{\nu,\mathfrak{a}}\in[0,1)$ by the condition

\nu\left(\sigma_{\mathfrak{a}}\begin{pmatrix}1&1\\ 0&1\end{pmatrix}\sigma_{\mathfrak{a}}^{-1}\right)=e(-\alpha_{\nu,\mathfrak{a}}).

(2.6)

The cusp $\mathfrak{a}$ is called singular if $\alpha_{\nu,\mathfrak{a}}=0$ . When $\mathfrak{a}=\infty$ , we drop the subscript, denote $\alpha_{\nu}\vcentcolon=\alpha_{\nu,\infty}$ , and define $\tilde{n}\vcentcolon=n-\alpha_{\nu}$ for $n\in{\mathbb{Z}}$ . The Kloosterman sum at the cusp pair $(\infty,\infty)$ with respect to the multiplier system $\nu$ is given by

S(m,n,c,\nu):=\!\!\!\sum_{\begin{subarray}{c}0\leq a,d<c\\ \gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma\end{subarray}}\!\!\!\overline{\nu}(\gamma)e\left(\frac{\tilde{m}a+\tilde{n}d}{c}\right)=\!\!\!\sum_{\begin{subarray}{c}\gamma\in\Gamma_{\infty}\setminus\Gamma/\Gamma_{\infty}\\ \gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\end{subarray}}\!\!\!\overline{\nu}(\gamma)e\left(\frac{\tilde{m}a+\tilde{n}d}{c}\right).

(2.7)

They have the relationships

\overline{S(m,n,c,\nu)}=\left\{\begin{array}[]{ll}S(1-m,1-n,c,\overline{\nu})&\ \ \text{if\ }\alpha_{\nu}>0,\\ S(-m,-n,c,\overline{\nu})&\ \ \text{if\ }\alpha_{\nu}=0,\end{array}\right.

(2.8)

because

n_{\overline{\nu}}=\left\{\begin{array}[]{ll}-(1-n)_{\nu}&\ \ \text{if\ }\alpha_{\nu}>0,\\ n&\ \ \text{if\ }\alpha_{\nu}=0.\end{array}\right.

(2.9)

2.2. Maass forms

We call a function $f:{\mathbb{H}}\rightarrow{\mathbb{C}}$ automorphic of weight $k$ and multiplier $\nu$ on $\Gamma$ if

f|_{k}\gamma=\nu(\gamma)f\quad\text{for all }\gamma\in\Gamma.

Let $\mathcal{A}_{k}(N,\nu)$ denote the linear space consisting of all such functions and ${\mathcal{L}}_{k}(N,\nu)\subset\mathcal{A}_{k}(N,\nu)$ denote the space of square-integrable functions on $\Gamma\setminus{\mathbb{H}}$ with respect to the measure

d\mu(z)=\frac{dxdy}{y^{2}}

and the Petersson inner product

\langle f,g\rangle\vcentcolon=\int_{\Gamma\setminus{\mathbb{H}}}f(z)\overline{g(z)}\frac{dxdy}{y^{2}}

for $f,g\in{\mathcal{L}}_{k}(N,\nu)$ . For $k\in{\mathbb{R}}$ , the Laplacian

\Delta_{k}\vcentcolon=y^{2}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)-iky\frac{\partial}{\partial x}

(2.10)

can be expressed as

	$\displaystyle\Delta_{k}$	$\displaystyle=-R_{k-2}L_{k}-\frac{k}{2}\left(1-\frac{k}{2}\right)$		(2.11)
		$\displaystyle=-L_{k+2}R_{k}+\frac{k}{2}\left(1+\frac{k}{2}\right)$		(2.12)

where $R_{k}$ is the Maass raising operator

R_{k}\vcentcolon=\frac{k}{2}+2iy\frac{\partial}{\partial z}=\frac{k}{2}+iy\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)

(2.13)

and $L_{k}$ is the Maass lowering operator

L_{k}\vcentcolon=\frac{k}{2}+2iy\frac{\partial}{\partial\bar{z}}=\frac{k}{2}+iy\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right).

(2.14)

These operators raise and lower the weight of an automorphic form as

(R_{k}f)|_{k+2}\;\gamma=R_{k}(f|_{k}\gamma),\quad(L_{k}f)|_{k-2}\;\gamma=L_{k}(f|_{k}\gamma),\quad\text{for\ }f\in\mathcal{A}_{k}(N,\nu)

and satisfy the commutative relations

R_{k}\Delta_{k}=\Delta_{k+2}R_{k},\quad L_{k}\Delta_{k}=\Delta_{k-2}L_{k}.

(2.15)

Moreover, $\Delta_{k}$ commutes with the weight $k$ slash operator for all $\gamma\in{\rm SL}_{2}({\mathbb{R}})$ .

We call a real analytic function $f:{\mathbb{H}}\rightarrow{\mathbb{C}}$ an eigenfunction of $\Delta_{k}$ with eigenvalue $\lambda\in{\mathbb{C}}$ if

\Delta_{k}f+\lambda f=0.

From (2.15), it is clear that an eigenvalue $\lambda$ for the weight $k$ Laplacian is also an eigenvalue for weight $k\pm 2$ . We call $f\in\mathcal{A}_{k}(N,\nu)$ a Maass form if it is an smooth eigenfunction of $\Delta_{k}$ and satisfies the growth condition

(f|_{k}\gamma)(x+iy)\ll y^{\sigma}+y^{1-\sigma}

for all $\gamma\in{\rm SL}_{2}({\mathbb{Z}})$ and some $\sigma$ depending on $\gamma$ when $y\rightarrow+\infty$ . Moreover, if a Maass form $f$ satisfies

\int_{0}^{1}(f|_{k}\sigma_{\mathfrak{a}})(x+iy)\;e(\alpha_{\nu,\mathfrak{a}}x)dx=0

for all cusps $\mathfrak{a}$ of $\Gamma$ , then $f\in{\mathcal{L}}_{k}(N,\nu)$ and we call $f$ a Maass cusp form. For details see [AA18, §2.3]

Let $\mathcal{B}_{k}(N,\nu)\subset{\mathcal{L}}_{k}(N,\nu)$ denote the space of smooth functions $f$ such that both $f$ and $\Delta_{k}f$ are bounded. One can show that $\mathcal{B}_{k}(N,\nu)$ is dense in ${\mathcal{L}}_{k}(N,\nu)$ and $\Delta_{k}$ is self-adjoint on $\mathcal{B}_{k}(N,\nu)$ . If we let $\lambda_{0}\vcentcolon=\lambda_{0}(k)=\tfrac{|k|}{2}(1-\tfrac{|k|}{2})$ , then for $f\in\mathcal{B}_{k}(N,\nu)$ ,

\langle f,-\Delta_{k}f\rangle\geq\lambda_{0}\langle f,f\rangle,

i.e. $-\Delta_{k}$ is bounded from below. By the Friedrichs extension theorem, $-\Delta_{k}$ can be extended to a self-adjoint operator on ${\mathcal{L}}_{k}(N,\nu)$ . The spectrum of $\Delta_{k}$ consists of two parts: the continuous spectrum $\lambda\in[\frac{1}{4},\infty)$ and a discrete spectrum of finite multiplicity contained in $[\lambda_{0},\infty)$ .

Let $\lambda_{\Delta}(G,\nu,k)$ denote the first eigenvalue larger than $\lambda_{0}$ in the discrete spectrum with respect to the congruence subgroup $G$ , weight $k$ and multiplier $\nu$ . For weight 0, Selberg [Sel65] showed that $\lambda_{\Delta}(\Gamma(N),\boldsymbol{1},0)\geq\frac{3}{16}$ for all $N$ and Selberg’s famous eigenvalue conjecture states that $\lambda_{\Delta}(G,\boldsymbol{1},0)\geq\frac{1}{4}$ for all $G$ . We introduce the hypothesis $H_{\theta}$ as

H_{\theta}:\quad\lambda_{\Delta}(\Gamma_{0}(N),\boldsymbol{1},0)\geq\tfrac{1}{4}-\theta^{2}\ \ \text{for all\ }N.

(2.16)

Selberg’s conjecture includes $H_{0}$ and the best progress known today is $H_{\frac{7}{64}}$ by [Kim03]. We denote $\lambda_{\Delta}(G,\nu,k)$ as $\lambda_{\Delta}$ when $(G,\nu,k)$ is clear from context.

Let ${\tilde{\mathcal{L}}}_{k}(N,\nu)\subset{\mathcal{L}}_{k}(N,\nu)$ denote the subspace spanned by eigenfunctions of $\Delta_{k}$ . For each eigenvalue $\lambda$ , we write

\lambda=\tfrac{1}{4}+r^{2}=s(1-s),\quad s=\tfrac{1}{2}+ir,\quad r\in i(0,\tfrac{1}{4}]\cup[0,\infty).

So $r\in i{\mathbb{R}}$ corresponds to $\lambda<\frac{1}{4}$ and any such eigenvalue $\lambda\in(\lambda_{0},\frac{1}{4})$ is called an exceptional eigenvalue. Set

r_{\Delta}(N,\nu,k)\vcentcolon=i\cdot\sqrt{\tfrac{1}{4}-\lambda_{\Delta}(\Gamma_{0}(N),\nu,k)}.

(2.17)

Let ${\tilde{\mathcal{L}}}_{k}(N,\nu,r)\subset{\tilde{\mathcal{L}}}_{k}(N,\nu)$ denote the subspace corresponding to the spectral parameter $r$ . Complex conjugation gives an isometry

{\tilde{\mathcal{L}}}_{k}(N,\nu,r)\longleftrightarrow{\tilde{\mathcal{L}}}_{-k}(N,\overline{\nu},r)

between normed spaces. For each $v\in{\tilde{\mathcal{L}}}_{k}(n,\nu,r)$ , we have the Fourier expansion

v(z)=v(x+iy)=c_{0}(y)+\sum_{\tilde{n}\neq 0}\rho(n)W_{\frac{k}{2}\operatorname{sgn}\tilde{n},\;ir}(4\pi|\tilde{n}|y)e(\tilde{n}x)

(2.18)

where $W_{\kappa,\mu}$ is the Whittaker function and

c_{0}(y)=\left\{\begin{array}[]{ll}0&\alpha_{\nu}\neq 0,\\ 0&\alpha_{\nu}=0\text{ and }r\geq 0,\\ \rho(0)y^{\frac{1}{2}+ir}+\rho^{\prime}(0)y^{\frac{1}{2}-ir}&\alpha_{\nu}=0\text{ and }r\in i(0,\frac{1}{4}].\end{array}\right.

Using the fact that $W_{\kappa,\mu}$ is a real function when $\kappa$ is real and $\mu\in{\mathbb{R}}\cup i{\mathbb{R}}$ [DLMF, (13.4.4), (13.14.3), (13.14.31)], if we denote the Fourier coefficient of $f_{c}\vcentcolon=\bar{f}$ as $\rho_{c}(n)$ , then

\rho_{c}(n)=\left\{\begin{array}[]{ll}\overline{\rho(1-n)},&\alpha_{\nu}>0,\ n\neq 0\\ \overline{\rho(-n)},&\alpha_{\nu}=0.\end{array}\right.

(2.19)

2.3. Holomorphic cusp forms of half-integral weight

For positive integers $N,l$ and a weight $k\in{\mathbb{Z}}+\frac{1}{2}$ multiplier $\nu$ on $\Gamma_{0}(N)$ , we know that $\nu$ is also a weight $k+2l$ multiplier system on $\Gamma_{0}(N)$ . For simplicity we denote $K=k+2l\in{\mathbb{Z}}+\frac{1}{2}$ . Let $S_{K}(N,\nu)$ (resp. $M_{K}(N,\nu)$ ) be the space of holomorphic cusp forms (resp. modular forms) on $\Gamma_{0}(N)$ which satisfy the transformation law

F(\gamma z)=\nu(\gamma)(cz+d)^{K}F(z),\quad\gamma\in\Gamma_{0}(N).

The inner product on $S_{K}(N,\nu)$ is defined as

\langle F,G\rangle_{H}\vcentcolon=\int_{\Gamma_{0}(N)\setminus{\mathbb{H}}}F(z)\overline{G(z)}y^{K}\frac{dxdy}{y^{2}},\quad f,g\in S_{K}(N,\nu).

It is known that (see e.g. [Ran77, §5]) $S_{K}(N,\nu)$ is a finite-dimensional Hilbert space under the inner product $\langle\cdot,\cdot\rangle_{H}$ . If we take an orthonormal basis $\{F_{j}(\cdot):\ 1\leq j\leq d\vcentcolon=\dim S_{K}(N,\nu)\}$ of $S_{K}(N,\nu)$ and write the Fourier expansion of $F_{j}$ as

F_{j}(z)=\sum_{n=1}^{\infty}a_{j}(n)e(\tilde{n}z),

then the sum

\frac{\Gamma(K-1)}{(4\pi\tilde{n})^{K-1}}\sum_{j=1}^{d}|a_{j}(n)|^{2}=1+2\pi i^{-K}\sum_{N|c}\frac{S(n,n,c,\nu)}{c}J_{K-1}\left(\frac{4\pi\tilde{n}}{c}\right)

(2.20)

is independent of the choice of the basis.

There is a one-to-one correspondence between all $f\in{\mathcal{L}}_{k}(N,\nu)$ with eigenvalue $\lambda_{0}$ and weight $k$ holomorphic modular forms $F$ by

f(z)=\left\{\begin{array}[]{ll}y^{\frac{k}{2}}F(z)&k\geq 0,\;\ F\in M_{k}(N,\nu),\\ y^{-\frac{k}{2}}\overline{F(z)}&k<0,\;\ F\in M_{-k}(N,\overline{\nu}).\end{array}\right.

(2.21)

If we write the Fourier expansion of $f$ as $f(z)=\sum_{n\in{\mathbb{Z}}}a_{y}(n)e(\tilde{n}x)$ , then

\left\{\begin{array}[]{lll}k\geq 0&\Rightarrow&a_{y}(n)=0\text{\ for\ }\tilde{n}<0,\\ k<0&\Rightarrow&a_{y}(n)=0\text{\ for\ }\tilde{n}>0.\end{array}\right.

(2.22)

3. Trace formula

Let $k\in{\mathbb{Z}}+\frac{1}{2}$ , $N$ be a positive integer, and $\mathfrak{a}$ be a singular cusp for the weight $k$ multiplier system $\nu$ on $\Gamma=\Gamma_{0}(N)$ . Define the Eisenstein series associated to $\mathfrak{a}$ by

E_{\mathfrak{a}}(\tau,s)\vcentcolon=\sum_{\gamma\in\Gamma_{\mathfrak{a}}\setminus\Gamma}\overline{\nu(\gamma)w(\sigma_{\mathfrak{a}}^{-1},\gamma)}(\operatorname{Im}\sigma_{\mathfrak{a}}^{-1}\gamma\tau)^{s}j(\sigma_{\mathfrak{a}}^{-1}\gamma,\tau)^{-k}.

(3.1)

We also define the Poincaré series for $m>0$ by

{\mathcal{U}}_{m}(\tau,s)\vcentcolon=\sum_{\gamma\in\Gamma_{\infty}\setminus\Gamma}\overline{\nu}(\gamma)(\operatorname{Im}\gamma\tau)^{s}j(\gamma,\tau)^{-k}e(\tilde{m}\gamma\tau).

Both of them are automorphic functions of weight $k$ . The properties of these two series can be found in [Pro05]. The Fourier expansion of ${\mathcal{U}}_{m}$ is

{\mathcal{U}}_{m}(x+iy,s)=y^{s}e(\tilde{m}\tau)+y^{s}\sum_{\ell\in{\mathbb{Z}}}\sum_{c>0}\frac{S(m,\ell,c,\nu)}{c^{2s}}B(c,\tilde{m},\tilde{\ell},y,s,k)e(\tilde{\ell}x)

(3.2)

where the function $B$ is in [AA18, (4.5)]. When $\operatorname{Re}s>1$ , ${\mathcal{U}}_{m}(\cdot,s)\in{\mathcal{L}}_{k}(N,\nu)$ . The Fourier expansion of $E_{\mathfrak{a}}$ at $s=\frac{1}{2}+ir$ for $r\in{\mathbb{R}}$ is

$\displaystyle E_{\mathfrak{a}}(x+iy,s)$	$\displaystyle=\delta_{\mathfrak{a}\infty}y^{s}$	$\displaystyle+\rho_{\mathfrak{a}}(0,r)y^{1-s}+\sum_{\ell\neq 0}\rho_{\mathfrak{a}}(\ell,r)W_{\frac{k}{2}\operatorname{sgn}\tilde{\ell},\;\frac{1}{2}-s}(4\pi\|\tilde{\ell}\|y)e(\tilde{\ell}x)$	(3.3)
	$\displaystyle=\delta_{\mathfrak{a}\infty}y^{s}$	$\displaystyle+\frac{\delta_{\alpha_{\nu}0}\cdot 4^{1-s}\Gamma(2s-1)}{e^{\pi ik/2}\Gamma(s+\frac{k}{2})\Gamma(s-\frac{k}{2})}y^{1-s}\varphi_{\mathfrak{a}0}(s)$
	$\displaystyle+\sum_{\ell\neq 0}\|\tilde{\ell}\|^{s-1}\frac{\pi^{s}W_{\frac{k}{2}\operatorname{sgn}\tilde{\ell},\;\frac{1}{2}-s}(4\pi\|\tilde{\ell}\|y)}{e^{\pi ik/2}\Gamma(s+\frac{k}{2}\operatorname{sgn}\tilde{\ell})}\varphi_{\mathfrak{a}\ell}(s)e(\tilde{\ell}x).$

Suppose $m$ and $n$ are positive integers and recall the definition of $\tilde{n}$ in §2.1. The following notations are very important in the remaining part of this paper:

Setting 3.1.

Let $a=4\pi\sqrt{\tilde{m}\tilde{n}}>0$ and $0<T\leq x/3$ with $T\asymp x^{1-\delta}$ where $\delta\in(0,\frac{1}{2})$ will be finally taken as $\frac{1}{3}$ .

Setting 3.2.

The test function $\phi\vcentcolon=\phi_{a,x,T}:[0,\infty)\rightarrow{\mathbb{R}}$ is a four times continuously differentiable function satisfying

(1)

$\phi(0)=\phi^{\prime}(0)=0$ and $\phi^{(j)}(x)\ll_{\varepsilon}x^{-2-\varepsilon}$ ( $j=0,\cdots,4$ ) as $x\rightarrow\infty$ .
(2)

$\phi(t)=1$ for $\frac{a}{2x}\leq t\leq\frac{a}{x}$ .
(3)

$\phi(t)=0$ for $t\leq\frac{a}{2x+2T}$ and $t\geq\frac{a}{x-T}$ .
(4)

$\phi^{\prime}(t)\ll\left(\frac{a}{x-T}-\frac{a}{x}\right)^{-1}\ll\frac{x^{2}}{aT}$ .
(5)

$\phi$ and $\phi^{\prime}$ are piecewise monotone on a fixed number of intevals.

This test function was used in [ST09] and [AA18]. We also define the following transformations of $\phi$ :

\widetilde{\phi}(r)=\int_{0}^{\infty}J_{r-1}(u)\phi(u)\frac{du}{u}

(3.4)

and for $k\geq 0$ ,

\widehat{\phi}(r)\vcentcolon=\pi^{2}e^{\frac{(1+k)\pi i}{2}}\frac{\int_{0}^{\infty}\left(\cos(\frac{k\pi}{2}+\pi ir)J_{2ir}(u)-\cos(\frac{k\pi}{2}-\pi ir)J_{-2ir}(u)\right)\phi(u)\frac{du}{u}}{\operatorname{sh}(\pi r)(\operatorname{ch}(2\pi r)+\cos\pi k)\Gamma(\frac{1}{2}-\frac{k}{2}+ir)\Gamma(\frac{1}{2}-\frac{k}{2}-ir)}

(3.5)

with the corrected version of [Blo08, (2.12)]

\widehat{\phi}\left(\tfrac{i}{4}\right)=\left\{\begin{array}[]{ll}e^{\frac{\pi i}{4}}\int_{0}^{\infty}\cos(u)\phi(u)u^{-\frac{3}{2}}du&\ \ k=\frac{1}{2},\vspace{4px}\\ \frac{1}{2}e^{\frac{3\pi i}{4}}\int_{0}^{\infty}\sin(u)\phi(u)u^{-\frac{3}{2}}du&\ \ k=\frac{3}{2}.\end{array}\right.

(3.6)

For an integer $l\geq 1$ , let $B_{l}$ denote an orthonormal basis for the space of holomorphic cusp forms $S_{k+2l}(N,\nu)$ and

\mathscr{B}_{k}\vcentcolon=\bigcup_{l=1}^{\infty}B_{l}.

Suppose that the Fourier expansion of each $F\in\mathscr{B}_{k}$ is given by

F(z)\vcentcolon=\sum_{n=1}^{\infty}a_{F}(n)e(\tilde{n}z).

(3.7)

Let $w_{F}$ denote the weight of $F\in\mathscr{B}_{k}$ . Here is the trace formula:

Theorem 3.3 ([Pro05, §6 Theorem]).

Suppose $\nu$ is a multiplier system of weight $k=\frac{1}{2}$ or $\frac{3}{2}$ on $\Gamma$ . Let $\{v_{j}(\cdot)\}$ be an orthonormal basis of ${\tilde{\mathcal{L}}}_{k}(N,\nu)$ and $E_{\mathfrak{a}}(\cdot,s)$ be the Eisenstein series associated to a singular cusp $\mathfrak{a}$ . Let $\rho_{j}(n)$ denote the $n$ -th Fourier coefficient of $v_{j}$ . Let $\varphi_{\mathfrak{a}n}(\frac{1}{2}+ir)$ or $\rho_{\mathfrak{a}}(n,r)$ denote the $n$ -th Fourier coefficient of $E_{\mathfrak{a}}(\cdot,\frac{1}{2}+ir)$ as in (3.3). Let $\mathscr{B}_{k}$ and $a_{F}(n)$ be defined as in (3.7). Then for $\tilde{m}>0$ and $\tilde{n}>0$ we have

\displaystyle\sum_{c>0}\frac{S(m,n,c,\nu)}{c}\phi\left(\frac{4\pi\sqrt{\tilde{m}\tilde{n}}}{c}\right)=\mathcal{U}_{k}+\mathcal{W}+\!\!\!\sum_{\mathrm{singular\ }\mathfrak{a}}\!\!\!\mathcal{E}_{\mathfrak{a}},

(3.8)

where

\mathcal{U}_{k}=\sum_{F\in\mathscr{B}_{k}}\frac{4\Gamma(w_{F})e^{\pi iw_{F}/2}}{(4\pi)^{w_{F}}(\tilde{m}\tilde{n})^{(w_{F}-1)/2}}\overline{a_{F}(m)}a_{F}(n)\widetilde{\phi}(w_{F}),

\mathcal{W}=4\sqrt{\tilde{m}\tilde{n}}\sum_{r_{j}}\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j}),

and

	$\displaystyle\mathcal{E}_{\mathfrak{a}}$	$\displaystyle=\int_{-\infty}^{\infty}\left(\frac{\tilde{m}}{\tilde{n}}\right)^{-ir}\frac{\overline{\varphi_{\mathfrak{a}m}\left(\frac{1}{2}+ir\right)}\varphi_{\mathfrak{a}n}\left(\frac{1}{2}+ir\right)\widehat{\phi}(r)dr}{\operatorname{ch}\pi r\;\|\Gamma(\frac{1}{2}+\frac{k}{2}+ir)\|^{2}}$
		$\displaystyle=4\sqrt{\tilde{m}\tilde{n}}\cdot\frac{1}{4\pi}\int_{-\infty}^{\infty}\overline{\rho_{\mathfrak{a}}\left(m,r\right)}\rho_{\mathfrak{a}}\left(n,r\right)\frac{\widehat{\phi}(r)}{\operatorname{ch}\pi r}dr.$

Remark.

We clarify two points in the theorem.

(1)

In the term $\mathcal{U}_{k}$ corresponding to holomorphic cusp forms, each function $F\in\mathscr{B}_{k}$ has weight $w_{F}=k+2l\geq\frac{5}{2}$ .

(2)

The equality of the two expressions in $\mathcal{E}_{\mathfrak{a}}$ is by (3.3):

\sqrt{\frac{\tilde{n}}{\pi}}\,\rho_{\mathfrak{a}}(n,r)=\frac{e^{-\frac{\pi ik}{2}}\pi^{ir}\tilde{n}^{ir}}{\Gamma\left(\frac{1}{2}+ir+\frac{k}{2}\operatorname{sgn}\tilde{n}\right)}\varphi_{\mathfrak{a}n}\left(\frac{1}{2}+ir\right).

3.1. Properties of admissible multipliers

Suppose $\nu$ is a weight $k$ admissible multiplier system on $\Gamma=\Gamma_{0}(N)$ (Definition 1.1) with parameters $B$ , $M$ and $D$ . Then we have the following propositions.

Proposition 3.4.

[Sun23, Proposition 5.7] Suppose that $\nu$ satisfies condition (1) of Definition 1.1 and assume $H_{\theta}$ (2.16). Then we have $2\operatorname{Im}r_{\Delta}\left(N,\nu,k\right)\leq\theta.$

Proposition 3.5.

Suppose that $\nu$ satisfies condition (1) of Definition 1.1 with $\nu^{\prime}=(\frac{|D|}{\cdot})\nu_{\theta}^{2k}$ . For $l\in{\mathbb{Z}}$ , let $K=k+2l\geq\frac{5}{2}$ . Suppose $\{F_{j,l}(\cdot)\}_{j}$ is an orthonormal basis of $S_{K}(N,\nu)$ and $\{G_{j,l}(\cdot)\}_{j}$ is an orthonormal basis of $S_{K}(M,\nu^{\prime})$ . Denote $a_{F,j,l}(n)$ as the Fourier coefficient of $F_{j,l}$ and $a_{G,j,l}(n)$ as the Fourier coefficient of $G_{j,l}$ . Then we have

\sum_{j=1}^{\dim S_{K}(N,\nu)}|a_{F,j,l}(n)|^{2}\ll_{N,\nu}\sum_{j=1}^{\dim S_{K}(M,\nu^{\prime})}|a_{G,j,l}(B\tilde{n})|^{2}.

(3.9)

Proof.

By condition (1) of Definition 1.1, we know that

\left\{[\Gamma_{0}(N):\Gamma_{0}(M)]^{-\frac{1}{2}}F_{j,l}(Bz):\ 1\leq j\leq\dim S_{K}(N,\nu)\right\}

(3.10)

is an orthonormal subset of $S_{K}(M,\nu^{\prime})$ . Since the left hand side of (2.20) is independent from the choice of basis, we expand (3.10) to an orthonormal basis of $S_{K}(M,\nu^{\prime})$ and get the result. ∎

Now we start to prove a bound for the right hand side of (3.9). First we have

Proposition 3.6 ([Wai17, Theorem 1]).

For $K\in{\mathbb{Z}}+\frac{1}{2}$ , $K\geq\frac{5}{2}$ and a quadratic character $\chi$ modulo $M$ , suppose

\left\{\Phi_{j}=\sum_{n=1}^{\infty}a_{j}(n)e(nz):\ 1\leq j\leq d\vcentcolon=\dim S_{K}(M,\chi\nu_{\theta}^{2K})\right\}

is an orthonormal basis of $S_{K}(M,\chi\nu_{\theta}^{2K})$ . For $n\geq 1$ , write $n=tu^{2}w^{2}$ with $t$ square-free, $u|M^{\infty}$ and $(w,M)=1$ . Then we have

\frac{\Gamma(K-1)}{(4\pi n)^{K-1}}\sum_{j=1}^{d}|a_{j}(n)|^{2}\ll_{K,M,\varepsilon}\left(n^{\frac{3}{7}}+u\right)n^{\varepsilon}.

Note that the implied constant in Waibel’s bound depends on $K$ when expressing Bessel functions (see [Wai17, after (8)] and [Iwa87, Theorem 1 and p. 400]). For our proof, it’s essential that the bound remains uniform across the weights. We modify the estimate and get the following proposition.

Proposition 3.7.

With the same setting as Proposition 3.6,

\frac{\Gamma(K-1)}{(4\pi n)^{K-1}}\sum_{j=1}^{d}|a_{j}(n)|^{2}\ll_{M,\varepsilon}(n^{\frac{19}{42}}+u)n^{\varepsilon}.

Proof.

We do the same preparation as [Wai17, around (8)] to estimate the right hand side of (2.20). Let $P>1+(\log 2nM)^{2}$ (finally chosen to be $\asymp_{M}n^{\frac{1}{7}}$ ) and define the set of prime numbers

\mathcal{P}\vcentcolon=\ \{p\text{ prime}:P<p\leq 2P,\ p\nmid 2nM\}.

Here we have $\#\mathcal{P}\asymp P/\log P$ .

For $\{\Phi_{j}\}_{j}$ a orthonormal basis of $S_{K}(M,\chi\nu_{\theta}^{2K})$ , the set $\{[\Gamma_{0}(M):\Gamma_{0}(pM)]^{-\frac{1}{2}}\Phi_{j}\}_{j}$ is an orthonormal subset of $S_{K}(pM,\chi\nu_{\theta}^{2K})$ . Recall (2.20) and we have

\frac{\Gamma(K-1)}{(4\pi n)^{K-1}}\sum_{j=1}^{d}\frac{|a_{j}(n)|^{2}}{[\Gamma_{0}(M):\Gamma_{0}(pM)]}\leq 1+2\pi i^{-K}\sum_{pM|c}\frac{S(n,n,c,\chi\nu_{\theta}^{2K})}{c}J_{K-1}\left(\frac{4\pi n}{c}\right).

(3.11)

For those $p\in\mathcal{P}$ , $[\Gamma_{0}(M):\Gamma_{0}(pM)]\leq p+1$ . Summing (3.11) on $p\in\mathcal{P}$ and dividing $\#\mathcal{P}$ we get

\frac{\Gamma(K-1)}{(4\pi n)^{K-1}}\sum_{j=1}^{d}|a_{j}(n)|^{2}\ll P+(\log P)\sum_{p\in\mathcal{P}}\left|\sum_{pM|c}\frac{S(n,n,c,\chi\nu_{\theta}^{2K})}{c}J_{K-1}\left(\frac{4\pi n}{c}\right)\right|.

(3.12)

The average estimate of

K_{pM}^{(\mu)}(x)\vcentcolon=\sum_{pM|c\leq x}\frac{S(n,n,c,\chi\nu_{\theta}^{2K})}{\sqrt{c}}e\left(\frac{2\mu n}{c}\right),\quad\mu\in\{-1,0,1\}

can be found in [Wai17, (19)] that for $\mu\in\{-1,0,1\}$ ,

\sum_{p\in\mathcal{P}}|K_{pM}^{(\mu)}(x)|\ll_{M,\varepsilon}\left(xun^{-\frac{1}{2}}+xP^{-\frac{1}{2}}+(x+n)^{\frac{5}{8}}\left(x^{\frac{1}{4}}P^{\frac{3}{8}}+n^{\frac{1}{8}}x^{\frac{1}{8}}P^{\frac{1}{4}}\right)\right)(nx)^{\varepsilon}.

(3.13)

We break the sum on $c\equiv 0\ (\mathrm{mod}\ pM)$ at the right hand side of (3.12) into $c\leq n$ and $c\geq n$ to estimate. The uniform bound of $J$ -Bessel functions is given by [Lan00]

|J_{\beta}(x)|\leq c_{0}x^{-\frac{1}{3}}\quad\text{for all }\beta>0\text{ and }x>0,

(3.14)

where $c_{0}=0.7857\cdots$ .

When $c\leq n$ , using (3.14) and [DLMF, (10.6.1)]

2J_{\beta-1}^{\prime}(x)=J_{\beta-2}(x)-J_{\beta}(x),

we find that

\left(x^{-\frac{1}{2}}J_{K-1}\left(\frac{4\pi n}{x}\right)\right)^{\prime}\ll n^{-\frac{1}{3}}x^{-\frac{7}{6}}+n^{\frac{2}{3}}x^{-\frac{13}{6}}.

(3.15)

Then a partial summation using (3.13), (3.15) and (3.14) yields

\sum_{p\in\mathcal{P}}\left|\sum_{pM|c\leq n}\frac{S(n,n,c,\chi\nu_{\theta}^{2K})}{c}J_{K-1}\left(\frac{4\pi n}{c}\right)\right|\ll_{M,\varepsilon}(n^{\frac{19}{42}}+u)n^{\varepsilon}.

(3.16)

When $c\geq n$ , we get another bound

\left(x^{-\frac{1}{2}}J_{K-1}\left(\frac{4\pi n}{x}\right)\right)^{\prime}\ll nx^{-\frac{5}{2}}\quad\text{for }x\geq n

(3.17)

by $|J_{\beta-1}(x)|\leq\frac{(x/2)^{\beta-1}}{\Gamma(\beta)}$ [DLMF, (10.14.4)] and $|J_{\beta}(x)|\leq 1$ [DLMF, (10.14.1)]. Remember $K\geq\frac{5}{2}$ here. We do a partial summation again using (3.13) and (3.17) and get

\sum_{p\in\mathcal{P}}\left|\sum_{pM|c\geq n}\frac{S(n,n,c,\chi\nu_{\theta}^{2K})}{c}J_{K-1}\left(\frac{4\pi n}{c}\right)\right|\ll_{M,\varepsilon}(n^{\frac{3}{7}}+u)n^{\varepsilon}.

(3.18)

From (3.16), (3.18), (3.12) and $P\asymp_{M}n^{\frac{1}{7}}$ , we finish the proof. ∎

Combining Proposition 3.5 and Proposition 3.7, one observes the following bound:

Proposition 3.8.

With the same setting as Proposition 3.5, we factor $B\tilde{n}=t_{n}u_{n}^{2}w_{n}^{2}$ with $t_{n}$ square-free, $u_{n}|M^{\infty}$ and $(w_{n},M)=1$ . Then

\frac{\Gamma(K-1)}{(4\pi\tilde{n})^{K-1}}\sum_{j=1}^{\dim S_{K}(N,\nu)}|a_{F,j,l}(n)|^{2}\ll_{N,\nu,\varepsilon}(\tilde{n}^{\frac{19}{42}}+u_{n})\tilde{n}^{\varepsilon}.

4. Bounds on $\widetilde{\phi}$ and $\widehat{\phi}$

In this section, all of the implied constants among the estimates for $\widetilde{\phi}$ and $\widehat{\phi}$ depend on $N$ and the multiplier system $\nu$ unless specified. Recall the definitions (3.4) and (3.5). To deal with the $\Gamma$ -function in the denominator of $\widehat{\phi}$ , we need [DLMF, (5.6.6-7)]

\frac{\Gamma(x)^{2}}{\operatorname{ch}(\pi r)}\leq|\Gamma(x+ir)|^{2}\leq\Gamma(x)^{2}\quad\text{for }x\geq 0\text{ and }r\in{\mathbb{R}}.

(4.1)

Recall (3.4) and (3.5) that we define $\widehat{\phi}$ for $k\geq 0$ . We also have

\displaystyle\begin{split}&\widehat{\phi}(r)=\frac{\pi^{2}e^{\frac{1+k}{2}\pi i}}{\operatorname{sh}(\pi r)(\operatorname{ch}(2\pi r)+\cos(\pi k))\Gamma(\frac{1}{2}-\frac{k}{2}+ir)\Gamma(\frac{1}{2}-\frac{k}{2}-ir)}\\ &\cdot\left\{\cos\tfrac{k\pi}{2}\operatorname{ch}(\pi r)\left(\widetilde{\phi}(1+2ir)-\widetilde{\phi}(1-2ir)\right)-i\sin\tfrac{k\pi}{2}\operatorname{sh}(\pi r)\left(\widetilde{\phi}(1+2ir)+\widetilde{\phi}(1-2ir)\right)\right\}.\end{split}

(4.2)

Like [AD20, after (5.3)], we define $\xi_{k}$ as

\xi_{k}(r)\vcentcolon=\frac{2i\pi^{2}e^{\frac{1+k}{2}\pi i}}{\Gamma(\frac{1}{2}-\frac{k}{2}+ir)\Gamma(\frac{1}{2}-\frac{k}{2}-ir)}.

(4.3)

Then

\xi_{k}(r)\left\{\begin{array}[]{ll}\ll 1&\quad\text{for }r\in[-1,1],\\ \asymp|r|^{k}e^{\pi|r|}&\quad\text{for }r\in(-\infty,-1]\cup[1,\infty).\end{array}\right.

(4.4)

We refer to [Dun90] for estimates on $J$ -Bessel functions. Denote

F_{\mu}(z)\vcentcolon=\frac{J_{\mu}(z)+J_{-\mu}(z)}{2\cos(\mu\pi/2)},\quad G_{\mu}(z)\vcentcolon=\frac{J_{\mu}(z)-J_{-\mu}(z)}{2\sin(\mu\pi/2)}.

As a result of the relationship $\overline{J_{2ir}(u)}=J_{-2ir}(u)$ for $r,u\in{\mathbb{R}}$ by [DLMF, (10.11.9)], we have

F_{2ir}(u)=\frac{\operatorname{Re}J_{2ir}(u)}{\operatorname{ch}(\pi r)}\in{\mathbb{R}},\quad G_{2ir}(u)=\frac{\operatorname{Im}J_{2ir}(u)}{\operatorname{sh}(\pi r)}\in{\mathbb{R}}.

Moreover, for $k\in{\mathbb{Z}}+\frac{1}{2}$ and $k\geq 0$ ,

\widehat{\phi}(r)=\frac{\xi_{k}(r)\operatorname{ch}(\pi r)}{\operatorname{ch}(2\pi r)}\int_{0}^{\infty}\left(G_{2ir}(u)\cos\frac{k\pi}{2}-F_{2ir}(u)\sin\frac{k\pi}{2}\right)\frac{\phi(u)}{u}du

(4.5)

and $\widehat{\phi}(r)=\widehat{\phi}(-r)$ for $r\in{\mathbb{R}}$ because $F_{\mu}(z)=F_{-\mu}(z)$ and $G_{\mu}(z)=G_{-\mu}(z)$ .

Lemma 4.1.

For $r\in[-1,1]$ , uniformly and with absolute implied constants we have

\displaystyle\begin{split}G_{2ir}(u)\ll\left\{\begin{array}[]{ll}\ln(\tfrac{u}{2}),&u\in[0,\frac{3}{2}],\vspace{4px}\\ u^{-\frac{3}{2}},&u\in[\frac{3}{2},\infty).\end{array}\right.\end{split}

(4.6)

Proof.

First we deal with the range $u\in[0,\frac{3}{2}]$ . The series expansion of $G_{ir}$ is given by [Dun90, (3.9), (3.16)]:

	$\displaystyle G_{2ir}(u)$	$\displaystyle=\left(\frac{4r\operatorname{ch}(\pi r)}{\pi\operatorname{sh}(\pi r)}\right)^{\frac{1}{2}}\sum_{\ell=0}^{\infty}\frac{(-1)^{\ell}(u^{2}/4)^{\ell}\sin(2r\ln(u/2)-\phi_{2r,\ell})}{\ell!\prod_{j=0}^{\ell^{2}}(j+4r^{2})^{1/2}}$
		$\displaystyle=\left(\frac{\operatorname{ch}(\pi r)}{\pi r\operatorname{sh}(\pi r)}\right)^{\frac{1}{2}}\sin\left(2r\ln\left(\frac{u}{2}\right)-\phi_{2r,0}\right)+O\left(\left(\frac{u}{2}\right)^{2}\right).$

where $\phi_{r,\ell}=\arg\Gamma(1+\ell+ir)$ . The implied constant in the second equation is absolute. As a function of $r$ , $\phi_{2r,0}\in C^{\infty}[0,1]$ and $\lim_{r\rightarrow 0}\phi_{2r,0}=0$ . Then $\phi_{2r,0}/r=O(1)$ and

	$\displaystyle G_{2ir}(u)$	$\displaystyle\ll r^{-1}O\left(\left\|2r\ln\left(\frac{u}{2}\right)\right\|+\|\phi_{2r,0}\|\right)+O\left(\left(\frac{u}{2}\right)^{2}\right)$
		$\displaystyle\ll\ln\left(\frac{u}{2}\right)+O(1).$

For the range $u\geq\frac{3}{2}$ , we check with [Dun90, (5.16)] where $U_{s}(p)$ for $s\geq 0$ are fixed polynomials of $p$ whose lowest degree term is $p^{s}$ :

	$\displaystyle G_{2ir}(u)$	$\displaystyle=\left(\frac{4/\pi^{2}}{4r^{2}+u^{2}}\right)^{\frac{1}{4}}\left(\frac{C}{\sqrt{4r^{2}+u^{2}}}+O\left(\frac{1}{4r^{2}+u^{2}}\right)\right)$
		$\displaystyle\ll\left(r^{2}+\frac{u^{2}}{4}\right)^{-\frac{3}{4}}+O\left(\left(r^{2}+\frac{u^{2}}{4}\right)^{-\frac{5}{4}}\right).$

Our claimed bound is clear as $r^{2}\geq 0$ . The implied constant above is absolute due to [Dun90, (3.3)] and [Olv97, Chapter 8, §13] or by [Olv97, Chapter 10, (3.04)].

∎

Lemma 4.2.

For $r\in[-1,1]$ , we have

|\widetilde{\phi}(1+2ir)|\ll 1,\quad|\widehat{\phi}(r)|\ll_{\varepsilon}(ax)^{\varepsilon}.

(4.7)

Proof.

A trivial bound of $J_{2ir}$ is given by the integral representation [DLMF, (10.9.4)]:

J_{\nu}(z)=\frac{(z/2)^{\nu}}{\sqrt{\pi}\Gamma(\nu+\frac{1}{2})}\int_{0}^{\pi}\cos(z\cos\theta)(\sin\theta)^{2\nu}d\theta,\quad\operatorname{Re}\nu>-\frac{1}{2}.

Then we have $|J_{2ir}(u)|\leq\frac{\sqrt{\pi}}{|\Gamma(\frac{1}{2}+2ir)|}$ and

|\widetilde{\phi}(1+2ir)|\ll\int_{\frac{3a}{8x}}^{\frac{3a}{2x}}\frac{du}{u}\leq\ln 4\quad\text{for }r\in[-1,1].

This implies that

\operatorname{ch}(\pi r)\int_{0}^{\infty}F_{2ir}(u)\frac{\phi(u)}{u}du\ll 1\quad\text{for }r\in[-1,1].

Let the closed interval $[\alpha,\beta]=\emptyset$ when $\alpha>\beta$ . With the help of (4.5), (4.4) and Lemma 4.1 we get

	$\displaystyle\|\widehat{\phi}(r)\|$	$\displaystyle\ll\frac{\operatorname{ch}(\pi r)}{\operatorname{ch}(2\pi r)}\left(\left\|\int_{0}^{\infty}G_{2ir}(u)\phi(u)\frac{du}{u}\right\|+\left\|\int_{0}^{\infty}F_{2ir}(u)\phi(u)\frac{du}{u}\right\|\right)$
		$\displaystyle\ll\int_{[\frac{3a}{8x},\frac{3}{2}]}\ln\left(\frac{u}{2}\right)\frac{du}{u}+\int_{[\frac{3}{2},\frac{3a}{2x}]}u^{-\frac{5}{2}}du+O(1)$
		$\displaystyle\ll\left(\ln\frac{3a}{16x}\right)^{2}+O(1)\ll(ax)^{\varepsilon}.$

The last inequality is because $a=4\pi\sqrt{\tilde{m}\tilde{n}}>0$ has a lower bound depending on $\nu$ . ∎

When we focus on the exceptional eigenvalues $\lambda_{j}\in[\frac{3}{16},\frac{1}{4})$ of $\Delta_{k}$ , recall that $\lambda_{j}=\frac{1}{4}+r_{j}^{2}$ for $r_{j}\in i(0,\frac{1}{4}]$ . By Proposition 3.4, if we write $t_{j}=\operatorname{Im}r_{j}$ , assuming $H_{\theta}$ (2.16) we have an upper bound $t_{j}\leq\frac{\theta}{2}$ when $r_{j}\neq\frac{i}{4}$ . Moreover, since the exceptional eigenvalues are discrete, we also have a largest eigenvalue less than $\frac{1}{4}$ , hence a lower bound $\underline{t}>0$ (depending on $N$ and $\nu$ ) such that $t_{j}\geq\underline{t}$ .

Lemma 4.3.

With the hypothesis $H_{\theta}$ (2.16) for $\theta\leq\frac{1}{6}$ , when $r=it$ and $t\in[\underline{t},\frac{\theta}{2}]$ , we have

\widetilde{\phi}(1\pm 2t)\ll\left(\frac{a}{x}\right)^{\pm 2t}\quad\text{and}\quad\widehat{\phi}(r)\ll\left(\frac{a}{x}\right)^{2t}+\left(\frac{x}{a}\right)^{2t}\ll\left(\frac{a}{x}\right)^{\theta}+\left(\frac{x}{a}\right)^{\theta}.

(4.8)

Moreover, for $r=\frac{i}{4}$ we have

\widetilde{\phi}\left(1\pm\frac{1}{2}\right)\ll\left(\frac{a}{x}\right)^{\pm\frac{1}{2}}\quad\text{and}\quad\widehat{\phi}\left(\frac{i}{4}\right)\ll\left\{\begin{array}[]{ll}(\frac{x}{a})^{\frac{1}{2}},&k=\frac{1}{2},\vspace{4px}\\ (\frac{a}{x})^{\frac{1}{2}},&k=\frac{3}{2}.\end{array}\right.

(4.9)

Proof.

As in the previous lemma, when $t\in[\underline{t},\frac{\theta}{2}]$ , the bound [DLMF, (10.9.4)] gives

|J_{\pm 2t}(u)|\ll\frac{u^{\pm 2t}}{\Gamma(\frac{1}{2}-\theta)}\quad\text{and}\quad|\widetilde{\phi}(1\pm 2t)|\ll\int_{\frac{3a}{8x}}^{\frac{3a}{2x}}u^{\pm 2t}\;\frac{du}{u}\ll\left(\frac{a}{x}\right)^{\pm 2t}.

The bound for $\widehat{\phi}$ follows from (4.2). When $r=\frac{i}{4}$ , by [DLMF, (10.16.1)] we have

J_{-\frac{1}{2}}(u)\ll u^{-\frac{1}{2}}\quad\text{and}\quad J_{\frac{1}{2}}(u)\ll u^{-\frac{1}{2}}\sin u\leq u^{\frac{1}{2}}.

The bounds for $\widetilde{\phi}(1\pm\frac{1}{2})$ and $\widehat{\phi}(\frac{i}{4})$ follow from the same process above with (3.4) and (3.6). ∎

For the range $|r|\geq 1$ we have

Lemma 4.4.

[AD20, Lemma 6.3] Let $k=\frac{1}{2}$ or $\frac{3}{2}$ . Then

\widehat{\phi}(r)\ll\left\{\begin{array}[]{ll}r^{k-\frac{3}{2}},&\ r\geq 1,\\ r^{k}\min(r^{-\frac{3}{2}},r^{-\frac{5}{2}}\frac{x}{T}),&\ r\geq\max(\frac{a}{x},1).\end{array}\right.

(4.10)

Remark.

In the original paper they stated the result for $k=\pm\frac{1}{2}$ . However, the power $r^{k}$ in the estimate above only arises from $\xi_{k}(r)e^{-\pi|r|}$ (4.3) and by (4.4) we get the above lemma on weight $k=\frac{3}{2}$ .

4.1. A special test function

Here we choose a special test function $\phi$ satisfying Setting 3.2 to compute the terms corresponding to the exceptional spectrum $r\in i(0,\frac{1}{4}]$ in Theorem 1.3.

For $k=\frac{1}{2}$ or $\frac{3}{2}$ , let $\lambda\in[\frac{3}{16},\frac{1}{4})$ be an exceptional eigenvalue of $\Delta_{k}$ on $\Gamma_{0}(N)$ , we set $\lambda=s(1-s)$ for $s\in(\frac{1}{2},\frac{3}{4}]$ and

t=\operatorname{Im}r=\sqrt{\tfrac{1}{4}-\lambda}=\sqrt{\tfrac{1}{4}-s(1-s)}=s-\tfrac{1}{2}.

Since the exceptional spectrum is discrete, let the lower bound for $t>0$ be $\underline{t}$ depending on $N$ and $\nu$ . Recall Setting 3.1. Let $0<T^{\prime}\leq T\leq\frac{x}{3}$ be $T^{\prime}\vcentcolon=Tx^{-\delta}\asymp x^{1-2\delta}$ .

Setting 4.5.

In addition to the requirement in Setting 3.2, when $\frac{a}{x-T}\leq 1.999$ , we pick $\phi$ as a smoothed function of this piecewise linear one

where

\left\{\begin{array}[]{ll}\phi^{\prime}(u)=\tfrac{2x(x+T)}{aT}&u\in(\tfrac{a}{2x+2T-2T^{\prime}},\;\tfrac{a}{2x+2T^{\prime}}),\\ \\ \phi^{\prime}(u)=-\tfrac{x(x-T)}{aT}&u\in(\tfrac{a}{x-T^{\prime}},\;\tfrac{a}{x-T+T^{\prime}}),\\ \\ 0\leq\phi^{\prime}(u)\leq\tfrac{4x(x+T)}{aT}&u\in(\tfrac{a}{2x+2T},\;\tfrac{a}{2x+2T-2T^{\prime}})\cup(\tfrac{a}{2x+2T^{\prime}},\;\tfrac{a}{2x}),\\ \\ 0\geq\phi^{\prime}(u)\geq-\tfrac{2x(x-T)}{aT}&u\in(\tfrac{a}{x},\;\tfrac{a}{x-T^{\prime}})\cup(\tfrac{a}{x-T+T^{\prime}},\;\tfrac{a}{x-T}),\vspace{10px}\\ \phi^{\prime}(u)=0&\text{otherwise}.\tfrac{}{}\end{array}\right.

(4.11)

The above choice for $\phi^{\prime}$ is possible because there is no requirement for $\phi^{\prime\prime}(u)$ when $u\leq 2$ but for $u\rightarrow\infty$ in Setting 3.2.

Now we take $r=it\in i(0,\frac{1}{4}]$ . When $u\leq 1.999$ , by the series expansion [DLMF, (10.2.2)]:

J_{\nu}(z)=\left(\frac{z}{2}\right)^{\nu}\sum_{j=0}^{\infty}\frac{(-1)^{j}}{j!\Gamma(j+1+\nu)}\left(\frac{z}{2}\right)^{2j},

we have

J_{\pm 2t}(u)=\frac{(u/2)^{\pm 2t}}{\Gamma(1\pm 2t)}+O\left(\left(\frac{u}{2}\right)^{2\pm 2t}\right),\qquad 0<u\leq 1.999.

(4.12)

The implied constant is absolute. Now we compute the bound for $\widetilde{\phi}$ and $\widehat{\phi}$ .

Lemma 4.6.

Assuming $H_{\theta}$ (2.16) for $\theta\leq\frac{1}{6}$ and with the choice of $\phi$ in Setting 4.5, when $r=it\in i(0,\frac{1}{4}]$ ,

\displaystyle\begin{split}\widetilde{\phi}(1-2t)&=\frac{1}{\Gamma(1-2t)}\int_{\frac{a}{2x}}^{\frac{a}{x}}\left(\frac{u}{2}\right)^{-2t}\frac{\phi(u)}{u}du+O\left(a^{-2t}x^{2t-\delta}+1\right)\\ &=\frac{2^{2t}(2^{2t}-1)}{2t\Gamma(1-2t)}\left(\frac{x}{a}\right)^{2t}+O\left(a^{-2t}x^{2t-\delta}+1\right),\end{split}

(4.13)

Proof.

When $1.999<\frac{a}{x-T}\leq\frac{3a}{2x}$ , we get $x\ll a$ and $\widetilde{\phi}(1-2t)=O(1)$ by Lemma 4.3, so the lemma is true in this case. When $\frac{a}{x-T}\leq 1.999$ , we have $a\ll x$ and with (4.12),

	$\displaystyle\widetilde{\phi}(1-2t)$	$\displaystyle=\int_{0}^{\infty}\frac{(u/2)^{-2t}}{\Gamma(1-2t)}\frac{\phi(u)}{u}du+O\left(\int_{0}^{\infty}\left(\frac{u}{2}\right)^{2-2t}\frac{\phi(u)}{u}du\right)$
		$\displaystyle=\frac{2^{2t}}{\Gamma(1-2t)}\int_{\frac{a}{2x}}^{\frac{a}{x}}u^{-2t-1}du+\frac{2^{2t}}{\Gamma(1-2t)}\left(\int_{\frac{a}{2x+2T}}^{\frac{a}{2x}}+\int_{\frac{a}{x}}^{\frac{a}{x-T}}\right)u^{-2t-1}\phi(u)du$
		$\displaystyle+O\left(\int_{0}^{\infty}u^{1-2t}\phi(u)du\right)$
		$\displaystyle=:\frac{2^{2t}(2^{2t}-1)}{2t\Gamma(1-2t)}\left(\frac{x}{a}\right)^{2t}+(I_{1}+I_{2})+O(I_{3}).$

Recall that we always have the lower bound $\underline{t}>0$ for $t=\operatorname{Im}r$ . A bound for $I_{1}$ and $I_{2}$ follows from the same process as [Sun23, Proof of Lemma 7.2]:

I_{1}+I_{2}\ll\left(\int_{\frac{a}{2x+2T}}^{\frac{a}{2x}}+\int_{\frac{a}{x}}^{\frac{a}{x-T}}\right)u^{-2t-1}\phi(u)du\ll a^{-2t}x^{2t-\delta}.

We also get

I_{3}\ll\int_{\frac{3a}{8x}}^{\frac{3a}{2x}}u^{1-2t}du\ll\left(\frac{a}{x}\right)^{2-2t}\ll 1

and finish the proof.

∎

Lemma 4.7.

Assume $H_{\theta}$ (2.16) for $\theta\leq\frac{1}{6}$ . For $r=it\in i(0,\frac{\theta}{2}]$ we have

\widehat{\phi}(r)=\frac{e^{\frac{k\pi i}{2}}\cos(\pi t)\Gamma(\frac{1}{2}+\frac{k}{2}+t)\Gamma(2t)}{\Gamma(\frac{1}{2}-\frac{k}{2}+t)2^{2t}\pi^{2t}(\tilde{m}\tilde{n})^{t}}\cdot\frac{(2^{2t}-1)x^{2t}}{2t}+O\left(\frac{x^{2t-\delta}}{a^{2t}}+\frac{a^{2t}}{x^{2t}}+1\right).

Moreover,

\widehat{\phi}(\tfrac{i}{4})=\left\{\begin{array}[]{lr}2e^{\frac{\pi i}{4}}(\sqrt{2}-1)(\frac{x}{a})^{\frac{1}{2}}+O(x^{-\delta}(\frac{x}{a})^{\frac{1}{2}}+1)&\text{for\ }k=\frac{1}{2},\vspace{4px}\\ e^{\frac{3\pi i}{4}}(1-\tfrac{1}{\sqrt{2}})(\frac{a}{x})^{\frac{1}{2}}+O(x^{-\delta}(\frac{a}{x})^{\frac{1}{2}}+1)&\text{for\ }k=\frac{3}{2}.\end{array}\right.

The implied constants only depend on $N$ and $\nu$ .

Proof.

When $t\in[\underline{t},\frac{\theta}{2}]$ , we substitute Lemma 4.6 into (3.5) and use Lemma 4.3 to get

	$\displaystyle\widehat{\phi}(it)$	$\displaystyle=\frac{i\pi^{2}e^{\frac{k\pi i}{2}}\left(\cos(\frac{k\pi}{2}-\pi t)\widetilde{\phi}(1-2t)-\cos(\frac{k\pi}{2}+\pi t)\widetilde{\phi}(1+2t)\right)}{i\sin(\pi t)\cos(2\pi t)\Gamma(\frac{1}{2}-\frac{k}{2}-t)\Gamma(\frac{1}{2}-\frac{k}{2}+t)}$
		$\displaystyle=\frac{\pi^{2}e^{\frac{k\pi i}{2}}\cos(\frac{k\pi}{2}-\pi t)2^{2t}(2^{2t}-1)(x/a)^{2t}}{\sin(\pi t)\cos(2\pi t)\Gamma(\frac{1}{2}-\frac{k}{2}-t)\Gamma(\frac{1}{2}-\frac{k}{2}+t)2t\Gamma(1-2t)}+O\left(\frac{x^{2t-\delta}}{a^{2t}}+\frac{a^{2t}}{x^{2t}}+1\right).$

With the help of the functional equation of the $\Gamma$ function

\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}\quad\text{for\ }z\in{\mathbb{C}}\setminus{\mathbb{Z}}

and the trigonometric identities

\sin(\tfrac{\pi}{2}-x)=\cos x,\quad 2\cos x\cos y=\cos(x+y)+\cos(x-y)\quad\text{for\ }x,y\in{\mathbb{R}},

we have

		$\displaystyle\frac{\pi}{\sin(\pi t)\Gamma(1-2t)}=2\cos(\pi t)\Gamma(2t),$
		$\displaystyle\frac{\pi}{\Gamma(\frac{1}{2}-\frac{k}{2}-t)}=\Gamma(\tfrac{1}{2}+\tfrac{k}{2}+t)\cos(\tfrac{k\pi}{2}+\pi t),$
	and	$\displaystyle 2\cos(\tfrac{k\pi}{2}-\pi t)\cos(\tfrac{k\pi}{2}+\pi t)=\cos(2\pi t).$

Then the first part of the lemma follows. The implied constant only depends on $N$ and $\nu$ because $t\in[\underline{t},\frac{\theta}{2}]$ is bounded above and below away from 0.

When $t=\frac{1}{4}$ , the process is similar to the proof of Lemma 4.6 with the help of (3.6). First we deal with the case $k=\frac{1}{2}$ with $\cos u=1+O(u^{2})$ for $u\in[0,\frac{\pi}{2}]$ . Thus, when $\frac{a}{x-T}>\frac{\pi}{2}$ , we have $x\ll a$ and $\widehat{\phi}(\frac{i}{4})=O(1)$ in this case. When $\frac{a}{x-T}\leq\frac{\pi}{2}$ , we have $a\ll x$ and

	$\displaystyle\widehat{\phi}(\tfrac{i}{4})$	$\displaystyle=e^{\frac{\pi i}{4}}\int_{0}^{\infty}\phi(u)u^{-\frac{3}{2}}du+O\left(\int_{0}^{\infty}\phi(u)u^{\frac{1}{2}}du\right)$
		$\displaystyle=e^{\frac{\pi i}{4}}\int_{\frac{a}{2x}}^{\frac{a}{x}}u^{-\frac{3}{2}}du+e^{\frac{\pi i}{4}}\left(\int_{\frac{a}{2x+2T}}^{\frac{a}{2x}}+\int_{\frac{a}{x}}^{\frac{a}{x-T}}\right)u^{-\frac{3}{2}}\phi(u)du+O(1)$
		$\displaystyle=e^{\frac{\pi i}{4}}(2\sqrt{2}-2)\left(\frac{x}{a}\right)^{\frac{1}{2}}+O\left(x^{-\delta}\left(\frac{x}{a}\right)^{\frac{1}{2}}\right)+O(1).$

The case for $k=\frac{3}{2}$ is similar using $\sin u=u+O(u^{3})$ for $u\in[0,\frac{\pi}{2}]$ . ∎

5. Proof of Theorem 1.3 and Theorem 1.5

The proof depends on the following two propositions for the Fourier coefficients of Maass forms, which were originally obtained for the discrete spectrum in [AD22, Theorem 4.1] and [AD19, Theorem 4.3]. The author proved the generalized propositions in [Sun23, §8] to include the continuous spectrum. Recall our notations in Settings 3.1 and 3.2. Suppose that for some $\beta\in(\frac{1}{2},1)$ ,

\sum_{N|c>0}\frac{|S(n,n,c,\nu)|}{c^{1+\beta}}\ll_{N,\nu,\varepsilon}|\tilde{n}|^{\varepsilon}

(5.1)

Then we have the following proposition:

Proposition 5.1 ([Sun23, Proposition 8.1]).

Suppose that $\nu$ is a multiplier on $\Gamma=\Gamma_{0}(N)$ of weight $k=\pm\frac{1}{2}$ which satisfies (5.1) for some $\beta\in(\frac{1}{2},1)$ . Let $\rho_{j}(n)$ denote the Fourier coefficients of an orthonormal basis $\{v_{j}(\cdot)\}$ of ${\tilde{\mathcal{L}}}_{k}(N,\nu)$ . For each singular cusp $\mathfrak{a}$ of $(\Gamma,\nu)$ , let $E_{\mathfrak{a}}(\cdot,s)$ be the associated Eisenstein series. Let $\rho_{\mathfrak{a}}(n,r)$ be defined as in (3.3). Then for $x>0$ we have

	$\displaystyle x^{k\operatorname{sgn}\tilde{n}}\|\tilde{n}\|\left(\sum_{x\leq r_{j}\leq 2x}\|\rho_{j}(n)\|^{2}e^{-\pi r_{j}}+\right.\sum_{\mathrm{singular\ }\mathfrak{a}}$	$\displaystyle\left.\int_{\|r\|\in[x,2x]}\|\rho_{\mathfrak{a}}(n,r)\|^{2}e^{-\pi\|r\|}dr\right)$
		$\displaystyle\ll_{N,\nu,\varepsilon}x^{2}+\|\tilde{n}\|^{\beta+\varepsilon}x^{1-2\beta}\log^{\beta}x.$

Remark.

In Definition 1.1, an admissible multiplier satisfies (5.1) with $\beta=\frac{1}{2}+\varepsilon$ for any $\varepsilon$ .

Proposition 5.2 ([Sun23, Proposition 8.2]).

Suppose that $\nu$ is a weight $k=\pm\frac{1}{2}$ admissible multiplier on $\Gamma=\Gamma_{0}(N)$ with $M$ , $D$ and $B$ given in Definition 1.1. Let $\rho_{j}(n)$ denote the Fourier coefficients of an orthonormal basis $\{v_{j}(\cdot)\}$ of ${\tilde{\mathcal{L}}}_{k}(N,\nu)$ . For each singular cusp $\mathfrak{a}$ of $(\Gamma,\nu)$ , let $E_{\mathfrak{a}}(\cdot,s)$ be the associated Eisenstein series. Let $\rho_{\mathfrak{a}}(n,r)$ be defined as in (3.3). Suppose $x\geq 1$ . For $n\neq 0$ we factor $B\tilde{n}=t_{n}u_{n}^{2}w_{n}^{2}$ where $t_{n}$ is square-free, $u_{n}|M^{\infty}$ is positive and ${(w_{n},M)=1}$ . Then we have

x^{k\operatorname{sgn}{\tilde{n}}}|{\tilde{n}}|\left(\sum_{|r_{j}|\leq x}\frac{|\rho_{j}(n)|^{2}}{\operatorname{ch}\pi r_{j}}+\sum_{\mathrm{singular\ }\mathfrak{a}}\int_{-x}^{x}\frac{|\rho_{\mathfrak{a}}(n,r)|^{2}}{\operatorname{ch}\pi r}dr\right)\ll_{N,\nu,\varepsilon}\left(|{\tilde{n}}|^{\frac{131}{294}}+u_{n}\right)x^{3}|\tilde{n}|^{\varepsilon}.

Remark.

We make some remarks about the weight $k$ :

•

The trace formula (Theorem 3.3) works for $k=\frac{1}{2}$ and $\frac{3}{2}$ .
•

The estimates on $\widehat{\phi}$ and $\widetilde{\phi}$ in the previous section work for $k=\frac{1}{2}$ and $\frac{3}{2}$ .
•

The above two propositions work for $k=\frac{1}{2}$ and ${-\tfrac{1}{2}}$ .

Therefore, in this section, we separate the proof of Theorem 1.3 into two cases $k=\frac{1}{2}$ and ${-\tfrac{1}{2}}$ . In the second case we will apply the Maass lowering operator $L_{\frac{3}{2}}$ (2.14) to connect the eigenforms of weight $\frac{3}{2}$ and weight $-\frac{1}{2}$ .

We declare that all implicit constants for the bounds in this section depend on $N$ , $\nu$ and $\varepsilon$ , and we drop the subscripts unless specified.

Since the exceptional spectral parameter $r_{j}\in i(0,\frac{1}{4}]$ of Laplacian $\Delta_{k}$ on $\Gamma=\Gamma_{0}(N)$ is discrete, $t_{j}=\operatorname{Im}r_{j}$ has a positive lower bound denoted as $\underline{t}>0$ depending on $N$ and $\nu$ . We also have $2\operatorname{Im}r_{\Delta}\leq\theta$ assuming $H_{\theta}$ (2.16) by Theorem 3.4. For simplicity let

A(m,n)\vcentcolon=(\tilde{m}^{\frac{131}{294}}+u_{m})^{\frac{1}{2}}(\tilde{n}^{\frac{131}{294}}+u_{n})^{\frac{1}{2}}\ll(\tilde{m}\tilde{n})^{\frac{131}{588}}+\tilde{m}^{\frac{131}{588}}u_{n}^{\frac{1}{2}}+\tilde{n}^{\frac{131}{588}}u_{m}^{\frac{1}{2}}+(u_{m}u_{n})^{\frac{1}{2}}

and

\displaystyle\begin{split}A_{u}(m,n)&\vcentcolon=A(m,n)^{\frac{1}{4}}(\tilde{m}\tilde{n})^{\frac{3}{16}}\\ &\ll(\tilde{m}\tilde{n})^{\frac{143}{588}}+\tilde{m}^{\frac{143}{588}}\tilde{n}^{\frac{3}{16}}\,u_{n}^{\frac{1}{8}}+\tilde{m}^{\frac{3}{16}}u_{m}^{\frac{1}{8}}\tilde{n}^{\frac{143}{588}}+(\tilde{m}\tilde{n})^{\frac{3}{16}}(u_{m}u_{n})^{\frac{1}{8}}.\end{split}

(5.2)

Recall the notations in Setting 3.1 and Setting 3.2. The following inequalities will be used later in the proof:

A(m,n)\ll A_{u}(m,n)\ll(\tilde{m}\tilde{n})^{\frac{1}{4}};

(5.3)

\left(\frac{a}{x}\right)^{\beta}A(m,n)\ll A_{u}(m,n)\quad\text{for }0\leq\beta\leq\frac{3}{2},\quad\text{when }x\gg A_{u}(m,n)^{2}.

(5.4)

5.1. On the case $k=\frac{1}{2}$

Let $\rho_{j}(n)$ denote the coefficients of an orthonormal basis $\{v_{j}(\cdot)\}$ of ${\tilde{\mathcal{L}}}_{\frac{1}{2}}(N,\nu)$ . For each singular cusp $\mathfrak{a}$ of $\Gamma=\Gamma_{0}(N)$ , let $\rho_{\mathfrak{a}}(n,r)$ be defined as in (3.3). Recall the definition of $\tau_{j}(m,n)$ in Theorem 1.3 and the notations in Settings 3.1 and 3.2. We claim the following proposition:

Proposition 5.3.

With the same setting as Theorem 1.3 for $k=\frac{1}{2}$ , when $2x\geq A_{u}(m,n)^{2}$ , we have

\displaystyle\begin{split}\sum_{\begin{subarray}{c}x<c\leq 2x\\ N|c\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{1}{4}]}(2^{2s_{j}-1}-1)\tau_{j}(m,n)\frac{x^{2s_{j}-1}}{2s_{j}-1}\ll\left(x^{\frac{1}{6}}+A_{u}(m,n)\right)(\tilde{m}\tilde{n}x)^{\varepsilon}.\end{split}

(5.5)

We first show that Proposition 5.3 implies Theorem 1.3 in the case $k=\frac{1}{2}$ , which follows from a similar process as [Sun23, after Proposition 9.1]. Recall that $2\operatorname{Im}r_{j}=2s_{j}-1$ for $r_{j}\in i(0,\frac{1}{4}]$ and that the corresponding exceptional eigenvalue $\lambda_{j}=\frac{1}{4}+r_{j}^{2}=s_{j}(1-s_{j})$ . The sum to be estimated is

\sum_{N|c\leq X}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{1}{4}]}\tau_{j}(m,n)\frac{X^{2\operatorname{Im}r_{j}}}{2\operatorname{Im}r_{j}},

(5.6)

where

\tau_{j}(m,n)=2i^{\frac{1}{2}}\overline{\rho_{j}(m)}\rho_{j}(n)\pi^{1-2s_{j}}(4\tilde{m}\tilde{n})^{1-s_{j}}\frac{\Gamma(s_{j}+\frac{1}{4})\Gamma(2s_{j}-1)}{\Gamma(s_{j}-\frac{1}{4})}.

Since $t_{j}=\operatorname{Im}r_{j}\in[\underline{t},\frac{1}{4}]$ and $s_{j}=\operatorname{Im}r_{j}+\frac{1}{2}\in[\underline{t}+\frac{1}{2},\frac{3}{4}]$ , the quantity

\pi^{1-2s_{j}}4^{1-s_{j}}\frac{\Gamma(s_{j}+\frac{1}{4})\Gamma(2s_{j}-1)}{\Gamma(s_{j}-\frac{1}{4})}

is bounded from above and below. By Proposition 5.2,

\frac{\tau_{j}(m,n)}{2s_{j}-1}\ll|\rho_{j}(m)\rho_{j}(n)|(\tilde{m}\tilde{n})^{1-s_{j}}\ll A(m,n)(\tilde{m}\tilde{n})^{\frac{1}{2}-s_{j}+\varepsilon}.

(5.7)

When $X\ll A_{u}(m,n)^{2}$ , since $A(m,n)\ll(\tilde{m}\tilde{n})^{\frac{1}{4}}$ by (5.3),

\displaystyle\begin{split}\tau_{j}(m,n)\frac{X^{2s_{j}-1}}{2s_{j}-1}&\ll A(m,n)|\tilde{m}\tilde{n}|^{\frac{1}{2}-s_{j}+\varepsilon}A_{u}(m,n)^{4s_{j}-2}\\ &=A(m,n)^{s_{j}+\frac{1}{2}}|\tilde{m}\tilde{n}|^{\frac{1}{8}-\frac{1}{4}s_{j}+\varepsilon}\ll A_{u}(m,n)(\tilde{m}\tilde{n})^{\varepsilon}.\end{split}

(5.8)

So in this case we get Theorem 1.3 where the $\tau_{j}$ terms are absorbed in the errors.

When $X\geq A_{u}(m,n)^{2}$ , the segment for summing Kloosterman sums on $1\leq c\leq A_{u}(m,n)^{2}$ contributes a $O_{\nu,\varepsilon}(A_{u}(m,n)|\tilde{m}\tilde{n}|^{\varepsilon})$ by condition (2) of Definition 1.1. The segment for $A_{u}(m,n)^{2}\leq c\leq X$ can be broken into no more than $O(\log X)$ dyadic intervals $x<c\leq 2x$ with $A_{u}(m,n)^{2}\leq x\leq\frac{X}{2}$ and we use Proposition 5.3 for both the Kloosterman sum and the $\tau_{j}$ terms. In summing dyadic intervals, for each $r_{j}\in i(0,\frac{1}{4}]$ , we get

	$\displaystyle\sum_{\ell=1}^{\left\lceil\log_{2}\left(X/A_{u}(m,n)^{2}\right)\right\rceil}$	$\displaystyle\frac{(2^{2s_{j}-1}-1)\tau_{j}(m,n)}{2s_{j}-1}\left(\frac{X}{2^{\ell}}\right)^{2s_{j}-1}$
		$\displaystyle\qquad=\frac{\tau_{j}(m,n)}{2s_{j}-1}X^{2s_{j}-1}\left(1-2^{(1-2s_{j})\left\lceil\log_{2}\left(X/A_{u}(m,n)^{2}\right)\right\rceil}\right).$

The difference between the above quantity and the quantity $\tau_{j}(m,n)\dfrac{X^{2s_{j}-1}}{2s_{j}-1}$ in (5.6) is

{\tau_{j}(m,n)}\frac{X^{2s_{j}-1}}{2s_{j}-1}\cdot 2^{(1-2s_{j})\left\lceil\log_{2}\left(X/A_{u}(m,n)^{2}\right)\right\rceil}\ll\frac{\tau_{j}(m,n)}{2s_{j}-1}A_{u}(m,n)^{4s_{j}-2}\ll A_{u}(m,n)

(5.9)

by (5.7). In conclusion, for $X\geq A_{u}(m,n)^{2}$ we get

	$\displaystyle\sum_{\begin{subarray}{c}N\|c\leq X\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{1}{4}]}\tau_{j}(m,n)\frac{X^{2s_{j}-1}}{2s_{j}-1}$
	$\displaystyle=\sum_{\begin{subarray}{c}A_{u}(m,n)^{2}<c\leq X\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{1}{4}]}\tau_{j}(m,n)\frac{X^{2s_{j}-1}}{2s_{j}-1}+O(A_{u}(m,n)\|\tilde{m}\tilde{n}\|^{\varepsilon})$
	$\displaystyle=\sum_{\ell=1}^{\left\lceil\log_{2}\left(X/A_{u}(m,n)^{2}\right)\right\rceil}\left(\sum_{\begin{subarray}{c}\frac{X}{2^{\ell}}<c\leq\frac{X}{2^{\ell-1}}\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{1}{4}]}\frac{(2^{2s_{j}-1}-1)\tau_{j}(m,n)}{2s_{j}-1}\left(\frac{X}{2^{\ell}}\right)^{2s_{j}-1}\right)$
	$\displaystyle\quad+O(A_{u}(m,n)\|\tilde{m}\tilde{n}\|^{\varepsilon})$
	$\displaystyle\ll\left(X^{\frac{1}{6}}+A_{u}(m,n)\right)\|\tilde{m}\tilde{n}X\|^{\varepsilon}$

where the second equality follows from (5.9) and the last inequality is by Proposition 5.3. Theorem 1.3 follows in the case $k=\frac{1}{2}$ .

The proof of Proposition 5.3 takes the rest of this subsection. For $r_{j}\in i(0,\frac{1}{4}]$ , by Proposition 5.2 we have

\sqrt{\tilde{m}\tilde{n}}\;\overline{\rho_{j}(m)}\rho_{j}(n)\ll A(m,n)(\tilde{m}\tilde{n})^{\varepsilon}.

Recall that $a=4\pi\sqrt{\tilde{m}\tilde{n}}$ and $\delta=\frac{1}{3}$ in Setting 3.1. Thanks to $H_{\frac{7}{64}}$ (2.16) and Proposition 3.4, when $r_{j}=it_{j}\in i(0,\frac{\theta}{2}]$ we have $2t_{j}<\delta=\frac{1}{3}$ . Since $2x\geq A_{u}(m,n)^{2}$ by hypothesis, it follows from (5.4) that

\displaystyle\sqrt{\tilde{m}\tilde{n}}\;\overline{\rho_{j}(m)}\rho_{j}(n)\left(\frac{x^{2t_{j}-\delta}}{a^{2t_{j}}}+\frac{a^{2t_{j}}}{x^{2t_{j}}}+1\right)\ll A_{u}(m,n)(\tilde{m}\tilde{n})^{\varepsilon}.

Applying Lemma 4.7 where $t_{j}\in[\underline{t},\frac{\theta}{2}]$ and recalling the definition of $\tau_{j}$ in Theorem 1.3, we get

\displaystyle\begin{split}4\sqrt{\tilde{m}\tilde{n}}&\;\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\\ &=(2^{2s_{j}-1}-1)\tau_{j}(m,n)\frac{x^{2s_{j}-1}}{2s_{j}-1}+O\left(A_{u}(m,n)(\tilde{m}\tilde{n})^{\varepsilon}\right).\end{split}

(5.10)

When $r_{j}=\frac{i}{4}$ and $k=\frac{1}{2}$ , Lemma 4.7 and (5.4) give

\displaystyle\begin{split}4\sqrt{\tilde{m}\tilde{n}}\;\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\cos\frac{\pi}{4}}\widehat{\phi}(\tfrac{i}{4})=2(\sqrt{2}-1)\tau_{j}(m,n)x^{\frac{1}{2}}+O\left(x^{\frac{1}{2}-\delta}(\tilde{m}\tilde{n})^{\varepsilon}\right).\end{split}

(5.11)

With the help of (5.10) and (5.11) we break up the left hand side of (5.5) to obtain the following analogue to [Sun23, (9.8)]:

\displaystyle\begin{split}&\left|\sum_{\begin{subarray}{c}x<c\leq 2x\\ N|c\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{1}{4}]}(2^{2s_{j}-1}-1)\tau_{j}(m,n)\frac{x^{2s_{j}-1}}{2s_{j}-1}\right|\\ \leq&\left|\sum_{\begin{subarray}{c}x<c\leq 2x\\ N|c\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{N|c>0}\frac{S(m,n,c,\nu)}{c}\phi\left(\frac{a}{c}\right)\right|+O\left(\left(x^{\frac{1}{2}-\delta}+A_{u}(m,n)\right)(\tilde{m}\tilde{n})^{\varepsilon}\right)\\ &+\left|\sum_{N|c>0}\frac{S(m,n,c,\nu)}{c}\phi\left(\frac{a}{c}\right)-4\sqrt{\tilde{m}\tilde{n}}\sum_{r_{j}\in i(0,\frac{1}{4}]}\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\\ =:&\;S_{1}+O\left(\left(x^{\frac{1}{2}-\delta}+A_{u}(m,n)\right)(\tilde{m}\tilde{n})^{\varepsilon}\right)+S_{2}.\end{split}

(5.12)

The first sum $S_{1}$ above can be estimated by condition (2) of Definition 1.1 as

\displaystyle\begin{split}S_{1}\leq\sum_{\begin{subarray}{c}x-T\leq c\leq x\\ 2x\leq c\leq 2x+2T\\ N|c\end{subarray}}\frac{|S(m,n,c,\nu)|}{c}\ll_{N,\nu,\delta,\varepsilon}x^{\frac{1}{2}-\delta}(\tilde{m}\tilde{n}x)^{\varepsilon}.\end{split}

(5.13)

We then prove a bound for $S_{2}$ . By Theorem 3.3, we have

S_{2}\ll|\mathcal{U}_{\frac{1}{2}}|+\left|\sqrt{\tilde{m}\tilde{n}}\sum_{r_{j}\geq 0}\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})+\sqrt{\tilde{m}\tilde{n}}\sum_{\mathrm{singular\ }\mathfrak{a}}\int_{-\infty}^{\infty}\overline{\rho_{\mathfrak{a}}(m,r)}\rho_{\mathfrak{a}}(n,r)\frac{\widehat{\phi}(r)}{\operatorname{ch}\pi r}dr\right|.

(5.14)

5.1.1. Contribution from holomorphic forms

For $k=\frac{1}{2}$ or $\frac{3}{2}$ , recall the notation $\mathscr{B}_{k}$ before Theorem 3.3. For $l\geq 1$ , let $\{F_{j,l}(\cdot)\}_{j}$ be an orthonormal basis of $S_{k+2l}(N,\nu)$ with Fourier coefficient $a_{F,j,l}$ . By Proposition 3.8, uniformly for every $l\geq 1$ with $d_{l}\vcentcolon=\dim S_{k+2l}(N,\nu)$ , we have $k+2l\geq\frac{5}{2}$ and

	$\displaystyle\frac{\Gamma(k+2l-1)}{(4\pi)^{k+2l-1}(\tilde{m}\tilde{n})^{\frac{k+2l-1}{2}}}\sum_{j=1}^{d_{l}}\overline{a_{F,j,l}(m)}a_{F,j,l}(n)$
	$\displaystyle\leq\left(\frac{\Gamma(k+2l-1)}{(4\pi\tilde{n})^{k+2l-1}}\sum_{j=1}^{d_{l}}\|a_{F,j,l}(m)\|^{2}\right)^{\frac{1}{2}}\left(\frac{\Gamma(k+2l-1)}{(4\pi\tilde{m})^{k+2l-1}}\sum_{j=1}^{d_{l}}\|a_{F,j,l}(n)\|^{2}\right)^{\frac{1}{2}}$
	$\displaystyle\ll(\tilde{m}^{\frac{19}{42}}+u_{m})^{\frac{1}{2}}(\tilde{n}^{\frac{19}{42}}+u_{n})^{\frac{1}{2}}(\tilde{m}\tilde{n})^{\varepsilon}.$

We also have

\sum_{l=1}^{\infty}(k+2l-1)\,|\widetilde{\phi}(k+2l)|\ll 1+\frac{a}{x}

by [Dun18, Lemma 5.1 and proof of Lemma 7.1] and Lemma 4.3. Note that [Dun18, Lemma 5.1] is only for $k=\frac{1}{2}$ , while the same process works for $k=\frac{3}{2}$ . Then the contribution from $\mathcal{U}_{k}$ is

	$\displaystyle\mathcal{U}_{k}$	$\displaystyle=\sum_{l=1}^{\infty}\frac{k+2l-1}{4\pi}\,\widetilde{\phi}\,(k+2l)\frac{\Gamma(k+2l-1)}{(4\pi)^{k+2l-1}(\tilde{m}\tilde{n})^{\frac{k+2l-1}{2}}}\sum_{j=1}^{d_{l}}\overline{a_{F,j,l}(m)}a_{F,j,l}(n)$
		$\displaystyle\ll\left(1+\frac{a}{x}\right)(\tilde{m}^{\frac{19}{42}}+u_{m})^{\frac{1}{2}}(\tilde{n}^{\frac{19}{42}}+u_{n})^{\frac{1}{2}}(\tilde{m}\tilde{n})^{\varepsilon}.$

Recall $a=4\pi\sqrt{\tilde{m}\tilde{n}}$ and (5.2) for the definition of $A_{u}(m,n)$ . Since

\tilde{n}^{\frac{19}{42}}\ll\tilde{n}^{\frac{131}{294}\cdot\frac{1}{4}+\frac{3}{8}}\leq(\tilde{n}^{\frac{131}{294}}+u_{n})^{\frac{1}{4}}\tilde{n}^{\frac{3}{8}}\quad\text{and}\quad u_{n}\ll u_{n}^{\frac{1}{4}}\tilde{n}^{\frac{3}{8}}\leq(\tilde{n}^{\frac{131}{294}}+u_{n})^{\frac{1}{4}}\tilde{n}^{\frac{3}{8}},

we get $(\tilde{m}^{\frac{19}{42}}+u_{m})^{\frac{1}{2}}(\tilde{n}^{\frac{19}{42}}+u_{n})^{\frac{1}{2}}\ll A_{u}(m,n)$ . Moreover, since

\tilde{n}^{\frac{19}{42}+1}\ll\tilde{n}^{\frac{131}{294}\cdot\frac{3}{4}+\frac{9}{8}}\leq(\tilde{n}^{\frac{131}{294}}+u_{n})^{\frac{3}{4}}\tilde{n}^{\frac{9}{8}},\ \quad nu_{n}\ll u_{n}^{\frac{3}{4}}\tilde{n}^{\frac{9}{8}}\leq(\tilde{n}^{\frac{131}{294}}+u_{n})^{\frac{3}{4}}\tilde{n}^{\frac{9}{8}},

and $2x\geq A_{u}(m,n)^{2}$ by hypothesis, we also get

(\tilde{m}^{\frac{19}{42}}+u_{m})^{\frac{1}{2}}(\tilde{n}^{\frac{19}{42}}+u_{n})^{\frac{1}{2}}\cdot\frac{a}{x}\ll(\tilde{m}^{\frac{19}{42}}+u_{m})^{\frac{1}{2}}(\tilde{n}^{\frac{19}{42}}+u_{n})^{\frac{1}{2}}\cdot\frac{a}{A_{u}(m,n)^{2}}\ll A_{u}(m,n).

Finally we conclude

\mathcal{U}_{k}\ll A_{u}(m,n)(\tilde{m}\tilde{n})^{\varepsilon}\qquad\text{for }k=\tfrac{1}{2}\text{ or }\tfrac{3}{2}.

(5.15)

5.1.2. Contribution from Maass cusp forms and Eisenstein series.

We combine the two propositions at the beginning of this section and bounds on $\widehat{\phi}$ in Section 4 to estimate the contribution from the remaining part of $S_{2}$ (5.14) other than $\mathcal{U}_{k}$ . The process is the same as [Sun23, §9.1] for $|r|\leq 1$ as $\widehat{\phi}$ shares the same bound as $\check{\phi}$ there. We record the bounds in the following equations.

Fix $k=\frac{1}{2}$ . In the following estimations we focus on the discrete spectrum $r_{j}\geq 0$ because each bound for $r_{j}\in[a,b]$ for any interval $[a,b]\subset{\mathbb{R}}$ is the same as the bound for $r\in[a,b]\cup[-b,-a]$ in the continuous spectrum. This is a direct result from Proposition 5.1 and Proposition 5.2. Recall that $2x\geq A_{u}(m,n)^{2}$ in the assumption of Proposition 5.3.

For $r\in[0,1)$ , we apply Lemma 4.2, Proposition 5.2 and Cauchy-Schwarz to get

\sqrt{\tilde{m}\tilde{n}}\sum_{r\in[0,1)}\left|\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\ll A(m,n)(\tilde{m}\tilde{n}x)^{\varepsilon}.

(5.16)

For $r\in[1,\frac{a}{x})$ , we apply Proposition 5.2 and $\widehat{\phi}(r)\ll r^{-1}$ from (4.10). Since

\displaystyle\begin{split}S(R)\vcentcolon=\sqrt{\tilde{m}\tilde{n}}\sum_{r\in[1,R]}\left|\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\right|\ll A(m,n)R^{\frac{5}{2}}(\tilde{m}\tilde{n})^{\varepsilon},\end{split}

(5.17)

with the help of (5.4) we have

\displaystyle\begin{split}\sqrt{\tilde{m}\tilde{n}}\sum_{r\in[1,\frac{a}{x})}\left|\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|&\ll r^{-1}S(r)\Big{|}_{r=1}^{\frac{a}{x}}+\int_{1}^{\frac{a}{x}}S(r)r^{-2}dr\\ &\ll A(m,n)\left(\frac{a}{x}\right)^{\frac{3}{2}}(\tilde{m}\tilde{n}x)^{\varepsilon}\ll A_{u}(m,n)(\tilde{m}\tilde{n}x)^{\varepsilon}.\end{split}

(5.18)

Let

P(m,n)\vcentcolon=2(\tilde{m}\tilde{n})^{\frac{1}{8}}A(m,n)^{-\frac{1}{2}}\geq 1.

Divide $r\geq\max(\frac{a}{x},1)$ into two parts: $\max\left(\frac{a}{x},1\right)\leq r<P(m,n)$ and $r\geq\max\left(\frac{a}{x},1,P(m,n)\right)$ . We apply Proposition 5.2 on the first range and $\widehat{\phi}(r)\ll r^{-1}$ from (4.10) to get

\sqrt{\tilde{m}\tilde{n}}\sum_{\max(\frac{a}{x},1)\leq r_{j}<P(m,n)}\left|\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\\ \ll A_{u}(m,n)(\tilde{m}\tilde{n}x)^{\varepsilon}

(5.19)

by partial summation as in (5.18). We divide the second range into dyadic intervals $C\leq r_{j}<2C$ . Applying Proposition 5.1 with $\beta=\frac{1}{2}+\varepsilon$ and $\widehat{\phi}(r)\ll\min(r^{-1},r^{-2}\frac{x}{T})$ from (4.10), we get

\displaystyle\begin{split}\sqrt{\tilde{m}\tilde{n}}&\sum_{C\leq r_{j}<2C}\left|\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\\ &\ll\min\left(C^{-1},C^{-2}\frac{x}{T}\right)C^{-\frac{1}{2}}\left(C^{2}+(\tilde{m}^{\frac{1}{4}}+\tilde{n}^{\frac{1}{4}})C+(\tilde{m}\tilde{n})^{\frac{1}{4}}\right)(\tilde{m}\tilde{n}x)^{\varepsilon}\\ &\ll\left(\min\left(C^{\frac{1}{2}},C^{-\frac{1}{2}}\frac{x}{T}\right)+(\tilde{m}^{\frac{1}{4}}+\tilde{n}^{\frac{1}{4}})C^{-\frac{1}{2}}+(\tilde{m}\tilde{n})^{\frac{1}{4}}C^{-\frac{3}{2}}\right)(\tilde{m}\tilde{n}x)^{\varepsilon}.\end{split}

(5.20)

Next we sum over dyadic intervals. For the first term $\min(C^{\frac{1}{2}},C^{-\frac{1}{2}}\frac{x}{T})$ , when

\min\left(C^{\frac{1}{2}},C^{-\frac{1}{2}}\frac{x}{T}\right)=C^{\frac{1}{2}}:\quad\sum_{\begin{subarray}{c}j\geq 1:\ 2^{j}C=\frac{x}{T}\\ C\geq P(m,n)\end{subarray}}C^{\frac{1}{2}}\leq\sum_{j=1}^{\infty}2^{-\frac{j}{2}}\left(\frac{x}{T}\right)^{\frac{1}{2}}\ll\left(\frac{x}{T}\right)^{\frac{1}{2}},

and when

\min\left(C^{\frac{1}{2}},C^{-\frac{1}{2}}\frac{x}{T}\right)=C^{-\frac{1}{2}}\frac{x}{T}:\quad\sum_{j\geq 0:\ C=2^{j}\frac{x}{T}}C^{-\frac{1}{2}}\frac{x}{T}\leq\sum_{j=0}^{\infty}2^{-\frac{j}{2}}\left(\frac{x}{T}\right)^{\frac{1}{2}}\ll\left(\frac{x}{T}\right)^{\frac{1}{2}}.

So after summing up from (5.20), recalling $T\asymp x^{1-\delta}$ in Setting 3.1, using $C\geq P(m,n)$ and (5.3), we have

\displaystyle\begin{split}\sqrt{\tilde{m}\tilde{n}}&\sum_{r_{j}\geq\max(\frac{a}{x},1,P(m,n))}\left|\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\\ &\ll\left(\left(\frac{x}{T}\right)^{\frac{1}{2}}+(\tilde{m}+\tilde{n})^{\frac{1}{4}}(\tilde{m}\tilde{n})^{-\frac{1}{16}}A(m,n)^{\frac{1}{4}}+(\tilde{m}\tilde{n})^{\frac{1}{16}}A(m,n)^{\frac{3}{4}}\right)(\tilde{m}\tilde{n}x)^{\varepsilon}\\ &\ll\left(x^{\frac{\delta}{2}}+(\tilde{m}\tilde{n})^{\frac{3}{16}}A(m,n)^{\frac{1}{4}}\right)(\tilde{m}\tilde{n}x)^{\varepsilon}.\end{split}

(5.21)

Combining (5.19) and (5.21) we have

\sqrt{\tilde{m}\tilde{n}}\sum_{r_{j}\geq\max(\frac{a}{x},1)}\left|\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\\ \ll\left(x^{\frac{\delta}{2}}+A_{u}(m,n)\right)(\tilde{m}\tilde{n}x)^{\varepsilon}.

(5.22)

From (5.12), (5.13), (5.14), (5.15), (5.16), (5.18), and (5.22), we get

	$\displaystyle\sum_{\begin{subarray}{c}x<c\leq 2x\\ N\|c\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{1}{4}]}(2^{2s_{j}-1}-1)$	$\displaystyle\tau_{j}(m,n)\frac{x^{2s_{j}-1}}{2s_{j}-1}$
		$\displaystyle\ll\left(x^{\frac{1}{2}-\delta}+x^{\frac{\delta}{2}}+A_{u}(m,n)\right)(\tilde{m}\tilde{n}x)^{\varepsilon}.$

Proposition 5.3 follows by choosing $\delta=\frac{1}{3}$ . We finish the proof of Theorem 1.3 in weight $\frac{1}{2}$ .

5.2. On the case $k=-\frac{1}{2}$

Recall the remark after Proposition 5.2. Let $\rho_{j}^{\prime}(n)$ denote the Fourier coefficients of an orthonormal basis $\{v_{j}^{\prime}(\cdot)\}$ of ${\tilde{\mathcal{L}}}_{\frac{3}{2}}(N,\nu)$ . For each singular cusp $\mathfrak{a}$ of $(\Gamma,\nu)$ , let $E_{\mathfrak{a}}^{\prime}(\cdot,s)$ be the associated Eisenstein series in weight $\frac{3}{2}$ . Let $\rho_{\mathfrak{a}}^{\prime}(n,r)$ be defined as in (3.3) associated with $E_{\mathfrak{a}}^{\prime}(z,\frac{1}{2}+ir)$ for $r\in{\mathbb{R}}$ .

Recall the definition of the Maass lowering operator $L_{k}$ in (2.14) and $H_{\theta}$ (2.16) for $\theta=\frac{7}{64}$ . By [DFI02, (4.52)] (where they used $\Lambda_{k}$ for the lowering operator and $\lambda(s)=s(1-s)$ ), the set

\left\{v_{j}\vcentcolon=\left(\tfrac{1}{16}+r_{j}^{2}\right)^{-\frac{1}{2}}L_{\frac{3}{2}}v_{j}^{\prime}:\ r_{j}\neq\tfrac{i}{4}\right\}\text{ is an orthonormal basis of }\bigoplus_{r_{j}\neq\frac{i}{4}}{\tilde{\mathcal{L}}}_{-\frac{1}{2}}(N,\nu,r_{j}).

Combining [DFI02, (4.36), (4.27) and the last equation of p. 502], for $r_{j}\neq\frac{i}{4}$ and $\tilde{n}>0$ , since

L_{\frac{3}{2}}\left(W_{\frac{3}{4}\tilde{n},\,\operatorname{Im}r}(4\pi\tilde{n}y)e(\tilde{n}x)\right)=-(\tfrac{1}{16}+r^{2})W_{-\frac{\tilde{n}}{4},\,\operatorname{Im}r}(4\pi\tilde{n}y)e(\tilde{n}x),

the Fourier coefficient $\rho_{j}(n)$ of $v_{j}$ satisfies

\rho_{j}(n)=-(\tfrac{1}{16}+r^{2})^{\frac{1}{2}}\rho_{j}^{\prime}(n)\qquad\text{for }r_{j}\neq\tfrac{i}{4},\ \tilde{n}>0,

(5.23)

and then

|\rho_{j}(n)|\asymp|\rho_{j}^{\prime}(n)|\quad\text{if }|r_{j}|\leq 1,\ \operatorname{Im}r_{j}\leq\tfrac{\theta}{2}\qquad\text{and}\quad|\rho_{j}(n)|\asymp r|\rho_{j}^{\prime}(n)|\quad\text{if }r_{j}\geq 1,

(5.24)

where the bound $2\operatorname{Im}r_{j}\leq\theta$ is from Proposition 3.4.

In the case $r_{j}=\frac{i}{4}$ , (2.21) and (2.22) show that $\rho_{j}(n)=0$ and

\tau_{j}(m,n)=0\qquad\text{for\ }\tilde{n}>0,\ r_{j}=\tfrac{i}{4}.

(5.25)

Moreover, by [DFI02, (4.48)], if $E_{\mathfrak{a}}(z,s)$ is the Eisenstein series defined in weight $-\frac{1}{2}$ , then

L_{\frac{3}{2}}E_{\mathfrak{a}}^{\prime}(z,\tfrac{1}{2}+ir)=(\tfrac{1}{4}-ir)E_{\mathfrak{a}}(z,s)\quad\text{and}\quad(\tfrac{1}{16}+r^{2})^{\frac{1}{2}}|\rho_{\mathfrak{a}}^{\prime}(n,r)|=|\rho_{\mathfrak{a}}(n,r)|.

We also get

|\rho_{\mathfrak{a}}(n,r)|\asymp|\rho_{\mathfrak{a}}^{\prime}(n,r)|\quad\text{if }r\in[-1,1]\qquad\text{and}\quad|\rho_{\mathfrak{a}}(n,r)|\asymp r|\rho_{\mathfrak{a}}^{\prime}(n,r)|\quad\text{if }|r|\geq 1.

(5.26)

We have the following proposition:

Proposition 5.4.

With the same setting as Theorem 1.3 for $k=-\frac{1}{2}$ , when $2x\geq A_{u}(m,n)^{2}$ , we have

\displaystyle\begin{split}\sum_{\begin{subarray}{c}x<c\leq 2x\\ N|c\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{\theta}{2}]}(2^{2s_{j}-1}-1)\tau_{j}(m,n)\frac{x^{2s_{j}-1}}{2s_{j}-1}\ll\left(x^{\frac{1}{6}}+A_{u}(m,n)\right)(\tilde{m}\tilde{n}x)^{\varepsilon},\end{split}

Note that here $\tau_{j}(m,n)$ is defined in weight $-\frac{1}{2}$ , i.e.

\tau_{j}(m,n)=2i^{-\frac{1}{2}}\overline{\rho_{j}(m)}\rho_{j}(n)\pi^{1-2s_{j}}(4\tilde{m}\tilde{n})^{1-s_{j}}\frac{\Gamma(s_{j}-\frac{1}{4})\Gamma(2s_{j}-1)}{\Gamma(s_{j}+\frac{1}{4})}

where $\rho_{j}(n)$ is from (5.24) as the Fourier coefficient of $v_{j}\in{\tilde{\mathcal{L}}}_{-\frac{1}{2}}(N,\nu,r_{j})$ .

The proof that Proposition 5.4 implies Theorem 1.3 in the case $k=-\frac{1}{2}$ is the same as the case of weight $\frac{1}{2}$ before. This is because $\tau_{j}(m,n)=0$ for $r_{j}=\frac{i}{4}$ (5.25) and because (5.7), (5.8) and (5.9) still hold for $r_{j}\in i(0,\frac{\theta}{2}]$ (the process only involves estimates on $\rho_{j}(n)$ with some applications of Proposition 5.2 in weight $-\frac{1}{2}$ ). In the rest of this subsection we prove Proposition 5.4.

First we show that the main terms corresponding to $r_{j}=it_{j}\in i(0,\frac{\theta}{2}]$ are the same when we shift the weight between $-\frac{1}{2}$ and $\frac{3}{2}$ . Recall $s_{j}=\frac{1}{2}+t_{j}$ . Let $\tau_{j}^{\prime}(m,n)$ denote the corresponding coefficients for $x^{2s_{j}-1}$ in weight $\frac{3}{2}$ :

\tau_{j}^{\prime}(m,n)=2e^{\frac{3\pi i}{4}}\overline{\rho_{j}^{\prime}(m)}\rho_{j}^{\prime}(n)\pi^{-2t_{j}}(4\tilde{m}\tilde{n})^{\frac{1}{2}-t_{j}}\frac{\Gamma(\frac{5}{4}+t_{j})\Gamma(2t_{j})}{\Gamma(t_{j}-\frac{1}{4})},

where $\rho_{j}^{\prime}(n)$ is defined at the beginning of this subsection.

We claim that

\tau_{j}^{\prime}(m,n)=\tau_{j}(m,n),\quad\text{for }\tilde{m},\tilde{n}>0\text{ and }r_{j}\in i(0,\tfrac{1}{4}].

(5.27)

When $r_{j}=\frac{i}{4}$ , this is true because both of them equal to zero by (5.25) and $\Gamma(0)=\infty$ . When $r_{j}\in i(0,\frac{\theta}{2}]$ ,

	$\displaystyle\tau_{j}(m,n)$	$\displaystyle=2e^{-\frac{\pi i}{4}}\overline{\rho_{j}(m)}\rho_{j}(n)\pi^{-2t_{j}}(4\tilde{m}\tilde{n})^{\frac{1}{2}-t_{j}}\frac{\Gamma(\frac{1}{4}+t_{j})\Gamma(2t_{j})}{\Gamma(\frac{3}{4}+t_{j})}$
		$\displaystyle=-2e^{\frac{3\pi i}{4}}\left(\frac{1}{16}-t_{j}^{2}\right)\overline{\rho_{j}^{\prime}(m)}\rho_{j}^{\prime}(n)\pi^{-2t_{j}}(4\tilde{m}\tilde{n})^{\frac{1}{2}-t_{j}}\frac{\Gamma(\frac{5}{4}+t_{j})/(\frac{1}{4}+t_{j})}{(-\frac{1}{4}+t_{j})\Gamma(-\frac{1}{4}+t_{j})}\Gamma(2t_{j})$
		$\displaystyle=\tau_{j}^{\prime}(m,n).$

Recall that the definition on $\widehat{\phi}$ (3.5) is for weight $k\geq 0$ and here we use $\widehat{\phi}$ for weight $\frac{3}{2}$ . We derive

4\sqrt{\tilde{m}\tilde{n}}\;\frac{\overline{\rho_{j}^{\prime}(m)}\rho_{j}^{\prime}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})=(2^{2s_{j}-1}-1)\tau_{j}^{\prime}(m,n)\frac{x^{2s_{j}-1}}{2s_{j}-1}+O\left(A_{u}(m,n)(\tilde{m}\tilde{n})^{\varepsilon}\right).

(5.28)

by the same process as we derive (5.10) above. Since $\tau_{j}^{\prime}(m,n)=0$ when $r_{j}=\frac{i}{4}$ , we have $2t_{j}\leq\theta<\delta$ (with $\theta=\frac{7}{64}$ (2.16) and $\delta=\frac{1}{3}$ chosen later) by Proposition 3.4 and still get

\displaystyle\begin{split}&\left|\sum_{\begin{subarray}{c}x<c\leq 2x\\ N|c\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{\theta}{2}]}(2^{2s_{j}-1}-1)\tau_{j}^{\prime}(m,n)\frac{x^{2s_{j}-1}}{2s_{j}-1}\right|\\ \leq&\left|\sum_{\begin{subarray}{c}x<c\leq 2x\\ N|c\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{N|c>0}\frac{S(m,n,c,\nu)}{c}\phi\left(\frac{a}{c}\right)\right|+O\left(A_{u}(m,n)(\tilde{m}\tilde{n})^{\varepsilon}\right)\\ &+\left|\sum_{N|c>0}\frac{S(m,n,c,\nu)}{c}\phi\left(\frac{a}{c}\right)-4\sqrt{\tilde{m}\tilde{n}}\sum_{r_{j}\in i(0,\frac{\theta}{2}]}\frac{\overline{\rho_{j}^{\prime}(m)}\rho_{j}^{\prime}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\\ =:&\;S_{3}+O\left(A_{u}(m,n)(\tilde{m}\tilde{n})^{\varepsilon}\right)+S_{4}.\end{split}

(5.29)

The first sum $S_{3}$ above can be estimated similarly by condition (2) of Definition 1.1 as

\displaystyle\begin{split}S_{3}\leq\sum_{\begin{subarray}{c}x-T\leq c\leq x\\ 2x\leq c\leq 2x+2T\\ N|c\end{subarray}}\frac{|S(m,n,c,\nu)|}{c}\ll_{N,\nu,\delta,\varepsilon}x^{\frac{1}{2}-\delta}(\tilde{m}\tilde{n}x)^{\varepsilon}.\end{split}

(5.30)

By Theorem 3.3,

S_{4}\ll|\mathcal{U}_{\frac{3}{2}}|+\left|\sqrt{\tilde{m}\tilde{n}}\sum_{r_{j}\geq 0}\frac{\overline{\rho_{j}^{\prime}(m)}\rho_{j}^{\prime}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})+\sqrt{\tilde{m}\tilde{n}}\sum_{\mathrm{singular\ }\mathfrak{a}}\int_{-\infty}^{\infty}\overline{\rho_{\mathfrak{a}}^{\prime}(m,r)}\rho_{\mathfrak{a}}^{\prime}(n,r)\frac{\widehat{\phi}(r)}{\operatorname{ch}\pi r}dr\right|.

The bound for $\mathcal{U}_{\frac{3}{2}}$ is done in (5.15). Estimates for the remaining part of $S_{4}$ follow from the same process as §5.1.2 in the case of weight $\frac{1}{2}$ , taking (5.24) and (5.26) into account. For the same reason as the beginning of §5.1.2, we just record the bounds with respect to the discrete spectrum here.

For $r\in[0,1)$ , we apply Proposition 5.2, (5.24) and (4.2) to get

\sqrt{\tilde{m}\tilde{n}}\sum_{r\in[0,1)}\left|\frac{\overline{\rho_{j}^{\prime}(m)}\rho_{j}^{\prime}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\ll\sqrt{\tilde{m}\tilde{n}}\sum_{r\in[0,1)}\left|\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\ll A(m,n)(\tilde{m}\tilde{n}x)^{\varepsilon}.

(5.31)

For $r\in[1,\frac{a}{x})$ , we apply Proposition 5.2, $\rho_{j}^{\prime}(n)\ll r_{j}^{-1}|\rho_{j}(n)|$ from (5.24), and $\widehat{\phi}(r)\ll 1$ from (4.10). Since

\displaystyle\begin{split}s(R)\vcentcolon=\sqrt{\tilde{m}\tilde{n}}\sum_{r\in[1,R]}\left|\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}\right|\ll A(m,n)R^{\frac{7}{2}}(\tilde{m}\tilde{n})^{\varepsilon}\end{split}

(5.32)

by Cauchy-Schwarz, with the help of (5.4) we have

\displaystyle\begin{split}\sqrt{\tilde{m}\tilde{n}}\sum_{r_{j}\in[1,\frac{a}{x})}\left|\frac{\overline{\rho_{j}^{\prime}(m)}\rho_{j}^{\prime}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|&\ll\sqrt{\tilde{m}\tilde{n}}\sum_{r_{j}\in[1,\frac{a}{x})}\left|\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}r_{j}^{-2}\right|\\ &\ll r^{-2}s(r)\Big{|}_{r=1}^{\frac{a}{x}}+\int_{1}^{\frac{a}{x}}s(r)r^{-3}dr\\ &\ll A(m,n)\left(\frac{a}{x}\right)^{\frac{3}{2}}(\tilde{m}\tilde{n}x)^{\varepsilon}\\ &\ll A_{u}(m,n)(\tilde{m}\tilde{n}x)^{\varepsilon}.\end{split}

(5.33)

We still let

P(m,n)=2(\tilde{m}\tilde{n})^{\frac{1}{8}}A(m,n)^{-\frac{1}{2}}\geq 1

and divide $r\geq\max(\frac{a}{x},1)$ into two parts: $\max\left(\frac{a}{x},1\right)\leq r<P(m,n)$ and $r\geq\max\left(\frac{a}{x},1,P(m,n)\right)$ . In the first range, we apply Proposition 5.2, (5.24) and $\widehat{\phi}(r)\ll 1$ from (4.10) to get

\sqrt{\tilde{m}\tilde{n}}\sum_{\max(\frac{a}{x},1)\leq r_{j}<P(m,n)}\left|\frac{\overline{\rho_{j}^{\prime}(m)}\rho_{j}^{\prime}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\\ \ll A_{u}(m,n)(\tilde{m}\tilde{n}x)^{\varepsilon}

(5.34)

by partial summation similar as (5.33). We divide the second range into dyadic intervals $C\leq r_{j}<2C$ and apply Proposition 5.1, (5.24) and $\widehat{\phi}(r)\ll\min(1,\frac{x}{rT})$ from (4.10):

\displaystyle\begin{split}\sqrt{\tilde{m}\tilde{n}}&\sum_{C\leq r_{j}<2C}\left|\frac{\overline{\rho_{j}^{\prime}(m)}\rho_{j}^{\prime}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\ll\sqrt{\tilde{m}\tilde{n}}\sum_{C\leq r_{j}<2C}\left|\frac{\overline{\rho_{j}(m)}\rho_{j}(n)}{\operatorname{ch}\pi r_{j}}r_{j}^{-2}\widehat{\phi}(r_{j})\right|\\ &\ll\min\left(1,\frac{x}{CT}\right)C^{-2}\left(C^{\frac{5}{2}}+(\tilde{m}^{\frac{1}{4}}+\tilde{n}^{\frac{1}{4}})C^{\frac{3}{2}}+(\tilde{m}\tilde{n})^{\frac{1}{4}}C^{\frac{1}{2}}\right)(\tilde{m}\tilde{n}x)^{\varepsilon}\\ &\ll\left(\min\left(C^{\frac{1}{2}},C^{-\frac{1}{2}}\frac{x}{T}\right)+(\tilde{m}^{\frac{1}{4}}+\tilde{n}^{\frac{1}{4}})C^{-\frac{1}{2}}+(\tilde{m}\tilde{n})^{\frac{1}{4}}C^{-\frac{3}{2}}\right)(\tilde{m}\tilde{n}x)^{\varepsilon}.\end{split}

(5.35)

Summing up from (5.35) similar as we did after (5.20) and recalling $T\asymp x^{1-\delta}$ in Setting 3.1, we have

\displaystyle\begin{split}\sqrt{\tilde{m}\tilde{n}}&\sum_{r_{j}\geq\max(\frac{a}{x},1,P(m,n))}\left|\frac{\overline{\rho_{j}^{\prime}(m)}\rho_{j}^{\prime}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\\ &\ll\left(\left(\frac{x}{T}\right)^{\frac{1}{2}}+(\tilde{m}+\tilde{n})^{\frac{1}{4}}(\tilde{m}\tilde{n})^{-\frac{1}{16}}A(m,n)^{\frac{1}{4}}+(\tilde{m}\tilde{n})^{\frac{1}{16}}A(m,n)^{\frac{3}{4}}\right)(\tilde{m}\tilde{n}x)^{\varepsilon}\\ &\ll\left(x^{\frac{\delta}{2}}+(\tilde{m}\tilde{n})^{\frac{3}{16}}A(m,n)^{\frac{1}{4}}\right)(\tilde{m}\tilde{n}x)^{\varepsilon}.\end{split}

(5.36)

From (5.34) and (5.36) we have

\sqrt{\tilde{m}\tilde{n}}\sum_{r_{j}\geq\max(\frac{a}{x},1)}\left|\frac{\overline{\rho_{j}^{\prime}(m)}\rho_{j}^{\prime}(n)}{\operatorname{ch}\pi r_{j}}\widehat{\phi}(r_{j})\right|\\ \ll\left(x^{\frac{\delta}{2}}+A_{u}(m,n)\right)(\tilde{m}\tilde{n}x)^{\varepsilon}.

(5.37)

Combining (5.29), (5.30), (5.15), (5.31), (5.33), and (5.37), we get

	$\displaystyle\sum_{\begin{subarray}{c}x<c\leq 2x\\ N\|c\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{1}{4}]}(2^{2s_{j}-1}-1)$	$\displaystyle\tau_{j}(m,n)\frac{x^{2s_{j}-1}}{2s_{j}-1}$
		$\displaystyle\ll\left(x^{\frac{1}{2}-\delta}+x^{\frac{\delta}{2}}+A_{u}(m,n)\right)(\tilde{m}\tilde{n}x)^{\varepsilon}.$

Proposition 5.4 follows by choosing $\delta=\frac{1}{3}$ and we finish the proof of Theorem 1.3.

Proof of Theorem 1.5.

The proof follows from the same process as [Sun23, §9.2]. Note that we need to restrict $\sum_{r_{j}=\frac{i}{4}}\tau_{j}(m,n)=0$ when $\tilde{m}>0$ , $\tilde{n}>0$ and $k=\frac{1}{2}$ (and the conjugate case $\tilde{m}<0$ , $\tilde{n}<0$ and $k=-\frac{1}{2}$ by (2.8)), otherwise the sum may not converge. ∎

Acknowledgement

The author thanks Professor Scott Ahlgren for his plenty of helpful discussions and suggestions.

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	$\displaystyle\sum_{\begin{subarray}{c}N\|c\leq X\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{1}{4}]}\tau_{j}(m,n)\frac{X^{2s_{j}-1}}{2s_{j}-1}$
	$\displaystyle=\sum_{\begin{subarray}{c}A_{u}(m,n)^{2}<c\leq X\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{1}{4}]}\tau_{j}(m,n)\frac{X^{2s_{j}-1}}{2s_{j}-1}+O(A_{u}(m,n)\|\tilde{m}\tilde{n}\|^{\varepsilon})$
	$\displaystyle=\sum_{\ell=1}^{\left\lceil\log_{2}\left(X/A_{u}(m,n)^{2}\right)\right\rceil}\left(\sum_{\begin{subarray}{c}\frac{X}{2^{\ell}}<c\leq\frac{X}{2^{\ell-1}}\end{subarray}}\frac{S(m,n,c,\nu)}{c}-\sum_{r_{j}\in i(0,\frac{1}{4}]}\frac{(2^{2s_{j}-1}-1)\tau_{j}(m,n)}{2s_{j}-1}\left(\frac{X}{2^{\ell}}\right)^{2s_{j}-1}\right)$
	$\displaystyle\quad+O(A_{u}(m,n)\|\tilde{m}\tilde{n}\|^{\varepsilon})$
	$\displaystyle\ll\left(X^{\frac{1}{6}}+A_{u}(m,n)\right)\|\tilde{m}\tilde{n}X\|^{\varepsilon}$

Uniform bounds for Kloosterman sums of half-integral weight, same-sign case

Abstract.

Key words and phrases:

1. Introduction

Definition 1.1.

Lemma 1.2.

Theorem 1.3.

Remark.

Conjecture 1.4.

Theorem 1.5.

1.1. Application

Corollary 1.6.

2. Background

2.1. Kloosterman sums

2.2. Maass forms

2.3. Holomorphic cusp forms of half-integral weight

3. Trace formula

Setting 3.1.

Setting 3.2.

Theorem 3.3 ([Pro05, §6 Theorem]).

Remark.

3.1. Properties of admissible multipliers

Proposition 3.4.

Proposition 3.5.

Proof.

Proposition 3.6 ([Wai17, Theorem 1]).

Proposition 3.7.

Proof.

Proposition 3.8.

4. Bounds on ϕ~\widetilde{\phi} and ϕ^\widehat{\phi}

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Lemma 4.4.

Remark.

4.1. A special test function

Setting 4.5.

Lemma 4.6.

Proof.

Lemma 4.7.

Proof.

5. Proof of Theorem 1.3 and Theorem 1.5

Proposition 5.1 ([Sun23, Proposition 8.1]).

Remark.

Proposition 5.2 ([Sun23, Proposition 8.2]).

Remark.

5.1. On the case k=12k=\frac{1}{2}

Proposition 5.3.

5.1.1. Contribution from holomorphic forms

5.1.2. Contribution from Maass cusp forms and Eisenstein series.

5.2. On the case k=−12k=-\frac{1}{2}

Proposition 5.4.

Proof of Theorem 1.5.

Acknowledgement

References

4. Bounds on $\widetilde{\phi}$ and $\widehat{\phi}$

5.1. On the case $k=\frac{1}{2}$

5.2. On the case $k=-\frac{1}{2}$