Second Moment of the Prime Geodesic Theorem for $\mathrm{PSL}(2, \mathbb{Z}[i])$

TL;DR

This paper investigates the average behavior of the error term in the Prime Geodesic Theorem for the Picard group, showing that an expected bound holds unconditionally through second moment analysis.

Contribution

It establishes an unconditional average bound for the error term in the Prime Geodesic Theorem for (2, 1[i]) by analyzing its second moment.

Findings

Unconditional second moment bound for the error term.

Average error term behaves as predicted under Lindelf6f hypothesis.

Improves understanding of prime geodesic distribution for (2, 1[i])

Abstract

The remainder $E_\Gamma(X)$ in the Prime Geodesic Theorem for the Picard group $\Gamma = \mathrm{PSL}(2,\mathbb{Z}[i])$ is known to be bounded by $O(X^{3/2+\epsilon})$ under the assumption of the Lindel\"of hypothesis for quadratic Dirichlet $L$-functions over Gaussian integers. By studying the second moment of $E_\Gamma(X)$, we show that on average the same bound holds unconditionally.

Equations

$${\pi }_{\Gamma }(X)\sim \mathrm{li}(X^2).$$

$${\psi }_{\Gamma }(X)=\sum _{N(P)\le X}{\Lambda }_{\Gamma }(N(P)).$$

$${\psi }_{\Gamma }(X)=\frac{X^2}{2}+E_{\Gamma }(X),E_{\Gamma }(X){\ll }_{ϵ}{X}^{5/3+ϵ}.$$

$$E_{\Gamma }(X){\ll }_{ϵ}{X}^{\eta +ϵ},\eta =\frac{3}{2}+\frac{103}{1024}.$$

$$\frac{1}{Y}{\int }_V^{V+Y}|E_{\Gamma }(X){|}^{2}dX\ll V^{3+ϵ}{\left(\frac{V}{Y}\right)}^{2/3}.$$

$$S(T,X)=\sum _{0<r_j\le T}{X}^{i{r}_j},$$

$$\frac{1}{Y}{\int }_V^{V+Y}|S(T,X){|}^{2}dX\ll T^{3+ϵ}{V}^{3/2+ϵ}{Y}^{-1}.$$

$$\frac{1}{V}{\int }_V^{2V}{\left|{\psi }_{\Gamma }(X)-{\psi }_{\Gamma }(X-\eta X)-\eta (1-\eta /2)X^2\right|}^{2}dX{\ll }_{ϵ}{V}^{3+ϵ},$$

$$\frac{1}{Y}{\int }_V^{V+Y}|E_{\Gamma }(X){|}^{2}dX\ll V^{16/5}{\left(\frac{V}{Y}\right)}^{2/5}(\log V{)}^{2/5}.$$

$$S_{{\mathbb{Q}}(i)}(n,m,c)=\sum_{a\in({\mathbb{Z}}[i]/(c))^{\times}}e\big{(}\langle m,a/c\rangle\big{)}e\big{(}\langle n,a^{*}/c\rangle\big{)},$$

$$|S_{\mathbb{Q}(i)}(n,m,c)|\le |(n,m,c)|d(c)N(c{)}^{1/2},$$

$$D+C=U+S,$$

$$\begin{array}{rl}D& =\sum _{j=1}^{\infty }\frac{r_j{\rho }_j(n)\overline{{\rho }_j(m)}}{\sinh \pi r_j}h(r_j),\\ C& =2\pi {\int }_{-\infty }^{\infty }\frac{{\sigma }_{ir}(n){\sigma }_{ir}(m)}{|mn{|}^{ir}|{\zeta }_K(1+ir){|}^2}dr,\\ U& =\frac{{\delta }_{m,n}+{\delta }_{m,-n}}{{\pi }^2}{\int }_{-\infty }^{\infty }{r}^{2}h(r)dr,\\ S& =\sum _{c\in \mathbb{Z}[i]\setminus \{0\}}\frac{S_{\mathbb{Q}(i)}(n,m,c)}{|c{|}^2}{\int }_{-\infty }^{\infty }\frac{i{r}^2}{\sinh \pi r}h(r)H_{ir}\left(\frac{2\pi \sqrt{\overline{mn}}}{c}\right)dr,\end{array}$$

$$H_{\nu }(z)=2^{-2\nu }|z{|}^{2\nu }{J}_{\nu }^{\ast }(z)J_{\nu }^{\ast }(\overline{z}),$$

$$J_{ir}(z)(z/2{)}^{-ir}=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{k!\Gamma (k+1+ir)}{\left(\frac{z}{2}\right)}^{2k}.$$

$$\sum _{r_j}{X}^{i{r}_j}{e}^{-r_j/T}.$$

$$\int_{V}^{V+Y}\Big{|}\sum_{r_{j}}X^{ir_{j}}e^{-r_{j}/T}\Big{|}^{2}dX\ll T^{3}V^{3/2+\epsilon},$$

$$\begin{split}\int_{V}^{V+Y}\!\!\!&|S(T,X)|^{2}dX\\ &\ll T^{\epsilon}\!\!\int_{|\xi|\leq 1}\int_{V}^{V+Y}\Big{|}\sum_{r_{j}}(Xe^{-2\pi\xi})^{ir_{j}}e^{-r_{j}/T}\Big{|}^{2}dX\min(T,|\xi|^{-1})d\xi+T^{4+\epsilon}Y\\ &\ll T^{3+\epsilon}V^{3/2+\epsilon}.\end{split}$$

$$h(r)=\frac{\sinh ((\pi +2i\alpha )r)}{\sinh \pi r},2\alpha =\log X+i/T,$$

$$\frac{1}{N}\sum _{n,r_j}f(|n|)h(r_j)|v_j(n){|}^2=\frac{1}{\pi iN}\sum _{r_j}h(r_j){\int }_{(3)}\tilde{f}(2s)L(s,u_j\otimes u_j)ds,$$

$$L(s,u_j\otimes u_j)=\sum _{n\in \mathbb{Z}[i]\setminus \{0\}}\frac{|v_j(n){|}^2}{N(n{)}^s},$$

$$\begin{array}{rl}\sum _{r_j}{X}^{i{r}_j}{e}^{-r_j/T}& =\frac{c_1}{N}\sum _{n\in \mathbb{Z}[i]\setminus \{0\}}f(|n|){\mathcal{S}}_n(\omega )\\ & +\frac{c_2}{N}{\int }_{(1/2)}\tilde{f}(2s)M_1(s)ds+O(T^2).\end{array}$$

$${\mathcal{S}}_n(\omega )=\sum _{c\in \mathbb{Z}[i]\backslash \{0\}}\frac{S_{\mathbb{Q}(i)}(n,n;c)}{N(c)}\omega \left(\frac{2\pi \bar{n}}{c}\right),$$

$$\omega (z)={\int }_{-\infty }^{+\infty }\frac{i{r}^2}{\sinh (\pi r)}{H}_{ir}(z)h(r)dr.$$

$$H_{\nu }(z)=2^{-2\nu }|z{|}^{2\nu }{J}_{\nu }^{\ast }(z)J_{\nu }^{\ast }(\overline{z}),$$

$$M_1(s)=\sum _{r_j}h(r_j)L(s,u_j\otimes u_j).$$

$$N\ll (TX{)}^A,$$

$${\int }_V^{V+Y}|{\mathcal{S}}_n(\omega ){|}^{2}dX\ll (N{V}^2+T^{3}Y)(NV{)}^{ϵ}.$$

$${\int }_V^{V+Y}|M_1(s){|}^{2}dX\ll |s{|}^A{T}^{6+ϵ}V,$$

$$\begin{split}\int_{V}^{V+Y}\Big{|}\sum_{r_{j}}X^{ir_{j}}e^{-r_{j}/T}\Big{|}^{2}dX&\ll\frac{1}{N}\sum_{N(n)\sim N}\int_{V}^{V+Y}|\mathcal{S}_{n}(\omega)|^{2}dX\\ &+\frac{1}{N}\int_{(1/2)}|s|^{-M}\int_{V}^{V+Y}|M_{1}(s)|^{2}dXds+O(T^{4}).\end{split}$$

Full text

The first author is currently supported by the Labex CEMPI (ANR-11-LABX-0007-01). He also wishes to thank the Mathematics department of King’s College London for the hospitality during the spring and the summer of 2017 and the financial support through the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 335141 Nodal. The second and third author were supported by the MTA Rényi Intézet Lendület Automorphic Research Group. The second author was also partially supported by a Ing. Giorgio Schirillo postdoctoral grant (2017–2018) from INdAM. The third author would also like to thank the Department of Mathematics and Statistics at McGill University and the Centre de Recherches Mathématiques for their hospitality and support.

1. Introduction

Let $M=\Gamma\backslash\mathbb{H}^{2}$ be a hyperbolic surface, with $\Gamma$ a cofinite Fuchsian group, and denote by $\pi_{\Gamma}$ the counting function of the primitive length spectrum of $M$, i.e. $\pi_{\Gamma}(X)$ is the number of primitive closed geodesics on $M$ of length at most $\log X$. The study of $\pi_{\Gamma}$ has a long history dating back to works of Huber [11, 12], Selberg [14, Chapter 10] and others. In particular, for surfaces of arithmetic type, much progress has been made in estimating the asymptotic error term related to $\pi_{\Gamma}$, see e.g. [13], [16], [21]. In three dimensions, that is $M=\Gamma\backslash\mathbb{H}^{3}$ where $\Gamma\subset\mathrm{PSL}(2,\mathbb{C})$ is a cofinite Kleinian group, we know that [8]

[TABLE]

In analogy with the classical prime number theory, it is more convenient to work with the hyperbolic analogue of the Chebyshev function, which is defined as

[TABLE]

Here the sum runs over hyperbolic and loxodromic conjugacy classes of $\Gamma$ of norm at most $X$ and $\Lambda_{\Gamma}$ denotes the hyperbolic von Mangoldt function. That is, $\Lambda_{\Gamma}(N(P))=\log N(P_{0})$, if $P$ is a power of a primitive hyperbolic (or loxodromic) conjugacy class $P_{0}$, and zero otherwise. The classical bound for the remainder term in (1.1) was given by Sarnak [20] in 1983. In the arithmetic case, $\Gamma=\mathrm{PSL}(2,\mathbb{Z}[i])$, his result says that

[TABLE]

The estimate (1.2) for the error term is actually valid for all cofinite Kleinian groups, provided that the contribution from possible small eigenvalues is included in the main term. Sarnak’s pointwise bound (1.2) has been improved for the Picard group in [15, 2, 3]. The current best unconditional bound is due to Balkanova and Frolenkov [3], who showed that

[TABLE]

By assuming the Lindelöf hypothesis for quadratic Dirichlet $L$-functions over Gaussian integers, they obtain $\eta=3/2$. It is not clear how far this is from the truth (see the discussion in Remarks 1.5 and 3.1 in [2]).

The main result of this paper is that the exponent $3/2+\epsilon$ holds on average. This is achieved by studying the second moment of the error term. Namely, we prove the following theorem.

Theorem 1.1**.**

Let $V\geq Y\gg 1$ and $\epsilon>0$. Then

[TABLE]

Theorem 1.1 follows from a short interval second moment estimate for the spectral exponential sum $S(T,X)$, which is defined as

[TABLE]

where $\lambda_{j}=1+r_{j}^{2}$ are the (embedded) eigenvalues of the Laplace–Beltrami operator $\Delta$ on $M$.

Theorem 1.2**.**

Let $V\geq Y\gg 1$ and $T\ll V^{1/2-\epsilon}$. Then

[TABLE]

The connection between the Prime Geodesic Theorem and $S(T,X)$ is given by the explicit formula of Nakasuji, see (6.1).

Remark 1.3**.**

Note that the bound for arbitrary $Y$ in Theorem 1.2 follows by positivity from the estimate over the dyadic interval $[V,2V]$. Nevertheless, Theorem 1.2 allows us to prove a nontrivial result in short intervals in Theorem 1.1 since the parameter $T$ can depend on $Y$. Despite this, we will carry out the proof of Theorem 1.2 in the interval $[V,V+Y]$ in order to highlight how the dependence in $Y$ gets absorbed into the final bound.

As a corollary of Theorem 1.1 we recover the pointwise bound $E_{\Gamma}(X)\ll_{\epsilon}X^{13/8+\epsilon}$ of [2, Theorem 1.1]. Furthermore, our second moment bound (1.4) has immediate consequences analogous to Corollary 1.3 and Equation (1.3) in [2], but we will not write them here explicitly. Finally, we observe that Theorem 1.1 implies that the short interval estimate

[TABLE]

is valid for all $0\leq\eta\leq 1$. In other words, the approximation $\psi_{\Gamma}(X)-\psi_{\Gamma}(X-\eta X)=\eta(1-\eta/2)X^{2}$ holds with the error term $O(X^{3/2+\epsilon})$ in a square mean sense.

A weaker second moment estimate, which is valid for all cofinite $\Gamma$, was obtained in [2, Theorem 1.2], where the authors showed for $V\geq Y\gg 1$ that

[TABLE]

This was proved by using the Selberg trace formula and it is of analogous strength to Sarnak’s bound (see Remark 1.5 in [2]). In our proof we will instead use the Kuznetsov trace formula (see [17, 18]) for $\mathrm{PSL}(2,\mathbb{Z}[i])$, which allows us to get stronger estimates. A key component of our proof is a careful analysis of integrals involving multiple Bessel functions. In particular, by relying on exact formulas, we avoid having to deal with the oscillatory integrals that appear in the proof of Koyama [15] for the pointwise bound. We also incorporate some ideas of [4] and [5] from two dimensions.

The paper is organized as follows. We begin by stating our main tool, the Kuznetsov formula, in section 2. Then, in section 3, we give a detailed outline of the proof of Theorem 1.2 under the assumption of two key estimates, which are stated as Propositions 3.1 and 3.2. In sections 4 and 5 we prove these two estimates. Finally, in section 6 we show how to recover Theorem 1.1 from Theorem 1.2.

2. Kuznetsov formula

The Kuznetsov trace formula relates the Fourier coefficients of cusp forms to Kloosterman sums. For Gaussian integers, Kloosterman sums are defined as

[TABLE]

where $m,n,c\in\mathbb{Z}[i]$, $c\neq 0$; $a^{*}$ denotes the inverse of $a$ modulo the ideal $(c)$; and $\langle x,y\rangle$ denotes the standard inner product on ${\mathbb{R}}^{2}\cong{\mathbb{C}}$. The Kloosterman sums obey Weil’s bound [18, (3.5)]

[TABLE]

which we will use repeatedly. Here $d(c)$ is the number of divisors of $c$.

Theorem 2.1** **(Kuznetsov formula

[17, 18]).

Let $h$ be an even function, holomorphic in $|\Im r|<1/2+\epsilon$, for some $\epsilon>0$, and assume that $h(r)=O((1+|r|)^{-3-\epsilon})$ in the strip. Then, for any non-zero $m,n\in{\mathbb{Z}}[i]$:

[TABLE]

with

[TABLE]

where $\sigma_{s}(n)=\sum_{d|n}N(d)^{s}$ is the divisor function,

[TABLE]

$J_{\nu}$* is the $J$-Bessel function of order $\nu$, and $J_{\nu}^{*}(z)=J_{\nu}(z)(z/2)^{-\nu}$.*

For the definition of the $\rho_{j}$, see the explanation after (3.3). We will also need the power series expansion [9, 8.402] for the $J$-Bessel function:

[TABLE]

3. Outline of proof of Theorem 1.2

In this section we outline the proof of Theorem 1.2. The result for the sharp sum $S(T,X)$ can be deduced from the corresponding result for the smooth sum

[TABLE]

Indeed, if we assume the inequality

[TABLE]

then using a standard Fourier analysis method (see [13, 16]) and the Cauchy–Schwarz inequality we can estimate, for $T<V^{1/2}$ and $Y\leq V$,

[TABLE]

To study the sum in (3.1), we approximate $X^{ir_{j}}e^{-r_{j}/T}$ by a more regular function that we borrow from [6], namely

[TABLE]

which satisfies $h(r)=X^{ir}e^{-r/T}+O(e^{-\pi r})$ [17, 18]. Before applying the Kuznetsov formula, we need to insert the Fourier coefficients into our spectral sum. We do this by means of an extra average and by using the fact that the (normalized) Rankin–Selberg $L$-function has a simple pole at $s=1$ with the residue being an absolute constant.

Consider a smooth function $f$, compactly supported on $[\sqrt{N},\sqrt{2N}]$, satisfying $|f^{(p)}(\xi)|\ll N^{-p/2}$ for every $p\geq 0$ and with mass $\int_{0}^{\infty}f(x)dx=N$. Let $\tilde{f}$ be the Mellin transform of $f$. Then

[TABLE]

where $L(s,u_{j}\otimes u_{j})$ is the Rankin–Selberg $L$-function

[TABLE]

where $v_{j}(n)$ are Fourier coefficients of cusp forms, normalized by the relation $v_{j}(n)\sqrt{\sinh(\pi r_{j})}=\rho_{j}(n)\sqrt{r_{j}}$. We apply the Kuznetsov formula on the left-hand side in (3.3), while on the right-hand side we move the line of integration to $\Re(s)=1/2$, picking up the residue at $s=1$. We obtain, for absolute constants $c_{1},c_{2}$,

[TABLE]

The quantities appearing in (3.4) are described as follows. The term $\mathcal{S}_{n}(\omega)$ is a weighted sum of Gaussian Kloosterman sums,

[TABLE]

where $\omega$ is the integral transform of $h$ that appears in Kuznetsov’s formula, that is,

[TABLE]

The kernel $H_{ir}(z)$ is given by

[TABLE]

with $J_{\nu}^{*}(z)=J_{\nu}(z)(z/2)^{-\nu}$, where $J_{\nu}$ denotes the Bessel function of the first kind.

The term $M_{1}(s)$ in (3.4) is a weighted first moment of Rankin–Selberg $L$-functions:

[TABLE]

Note that the integral on the half line in (3.4) is absolutely convergent since $\tilde{f}(2s)\ll N^{1/2}|s|^{-M}$, for arbitrarily large $M$ (when $\Re(s)=1/2$), and $L(s,u_{j}\otimes u_{j})$ is polynomially bounded in $|s|$. Finally, the term $O(T^{2})$ in (3.4) comes from the identity element and the continuous spectrum in the Kuznetsov formula.

In sections 4 and 5 we will prove the following two estimates that we state as separate propositions. In order to simplify the exposition, we assume that $N$ is bounded polynomially in $X$ and $T$, i.e.

[TABLE]

for some arbitrary $A>0$. Our final choice of $N$ satisfies this condition and thus (3.7) is not restrictive.

Proposition 3.1**.**

Let $V\geq Y\gg 1$, and $V^{\epsilon}\ll T\ll V^{1/2-\epsilon}$. Let $N\geq 1$ be chosen so that (3.7) holds, and suppose $n\in{\mathbb{Z}}[i]$ with $N(n)\asymp N$. Then

[TABLE]

In our proof, the first term in (3.8) will be the dominant one. Since this term does not depend on $Y$, the most interesting result is obtained on the full dyadic interval $[V,2V]$. The same observation was made in Remark 1.3 and it applies to the next proposition as well.

Proposition 3.2**.**

Let $V\geq Y\gg 1$, and $T\gg 1$. Then

[TABLE]

for some absolute constant $A$.

Let us show that Theorem 1.2 follows from the above two propositions. By using the Cauchy–Schwarz inequality and integrating in $X$ in (3.4), we get

[TABLE]

Applying Proposition 3.1 and Proposition 3.2 yields

[TABLE]

We pick $N=T^{3}V^{-1/2}$ and thus arrive at the inequality

[TABLE]

since $T\leq V^{1/2-\epsilon}$. Note that $N\geq 1$ only if $T\geq V^{1/6}$. For $T<V^{1/6}$, the bound (3.9) follows from the trivial estimate $S(T,X)\ll T^{3}$. This proves (3.9) for every value of $T\leq V^{1/2-\epsilon}$, which concludes the proof of Theorem 1.2. It remains to prove Propositions 3.1 and 3.2.

4. Second moment of sums of Kloosterman sums

Next we want to prove Proposition 3.1. In order to do this, we will need to simplify expressions involving $\omega(z)$ according to the size of $\lvert z\rvert$. We will first prove a number of auxiliary lemmas, which are then used in different ranges of the summation in $\mathcal{S}_{n}(\omega)$. Throughout this section we shall assume that $N\geq 1$, $n\in{\mathbb{Z}}[i]$ satisfies $N(n)\asymp N$, and that $T$, $X$, $V$ and $Y$ are real numbers satisfying the inequalities

[TABLE]

Moreover, we recall the mild assumption (3.7) on $N$.

We begin the proof by simplifying the expression defining $\mathcal{S}_{n}(\omega)$. After removing the initial part of the sum, we can replace the weight function $\omega$ by a simpler function $\omega_{0}$ given by

[TABLE]

These two simplifications come at the cost of an admissible error term, as demonstrated in the following lemma.

Lemma 4.1**.**

Let $\mathcal{S}_{n}(\omega)$ be as in (3.5), and let $\omega_{0}$ be as in (4.2). Then

[TABLE]

where

[TABLE]

Proof.

Let us focus first on the portion of the sum where $N(c)\leq 4\pi^{2}N(n)$, i.e. when the complex number $z=2\pi\bar{n}/c$ satisfies $|z|\geq 1$. We start from the definition of $\omega(z)$, see (3.6), and apply an integral representation for the kernel $H_{ir}(z)$ (see [18, Equation (2.10)]). Writing $z=|z|e^{i\theta}$, we have

[TABLE]

When $\lvert r\rvert$ is bounded, we estimate $h(r)$ trivially and use the fact that $K_{2ir}(x)\ll x^{-1/2}$ for all $r\in\mathbb{R}$ and $x>0$ real. Thus the integral over $r$ contributes $O(|z|^{-1/2})$ in this range. Now, for $r$ bounded away from zero, we approximate $h(r)$ by

[TABLE]

and the error contributes again $O(|z|^{-1/2})$. Note also that, after integrating over $\lvert r\rvert=O(1)$, the fraction in (4.4) is bounded by $O(|z|^{-1/2})$. The remaining integral reads

[TABLE]

The integral over $r$ can be evaluated exactly by using the formula [9, 6.795.1]. This gives

[TABLE]

where $a:=2|z|\cos\phi$ and $b:=\alpha-i\pi/2$ (in particular, $|\Im(b)|<\pi/2$). Hence, we arrive at the expression

[TABLE]

Observe that $\Re(\cosh b)\asymp X^{1/2}T^{-1}$ and $|\cosh b|\asymp|\sinh b|\asymp X^{1/2}$. In the range $\cos\phi>\log^{2}T/|z|\Re(\cosh b)$ we bound the integrand in absolute value. Since the exponential is $O(T^{-p})$ for arbitrarily large $p$, the integral contributes $O(|z|^{2}XT^{-p})$. On the other hand, when $\cos\phi\leq\log^{2}T/|z|\Re(\cosh b)$, we integrate by parts in $\phi$ once. This gives a factor $1/|z|\cosh b$ from the exponential and thus the contribution from the integral is $O(T^{2+\epsilon}/|z|X^{1/2})$. All in all, we have proved that, for $|z|\geq 1$, we can estimate

[TABLE]

Summing this for $z=2\pi\bar{n}/c$ and $N(c)\leq 4\pi^{2}N(n)$, and using Weil’s bound to estimate the Kloosterman sums, we get a quantity not bigger than

[TABLE]

This is absorbed in the error in (4.3) since $V^{\epsilon}\ll T\leq V^{1/2-\epsilon}$ and $N$ satisfies (3.7) (in fact, this is the only place where we use this assumption).

It remains to estimate the portion of $\mathcal{S}_{n}(\omega)$ where $N(c)>4\pi^{2}N(n)$, i.e. when $|z|<1$. In this range we expand the $J$-Bessel functions in the definition of $\omega$, (3.6), into power series (see (2.2)). We get

[TABLE]

By Stirling’s formula it follows that, for any $k\geq 0$, we have

[TABLE]

Using (4.6) and the fact that $h(r)\ll e^{-|r|/T}$, we can bound all but the initial part of the double sum in (4.5) (that is, when $k_{1}+k_{2}\geq 1$) by

[TABLE]

By Weil’s bound, summing this for $N(c)>4\pi^{2}N(n)$ gives a contribution of at most $O(N^{1/2+\epsilon}T)$, which is absorbed in the error term in (4.3). We are thus left with the term associated to $k_{1}=k_{2}=0$, which is precisely $\omega_{0}(z)$. ∎

Next we evaluate $\omega_{0}$ with an explicit error term. It turns out that a simple closed formula for $\omega_{0}$ can be given in terms of the $K$-Bessel function of order zero. Estimates for $K_{0}$ and its derivative are collected in the following lemma.

Lemma 4.2**.**

Let $K_{0}(w)$ be the $K$-Bessel function of order zero and let $w\in{\mathbb{C}}\setminus\{0\}$ with $\Re(w)\geq 0$. Then

[TABLE]

Moreover, setting $f(w)=e^{w}K_{0}(w)$, we have

[TABLE]

Proof.

The integral representation [9, 8.432.8]

[TABLE]

holds for $|\arg(w)|<\pi$ and $\Re(\nu)>-1/2$. Since $\Re(w)\geq 0$, the estimate (4.7) follows after bounding the integrand in absolute value. Multiplying both sides by $e^{w}$, differentiating in $w$ and bounding the result yields (4.8). ∎

The most important consequence of the simple closed formula for $\omega_{0}$ is being able to see that $\omega_{0}(z)\ll|z|^{1/2+\epsilon}$ as $|z|\to 0$. Such a decay is guaranteed a priori by Kuznetsov’s formula, but is not directly visible in the definition of $\omega_{0}$ in (4.2). The behaviour of $\omega_{0}(z)$ for small $z$ is made explicit in the following lemma, which in turn allows us to replace the infinite sum over $c$ by a finite sum.

Lemma 4.3**.**

Let $N\geq 1$ and $n\in{\mathbb{Z}}[i]$ with $N(n)\asymp N$, and let $\omega_{0}$ be as in (4.2). Then

[TABLE]

where $M=e^{-i(\pi/2-1/2T)}$. In particular, we have

[TABLE]

where $\mathcal{S}_{n}^{\ddagger}$ is a finite weighted sum of Kloosterman sums given by

[TABLE]

with $C_{1}=N(n)V/T^{2}\log^{2}T$ and $C_{2}=N(n)V$.

Proof.

Let $2A=M|z|X^{1/2}$ and $2B=M|z|X^{-1/2}$. Then, from the definition of $h(r)$, together with the relation

[TABLE]

we deduce the identity

[TABLE]

The second equality follows from [7, §7.3 (17)] (see also [9, 17.43.32]). Note that $\Re(2A),\Re(2B)\geq 0$. Using (4.7), and bounding the exponential crudely by one, we see that

[TABLE]

This proves (4.9). Also, summing (4.12) over $N(c)>4\pi^{2}N(n)$ gives a quantity bounded by $O(N^{1/2+\epsilon}X^{-1/4}),$ which is absorbed in the error term in (4.10). Similarly, from (4.7) we see that

[TABLE]

Therefore, on summing over $4\pi^{2}N(n)<N(c)\leq C_{1}$ and $N(c)\geq C_{2}$ we obtain a quantity bounded by

[TABLE]

We have now reduced the problem of estimating $\mathcal{S}_{n}(\omega)$ to a matter of understanding a finite sum of Kloosterman sums, $\mathcal{S}_{n}^{\ddagger}(K_{0})$, weighted by a $K$-Bessel function of order zero.

Remark 4.4**.**

Notice that if we use the estimate (4.7) also in the remaining range, $C_{1}<N(c)\leq C_{2}$, we obtain $\mathcal{S}_{n}^{\ddagger}(K_{0})\ll(NTX)^{1/2+\epsilon}$. Collecting the errors from Lemma 4.1 and Lemma 4.3, we see that this contribution dominates. Therefore we would have

[TABLE]

which recovers the pointwise bound that appears in [15, p.792]. The method in our proof is slightly different at places and provides additional details compared to [15]. Moreover, Lemma 4.3 bypasses the use of the method of stationary phase, giving instead a closed formula for the weight function.

We will now study the second moment of $\mathcal{S}_{n}^{\ddagger}(K_{0})$. By exploiting the oscillation in the weight function $K_{0}(M|z|X^{1/2})$, we can obtain additional decay when integrating in $X$.

Lemma 4.5**.**

Let $M$ be as in Lemma 4.3, and let $x_{1},x_{2}$ be positive real numbers satisfying $1\gg x_{1},x_{2}\gg V^{-1/2}$. Then

[TABLE]

where $L=|x_{1}-x_{2}|$.

Proof.

Consider the function $f(w)=e^{w}K_{0}(w)$. From Lemma 4.2 we have

[TABLE]

for $\Re(w)\geq 0$ and $|w|$ bounded away from zero. The integral in (4.14) can be written as

[TABLE]

Bounding the integrand in absolute value and applying (4.15) leads to the first term in the minimum in (4.14). The second term in (4.14) follows from integration by parts and (4.15). ∎

We are now ready to prove Proposition 3.1.

4.1. Proof of Proposition 3.1

By Lemma 4.1 and Lemma 4.3, we have

[TABLE]

where $\mathcal{S}_{n}^{\ddagger}(K_{0})$ is as given in (4.11). From a dyadic decomposition and the Cauchy–Schwarz inequality it follows that we can bound the integral on the right-hand side by

[TABLE]

where the sum over $c_{1},c_{2}$ is restricted to $R<N(c_{1}),N(c_{2})\leq 2R$. Note that in this range the numbers $x_{j}=2\pi|n|/|c_{j}|$ satisfy the inequality $1\gg x_{j}\gg V^{-1/2}$, for $j=1,2$, and we can therefore bound the integral in (4.18) by using Lemma 4.5. For $x_{1}=x_{2}$, i.e. $|c_{1}|=|c_{2}|$, we use the factor $YV^{-1/2}$ in the minimum in (4.14). This, coupled with Weil’s bound for $S_{{\mathbb{Q}}(i)}(n,n,c)$, leads to the following estimate for the diagonal part of the sum:

[TABLE]

Here we use $r_{2}(n)$ to denote the number of ways of writing $n$ as a sum of two squares, along with the standard estimate $r_{2}(n)\ll n^{\epsilon}$. For the off-diagonal terms in (4.18) (when $|c_{1}|\neq|c_{2}|$) we use again Lemma 4.5. For technical convenience we interpolate the two bounds in the minimum with the exponents $(\epsilon,1-\epsilon)$, which gives

[TABLE]

Inserting this into (4.18), we can estimate the double sum over $|c_{1}|\neq|c_{2}|$ by

[TABLE]

where $\ell_{j}=N(c_{j})$ and the coefficients $a_{\ell}$ are given by

[TABLE]

Using the Hardy–Littlewood–Pólya inequality [10, Th. 381, p. 288] and Weil’s bound (2.1) for the Kloosterman sums, we can bound (4.20) by

[TABLE]

uniformly in $R$. Since the above bound dominates the last term in (4.17), combining (4.21) with (4.19) we deduce that

[TABLE]

which is what we wanted to prove. ∎

5. Average of Rankin–Selberg $L$-functions

In this section we prove Proposition 3.2. As in section 4, we take real numbers $T$, $X$, $V$ and $Y$ satisfying the inequalities in (4.1). Moreover, we assume that $s$ is a complex number with $\Re(s)=1/2$.

First, we note that $h(r)=X^{ir}e^{-r/T}+O(e^{-\pi r})$. Therefore, using the fact that the Rankin–Selberg $L$-function is bounded polynomially in both $r_{j}$ and $s$, we can write

[TABLE]

We decompose the sum on the right-hand side into intervals of length $T$ and use the Cauchy–Schwarz inequality to get

[TABLE]

where we use the shorthand $L_{j}=L(s,u_{j}\otimes u_{j})$. We want to integrate over $X\in[V,V+Y]$ in (5.1). Thus we need to understand the integral

[TABLE]

Opening up the square and integrating directly yields

[TABLE]

The Weyl law (with remainder) on $\Gamma\backslash\mathbb{H}^{3}$ [1, Theorem 2] implies the estimate $\#\{T<r_{j}\leq T+1\}\ll T^{2}$ in unit intervals (see [2, (2.1)]). This gives, for $r_{j}\leq mT$,

[TABLE]

Now, in order to estimate the remaining sum over $r_{j}$ in (5), we use the relation between the Rankin–Selberg $L$-function and the symmetric square $L$-function, i.e.

[TABLE]

where $\zeta_{{\mathbb{Q}}(i)}$ is the Dedekind zeta function of ${\mathbb{Q}}(i)$. The bound $v_{j}(1)\ll r_{j}^{\epsilon}$ (see [15, Proposition 3.1]) together with the second moment estimate [2, Theorem 3.3] give

[TABLE]

After substituting the estimates (5.3) and (5.4) into (5), we finally obtain

[TABLE]

Using this with (5.1) and summing over $m$ leads us to the desired bound. ∎

6. Recovering Theorem 1.1

Finally, we show how to recover Theorem 1.1 from Theorem 1.2. The error term $E_{\Gamma}(X)$ is related to the spectral exponential sum $S(T,X)$ via the explicit formula. For $\mathrm{PSL}(2,\mathbb{Z}[i])$, this was proved by Nakasuji [19, Thm. 4.1], who showed that

[TABLE]

for $1\leq T<X^{1/2}$. Thus, we can estimate the second moment of $E_{\Gamma}(X)$ as

[TABLE]

Using partial summation, we write the exponential sum as

[TABLE]

and by a repeated use of the Cauchy–Schwarz inequality we obtain

[TABLE]

We apply Theorem 1.2 and bound the right-hand side by

[TABLE]

Thus

[TABLE]

Balancing with $T=V^{1/6}Y^{1/3}$ completes the proof of Theorem 1.1.∎