# Prime geodesic theorem and closed geodesics for large genus

Yunhui Wu and Yuhao Xue  Yau Mathematical Sciences Center and Department of
Mathematics, Tsinghua University, Beijing, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{M}_{g}$ be the moduli space of hyperbolic surfaces of genus $g$
endowed with the Weil-Petersson metric. In this paper, we show that for any
$\epsilon>0$, as $g\to\infty$, for a generic surface in $\mathcal{M}_{g}$, the
error term in the Prime Geodesic Theorem is bounded from above by $g\cdot
t^{\frac{3}{4}+\epsilon}$, up to a uniform constant multiplication. The
expected value of the error term in the Prime Geodesic Theorem over
$\mathcal{M}_{g}$ is also studied. As an application, we show that as
$g\to\infty$, on a generic hyperbolic surface in $\mathcal{M}_{g}$ most
geodesics of length significantly less than $\sqrt{g}$ are simple and non-
separating, and most geodesics of length significantly greater than $\sqrt{g}$
are not simple, which confirms a conjecture of Lipnowski-Wright.

A novel effective upper bound for intersection numbers on $\mathcal{M}_{g,n}$
is also established, when certain indices are large compared to $\sqrt{g+n}$.

##  1\. Introduction

Let $X_{g}$ be a closed hyperbolic surface of genus $g$ $(g\geq 2)$. The
spectrum of $X$ is a discrete closed subset in $\mathbb{R}^{\geq 0}$ and
consists of eigenvalues with finite multiplicity. We enumerate them, counted
with multiplicity, in the following increasing order

$0=\lambda_{0}(X_{g})<\lambda_{1}(X_{g})\leq\lambda_{2}(X_{g})\leq\cdots\to\infty.$

For any $t>1$, we denote by $\pi_{X_{g}}(t)$ the number of oriented primitive
closed geodesics of length $\leq\ln t$. A remarkable application of Selberg’s
trace formula [Sel56] is the Prime Geodesic Theorem (PGT) (see e.g. [Hub61,
Hej76, Ran77]) which states

(1) $\pi_{X_{g}}(t)=\mathop{\rm
Li}(t)+\sum\limits_{\frac{3}{4}<s_{j}<1}\mathop{\rm
Li}(t^{s_{j}})+\mathrm{O}_{X_{g}}\left(\frac{t^{\frac{3}{4}}}{\ln t}\right)$

where $\mathop{\rm Li}(t)=\int_{2}^{t}\frac{dx}{\ln x}\sim\frac{t}{\ln t}\
\text{as}\ t\to\infty$,
$s_{j}=\frac{1}{2}+\sqrt{\frac{1}{4}-\lambda_{j}(X_{g})}$ and the implied
constant depends on the geometry of $X_{g}$. As an analog of the Riemann
hypothesis for closed hyperbolic surfaces, it is a well-known conjecture that
_the $\frac{3}{4}$ in (1) can be replaced by $\frac{1}{2}+\epsilon$_. Rewrite
(1) as

(2) $\pi_{X_{g}}(t)=\mathop{\rm Li}(t)+\mathrm{Er}_{X_{g}}(t).$

So the error term $\mathrm{Er}_{X_{g}}(t)$ depends on the geometry and first
non-zero eigenvalue of $X_{g}$. There are many works on the study of the error
term $\mathrm{Er}_{X_{g}}(t)$, especially for special arithmetic surfaces. See
[Sar80, Iwa84, LRS95, Cai02, SY13] and the references therein.

Let $\mathcal{M}_{g}$ be the moduli space of closed hyperbolic surfaces of
genus $g$ endowed with the Weil-Petersson metric. Based on her celebrated
thesis works [Mir07a, Mir07b], Mirzakhani in [Mir13] initiated the study of
random hyperbolic surfaces for large genus. In this paper, our aim is to study
the average behavior of the error terms in (1) and (2) on $\mathcal{M}_{g}$
for large $g$. More precisely, let $\mathop{\rm Prob}\nolimits_{\rm WP}^{g}$
and $\mathbb{E}_{\rm WP}^{g}$ be the probability measure and expectation on
$\mathcal{M}_{g}$ respectively given by the Weil-Petersson metric. We apply
recent results of [Mon22] and [WX22, LW21] to show that as $g\to\infty$, for a
generic $X_{g}\in\mathcal{M}_{g}$, the error term $\mathrm{Er}_{X_{g}}(t)$ in
(2) can be bounded from above by $g\cdot t^{\frac{3}{4}+\epsilon}$ up to a
universal constant multiplication. More precisely,

######  Theorem 1.

There exists a universal constant $c>0$ such that for any $\epsilon>0$ and all
$t=t(g)>2$ which may depend on $g$,

$\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\ \left|\pi_{X_{g}}(t)-\mathop{\rm
Li}(t)\right|\leq c\cdot g\cdot t^{\frac{3}{4}+\epsilon}\right)=1.$

Selberg’s theory is important in the proof of Theorem 1. One may see [GLMST21,
LS20, Mon22, Tho20, WX22, LW21, Hid21, Rud22] and the references therein for
related topics on the same spirit. As an analog of the Riemann hypothesis for
Weil-Petersson random hyperbolic surfaces, we conjecture

######  Conjecture.

Theorem 1 still holds with replacing $\frac{3}{4}$ by $\frac{1}{2}$.

We also study the expected value $\mathbb{E}_{\rm
WP}^{g}\left[\left|\pi_{X_{g}}(t)-\mathop{\rm Li}(t)\right|\right]$ for large
genus. In light of (1), we need to study the term
$\sum\limits_{\frac{3}{4}<s_{j}<1}\mathop{\rm
Li}(t^{s_{j}})+\mathrm{O}_{X_{g}}\left(\frac{t^{\frac{3}{4}}}{\ln t}\right)$
for large genus. Motivated by recent work [BP22] of Bourque-Petri, we first
prove the following result in which the error term
$\mathrm{O}_{X_{g}}\left(\frac{t^{\frac{3}{4}}}{\ln t}\right)$ in (1) for a
single surface $X_{g}\in\mathcal{M}_{g}$ is explicitly given in terms of the
geometry of $X_{g}$. More precisely,

######  Theorem 2.

For any $X_{g}\in\mathcal{M}_{g}$ and all $t=t(g)>2$ which may depend on $g$,

$\displaystyle\pi_{X_{g}}(t)$ $\displaystyle=$ $\displaystyle\mathop{\rm
Li}(t)+\sum_{0<\lambda_{j}(X_{g})\leq\frac{1}{4}}\mathop{\rm Li}(t^{s_{j}})$
$\displaystyle+$ $\displaystyle O\left(g\frac{t^{\frac{5}{6}}}{\ln
t}+g\frac{t^{\frac{2}{3}}}{\ln t}\max\left\\{0,\ln\left(\frac{1}{\mathop{\rm
sys}(X_{g})t^{\frac{1}{6}}}\right)\right\\}\right)$

where $\mathop{\rm sys}(X_{g})$ is the length of shortest closed geodesic in
$X_{g}$, and the implied constant is universal.

With the help of our previous work [WX22], we will get an effective upper
bound of $\mathbb{E}_{\rm
WP}^{g}\left[\sum\limits_{0<\lambda_{j}(X_{g})\leq\frac{1}{4}}\mathop{\rm
Li}(t^{s_{j}})\right]$ for large genus. Then we apply Theorem 2 to prove the
following bound on $\mathbb{E}_{\rm WP}^{g}\left[\pi_{X_{g}}(t)\right]$ for
large $g$ and suitable $t=t(g)$. More precisely,

######  Theorem 3.

Let $t(g)$ be a function such that $t(g)>g^{12}$. Then for any $0<\epsilon<1$
and large enough $g$,

$\mathbb{E}_{\rm WP}^{g}\left[\ \left|1-\frac{\pi_{X_{g}}(t(g))}{\mathop{\rm
Li}(t(g))}\right|\ \right]=O_{\epsilon}(g^{-1+\epsilon})$

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

For any $L>0$, denote by $N(X_{g},L)$ the number of unoriented primitive
closed geodesics in $X_{g}$ of length $\leq L$. So
$N(X_{g},L)=\frac{\pi_{X_{g}}(e^{L})}{2}.$ As introduced above, by the Prime
Geodesic Theorem

$N(X_{g},L)\sim\frac{1}{2}\frac{e^{L}}{L}\ \textit{as $L\to\infty$}.$

Denote by $N^{s}(X_{g},L)$, $N^{s}_{nsep}(X_{g},L)$ and $N^{ns}(X_{g},L)$ the
number of unoriented primitive _simple_ , _simple and non-separating_ , and
_non-simple_ closed geodesics in $X_{g}$ of length $\leq L$ respectively. A
celebrated result of Mirzakhani [Mir08] states that

$N^{s}(X_{g},L)\sim C(X_{g})\cdot L^{6g-6}\ \textit{as $L\to\infty$}$

where $C(X_{g})$ is a constant depending on $X_{g}$, and as a function on
$\mathcal{M}_{g}$, $C:\mathcal{M}_{g}\to\mathbb{R}$ is proper.

Based on Mirzakhani’s Integration Formula [Mir07a], it is not hard to see that
as $g\to\infty$, the expected values of $N^{s}(X_{g},L)$ and
$N^{s}_{nsep}(X_{g},L)$ over $\mathcal{M}_{g}$ satisfy that

$\lim\limits_{g\to\infty}\mathbb{E}_{\rm
WP}^{g}[N^{s}(X_{g},L)]\sim\lim\limits_{g\to\infty}\mathbb{E}_{\rm
WP}^{g}[N^{s}_{nsep}(X_{g},L)]\sim\frac{1}{2}\frac{e^{L}}{L}\ \textit{as
$L\to\infty$}.$

It is quite mysterious that both $N(X_{g},L)$ and
$\lim\limits_{g\to\infty}\mathbb{E}_{\rm WP}^{g}[N^{s}_{nsep}(X_{g},L)]$ have
the same asymptotic leading term $\frac{1}{2}\frac{e^{L}}{L}$ as $L\to\infty$.
This suggests that for suitable $L=L(g)$ depending on $g$, as $g\to\infty$, in
the set of closed geodesics of length $\leq L$ in a generic surface in
$\mathcal{M}_{g}$, simple and non-separating ones are dominant. Meanwhile, as
introduced above, $N^{s}(X_{g},L)\sim C(X_{g})\cdot L^{6g-6}$. This also seems
to suggest that for suitable $L=L(g)$ depending on $g$, as $g\to\infty$, in
the set of closed geodesics of length $\geq L$ in a generic surface in
$\mathcal{M}_{g}$, non-simple ones are major part. Actually it was conjectured
in [LW21, Conjecture $1.2$] that

######  Conjecture (Lipnowski-Wright).

As $g\to\infty$, on most closed hyperbolic surfaces in $\mathcal{M}_{g}$ most
geodesics of length significantly less than $\sqrt{g}$ are simple and non-
separating, and most geodesics of length significantly greater than $\sqrt{g}$
are not simple.

As an application of Theorem 1 and 3, we confirm the conjecture above. More
precisely, we prove

######  Theorem 4.

The following two probabilities hold:

  1. (1)

if $L(g)$ satisfies

$L(g)>12\ln g\text{ and }\lim\limits_{g\to\infty}\frac{L^{2}(g)}{g}=0,$

then there exists a function $\delta(g)>0$ satisfying
$\lim\limits_{g\to\infty}\delta(g)=0$ such that

$\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\left|1-\frac{N_{nsep}^{s}(X_{g},L(g))}{N(X_{g},L(g))}\right|<\delta(g)\right)=1.$

  2. (2)

If $L(g)$ satisfies

$\lim\limits_{g\to\infty}\frac{L^{2}(g)}{g}=\infty,$

then

$\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\left|1-\frac{N^{ns}(X_{g},L(g))}{N(X_{g},L(g))}\right|<\frac{g}{L(g)^{2}}\right)=1.$

Recall that a closed geodesic $\gamma\subset X_{g}\in\mathcal{M}_{g}$ is
called _filling_ if each component of the complement $X_{g}\setminus\gamma$ of
$\gamma$ in $X_{g}$ is homeomorphic to a disk. It is known by elementary
isoperimetric inequality that the length of each filling closed geodesic in
$X_{g}$ grows at least like $2\pi(g-1)$. We propose the following interesting
question:

######  Question.

As $g\to\infty$, on a generic closed hyperbolic surface in $\mathcal{M}_{g}$,
are most geodesics of length significantly greater than $g$ filling?

In the proof of Part $(2)$ of Theorem 4, we establish a new bound on the Weil-
Petersson volume $V_{g,n}(x_{1},\cdots,x_{n})$ of moduli space of bordered
hyperbolic surfaces with geodesic boundary components of length
$x_{1},\cdots,x_{n}$, which relies on a novel bound on intersection numbers in
this paper. Recall that on $\overline{\mathcal{M}}_{g,n}$ the compactified
moduli space of surfaces of genus $g$ with $n$ punctures, there are $n$
tautological line bundles $\\{\mathcal{L}_{i}\\}_{i=1}^{n}$. Let
$\psi_{i}=c_{1}(\mathcal{L}_{i})\in
H_{2}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$ be the first Chern class of
$\mathcal{L}_{i}$ and $\omega$ be the Weil-Petersson symplectic form on
$\overline{\mathcal{M}}_{g,n}$. For any $d=(d_{1},\cdots,d_{n})$ with each
$d_{i}\in\mathbb{Z}_{\geq 0}$ and $|d|=d_{1}+\cdots+d_{n}\leq 3g-3+n$, the
_intersection number_ $\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}$ is
defined as

$\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}=\frac{\prod_{i=1}^{n}2^{2d_{i}}(2d_{i}+1)!!}{(3g-3+n-|d|)!}\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}\omega^{3g-3+n-|d|}.$

In particular, $\left[\tau_{0}\cdots\tau_{0}\right]_{g,n}=\mathop{\rm
Vol}(\mathcal{M}_{g,n})$, which is also denoted by $V_{g,n}$. There are deep
connections between intersection numbers and Weil-Petersson volumes,
especially for their asymptotics. Actually Mirzakhani [Mir07b] showed that
Weil-Petersson volumes can be expressed in terms of intersection numbers. For
asymptotic behaviors, one may see [Mir13, LX14, MZ15, DGZZ20, Agg21] and the
references therein for related results. Based on her recursive formula,
Mirzakhani in [Mir13, Page 286] showed that there exists a constant $c(n)>0$
independent of $g$ and $d_{i}$’s such that

$1-c(n)\frac{|d|^{2}}{g}\leq\frac{\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}}{V_{g,n}}\leq
1.$

In particular, for fixed $n$ and $|d|=o(\sqrt{g})$,

$\frac{\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}}{V_{g,n}}\sim 1\text{
\ as $g\to\infty$}.$

In this paper, we study the case that $\max\limits_{1\leq i\leq
n}\\{d_{i}^{2}\\}$ is large compared to $2g+n$, which seems to have not been
presented generally before. We prove

######  Theorem 5.

There exists a universal constant $c^{\prime}>0$ such that

$\frac{\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}}{V_{g,n}}\leq
c^{\prime}\frac{2g-2+n}{\max\limits_{1\leq i\leq n}\\{d_{i}^{2}\\}}.$

It is known by [MP19, Proposition 3.1] and [NWX20, Lemma 19] that there exists
a constant $c(n)>0$ only depending on $n$ such that

$\left(1-c(n)\frac{\sum_{i=1}^{n}x_{i}^{2}}{g}\right)\prod_{i=1}^{n}\frac{\sinh(x_{i}/2)}{x_{i}/2}\leq\frac{V_{g,n}(x_{1},\cdots,x_{n})}{V_{g,n}}\leq\prod_{i=1}^{n}\frac{\sinh(x_{i}/2)}{x_{i}/2}.$

One may also see recent work [AM22] of Anantharaman-Monk for sharper results.
In particular, for fixed $n$ and $\sum x_{i}^{2}=o(g)$,

$\frac{V_{g,n}(x_{1},\cdots,x_{n})}{V_{g,n}}\sim\prod_{i=1}^{n}\frac{\sinh(x_{i}/2)}{x_{i}/2}\text{
\ as $g\to\infty$}.$

In this paper we apply Theorem 5 to study $V_{g,n}(x_{1},\cdots,x_{n})$ when
$\max\limits_{1\leq i\leq n}\\{x_{i}^{2}\\}$ is large compared to $2g+n$. The
following bound is essential in the proof of Part $(2)$ of Theorem 4.

######  Theorem 6.

For any $k\geq 1$, there exists a constant $c(k)>0$ independent of $g,n$ and
$x_{i}$’s such that

$\frac{V_{g,n}(x_{1},\cdots,x_{n})}{V_{g,n}}\leq\left(\prod_{i=1}^{n}\min\left\\{c(k)\frac{2g-2+n}{x_{i}^{2}},1\right\\}\right)^{k}\prod_{i=1}^{n}\frac{\sinh(x_{i}/2)}{x_{i}/2}.$

We believe that both Theorem 5 and 6 will find further other applications on
related problems.

### Notations.

We say two positive functions

$f_{1}(g)\prec f_{2}(g)\quad\emph{or}\quad f_{2}(g)\succ f_{1}(g)$

if there exists a universal constant $C>0$, independent of $g$, such that
$f_{1}(g)\leq C\cdot f_{2}(g)$; and we say

$f_{1}(g)\asymp f_{2}(g)$

if $f_{1}(g)\prec f_{2}(g)$ and $f_{2}(g)\prec f_{1}(g)$. And
$f_{1}(g)\prec_{*}f_{2}(g)$ means that the constant $C$ depends on $*$.

We also say two functions

$h_{1}=O\left(h_{2}\right),$

where $h_{1}$ may not be positive, if there exists a universal constant $C>0$
such that $|h_{1}|\leq C\cdot h_{2}$. And $h_{1}=O_{*}(h_{2})$ means that $C$
depends on $*$.

### Plan of the paper.

Section 2 will provide a review of relevant background materials. The proofs
of Theorem 1 and 2 are independent. We will first prove Theorem 2 in Section
3, and then apply it to prove Theorem 3 in Section 4. In Section 5 we complete
the proof of Theorem 1. Then in Section 6 we establish the two new bounds on
intersection numbers and Weil-Petersson volumes, i.e., prove Theorem 5 and 6.
In the last Section, we apply Theorem 1, 3, 5 and 6 to prove Theorem 4.

### Acknowledgement

We would like to thank all the participants in our seminar on Teichmüller
theory for helpful discussions and comments on this project. We also would
like to thank Zeév Rudnick for his useful comments. The first named author is
partially supported by the NSFC grant No. $12171263$.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 PGT for general hyperbolic surfaces
  4. 4 Expectation of the error term in PGT
  5. 5 PGT for random hyperbolic surfaces
    1. 5.1 First part of Theorem 28
    2. 5.2 Second part of Theorem 28
  6. 6 Intersection numbers and Weil-Petersson volumes
    1. 6.1 Intersection numbers
    2. 6.2 Weil-Petersson volumes
  7. 7 Proof of Theorem 4
    1. 7.1 Geodesics of length significantly less than $\sqrt{g}$
    2. 7.2 Geodesics of length significantly greater than $\sqrt{g}$

##  2\. Preliminaries

In this section, we include certain necessary backgrounds and tools used in
this paper.

### The Weil-Petersson metric

We denote $S_{g,n}$ to be an oriented topological surface with $g$ genus and
$n$ punctures or boundaries. Require $2g+n\geq 3$ so that $S_{g,n}$ admits
hyperbolic metrics. Let $\mathcal{T}_{g,n}$ be the Teichmüller space of
surfaces with $g$ genus and $n$ punctures. The moduli space of Riemann
surfaces is $\mathcal{M}_{g,n}=\mathcal{T}_{g,n}/\mathop{\rm Mod}_{g,n}$ where
$\mathop{\rm Mod}_{g,n}$ is the mapping class group of $S_{g,n}$ which fix the
order of punctures. If $n=0$, write $\mathcal{T}_{g}=\mathcal{T}_{g,0}$ and
$\mathcal{M}_{g}=\mathcal{M}_{g,0}$ for simplicity. Given
$L=(L_{1},\cdots,L_{n})\in\mathbb{R}^{n}_{\geq 0}$, the Teichmüller space
$\mathcal{T}_{g,n}(L)$ consists of bordered hyperbolic surfaces with geodesic
boundary components of length $L_{1},\cdots,L_{n}$. Note that a geodesic
boundary component of length $0$ is a puncture, so
$\mathcal{T}_{g,n}(0,\cdots,0)=\mathcal{T}_{g,n}$. The weighted moduli space
is $\mathcal{M}_{g,n}(L)=\mathcal{T}_{g,n}(L)/\mathop{\rm Mod}_{g,n}$.

Fix a pants decomposition $\\{\alpha_{i}\\}_{i=1}^{3g-3+n}$ of $S_{g,n}$, the
Fenchel-Nielsen coordinates
$X\mapsto(\ell_{\alpha_{i}}(X),\tau_{\alpha_{i}}(X))_{i=1}^{3g-3+n}$ is a
global coordinates for the Teichmüller space $\mathcal{T}_{g,n}(L)$. Where
$\ell_{\alpha_{i}}(X)$ is the length of $\alpha_{i}$ on $X$ and
$\tau_{\alpha_{i}}(X)$ is the twist along $\alpha_{i}$ (measured by length).
Wolpert [Wol82] showed that the Weil-Petersson symplectic form of
$\mathcal{T}_{g,n}(L)$ has a good expression under Fenchel-Nielsen
coordinates:

######  Theorem 7 (Wolpert).

The Weil-Petersson symplectic form $\omega_{WP}$ on $\mathcal{T}_{g,n}(L)$ is
given by

$\omega_{WP}=\sum_{i=1}^{3g-3+n}d\ell_{\alpha_{i}}\wedge d\tau_{\alpha_{i}}.$

The Weil-Petersson volume form is

$d\mathop{\rm
Vol}\nolimits_{WP}=\tfrac{1}{(3g-3+n)!}\underbrace{\omega_{WP}\wedge\cdots\wedge\omega_{WP}}_{3g-3+n\
\text{copies}}.$

This is a mapping class group invariant measure on $\mathcal{T}_{g,n}(L)$, and
hence a measure on $\mathcal{M}_{g,n}(L)$, which still denoted by
$d\mathop{\rm Vol}_{WP}$. The total volume of $\mathcal{M}_{g,n}(L)$ under the
Weil-Petersson metric is finite and denoted by $V_{g,n}(L)$. This also implies
a probability measure $\mathop{\rm Prob}\nolimits_{\rm WP}^{g}$ on
$\mathcal{M}_{g}$:

$\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}(\mathcal{A}):=\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\mathbf{1}_{\mathcal{A}}dX$

where $\mathcal{A}\subset\mathcal{M}_{g}$ is a Borel subset,
$\mathbf{1}_{\mathcal{A}}:\mathcal{M}_{g}\to\\{0,1\\}$ is its characteristic
function, and $dX$ is short for $d\mathop{\rm Vol}_{WP}(X)$. One may see
[Mir13, GPY11, MP19, NWX20, PWX21, SW22, HW22] and the references therein for
recent studies of the geometry of Weil-Petersson random hyperbolic surfaces.

### Mirzakhani’s integration formula

Now we introduce an integration formula in [Mir07a, Mir13], which is an
essential tool in the study of random hyperbolic surfaces of Weil-Petersson
model.

Given any non-peripheral closed curve $\gamma$ on topological surface
$S_{g,n}$ and $X\in\mathcal{T}_{g,n}$, denote $\ell(\gamma)=\ell_{\gamma}(X)$
to be the hyperbolic length of the unique closed geodesic in the homotopy
class $\gamma$ on $X$. Let $\Gamma=(\gamma_{1},\cdots,\gamma_{k})$ be an
ordered k-tuple where the $\gamma_{i}$’s are distinct disjoint homotopy
classes of nontrivial, non-peripheral, unoriented simple closed curves on
$S_{g,n}$. Consider the orbit containing $\Gamma$ under $\mathop{\rm
Mod}_{g,n}$-action

$\mathcal{O}_{\Gamma}=\\{(h\cdot\gamma_{1},\cdots,h\cdot\gamma_{k});\
h\in\mathop{\rm Mod}\nolimits_{g,n}\\}.$

Given a function $F:\mathbb{R}_{\geq 0}^{k}\to\mathbb{R}$, we define a
function on $\mathcal{M}_{g,n}$

$\displaystyle F^{\Gamma}:\mathcal{M}_{g,n}$ $\displaystyle\to$
$\displaystyle\mathbb{R}$ $\displaystyle X$ $\displaystyle\mapsto$
$\displaystyle\sum_{(\alpha_{1},\cdots,\alpha_{k})\in\mathcal{O}_{\Gamma}}F(\ell_{\alpha_{1}}(X),\cdots,\ell_{\alpha_{k}}(X)).$

Assume $S_{g,n}-\cup_{j=1}^{k}\gamma_{j}=\cup_{i=1}^{s}S_{g_{i},n_{i}}$. For
any $x=(x_{1},\cdots,x_{k})\in\mathbb{R}_{\geq 0}^{k}$, we consider the moduli
space $\mathcal{M}(S_{g,n}(\Gamma);\ell_{\Gamma}=x)$ of the hyperbolic
surfaces (possibly disconnected) homeomorphic to
$S_{g,n}-\cup_{j=1}^{k}\gamma_{j}$ with
$\ell_{\gamma_{i}^{1}}=\ell_{\gamma_{i}^{2}}=x_{i}$ for every $i=1,\cdots,k$,
where $\gamma_{i}^{1}$ and $\gamma_{i}^{2}$ are two boundary components of
$S_{g,n}-\cup_{j=1}^{k}\gamma_{j}$ given by cutting along $\gamma_{i}$.
Consider the Weil-Petersson volume

$V_{g,n}(\Gamma,x)=\mathop{\rm
Vol}\nolimits_{WP}\left(\mathcal{M}(S_{g,n}(\Gamma);\ell_{\Gamma}=x)\right)=\prod_{i=1}^{s}V_{g_{i},n_{i}}(x^{(i)})$

where $x^{(i)}$ is the list of those coordinates $x_{j}$ of $x$ such that
$\gamma_{j}$ is a boundary component of $S_{g_{i},n_{i}}$. Using Theorem 7,
Mirzakhani get the following integration formula. See [Mir07a, Theorem 7.1],
[Mir13, Theorem 2.2], [MP19, Theorem 2.2], [Wri20, Theorem 4.1] for different
versions.

######  Theorem 8 (Mirzakhani).

For any $\Gamma=(\gamma_{1},\cdots,\gamma_{k})$, the integral of $F^{\Gamma}$
over $M_{g,n}$ with respect to Weil-Petersson metric is given by

$\int_{\mathcal{M}_{g,n}}F^{\Gamma}dX=C_{\Gamma}\int_{\mathbb{R}_{\geq
0}^{k}}F(x_{1},\cdots,x_{k})V_{g,n}(\Gamma,x)x_{1}\cdots x_{k}dx_{1}\cdots
dx_{k}$

where the constant $C_{\Gamma}\in(0,1]$ only depend on $\Gamma$. Moreover,
$C_{\Gamma}=\frac{1}{2}$ if $g>2$ and $\Gamma$ is a simple non-separating
closed curve.

### The Weil-Petersson volume

We denote $V_{g,n}(x_{1},\cdots,x_{n})$ to be the Weil-Petersson volume of
$\mathcal{M}_{g,n}(x_{1},\cdots,x_{n})$ and $V_{g,n}=V_{g,n}(0,\cdots,0)$.
Recall that the intersection numbers
$\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}$ are integral of first
Chern classes of tautological line bundles and Weil-Petersson symplectic form
over $\overline{\mathcal{M}}_{g,n}$. Mirzakhani [Mir07b] showed that this
volume can be expressed by intersection numbers:

######  Theorem 9.

$V_{g,n}(2x_{1},\cdots,2x_{n})=\sum_{|d|\leq
3g-3+n}\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}\prod_{i=1}^{n}\frac{x_{i}^{2d_{i}}}{(2d_{i}+1)!}.$

Mirzakhani gave some estimates about $V_{g,n}$:

######  Theorem 10.

  1. (1)

[Mir13, Lemma 3.2] For any $g,n\geq 0$,

$b_{0}<\frac{(2g-2+n)V_{g,n}}{V_{g,n+1}}<b_{1}$

where $b_{0}=\frac{10-\pi^{2}}{120}\approx 0.001$ and
$b_{1}=\sum_{l=1}^{\infty}\frac{l\pi^{2l-2}}{(2l+1)!}\approx 0.401$.

  2. (2)

[Mir13, Theorem 3.5]

$\frac{(2g-2+n)V_{g,n}}{V_{g,n+1}}=\frac{1}{4\pi^{2}}+O\left(\frac{1}{g}\right),$
$\frac{V_{g,n}}{V_{g-1,n+2}}=1+O\left(\frac{1}{g}\right).$

Where the implied constants are related to $n$ and independent of $g$.

######  Theorem 11 ([Mir13, Corollary 3.7]).

$\sum_{\tiny\begin{array}[]{c}g_{1}+g_{2}=g\\\ 1\leq g_{1}\leq
g_{2}\end{array}}\frac{V_{g_{1},1}V_{g_{2},1}}{V_{g}}\asymp\frac{1}{g}.$

The implied constants are independent of $g$.

The asymptotic behavior of $V_{g,n}(x_{1},\cdots,x_{n})$ was first studied in
[MP19], also see [NWX20] and refined results in [AM22].

######  Theorem 12.

There exists a constant $c(n)>0$ independent of $g$ and $x_{i}$’s such that

$\left(1-c(n)\frac{\sum_{i=1}^{n}x_{i}^{2}}{g}\right)\prod_{i=1}^{n}\frac{\sinh(x_{i}/2)}{x_{i}/2}\leq\frac{V_{g,n}(x_{1},\cdots,x_{n})}{V_{g,n}}\leq\prod_{i=1}^{n}\frac{\sinh(x_{i}/2)}{x_{i}/2}.$

### Counting closed geodesics

Here we introduce some counting results. We want some upper bounds for the
number of closed geodesics of length $\leq L$. First, the collar lemma (see
e.g. [Bus10, Theorem 4.1.6]) states that:

######  Theorem 13.

In a hyperbolic surface $X_{g}\in\mathcal{M}_{g}$, there are at most $3g-3$
unoriented primitive closed geodesics of length $\leq 2\mathop{\rm
arcsinh}1\approx 1.7627$. Moreover, they are simple and disjoint with each
other.

Buser showed that:

######  Theorem 14 ([Bus10, Theorem 6.6.4]).

Let $X_{g}\in\mathcal{M}_{g}$ and $L>0$. In $X_{g}$ there are at most
$(g-1)e^{L+6}$ oriented closed geodesics of length $\leq L$ which are not
iterates of closed geodesics of length $\leq 2\mathop{\rm arcsinh}1$.

For a hyperbolic surface $X_{g}\in\mathcal{M}_{g}$, denote
$\mathcal{P}(X_{g})$ to be the set of all oriented primitive closed geodesics
in $X_{g}$. Then combine Theorem 13 and 14 we have:

######  Corollary 15.

For any $X_{g}\in\mathcal{M}_{g}$ and $L>0$,

$\\#\left\\{\gamma\in\mathcal{P}(X_{g});\ \ell(\gamma)\leq
L\right\\}\leq(g-1)e^{L+7}.$

### Small eigenvalues

For $X_{g}\in\mathcal{M}_{g}$, the spectrum of the Laplacian operator consists
of discrete eigenvalues

$0=\lambda_{0}(X_{g})<\lambda_{1}(X_{g})\leq\lambda_{2}(X_{g})\leq\cdots\to\infty.$

Spectrum of hyperbolic surfaces are widely studied in the past century, see
[Cha84, Bus10, Ber16] for example. Eigenvalues that are $\leq\frac{1}{4}$ are
called small eigenvalues. Buser [Bus77] showed that
$\lambda_{4g-2}(X_{g})>\frac{1}{4}$ and constructed
$\mathcal{X}_{g}\in\mathcal{M}_{g}$ which admits arbitrary small
$\lambda_{2g-3}(\mathcal{X}_{g})$. Later, Otal and Rosas [OR09] improved that
$\lambda_{2g-2}(X_{g})>\frac{1}{4}$. Here $2g-2$ is sharp. One may also see
Ballmann-Matthiesen-Mondal [BMM16, BMM17] for more general statements on
$\lambda_{2g-2}(X_{g})$.

######  Theorem 16 ([OR09]).

There are at most $2g-2$ eigenvalues in $[0,\frac{1}{4}]$ for any
$X_{g}\in\mathcal{M}_{g}$.

### Selberg’s trace formula

We introduce the Selberg’s trace formula [Sel56] of closed hyperbolic surfaces
here, which is a useful tool when consider the spectrum of Laplacian and
length spectrum. One may see [Bus10, Theorem 9.5.3] or [Ber16, Theorem 5.6]
for the version we describe here.

Given a smooth function with compact support $\varphi\in
C_{c}^{\infty}(\mathbb{R})$, its Fourier transform is defined as

$\widehat{\varphi}(z)=\int_{\mathbb{R}}\varphi(t)e^{-\mathbf{i}tz}dt$

for any $z\in\mathbb{C}$.

On a hyperbolic surface, a closed geodesic is called primitive if it is not an
iterate of any other closed geodesic at least twice. For any hyperbolic
surface $X_{g}$, we denote $\mathcal{P}(X_{g})$ to be the set of all oriented
primitive closed geodesics on $X_{g}$.

For $k\in\mathbb{Z}_{\geq 0}$, denote

(3) $r_{k}(X_{g})=\begin{cases}\sqrt{\lambda_{k}(X_{g})-\frac{1}{4}}\ &\
\text{if}\ \lambda_{k}(X_{g})>\frac{1}{4}\\\
\mathbf{i}\cdot\sqrt{\frac{1}{4}-\lambda_{k}(X_{g})}\ &\ \text{if}\
\lambda_{k}(X_{g})\leq\frac{1}{4}\end{cases}$

where $\lambda_{k}(X_{g})$ is the $k$-th eigenvalue of $X_{g}$. Then the
remarkable trace formula of Selberg states:

######  Theorem 17 (Selberg’s trace formula).

Let $X_{g}\in\mathcal{M}_{g}$ be a closed hyperbolic surface of genus $g$. For
any even function $\varphi\in C_{c}^{\infty}(\mathbb{R})$, we have

$\displaystyle\sum_{k=0}^{\infty}\widehat{\varphi}(r_{k}(X_{g}))$
$\displaystyle=$ $\displaystyle(g-1)\int_{\mathbb{R}}r\tanh(\pi
r)\widehat{\varphi}(r)dr$
$\displaystyle+\sum_{k=1}^{\infty}\sum_{\gamma\in\mathcal{P}(X_{g})}\frac{\ell(\gamma)}{2\sinh\left(\frac{k\ell(\gamma)}{2}\right)}\varphi(k\ell(\gamma)).$

And both sides of the formula are absolutely convergent.

##  3\. PGT for general hyperbolic surfaces

In this section we finish the proof of Theorem 2. Actually we show

######  Theorem 18.

For any $X_{g}\in\mathcal{M}_{g}$ and $t>2$,

$\pi_{X_{g}}(t)=\mathop{\rm
Li}(t)+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(t^{s_{j}})+O\left(g\frac{t^{\frac{5}{6}}}{\ln
t}+\frac{t^{\frac{2}{3}}}{\ln
t}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<t^{-1/6}\end{array}}\ln\left(\frac{1}{\ell(\gamma)t^{\frac{1}{6}}}\right)\right)$

where $s_{j}=s_{j}(X_{g})=\frac{1}{2}+\sqrt{\frac{1}{4}-\lambda_{j}(X_{g})}$,
$\mathcal{P}(X_{g})$ is the set of all oriented primitive closed geodesics in
$X_{g}$, and the implied constant is universal.

Theorem 2 is a direct consequence of Theorem 18 above.

###### Proof of Theorem 2.

By Theorem 13 we know that there exist at most $(3g-3)$ mutually disjoint
primitive closed geodesics in $X_{g}$ of length $\leq 2^{-1/6}$. So we have

$\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<t^{-\frac{1}{6}}\end{array}}\ln\left(\frac{1}{\ell(\gamma)t^{\frac{1}{6}}}\right)\prec
g\cdot\max\left\\{0,\ln\left(\frac{1}{\mathop{\rm
sys}(X_{g})t^{\frac{1}{6}}}\right)\right\\}$

which together with Theorem 18 implies the conclusion. ∎

The error term in Theorem 18 is explicitly given in terms of the geometry of
$X_{g}$. In order to achieve it, we will choose suitable test functions for
Selberg’s trace formula such that we are able to distinguish all the geometric
terms as shown in Theorem 18. Our method is motivated by the proof of [BP22,
Corollary 1.4] by Bourque and Petri.

Before showing Theorem 18, we first make the following application.

######  Corollary 19.

For any $X_{g}\in\mathcal{M}_{g}$, if $L$ satisfies
$e^{e^{\frac{1}{3}L}}>\frac{1}{\mathop{\rm sys}(X_{g})}$, then

$\\#\left\\{\gamma\in\mathcal{P}(X_{g});\ \ell(\gamma)\leq L\right\\}\prec
g\frac{e^{L}}{L}.$

###### Proof.

By Theorem 18 and 16,

$\\#\left\\{\gamma\in\mathcal{P}(X_{g});\ \ell(\gamma)\leq L\right\\}\prec
g\frac{e^{L}}{L}+\frac{e^{\frac{2}{3}L}}{L}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<e^{-L/6}\end{array}}\ln\left(\frac{1}{\ell(\gamma)e^{\frac{1}{6}L}}\right).$

When $e^{e^{\frac{1}{3}L}}>\frac{1}{\mathop{\rm sys}(X_{g})}$, we have

$\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<e^{-L/6}\end{array}}\ln\left(\frac{1}{\ell(\gamma)e^{\frac{1}{6}L}}\right)\prec\max\left\\{0,g\cdot\ln\left(\frac{1}{\mathop{\rm
sys}(X_{g})}\right)\right\\}\leq ge^{\frac{1}{3}L}.$

Then the conclusion follows by the two inequalities above. ∎

Now we return to prove Theorem 18.

For bounded $t$, the theorem holds naturally. So we only consider sufficiently
large $t$. Let $\eta(x)\geq 0$ be a non-negative smooth even function with
$supp(\eta)\subset(-\frac{1}{2},\frac{1}{2})$, $\int_{\mathbb{R}}\eta(x)dx=1$
and $\eta(0)=\max\limits_{x\in\mathbb{R}}\eta(x)=2$. Let
$\eta_{\epsilon}(x)=\frac{1}{\epsilon}\eta(\frac{x}{\epsilon})$. For any
$0<\epsilon<0.01$ and $L>1$, let

(4) $f_{\epsilon}(x)=(\eta_{\epsilon}*\eta_{\epsilon})(x),$ (5)
$\varphi_{L,\epsilon}^{\pm}(x)=\frac{1}{2}(f_{\epsilon}(x-L)+f_{\epsilon}(x+L))\pm
f_{\epsilon}(x).$

We will use $\varphi_{L,\epsilon}^{+}$ and $\varphi_{L,\epsilon}^{-}$ as test
functions to show the lower bound and upper bound of $\pi_{X_{g}}(t)$
respectively. By direct calculation, we have

$0\leq
f_{\epsilon}(x)=\int_{\mathbb{R}}\eta_{\epsilon}(y)\eta_{\epsilon}(x-y)dy\leq\frac{2}{\epsilon}\int_{\mathbb{R}}\eta_{\epsilon}(y)dy=\frac{2}{\epsilon}.$

So

(6) $0\leq\varphi_{L,\epsilon}^{+}(x)\leq\frac{4}{\epsilon}\ \ \ \text{and}\ \
-\frac{2}{\epsilon}\leq\varphi_{L,\epsilon}^{-}(x)\leq\frac{2}{\epsilon}.$

It is clear that both $f_{\epsilon}$ and $\varphi_{L,\epsilon}^{\pm}$ are
smooth even functions with compact supports. And their Fourier transforms
satisfy

(7) $\widehat{f_{\epsilon}}(r)=(\widehat{\eta}(\epsilon r))^{2}\geq 0,$ (8)
$\widehat{\varphi_{L,\epsilon}^{+}}(r)=(\cos(Lr)+1)(\widehat{\eta}(\epsilon
r))^{2}\geq 0$

and

(9)
$\widehat{\varphi_{L,\epsilon}^{-}}(r)=(\cos(Lr)-1)(\widehat{\eta}(\epsilon
r))^{2}\leq 0.$

######  Lemma 20.

For any $L>1$ and $\epsilon<0.01$ we have

$\\#\left\\{\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}\right\\}=2\int_{1-\epsilon}^{L+\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}}\varphi_{\tau,\epsilon}^{\pm}(\ell(\gamma))d\tau.$

###### Proof.

For any $x>1$ and $0<\epsilon<0.01$, since
$supp(f_{\epsilon})\subset(-\epsilon,\epsilon)$ we have

$\displaystyle\int_{0}^{\infty}\varphi_{\tau,\epsilon}^{\pm}(x)\mathbf{1}_{[-\epsilon,\epsilon]}(x-\tau)d\tau$
$\displaystyle=\int_{x-\epsilon}^{x+\epsilon}\frac{1}{2}f_{\epsilon}(x-\tau)d\tau+\int_{x-\epsilon}^{x+\epsilon}\frac{1}{2}f_{\epsilon}(x+\tau)d\tau$
$\displaystyle\pm\int_{x-\epsilon}^{x+\epsilon}f_{\epsilon}(x)d\tau=\int_{-\epsilon}^{\epsilon}\frac{1}{2}f_{\epsilon}(\tau)d\tau=\frac{1}{2}.$

So

(14)
$\displaystyle\\#\left\\{\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq L\end{array}\right\\}$ $\displaystyle=$
$\displaystyle\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq L\end{array}}1$ (17) $\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}}\varphi_{\tau,\epsilon}^{\pm}(\ell(\gamma))\mathbf{1}_{[-\epsilon,\epsilon]}(\ell(\gamma)-\tau)d\tau.$

Since
$\varphi_{\tau,\epsilon}^{\pm}(\ell(\gamma))=\frac{1}{2}f_{\epsilon}(\ell(\gamma)-\tau)$
when $1<\ell(\gamma)\leq L$ and $\tau\geq 0$, we have

(22)
$\displaystyle\\#\left\\{\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq L\end{array}\right\\}$ $\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}}\frac{1}{2}f_{\epsilon}(\ell(\gamma)-\tau)\mathbf{1}_{[-\epsilon,\epsilon]}(\ell(\gamma)-\tau)d\tau$
(25) $\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}}\frac{1}{2}f_{\epsilon}(\ell(\gamma)-\tau)d\tau$ (28)
$\displaystyle=$ $\displaystyle
2\int_{1-\epsilon}^{L+\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}}\frac{1}{2}f_{\epsilon}(\ell(\gamma)-\tau)d\tau$ (31)
$\displaystyle=$ $\displaystyle
2\int_{1-\epsilon}^{L+\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}}\varphi_{\tau,\epsilon}^{\pm}(\ell(\gamma))d\tau.$

Which finish the proof. ∎

By Selberg’s trace formula, i.e., Theorem 17,

(32)
$\displaystyle\sum_{k=1}^{\infty}\sum_{\gamma\in\mathcal{P}(X_{g})}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}\varphi_{\tau,\epsilon}^{\pm}(k\ell(\gamma))$
$\displaystyle=$
$\displaystyle\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\widehat{\varphi_{\tau,\epsilon}^{\pm}}(r_{j})+\sum_{\lambda_{j}>\frac{1}{4}}\widehat{\varphi_{\tau,\epsilon}^{\pm}}(r_{j})$
$\displaystyle-(g-1)\int_{\mathbb{R}}r\tanh(\pi
r)\widehat{\varphi_{\tau,\epsilon}^{\pm}}(r)dr.$

We will estimate each term and give an upper bound for
$\sum\varphi_{\tau,\epsilon}^{-}(\ell(\gamma))$ and a lower bound for
$\sum\varphi_{\tau,\epsilon}^{+}(\ell(\gamma))$ which appear in Lemma 20.

######  Lemma 21.

For $\tau\in[1-\epsilon,L+\epsilon]$ where $L>1$ and $0<\epsilon<0.01$, we
have

$\displaystyle\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq L\end{array}}\varphi_{\tau,\epsilon}^{-}(\ell(\gamma))$
$\displaystyle\leq\frac{1}{2}\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}\tau}}{\tau}+O_{\eta}\left(g\epsilon\frac{e^{\tau}}{\tau}+\frac{g}{\epsilon^{2}}\frac{e^{\frac{1}{2}\tau}}{\tau}+\frac{1}{\epsilon}\frac{e^{\frac{1}{2}\tau}}{\tau}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right).$

###### Proof.

Since $\varphi_{\tau,\epsilon}^{-}(x)\geq 0$ when $x>\epsilon$ and
$\varphi_{\tau,\epsilon}^{-}(x)=0$ when $x\geq\tau+\epsilon$, by applying
$\varphi_{\tau,\epsilon}^{-}(x)\geq-\frac{2}{\epsilon}$ in (6) and the fact
that the function $\frac{x}{2\sinh\frac{x}{2}}$ is decreasing on $(0,\infty)$,
we have

(35)
$\displaystyle\sum_{k=1}^{\infty}\sum_{\gamma\in\mathcal{P}(X_{g})}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}\varphi_{\tau,\epsilon}^{-}(k\ell(\gamma))$
(40) $\displaystyle\geq$
$\displaystyle\frac{\tau+\epsilon}{2\sinh\frac{\tau+\epsilon}{2}}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)>\epsilon\end{array}}\varphi_{\tau,\epsilon}^{-}(\ell(\gamma))-\frac{2}{\epsilon}\sum_{k=1}^{\infty}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
k\ell(\gamma)\leq\epsilon\end{array}}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}$
(45) $\displaystyle\geq$
$\displaystyle\frac{\tau+\epsilon}{2\sinh\frac{\tau+\epsilon}{2}}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}}\varphi_{\tau,\epsilon}^{-}(\ell(\gamma))-\frac{2}{\epsilon}\sum_{k=1}^{\infty}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
k\ell(\gamma)\leq\epsilon\end{array}}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}.$

Denote $\\{\gamma_{1},\cdots,\gamma_{m}\\}$ to be the set of all oriented
primitive closed geodesics with length $\leq\epsilon$. By Theorem 13, $m\leq
6g-6$. Then

(46) $\displaystyle\frac{2}{\epsilon}\sum_{k=1}^{\infty}\
\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
k\ell(\gamma)\leq\epsilon\end{array}}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}=\frac{2}{\epsilon}\sum_{j=1}^{m}\sum_{k=1}^{[\epsilon/\ell(\gamma_{j})]}\frac{\ell(\gamma_{j})}{2\sinh\frac{k\ell(\gamma_{j})}{2}}$
$\displaystyle\leq\frac{2}{\epsilon}\sum_{j=1}^{m}\sum_{k=1}^{[\epsilon/\ell(\gamma_{j})]}\frac{1}{k}\prec\frac{1}{\epsilon}\sum_{j=1}^{m}\left(1+\ln\frac{\epsilon}{\ell(\gamma_{j})}\right)$
$\displaystyle\prec\frac{g}{\epsilon}+\frac{1}{\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}.$

Since $\widehat{\varphi_{\tau,\epsilon}^{-}}(r)\leq 0$ for $r\in\mathbb{R}$,
we have

(47)
$\sum_{\lambda_{j}>\frac{1}{4}}\widehat{\varphi_{\tau,\epsilon}^{-}}(r_{j})\leq
0.$

For each small eigenvalue $\lambda_{j}\in[0,\frac{1}{4}]$,

$\displaystyle\widehat{\varphi_{\tau,\epsilon}^{-}}(r_{j})$ $\displaystyle=$
$\displaystyle\left(\cosh(\tau\sqrt{\frac{1}{4}-\lambda_{j}})-1\right)\left(\widehat{\eta}({\bf
i}\epsilon\sqrt{\frac{1}{4}-\lambda_{j}})\right)^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}e^{(s_{j}-\frac{1}{2})\tau}\left(1+O_{\eta}(\epsilon^{2})\right)$

where $\widehat{\eta}({\bf
i}\epsilon\sqrt{\frac{1}{4}-\lambda_{j}})=1+O_{\eta}(\epsilon^{2})$ since
$\widehat{\eta}(0)=1$ and $\widehat{\eta}$ is even. And hence combine with
Theorem 16 we have

(48)
$\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\widehat{\varphi_{\tau,\epsilon}^{-}}(r_{j})\leq\frac{1}{2}\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}e^{(s_{j}-\frac{1}{2})\tau}+O_{\eta}\left(g\epsilon^{2}e^{\frac{1}{2}\tau}\right).$

For the integral term,

(49) $\displaystyle\left|(g-1)\int_{\mathbb{R}}r\tanh(\pi
r)\widehat{\varphi_{\tau,\epsilon}^{-}}(r)dr\right|$
$\displaystyle=\left|2(g-1)\int_{0}^{\infty}r\tanh(\pi
r)(\cos(r\tau)-1)(\widehat{\eta}(\epsilon r))^{2}dr\right|$ $\displaystyle\leq
4(g-1)\int_{0}^{\infty}r\cdot(\widehat{\eta}(\epsilon r))^{2}dr$
$\displaystyle=4(g-1)\frac{1}{\epsilon^{2}}\int_{0}^{\infty}r\cdot(\widehat{\eta}(r))^{2}dr$
$\displaystyle\prec_{\eta}\frac{g}{\epsilon^{2}}.$

Then plug (35), (46), (47), (48) and (49) into (32) we have

(52)
$\displaystyle\frac{\tau+\epsilon}{2\sinh\frac{\tau+\epsilon}{2}}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq L\end{array}}\varphi_{\tau,\epsilon}^{-}(\ell(\gamma))$
(55) $\displaystyle\leq$
$\displaystyle\frac{1}{2}\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}e^{(s_{j}-\frac{1}{2})\tau}+O_{\eta}\left(g\epsilon^{2}e^{\frac{1}{2}\tau}+\frac{g}{\epsilon^{2}}+\frac{1}{\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right).$

Since
$\frac{2\sinh\frac{\tau+\epsilon}{2}}{\tau+\epsilon}\leq\frac{e^{\tau/2}}{\tau}e^{\epsilon/2}=\frac{e^{\tau/2}}{\tau}(1+O(\epsilon))$,
combining with Theorem 16 we get

(58)
$\displaystyle\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq L\end{array}}\varphi_{\tau,\epsilon}^{-}(\ell(\gamma))$
(61) $\displaystyle\leq$
$\displaystyle\frac{1}{2}\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}\tau}}{\tau}+O_{\eta}\left(g\epsilon\frac{e^{\tau}}{\tau}+\frac{g}{\epsilon^{2}}\frac{e^{\frac{1}{2}\tau}}{\tau}+\frac{1}{\epsilon}\frac{e^{\frac{1}{2}\tau}}{\tau}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right)$

as desired. ∎

######  Lemma 22.

For $\tau\in[1+\epsilon,L-\epsilon]$ where $L>1$ and $0<\epsilon<0.01$, we
have

$\displaystyle\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq L\end{array}}\varphi_{\tau,\epsilon}^{+}(\ell(\gamma))$
$\displaystyle\geq\frac{1}{2}\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}\tau}}{\tau}-O_{\eta}\left(g\epsilon\frac{e^{\tau}}{\tau}+g\frac{(1+\epsilon\tau)}{\epsilon^{2}}\frac{e^{\frac{1}{2}\tau}}{\tau}+\frac{1}{\epsilon}\frac{e^{\frac{1}{2}\tau}}{\tau}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right).$

###### Proof.

Since $\varphi_{\tau,\epsilon}^{+}(x)=0$ when
$x\in[\epsilon,\tau-\epsilon]\cup[\tau+\epsilon,+\infty]$, by applying
$\varphi_{\tau,\epsilon}^{+}(x)\leq\frac{4}{\epsilon}$ in (6) and the fact
that the function $\frac{x}{2\sinh\frac{x}{2}}$ is decreasing on $(0,\infty)$,
we have

(64)
$\displaystyle\sum_{k=1}^{\infty}\sum_{\gamma\in\mathcal{P}(X_{g})}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}\varphi_{\tau,\epsilon}^{+}(k\ell(\gamma))$
(72) $\displaystyle\leq$
$\displaystyle\frac{\tau-\epsilon}{2\sinh\frac{\tau-\epsilon}{2}}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)\in(\tau-\epsilon,\tau+\epsilon)\end{array}}\varphi_{\tau,\epsilon}^{+}(\ell(\gamma))$
$\displaystyle+\frac{4}{\epsilon}\sum_{k=2}^{\infty}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
k\ell(\gamma)\in(\tau-\epsilon,\tau+\epsilon)\end{array}}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}+\frac{4}{\epsilon}\sum_{k=1}^{\infty}\
\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
k\ell(\gamma)\leq\epsilon\end{array}}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}$
(80) $\displaystyle\leq$
$\displaystyle\frac{\tau-\epsilon}{2\sinh\frac{\tau-\epsilon}{2}}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq L\end{array}}\varphi_{\tau,\epsilon}^{+}(\ell(\gamma))$
$\displaystyle+\frac{4}{\epsilon}\sum_{k=2}^{\infty}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
k\ell(\gamma)\in(\tau-\epsilon,\tau+\epsilon)\end{array}}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}+\frac{4}{\epsilon}\sum_{k=1}^{\infty}\
\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
k\ell(\gamma)\leq\epsilon\end{array}}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}.$

With the same argument as (46),

(81) $\frac{4}{\epsilon}\sum_{k=1}^{\infty}\
\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
k\ell(\gamma)\leq\epsilon\end{array}}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}\prec\frac{g}{\epsilon}+\frac{1}{\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}.$

By Corollary 15,

(82) $\displaystyle\frac{4}{\epsilon}\sum_{k=2}^{\infty}\
\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
k\ell(\gamma)\in(\tau-\epsilon,\tau+\epsilon)\end{array}}\frac{\ell(\gamma)}{2\sinh\frac{k\ell(\gamma)}{2}}\leq\frac{4}{\epsilon}\frac{1}{2\sinh\frac{\tau-\epsilon}{2}}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\frac{\tau+\epsilon}{2}\end{array}}\sum_{k=\lceil\frac{\tau-\epsilon}{\ell(\gamma)}\rceil}^{[\frac{\tau+\epsilon}{\ell(\gamma)}]}\ell(\gamma)$
$\displaystyle\prec\frac{1}{\epsilon}e^{-\frac{\tau}{2}}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\ell(\gamma)<\frac{\tau+\epsilon}{2}\end{array}}\left(1+\frac{2\epsilon}{\ell(\gamma)}\right)\ell(\gamma)\prec\frac{1}{\epsilon}e^{-\frac{\tau}{2}}\left((g-1)e^{\frac{\tau+\epsilon}{2}+7}\right)\tau$
$\displaystyle\prec g\frac{\tau}{\epsilon}.$

Since $\widehat{\varphi_{\tau,\epsilon}^{+}}(r)\geq 0$ for $r\in\mathbb{R}$,
we have

(83)
$\sum_{\lambda_{j}>\frac{1}{4}}\widehat{\varphi_{\tau,\epsilon}^{+}}(r_{j})\geq
0.$

For each small eigenvalue $\lambda_{j}\in[0,\frac{1}{4}]$,

$\displaystyle\widehat{\varphi_{\tau,\epsilon}^{+}}(r_{j})$ $\displaystyle=$
$\displaystyle\left(\cosh(\tau\sqrt{\frac{1}{4}-\lambda_{j}})+1\right)\left(\widehat{\eta}({\bf
i}\epsilon\sqrt{\frac{1}{4}-\lambda_{j}})\right)^{2}$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}e^{(s_{j}-\frac{1}{2})\tau}\left(1+O_{\eta}(\epsilon^{2})\right)$

where $\widehat{\eta}({\bf
i}\epsilon\sqrt{\frac{1}{4}-\lambda_{j}})=1+O_{\eta}(\epsilon^{2})$ since
$\widehat{\eta}(0)=1$ and $\widehat{\eta}$ is even. And hence together with
Theorem 16,

(84)
$\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\widehat{\varphi_{\tau,\epsilon}^{+}}(r_{j})\geq\frac{1}{2}\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}e^{(s_{j}-\frac{1}{2})\tau}-O_{\eta}\left(g\epsilon^{2}e^{\frac{1}{2}\tau}\right).$

For the integral term, similar as (49) we have

(85) $\displaystyle\left|(g-1)\int_{\mathbb{R}}r\tanh(\pi
r)\widehat{\varphi_{\tau,\epsilon}^{+}}(r)dr\right|$
$\displaystyle=\left|2(g-1)\int_{0}^{\infty}r\tanh(\pi
r)(\cos(r\tau)+1)(\widehat{\eta}(\epsilon r))^{2}dr\right|$ $\displaystyle\leq
4(g-1)\int_{0}^{\infty}r\cdot(\widehat{\eta}(\epsilon r))^{2}dr$
$\displaystyle=4(g-1)\frac{1}{\epsilon^{2}}\int_{0}^{\infty}r\cdot(\widehat{\eta}(r))^{2}dr$
$\displaystyle\prec_{\eta}\frac{g}{\epsilon^{2}}.$

Then plug (64), (81), (82), (83), (84) and (85) into the trace formula (32),
we have

(88)
$\displaystyle\frac{\tau-\epsilon}{2\sinh\frac{\tau-\epsilon}{2}}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq L\end{array}}\varphi_{\tau,\epsilon}^{+}(\ell(\gamma))$
(91) $\displaystyle\geq$
$\displaystyle\frac{1}{2}\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}e^{(s_{j}-\frac{1}{2})\tau}-O_{\eta}\left(g\epsilon^{2}e^{\frac{1}{2}\tau}+\frac{g}{\epsilon^{2}}+g\frac{\tau}{\epsilon}+\frac{1}{\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right).$

Since
$\frac{2\sinh\frac{\tau-\epsilon}{2}}{\tau-\epsilon}\geq\frac{e^{\tau/2}}{\tau}(1-O(\epsilon+e^{-\tau}))$,
combining with Theorem 16 we get

(94)
$\displaystyle\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}}\varphi_{\tau,\epsilon}^{+}(\ell(\gamma))\geq\frac{1}{2}\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}\tau}}{\tau}$
(97) $\displaystyle-
O_{\eta}\left(g\epsilon\frac{e^{\tau}}{\tau}+g\frac{(1+\epsilon\tau)}{\epsilon^{2}}\frac{e^{\frac{1}{2}\tau}}{\tau}+\frac{1}{\epsilon}\frac{e^{\frac{1}{2}\tau}}{\tau}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right)$

as desired. ∎

Now we are ready to prove Theorem 18.

###### Proof of Theorem 18.

Lemma 20 and 21 give the upper bound:

(102)
$\displaystyle\\#\left\\{\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}\right\\}=2\int_{1-\epsilon}^{L+\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}}\varphi_{\tau,\epsilon}^{-}(\ell(\gamma))d\tau$ (105)
$\displaystyle\leq$
$\displaystyle\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\int_{1-\epsilon}^{L+\epsilon}\frac{e^{s_{j}\tau}}{\tau}d\tau+O_{\eta}\left(g\epsilon
e^{\frac{L}{2}}+\frac{g}{\epsilon^{2}}+\frac{1}{\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right)\cdot\frac{e^{\frac{L}{2}}}{L}$
(108) $\displaystyle=$
$\displaystyle\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(e^{s_{j}(L+\epsilon)})+O_{\eta}\left(g\epsilon
e^{\frac{L}{2}}+\frac{g}{\epsilon^{2}}+\frac{1}{\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right)\cdot\frac{e^{\frac{L}{2}}}{L}$
(111) $\displaystyle=$
$\displaystyle\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(e^{s_{j}L})+O_{\eta}\left(g\epsilon
e^{\frac{L}{2}}+\frac{g}{\epsilon^{2}}+\frac{1}{\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right)\cdot\frac{e^{\frac{L}{2}}}{L}$

where in the last equality we apply $\mathop{\rm
Li}(e^{s_{j}(L+\epsilon)})=\mathop{\rm
Li}(e^{s_{j}L})+O\left(\epsilon\frac{e^{L}}{L}\right)$.

For the other direction, Lemma 20 and 22 give the lower bound:

(116)
$\displaystyle\\#\left\\{\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}\right\\}=2\int_{1-\epsilon}^{L+\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}}\varphi_{\tau,\epsilon}^{+}(\ell(\gamma))d\tau$ (119)
$\displaystyle\geq
2\int_{1+\epsilon}^{L-\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}}\varphi_{\tau,\epsilon}^{+}(\ell(\gamma))d\tau\geq\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\int_{1+\epsilon}^{L-\epsilon}\frac{e^{s_{j}\tau}}{\tau}d\tau$
(122) $\displaystyle\quad-O_{\eta}\left(g\epsilon
e^{\frac{L}{2}}+g\frac{1+\epsilon
L}{\epsilon^{2}}+\frac{1}{\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right)\cdot\frac{e^{\frac{L}{2}}}{L}$
$\displaystyle=\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(e^{s_{j}(L-\epsilon)})$ (125) $\displaystyle\quad-O_{\eta}\left(g\epsilon
e^{\frac{L}{2}}+g\frac{1+\epsilon
L}{\epsilon^{2}}+\frac{1}{\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right)\cdot\frac{e^{\frac{L}{2}}}{L}$
(128) $\displaystyle=\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(e^{s_{j}L})-O_{\eta}\left(g\epsilon e^{\frac{L}{2}}+g\frac{1+\epsilon
L}{\epsilon^{2}}+\frac{1}{\epsilon}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<\epsilon\end{array}}\ln\frac{\epsilon}{\ell(\gamma)}\right)\cdot\frac{e^{\frac{L}{2}}}{L}$

where in the last equality we apply $\mathop{\rm
Li}(e^{s_{j}(L-\epsilon)})=\mathop{\rm
Li}(e^{s_{j}L})+O\left(\epsilon\frac{e^{L}}{L}\right)$. For sufficiently large
$L$, take $\epsilon=e^{-\frac{1}{6}L}<0.01$ and fix a function $\eta$, we have

$\displaystyle\\#\left\\{\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
1<\ell(\gamma)\leq
L\end{array}\right\\}=\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(e^{s_{j}L})$
$\displaystyle+O\left(g\frac{e^{\frac{5}{6}L}}{L}+\frac{e^{\frac{2}{3}L}}{L}\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<e^{-L/6}\end{array}}\ln\frac{1}{\ell(\gamma)e^{\frac{1}{6}L}}\right).$

Then the proof is completed by taking $t=e^{L}$. ∎

##  4\. Expectation of the error term in PGT

In this section we study the expected value of $\pi_{X_{g}}(t)$ over
$\mathcal{M}_{g}$ for large genus. We will apply Theorem 18 and our previous
work [WX22] to complete the proof of Theorem 3.

We first see the following consequence of Theorem 18.

######  Corollary 23.

For any $t>2$,

$\mathbb{E}_{\rm WP}^{g}\left[\ \left|\pi_{X_{g}}(t)-\mathop{\rm
Li}(t)-\sum_{0<\lambda_{j}\leq\frac{1}{4}}\mathop{\rm Li}(t^{s_{j}})\right|\
\right]\prec g\frac{t^{\frac{5}{6}}}{\ln t}.$

###### Proof.

By Theorem 13 we know that there are at most $6g-6$ oriented primitive closed
geodesics with length $\leq 1$. So for $t>2$, we have

$\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<t^{-1/6}\end{array}}\ln\left(\frac{1}{\ell(\gamma)t^{\frac{1}{6}}}\right)\prec\max\left\\{0,g\cdot\ln\left(\frac{1}{\mathop{\rm
sys}(X_{g})t^{\frac{1}{6}}}\right)\right\\}\leq\frac{g}{\mathop{\rm
sys}(X_{g})t^{\frac{1}{6}}}.$

It was shown by Mirzakhani in [Mir13, Corollary 4.3] that

$\mathbb{E}_{\rm WP}^{g}\left[\frac{1}{\mathop{\rm sys}(X_{g})}\right]\asymp
1.$

Then the conclusion clearly follows by Theorem 18 and the two estimates above.
∎

To prove Theorem 3, in light of Corollary 23 we need to estimate the term

$\mathbb{E}_{\rm WP}^{g}\left[\sum_{0<\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(t^{s_{j}})\right].$

We will apply our previous work [WX22] to control the term above. In [WX22] we
proved

######  Theorem 24.

For any $\epsilon>0$,

$\lim_{g\rightarrow\infty}\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\lambda_{1}(X_{g})>\frac{3}{16}-\epsilon\right)=1.$

One can also see an independent proof by Lipnowski-Wright in [LW21]. The first
positive lower bound $0.0024$ was established by Mirzakhani in [Mir13]. One
may also see related results for random covers of a fixed closed hyperbolic
surface in [MNP22] by Magee-Naud-Puder; and see a recent breakthrough by Hide-
Magee [HM21] for optimal relative spectral gap of random covers of a fixed
cusped hyperbolic surface.

Let $\phi_{T}$ be the test function on [WX22, Page 355]. Its Fourier transform
satisfies

(131) $\widehat{\phi_{T}}(z)\geq 0\text{ when $z\in\mathbb{R}\cup{\bf
i}\cdot\mathbb{R}$.}$

The proof of [WX22, Lemma 22] also gives that for any $j$ with
$\lambda_{j}(X_{g})<\frac{1}{4}$ and any $0<\epsilon<1$ there exists a
constant $C_{\epsilon}>0$ only depending on $\epsilon$ such that

(132) $\widehat{\phi_{4\ln g}}(r_{j}(X_{g}))\geq
C_{\epsilon}g^{4(1-\epsilon)\sqrt{\frac{1}{4}-\lambda_{j}(X_{g})}}\ln g.$

For the proof of Theorem 24, the essential part is the following estimation.

######  Theorem 25.

Let $\phi_{T}$ be the test function on [WX22, Page 355]. Then for any
$\epsilon_{1}>0$, there exists a constant $C(\epsilon_{1})>0$ only depending
on $\epsilon_{1}$ such that for all $g>C(\epsilon_{1})$,

$\mathbb{E}_{\rm
WP}^{g}\left[\sum_{0<\lambda_{j}\leq\frac{1}{4}}\widehat{\phi_{4\ln
g}}(r_{j}(X_{g}))\right]\leq g^{1+4\epsilon_{1}}(\ln g)^{68}.$

###### Proof.

This is exactly [WX22, Equation (56)]. ∎

Now we apply Theorem 25 to prove the following result which is slightly
stronger than Theorem 24 and essential in the proof of Theorem 3.

######  Theorem 26.

For any $\epsilon>0$, $0\leq r<\frac{1}{2}$ and $1\leq j\leq 2g-3$, there
exists a constant $c(\epsilon)>0$ independent of $g$, $r$ and $j$ such that
for $g>0$ large enough,

$\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\lambda_{j}(X_{g})\leq\frac{1}{4}-r^{2}\right)\leq
c(\epsilon)\frac{1}{j}g^{1-4r+\epsilon}.$

###### Proof.

By (131) and (132) we know that for any $0<\epsilon<1$ and any $0\leq
r<\frac{1}{2}$ there exists a constant $C_{\epsilon}>0$ only depending on
$\epsilon$ such that

$\displaystyle\mathbb{E}_{\rm
WP}^{g}\left[\sum_{0<\lambda_{j}\leq\frac{1}{4}}\widehat{\phi_{4\ln
g}}(r_{j}(X_{g}))\right]$ $\displaystyle\geq\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\lambda_{j}(X_{g})\leq\frac{1}{4}-r^{2}\right)\cdot j\cdot
C_{\epsilon}g^{4(1-\epsilon)r}\ln g.$

Which together with Theorem 25 implies that for any $\epsilon_{1}>0$ and $g>0$
large enough we have

$\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\lambda_{j}(X_{g})\leq\frac{1}{4}-r^{2}\right)\leq\frac{(\ln g)^{67}}{j\cdot
C_{\epsilon}}\cdot g^{1-4r+4(\epsilon_{1}+r\cdot\epsilon)}.$

Then the conclusion follows by choosing suitable $\epsilon,\epsilon_{1}$ and
$c(\epsilon)$. ∎

######  Remark.

If we choose $j=1$, $r=\frac{1}{4}+\epsilon$, by letting $g\to\infty$ Theorem
26 implies Theorem 24.

Now we apply Theorem 26 to estimate $\mathbb{E}_{\rm
WP}^{g}\left[\sum_{0<\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(t^{s_{j}})\right]$ for large $t=t(g)$. More precisely,

######  Theorem 27.

For any $\epsilon>0$ and $t>2g^{4}$, we have

$\mathbb{E}_{\rm WP}^{g}\left[\sum_{0<\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(t^{s_{j}})\right]\prec_{\epsilon}g^{-1+3\epsilon}\mathop{\rm Li}(t).$

###### Proof.

For a non-negative random variable $\xi(X)$, the expectation is

$\displaystyle\mathbb{E}_{\rm WP}^{g}[\xi]$ $\displaystyle=$
$\displaystyle\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\xi(X)dX$ $\displaystyle=$
$\displaystyle\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\int_{0}^{\infty}\mathbf{1}_{\\{\xi(X)\geq
t\\}}dt\ dX$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}\mathop{\rm
Prob}\nolimits_{\rm WP}^{g}(\xi(X)\geq t)dt.$

So

$\displaystyle\mathbb{E}_{\rm WP}^{g}\left[\mathop{\rm
Li}(t^{s_{j}})\mathbf{1}_{\\{\frac{1}{2}\leq s_{j}<1\\}}\right]$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\mathop{\rm
Prob}\nolimits_{\rm WP}^{g}\left(\mathop{\rm
Li}(t^{s_{j}})\mathbf{1}_{\\{\frac{1}{2}\leq s_{j}<1\\}}\geq\rho\right)d\rho$
$\displaystyle\leq$ $\displaystyle\mathop{\rm
Li}(t^{\frac{1}{2}})+\int_{\mathop{\rm Li}(t^{\frac{1}{2}})}^{\mathop{\rm
Li}(t)}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(\mathop{\rm
Li}(t^{s_{j}})\mathbf{1}_{\\{\frac{1}{2}\leq s_{j}<1\\}}\geq\rho\right)d\rho$
$\displaystyle=$ $\displaystyle\mathop{\rm
Li}(t^{\frac{1}{2}})+\int_{\frac{1}{2}}^{1}\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(s\leq s_{j}<1\right)\left(\frac{d}{ds}\mathop{\rm
Li}(t^{s})\right)ds$ $\displaystyle=$ $\displaystyle\mathop{\rm
Li}(t^{\frac{1}{2}})+\int_{\frac{1}{2}}^{1}\frac{t^{s}}{s}\mathop{\rm
Prob}\nolimits_{\rm WP}^{g}\left(s_{j}\geq s\right)ds$

where in the last equation we apply $\frac{d}{ds}\mathop{\rm
Li}(t^{s})=\frac{d}{ds}\left(\int_{2}^{t^{s}}\frac{1}{\ln
x}dx\right)=\frac{t^{s}}{s}$. By Theorem 26, for any $\epsilon>0$,
$\frac{1}{2}\leq s<1$ and $1\leq j\leq 2g-3$, we have

$\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
s_{j}(X_{g})\geq s\right)\leq c(\epsilon)\frac{1}{j}g^{3-4s+\epsilon}.$

Then it follows by Theorem 16 that for $t>2g^{4}$,

$\displaystyle\mathbb{E}_{\rm
WP}^{g}\left[\sum_{0<\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(t^{s_{j}})\right]$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2g-3}\mathbb{E}_{\rm WP}^{g}\left[\mathop{\rm
Li}(t^{s_{j}})\mathbf{1}_{\\{\frac{1}{2}\leq s_{j}<1\\}}\right]$
$\displaystyle\leq$ $\displaystyle(2g-3)\mathop{\rm
Li}(t^{\frac{1}{2}})+\sum_{j=1}^{2g-3}\int_{\frac{1}{2}}^{1}\frac{t^{s}}{s}c(\epsilon)\frac{1}{j}g^{3-4s+\epsilon}ds$
$\displaystyle\asymp$ $\displaystyle g\mathop{\rm
Li}(t^{\frac{1}{2}})+c(\epsilon)\cdot
g^{3-\epsilon}\cdot\left(\sum_{j=1}^{2g-3}\frac{1}{j}\right)\cdot\int_{\frac{1}{2}}^{1}\left(\frac{t}{g^{4}}\right)^{s}ds$
$\displaystyle\prec$ $\displaystyle g\mathop{\rm
Li}(t^{\frac{1}{2}})+c(\epsilon)g^{-1+\epsilon}\ln(g)\frac{t}{\ln t-4\ln g}$
$\displaystyle\prec_{\epsilon}$ $\displaystyle g\mathop{\rm
Li}(t^{\frac{1}{2}})+g^{-1+2\epsilon}\mathop{\rm Li}(t)\ln(g)$
$\displaystyle\prec_{\epsilon}$ $\displaystyle g^{-1+3\epsilon}\mathop{\rm
Li}(t).$

The proof is complete. ∎

Now we complete the proof of Theorem 3.

###### Proof of Theorem 3.

By triangle inequality and Corollary 23 we know that for $t>g^{12}$,

$\mathbb{E}_{\rm WP}^{g}\left[\left|\pi_{X_{g}}(t)-\mathop{\rm
Li}(t)\right|\right]\prec\left(\mathbb{E}_{\rm
WP}^{g}\left[\sum_{0<\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(t^{s_{j}})\right]+\frac{g}{t^{\frac{1}{6}}}\cdot\mathop{\rm Li}(t)\right)$

which together with Theorem 27 implies that for any $\epsilon>0$,

$\mathbb{E}_{\rm WP}^{g}\left[\ \left|1-\frac{\pi_{X_{g}}(t)}{\mathop{\rm
Li}(t)}\right|\
\right]\prec_{\epsilon}\left(\frac{1}{g^{1-3\epsilon}}+\frac{g}{t^{\frac{1}{6}}}\right).$

Then the conclusion clearly follows because $t>g^{12}$. ∎

##  5\. PGT for random hyperbolic surfaces

In this section, our main goal is to show that

######  Theorem 28.

There exists a subset $\mathcal{A}_{g}\subset\mathcal{M}_{g}$ with
$\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}(\mathcal{A}_{g})=1$ such that for any $X_{g}\in\mathcal{A}_{g}$ and
$t=t(g)>2$ which may depend on $g$,

$\pi_{X_{g}}(t)=\mathop{\rm
Li}(t)+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(t^{s_{j}})+O\left(g\frac{t^{\frac{3}{4}}}{\ln t}\right)$

where the implied constant is independent of $t$, $g$ and $X_{g}$.

Theorem 1 is as a direct consequence.

###### Proof of Theorem 1.

Since $\ln t>\ln 2$, it clearly follows by Theorem 16, 24 and 28. ∎

For the proof of Theorem 28, we split into two parts. In the first subsection,
with an assumption on Weyl’s law which is based on the work of Monk [Mon22],
we will show a Prime Geodesic Theorem with a complicated remainder. Then in
the second subsection we will simplify the remainder to the form as shown in
Theorem 28 for random hyperbolic surfaces.

###  5.1. First part of Theorem 28

We follow the strategy by Randol in [Cha84, Chapter XI].

For any $0\leq a\leq b$, denote $N_{X_{g}}^{\Delta}(a,b)$ to be the number of
eigenvalues of $X_{g}$ in the interval $[a,b]$. The standard Weyl’s law says
that

$\lim_{b\to\infty}\frac{N_{X_{g}}^{\Delta}(0,b)}{(g-1)b}=1$

for any fixed hyperbolic surface $X_{g}\in\mathcal{M}_{g}$. Monk [Mon22] gave
a bound of the remainder of Weyl’s law for random hyperbolic surfaces.

######  Theorem 29 ([Mon22, Theorem 5]).

There exists a universal constant $C>0$ and a subset
$\mathcal{B}_{g}\subset\mathcal{M}_{g}$ such that $1-\mathop{\rm
Prob}\nolimits_{\rm WP}^{g}(\mathcal{B}_{g})=O\left(g^{-\frac{1}{12}}(\ln
g)^{\frac{9}{8}}\right)$. And for any large enough $g$, any $0\leq a\leq b$
and any surface $X_{g}\in\mathcal{B}_{g}$, one can write

$\frac{N_{X_{g}}^{\Delta}(a,b)}{4\pi(g-1)}=\frac{1}{4\pi}\int_{\frac{1}{4}}^{\infty}\mathbf{1}_{[a,b]}(\lambda)\tanh\left(\pi\sqrt{\lambda-\frac{1}{4}}\right)d\lambda+R(X_{g},a,b)$

where

$-C\sqrt{\frac{b+1}{\ln g}}\leq R(X_{g},a,b)\leq C\sqrt{\frac{b+1}{\ln
g}}\ln\left(2+(b-a)\sqrt{\frac{\ln g}{b+1}}\right)^{\frac{1}{2}}.$

Now we make the following definition.

######  Definition.

For a constant $A\geq 1$, we say that a hyperbolic surface
$X_{g}\in\mathcal{M}_{g}$ satisfies the _condition $\mathcal{P}_{A}$_ if for
any $T>0$,

(133) $\\#\left\\{k;\
\frac{1}{4}<\lambda_{k}(X_{g})\leq\frac{1}{4}+T\right\\}\leq Ag(1+T).$

If we choose $a=\frac{1}{4}$ and $b=\frac{1}{4}+T$ in Theorem 29, we know that
there exists $A\geq 1$ such that as $g\to\infty$, the condition
$\mathcal{P}_{A}$ holds for a generic closed hyperbolic surface
$X_{g}\in\mathcal{M}_{g}$.

In this subsection we aim to show that

######  Theorem 30.

If $X_{g}\in\mathcal{M}_{g}$ satisfies the condition $\mathcal{P}_{A}$ for
some $A\geq 1$, then for any $t>2$,

$\pi_{X_{g}}(t)=\mathop{\rm
Li}(t)+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(t^{s_{j}})+O\left(g\sqrt{A}\frac{t^{\frac{3}{4}}}{\ln
t}+\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<1\end{array}}\ln\left(\frac{1}{\ell(\gamma)}\right)\right)$

where the implied constant is independent of $t$, $A$ and $X_{g}$.

For $X_{g}\in\mathcal{M}_{g}$ and $T>0$, denote

(134) $\displaystyle H(T)=H(X_{g},T)$ $\displaystyle:=\sum_{\ell(\gamma)\leq
T}\frac{\Lambda(\gamma)}{2\sinh(\frac{1}{2}\ell(\gamma))}2\cosh(\frac{1}{2}\ell(\gamma))$
$\displaystyle=\sum_{\ell(\gamma)\leq
T}\Lambda(\gamma)\frac{1+e^{-\ell(\gamma)}}{1-e^{-\ell(\gamma)}}$

where the sum is taken over all oriented closed geodesics in $X_{g}$ with
length $\leq T$, the number $\Lambda(\gamma)=\ell(\gamma_{0})$ where
$\gamma_{0}$ is the unique oriented primitive closed geodesic such that
$\gamma$ is an iterate of $\gamma_{0}$. Let

(135) $\varphi_{T}(x):=2\cosh(\frac{x}{2})\cdot\mathbf{1}_{[-T,T]}(x)$

where $\mathbf{1}_{[-T,T]}(x)$ is the characteristic function on $[-T,T]$.
Since Selberg’s trace formula only admits smooth functions, we do convolution
to $\varphi_{T}(x)$. Let $\eta(x)\geq 0$ be a non-negative smooth even
function satisfying that $supp(\eta)\subset(-1,1)$ and
$\int_{\mathbb{R}}\eta(x)dx=1$. For any $0<\epsilon<1$, let

(136) $\eta_{\epsilon}(x):=\frac{1}{\epsilon}\eta(\frac{x}{\epsilon})$

and

(137)
$\varphi_{T}^{\epsilon}(x):=\varphi_{T}*\eta_{\epsilon}(x)=\int_{\mathbb{R}}2\cosh(\frac{x-y}{2})\mathbf{1}_{[-T,T]}(x-y)\eta_{\epsilon}(y)dy.$

Then $\varphi_{T}^{\epsilon}(x)$ is a smooth even function with
$supp(\varphi_{T}^{\epsilon})\subset(-T-\epsilon,T+\epsilon)$. Their Fourier
transforms satisfy

(138) $\displaystyle\widehat{\varphi_{T}}(r)$
$\displaystyle=\int_{-T}^{T}e^{-{\bf i}rx}2\cosh(\frac{x}{2})dx$
$\displaystyle=\frac{2\sinh(T(\frac{1}{2}+{\bf i}r))}{\frac{1}{2}+{\bf
i}r}+\frac{2\sinh(T(\frac{1}{2}-{\bf i}r))}{\frac{1}{2}-{\bf i}r}$
$\displaystyle=\frac{8}{1+4r^{2}}\left(2r\sin(rT)\cosh(\frac{T}{2})+\cos(rT)\sinh(\frac{T}{2})\right)$

and

(139)
$\widehat{\varphi_{T}^{\epsilon}}(r)=\widehat{\varphi_{T}}(r)\widehat{\eta_{\epsilon}}(r)=\widehat{\varphi_{T}}(r)\widehat{\eta}(\epsilon
r).$

Denote

(140)
$H_{\epsilon}(T)=H_{\epsilon}(X_{g},T):=\sum_{\gamma}\frac{\Lambda(\gamma)}{2\sinh(\frac{1}{2}\ell(\gamma))}\varphi_{T}^{\epsilon}(\ell(\gamma))$

where the sum is taken over all oriented closed geodesics in $X_{g}$.

######  Proposition 31.

If $X_{g}\in\mathcal{M}_{g}$ satisfies the condition $\mathcal{P}_{A}$ defined
in (133) for some $A\geq 1$, then

$H_{\epsilon}(T)=e^{T}+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}T}}{s_{j}}+O_{\eta}\left(Ag\frac{1}{\epsilon}e^{\frac{1}{2}T}+\epsilon^{2}ge^{T}\right).$

###### Proof.

Plug $\varphi_{T}^{\epsilon}$ into Selberg’s trace formula, i.e., Theorem 17,
we have

(141)
$H_{\epsilon}(T)=\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\widehat{\varphi_{T}^{\epsilon}}(r_{j})+\sum_{\lambda_{j}>\frac{1}{4}}\widehat{\varphi_{T}^{\epsilon}}(r_{j})-(g-1)\int_{\mathbb{R}}r\widehat{\varphi_{T}^{\epsilon}}(r)\tanh(\pi
r)dr.$

Now we estimate these three terms separately.

First we estimate the term
$\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\widehat{\varphi_{T}^{\epsilon}}(r_{j})$.
Note that $\widehat{\eta}(r)=1+O_{\eta}(r^{2})$ since
$\int_{\mathbb{R}}\eta(x)dx=1$ and $\widehat{\eta}$ is even. For
$0\leq\lambda_{j}\leq\frac{1}{4}$ we have

(142)
$\displaystyle\widehat{\varphi_{T}^{\epsilon}}(r_{j})=\widehat{\varphi_{T}}(r_{j})\widehat{\eta}(\epsilon
r_{j})$ $\displaystyle=\widehat{\eta}(\epsilon
r_{j})\left(\frac{2\sinh(T(\frac{1}{2}+\sqrt{\frac{1}{4}-\lambda_{j}}))}{\frac{1}{2}+\sqrt{\frac{1}{4}-\lambda_{j}}}+\frac{2\sinh(T(\frac{1}{2}-\sqrt{\frac{1}{4}-\lambda_{j}}))}{\frac{1}{2}-\sqrt{\frac{1}{4}-\lambda_{j}}}\right)$
$\displaystyle=\left(1+O_{\eta}(\epsilon^{2})\right)\left(\frac{e^{s_{j}T}}{s_{j}}+O(e^{\frac{1}{2}T})\right)$
$\displaystyle=\frac{e^{s_{j}T}}{s_{j}}+O_{\eta}\left(e^{\frac{1}{2}T}+\epsilon^{2}e^{s_{j}T}\right).$

By Theorem 16, there are at most $2g-2$ eigenvalues in $[0,\frac{1}{4}]$. So

(143)
$\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\widehat{\varphi_{T}^{\epsilon}}(r_{j})=e^{T}+\sum_{0<\lambda_{j}\frac{1}{4}}\frac{e^{s_{j}T}}{s_{j}}+O_{\eta}\left(ge^{\frac{1}{2}T}+\epsilon^{2}ge^{T}\right).$

Next we estimate the integral term
$(g-1)\int_{\mathbb{R}}r\widehat{\varphi_{T}^{\epsilon}}(r)\tanh(\pi r)dr$.
For real $r$, since
$|\widehat{\varphi_{T}}(r)|=O(\frac{1}{1+|r|}e^{\frac{1}{2}T})$ and
$|\widehat{\eta}(\epsilon r)|=O_{\eta}(\frac{1}{1+|\epsilon r|^{2}})$, we have

(144)
$\displaystyle\left|(g-1)\int_{\mathbb{R}}r\widehat{\varphi_{T}^{\epsilon}}(r)\tanh(\pi
r)dr\right|$
$\displaystyle\prec_{\eta}ge^{\frac{1}{2}T}\int_{\mathbb{R}}\frac{1}{1+|\epsilon
r|^{2}}dr$ $\displaystyle\prec g\frac{1}{\epsilon}e^{\frac{1}{2}T}.$

For the term
$\sum_{\lambda_{j}>\frac{1}{4}}\widehat{\varphi_{T}^{\epsilon}}(r_{j})$
related to large eigenvalues, we first denote $N(r)=\\#\left\\{j;\
\frac{1}{4}<\lambda_{j}(X_{g})\leq\frac{1}{4}+r^{2}\right\\}$ and
$dN(r)=\sum_{\lambda_{j}>\frac{1}{4}}\delta_{r_{j}}(r)dr$ to be a discrete
measure where $r_{j}=\sqrt{\lambda_{j}-\frac{1}{4}}$ and $\delta(\cdot)$ is
the Dirac function. Then

$\sum_{\lambda_{j}>\frac{1}{4}}\widehat{\varphi_{T}^{\epsilon}}(r_{j})=\int_{0}^{\infty}\widehat{\varphi_{T}^{\epsilon}}(r)dN(r).$

Since $X_{g}$ satisfies the condition $\mathcal{P}_{A}$,

$N(r)\leq A\cdot g\cdot(1+r^{2}).$

So we have $N(0)=0$ and
$\lim\limits_{r\to\infty}\frac{N(r)}{(1+r)\cdot(1+\epsilon^{2}r^{2})}=0$. Then
using Integration by Parts we have

$\displaystyle\left|\sum_{\lambda_{j}>\frac{1}{4}}\widehat{\varphi_{T}^{\epsilon}}(r_{j})\right|$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}|\widehat{\varphi_{T}^{\epsilon}}(r)|dN(r)$
$\displaystyle\prec_{\eta}$
$\displaystyle\int_{0}^{\infty}\frac{e^{T/2}}{1+r}\frac{1}{1+|\epsilon
r|^{2}}dN(r)$ $\displaystyle=$
$\displaystyle-e^{T/2}\int_{0}^{\infty}N(r)\frac{d}{dr}\left(\frac{1}{(1+r)(1+\epsilon^{2}r^{2})}\right)dr$
$\displaystyle=$ $\displaystyle
e^{T/2}\int_{0}^{\infty}N(r)\left(\frac{1}{(1+r)^{2}(1+\epsilon^{2}r^{2})}+\frac{2\epsilon^{2}r}{(1+r)(1+\epsilon^{2}r^{2})^{2}}\right)dr$
$\displaystyle\prec$ $\displaystyle
e^{T/2}\int_{0}^{\infty}\frac{N(r)}{1+r^{2}}\frac{1}{1+\epsilon^{2}r^{2}}dr.$

So if $X_{g}\in\mathcal{M}_{g}$ satisfies the condition $\mathcal{P}_{A}$
defined in (133), then

(145)
$\left|\sum_{\lambda_{j}>\frac{1}{4}}\widehat{\varphi_{T}^{\epsilon}}(r_{j})\right|\prec_{\eta}Ag\frac{1}{\epsilon}e^{\frac{1}{2}T}.$

Then the conclusion follows by (141), (143), (144)and (145). ∎

######  Lemma 32.

For any $0<\epsilon<1$, $T>\epsilon$, let $\varphi_{T}$ and
$\varphi_{T}^{\epsilon}$ be functions as above, then we have that for all
$x>0$,

$\frac{1}{\cosh\frac{\epsilon}{2}}\varphi_{T-\epsilon}^{\epsilon}(x)\leq\varphi_{T}(x)\leq\varphi_{T+\epsilon}^{\epsilon}(x).$

###### Proof.

If $x>T$, the conclusion clearly follows because

$\varphi_{T-\epsilon}^{\epsilon}(x)=\varphi_{T}(x)=0\text{ and
}\varphi_{T+\epsilon}^{\epsilon}(x)\geq 0.$

Now we assume $x\in[0,T]$. By definition

$\varphi_{T}(x)=2\cosh\frac{x}{2}.$

Since $\eta$ is even,

$\displaystyle\int_{-\epsilon}^{\epsilon}2\cosh\frac{x-y}{2}\eta_{\epsilon}(y)dy$
$\displaystyle=\int_{0}^{\epsilon}2(\cosh\frac{x-y}{2}+\cosh\frac{x+y}{2})\eta_{\epsilon}(y)dy$
$\displaystyle=4\int_{0}^{\epsilon}\cosh\frac{x}{2}\cosh\frac{y}{2}\eta_{\epsilon}(y)dy$
$\displaystyle=2\cosh\frac{x}{2}\int_{-\epsilon}^{\epsilon}\cosh\frac{y}{2}\eta_{\epsilon}(y)dy.$

Then we have

$\displaystyle\varphi_{T+\epsilon}^{\epsilon}(x)$ $\displaystyle=$
$\displaystyle\int_{-\epsilon}^{\epsilon}2\cosh\frac{x-y}{2}\eta_{\epsilon}(y)dy$
$\displaystyle>$ $\displaystyle 2\cosh\frac{x}{2}$

and

$\displaystyle\varphi_{T-\epsilon}^{\epsilon}(x)$ $\displaystyle\leq$
$\displaystyle\int_{-\epsilon}^{\epsilon}2\cosh\frac{x-y}{2}\eta_{\epsilon}(y)dy$
$\displaystyle<$ $\displaystyle 2\cosh\frac{x}{2}\cosh\frac{\epsilon}{2}.$

The proof is complete. ∎

######  Proposition 33.

If $X_{g}\in\mathcal{M}_{g}$ satisfies the condition $\mathcal{P}_{A}$ for
some $A\geq 1$, then for $T>0$ large enough,

$H(T)=e^{T}+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}T}}{s_{j}}+O\left(g\sqrt{A}e^{\frac{3}{4}T}\right).$

###### Proof.

By the above Lemma 32,

$\frac{1}{\cosh\frac{\epsilon}{2}}H_{\epsilon}(T-\epsilon)\leq H(T)\leq
H_{\epsilon}(T+\epsilon).$

Since $\frac{1}{\cosh(\epsilon/2)}=1+O(\epsilon^{2})$ and $H_{\epsilon}(T)$ is
increasing with respect to $T$,

$\displaystyle\left|H(T)-H_{\epsilon}(T)\right|$ $\displaystyle\leq$
$\displaystyle\left|H_{\epsilon}(T+\epsilon)-H_{\epsilon}(T)\right|+\left|H_{\epsilon}(T+\epsilon)-\frac{1}{\cosh\frac{\epsilon}{2}}H_{\epsilon}(T-\epsilon)\right|$
$\displaystyle\leq$ $\displaystyle
2\left|H_{\epsilon}(T+\epsilon)-H_{\epsilon}(T-\epsilon)\right|+O(\epsilon^{2})H_{\epsilon}(T-\epsilon).$

Recall that $s_{j}\in[\frac{1}{2},1]$. Since
$\frac{e^{s_{j}(T\pm\epsilon)}}{s_{j}}=\frac{e^{s_{j}T}}{s_{j}}(1+O(\epsilon))$,
it follows by Proposition 31 and Theorem 16 that for $X_{g}\in\mathcal{M}_{g}$
satisfying the condition $\mathcal{P}_{A}$ defined in (133) we have

$\epsilon^{2}H_{\epsilon}(T-\epsilon)\prec Ag\epsilon
e^{\frac{1}{2}T}+g\epsilon^{2}e^{T}$

and

$\displaystyle\left|H_{\epsilon}(T+\epsilon)-H_{\epsilon}(T-\epsilon)\right|$
$\displaystyle=$
$\displaystyle\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}(T+\epsilon)}}{s_{j}}-\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}(T-\epsilon)}}{s_{j}}+O_{\eta}\left(Ag\frac{1}{\epsilon}e^{\frac{1}{2}T}+g\epsilon^{2}e^{T}\right)$
$\displaystyle=$ $\displaystyle
O(\epsilon)\sum_{0\leq\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}T}}{s_{j}}+O_{\eta}\left(Ag\frac{1}{\epsilon}e^{\frac{1}{2}T}+g\epsilon^{2}e^{T}\right)$
$\displaystyle=$ $\displaystyle
O_{\eta}\left(Ag\frac{1}{\epsilon}e^{\frac{1}{2}T}+g\epsilon e^{T}\right).$

So

$\left|H(T)-H_{\epsilon}(T)\right|\prec_{\eta}\left(Ag\frac{1}{\epsilon}e^{\frac{1}{2}T}+g\epsilon
e^{T}\right).$

And hence

$H(T)=e^{T}+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}T}}{s_{j}}+O_{\eta}\left(Ag\frac{1}{\epsilon}e^{\frac{1}{2}T}+g\epsilon
e^{T}\right).$

Then the conclusion follows by taking $\epsilon=\sqrt{A}e^{-\frac{1}{4}T}$ and
fixing $\eta$. ∎

Now we transfer to prove for $\pi_{X_{g}}(t)$. Denote

(146) $\psi(T)=\psi(X_{g},T):=\sum_{\ell(\gamma)\leq T}\Lambda(\gamma)$

where the sum is taken over all oriented closed geodesics in $X_{g}$ with
length $\leq T$, and the number $\Lambda(\gamma)=\ell(\gamma_{0})$ where
$\gamma_{0}$ is the unique oriented primitive closed geodesic such that
$\gamma$ is an iterate of $\gamma_{0}$. Then we have

######  Lemma 34.

$0\leq H(T)-\psi(T)\prec
g\cdot(1+T^{2})+\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<1\end{array}}\ln\left(\frac{1}{\ell(\gamma)}\right)$

where $\mathcal{P}(X_{g})$ is the set of all oriented primitive closed
geodesics in $X_{g}$.

###### Proof.

Rewrite

$\displaystyle H(T)$ $\displaystyle=$ $\displaystyle\sum_{\ell(\gamma)\leq
T}\Lambda(\gamma)\frac{1+e^{-\ell(\gamma)}}{1-e^{-\ell(\gamma)}}$
$\displaystyle=$ $\displaystyle\psi(T)+\sum_{\ell(\gamma)\leq
T}\Lambda(\gamma)\frac{2}{e^{\ell(\gamma)}-1}.$

So

$0\leq H(T)-\psi(T)=\sum_{\tiny\begin{array}[]{c}\ell(\gamma)\leq T\\\
\Lambda(\gamma)>2\mathop{\rm
arcsinh}1\end{array}}\frac{2\Lambda(\gamma)}{e^{\ell(\gamma)}-1}+\sum_{\tiny\begin{array}[]{c}\ell(\gamma)\leq
T\\\ \Lambda(\gamma)\leq 2\mathop{\rm
arcsinh}1\end{array}}\frac{2\Lambda(\gamma)}{e^{\ell(\gamma)}-1}.$

For the first term, by Theorem 14,

$\displaystyle\sum_{\tiny\begin{array}[]{c}\ell(\gamma)\leq T\\\
\Lambda(\gamma)>2\mathop{\rm
arcsinh}1\end{array}}\frac{2\Lambda(\gamma)}{e^{\ell(\gamma)}-1}$
$\displaystyle\prec$
$\displaystyle\sum_{k=1}^{[T]}\sum_{\tiny\begin{array}[]{c}k<\ell(\gamma)\leq
k+1\\\ \Lambda(\gamma)>2\mathop{\rm
arcsinh}1\end{array}}\frac{\ell(\gamma)}{e^{\ell(\gamma)}}$
$\displaystyle\prec$ $\displaystyle\sum_{k=1}^{[T]}ge^{k}\frac{k}{e^{k}}$
$\displaystyle\prec$ $\displaystyle gT^{2}.$

For the second term, let $\\{\gamma_{1},\cdots,\gamma_{m}\\}$ be the set of
all oriented primitive closed geodesics in $X_{g}$ with length $\leq
2\mathop{\rm arcsinh}1$. By Theorem 13, $m\leq 6g-6$. Then we have

$\displaystyle\sum_{\tiny\begin{array}[]{c}\ell(\gamma)\leq T\\\
\Lambda(\gamma)\leq 2\mathop{\rm
arcsinh}1\end{array}}\frac{2\Lambda(\gamma)}{e^{\ell(\gamma)}-1}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{m}\sum_{k=1}^{[T/\ell(\gamma_{j})]}\frac{2\ell(\gamma_{j})}{e^{k\ell(\gamma_{j})}-1}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{m}\ell(\gamma_{j})\left(\frac{2}{e^{\ell(\gamma_{j})}-1}+\int_{1}^{\infty}\frac{2}{e^{x\ell(\gamma_{j})}-1}dx\right)$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{m}\ell(\gamma_{j})\left(\frac{2}{\ell(\gamma_{j})}+\frac{2}{\ell(\gamma_{j})}\ln(\frac{1}{1-e^{-\ell(\gamma_{j})}})\right)$
$\displaystyle\prec$
$\displaystyle\sum_{j=1}^{m}\left(1+\max\left\\{1,\ln\frac{1}{\ell(\gamma_{j})}\right\\}\right)$
$\displaystyle\prec$ $\displaystyle
g+\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<1\end{array}}\ln\left(\frac{1}{\ell(\gamma)}\right).$

Then combining the two parts above, we finish the proof. ∎

Now denote

(150)
$\nu(T)=\nu(X_{g},T):=\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)\leq T\end{array}}\ell(\gamma)$

where $\mathcal{P}(X_{g})$ is the set of all oriented primitive closed
geodesics in $X_{g}$. Then we have

######  Lemma 35.

$0\leq\psi(T)-\nu(T)\prec gTe^{\frac{1}{2}T}.$

###### Proof.

By definitions,

$\displaystyle 0\leq\psi(T)-\nu(T)$ $\displaystyle=$
$\displaystyle\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)\leq T/2\end{array}}\sum_{k=2}^{[T/\ell(\gamma)]}\ell(\gamma)$
$\displaystyle\leq$ $\displaystyle
T\cdot\\#\left\\{\gamma\in\mathcal{P}(X_{g});\
\ell(\gamma)\leq\frac{T}{2}\right\\}.$

By Corollary 15,

$\\#\left\\{\gamma\in\mathcal{P}(X_{g});\
\ell(\gamma)\leq\frac{T}{2}\right\\}\prec ge^{\frac{1}{2}T}.$

Then the conclusion follows. ∎

Now we are ready to prove Theorem 30.

###### Proof of Theorem 30.

First combining Proposition 33, Lemma 34 and 35 we have

(152)
$\nu(T)=e^{T}+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}T}}{s_{j}}+O\left(g\sqrt{A}e^{\frac{3}{4}T}+\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<1\end{array}}\ln\left(\frac{1}{\ell(\gamma)}\right)\right).$

We know that

(153) $\displaystyle\pi_{X_{g}}(t)$
$\displaystyle=\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)\leq\ln t\end{array}}\frac{1}{\ell(\gamma)}\ell(\gamma)$
$\displaystyle=\\#\\{\gamma\in\mathcal{P}(X_{g});\ \ell(\gamma)\leq\ln
2\\}+\int_{T\in(\ln 2,\ln t]}\frac{1}{T}d\nu(T)$

where $d\nu(T)=\sum\ell(\gamma)\delta_{\ell(\gamma)}(T)dT$ is a discrete
measure, and $\delta(\cdot)$ is the Dirac function. By using (152),
Integration by Parts and Theorem 16, the second term satisfies

(154) $\displaystyle\int_{(\ln 2,\ln t]}\frac{d\nu(T)}{T}=\frac{\nu(\ln
t)}{\ln t}-\frac{\nu(\ln 2)}{\ln 2}+\int_{\ln 2}^{\ln
t}\nu(T)\frac{1}{T^{2}}dT$ $\displaystyle=\frac{t}{\ln
t}+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\frac{t^{s_{j}}}{s_{j}\ln
t}-\frac{2}{\ln
2}-\sum_{0<\lambda_{j}\leq\frac{1}{4}}\frac{2^{s_{j}}}{s_{j}\ln 2}$
$\displaystyle+\int_{\ln 2}^{\ln
t}\left(e^{T}+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\frac{e^{s_{j}T}}{s_{j}}\right)\frac{1}{T^{2}}dT$
$\displaystyle+O\left(g\sqrt{A}\frac{t^{\frac{3}{4}}}{\ln
t}+\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<1\end{array}}\ln\frac{1}{\ell(\gamma)}\right)$
$\displaystyle=\int_{\ln 2}^{\ln
t}\frac{e^{T}}{T}dT+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\int_{\ln 2}^{\ln
t}\frac{e^{s_{j}T}}{T}dT$
$\displaystyle+O\left(g\sqrt{A}\frac{t^{\frac{3}{4}}}{\ln
t}+\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<1\end{array}}\ln\frac{1}{\ell(\gamma)}\right)$
$\displaystyle=\mathop{\rm
Li}(t)+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(t^{s_{j}})+O\left(g\sqrt{A}\frac{t^{\frac{3}{4}}}{\ln
t}+\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<1\end{array}}\ln\frac{1}{\ell(\gamma)}\right).$

For the first term, it follows by Theorem 13 that for $t>2$,

(155) $\\#\\{\gamma\in\mathcal{P}(X_{g});\ \ell(\gamma)\leq\ln 2\\}\leq
6g-6\prec g\sqrt{A}\frac{t^{\frac{3}{4}}}{\ln t}.$

Then the conclusion follows by (152), (153), (154) and (155). ∎

###  5.2. Second part of Theorem 28

In this subsection we aim to finish the proof of Theorem 28. For
$X_{g}\in\mathcal{B}_{g}\subset\mathcal{M}_{g}$ given in Theorem 29, there
exists a universal constant $A>1$ such that the condition $\mathcal{P}_{A}$
(133) holds. So using Theorem 30, it remain to estimate
$\sum_{\ell(\gamma)<1,\gamma\in\mathcal{P}(X_{g})}\ln\frac{1}{\ell(\gamma)}$
for random hyperbolic surfaces.

######  Lemma 36.

$\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<1\end{array}}\ln\frac{1}{\ell(\gamma)}<g\right)=1-O\left(\frac{1}{g^{0.99}}\right).$

###### Proof.

For a hyperbolic surface $X_{g}\in\mathcal{M}_{g}$, denote
$\\{\gamma_{1},\cdots,\gamma_{m}\\}$ to be the set of all unoriented primitive
closed geodesics with length $<1$. By Theorem 13, they are all simple and
$m\leq 3g-3$. By [Mir13, Theorem 4.4],

$\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\ell_{sys}^{s}(X_{g})<0.01\ln g\right)=O\left(\frac{(\ln
g)^{3}}{g^{0.995}}\right)=O\left(\frac{1}{g^{0.99}}\right)$

where $\ell_{sys}^{s}(X_{g})$ is the length of the shortest simple separating
geodesic in $X_{g}$. So for sufficiently large $g$,

(156) $\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\begin{array}[]{c}\gamma_{1},\cdots,\gamma_{m}\ \text{are all}\\\ \text{simple
non-separating}\end{array}\right)=1-O\left(\frac{1}{g^{0.99}}\right).$

Then we consider unoriented primitive simple non-separating closed geodesics,
which are all in the same orbit of $\mathop{\rm Mod}_{g}$. By using
Mirzakhani’s Integration Formula, i.e., Theorem 8, and Theorem 12 and 10, we
have

$\displaystyle\mathbb{E}_{\rm
WP}^{g}\left[\sideset{}{{}^{nsep}}{\sum}_{\ell(\gamma)<1}\ln\frac{1}{\ell(\gamma)}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{V_{g}}\frac{1}{2}\int_{\mathbb{R}}\left(\ln\frac{1}{x}\right)\mathbf{1}_{(0,1)}(x)xV_{g-1,2}(x,x)dx$
$\displaystyle\leq$
$\displaystyle\frac{V_{g-1,2}}{V_{g}}\int_{0}^{1}\left(\ln\frac{1}{x}\right)\frac{2}{x}(\sinh(x/2))^{2}dx$
$\displaystyle\prec$ $\displaystyle 1$

where the sum $\sum^{nsep}$ is taken over all unoriented primitive simple non-
separating closed geodesics. Then by Markov’s inequality we have

(157) $\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\sideset{}{{}^{nsep}}{\sum}_{\ell(\gamma)<1}\ln\frac{1}{\ell(\gamma)}\geq\frac{1}{2}g\right)=O\left(\frac{1}{g}\right).$

Together with (156) we finish the proof. ∎

Now we are ready to prove Theorem 28.

###### Proof of Theorem 28.

Take

$\mathcal{A}_{g}=\mathcal{B}_{g}\cap\left\\{X_{g}\in\mathcal{M}_{g};\
\sum_{\tiny\begin{array}[]{c}\gamma\in\mathcal{P}(X_{g}),\\\
\ell(\gamma)<1\end{array}}\ln\frac{1}{\ell(\gamma)}<g\right\\}$

where $\mathcal{B}_{g}$ is the subset given in Theorem 29. By Theorem 29 and
Lemma 36,

$\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}(\mathcal{A}_{g})=1-O\left(g^{-\frac{1}{12}}(\ln
g)^{\frac{9}{8}}\right).$

And by Theorem 30 it is clear that for any $X_{g}\in\mathcal{A}_{g}$ and
$t>2$,

$\pi_{X_{g}}(t)=\mathop{\rm
Li}(t)+\sum_{0<\lambda_{j}\leq\frac{1}{4}}\mathop{\rm
Li}(t^{s_{j}})+O\left(g\frac{t^{\frac{3}{4}}}{\ln t}\right).$

The proof is complete. ∎

##  6\. Intersection numbers and Weil-Petersson volumes

In this section, based on Mirzakhani’s recursive formula [Mir07a] we prove
Theorem 5 and 6, which are essential in the proof of Theorem 4.

###  6.1. Intersection numbers

For any $d=(d_{1},\cdots,d_{n})$ with each $d_{i}\in\mathbb{Z}_{\geq 0}$ and
$|d|=d_{1}+\cdots+d_{n}\leq 3g-3+n$, Mirzakhani [Mir07a] gave a recursive
formula as follow. Let

$a_{0}=\frac{1}{2},\ a_{n}=\zeta(2n)(1-2^{1-2n})\ \text{for}\ n\geq 1.$

where $\zeta(\cdot)$ is the Riemann zeta function. Then

(158)
$\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}=\sum_{j=2}^{n}\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}^{A,j}+\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}^{B}+\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}^{C}.$

where

(159) $\displaystyle\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}^{A,j}$
$\displaystyle=$ $\displaystyle
8\sum_{L=0}^{3g-3+n-|d|}(2d_{j}+1)a_{L}\left[\tau_{d_{1}+d_{j}+L-1}\prod_{i\neq
1,j}\tau_{d_{i}}\right]_{g,n-1}$ $\displaystyle=$ $\displaystyle
8\sum_{L=d_{1}+d_{j}-1}^{3g-4+n-|d|+d_{1}+d_{j}}(2d_{j}+1)a_{L-d_{1}-d_{2}+1}\left[\tau_{L}\prod_{i\neq
1,j}\tau_{d_{i}}\right]_{g,n-1},$ (160)
$\displaystyle\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}^{B}$
$\displaystyle=$ $\displaystyle
16\sum_{L=0}^{3g-3+n-|d|}\sum_{k_{1}+k_{2}=L+d_{1}-2}a_{L}\left[\tau_{k_{1}}\tau_{k_{2}}\prod_{i\neq
1}\tau_{d_{i}}\right]_{g-1,n+1}$ $\displaystyle=$ $\displaystyle
16\sum_{L=d_{1}-2}^{3g-5+n-|d|+d_{1}}\sum_{k_{1}+k_{2}=L}a_{L-d_{1}+2}\left[\tau_{k_{1}}\tau_{k_{2}}\prod_{i\neq
1}\tau_{d_{i}}\right]_{g-1,n+1},$ (161)
$\displaystyle\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}^{C}$
$\displaystyle=$ $\displaystyle 16\sum_{I\amalg
J=\\{2,\cdots,n\\}}\sum_{g_{1}+g_{2}=g}\sum_{L=0}^{3g-3+n-|d|}\sum_{k_{1}+k_{2}=L+d_{1}-2}$
$\displaystyle a_{L}\left[\tau_{k_{1}}\prod_{i\in
I}\tau_{d_{i}}\right]_{g_{1},|I|+1}\times\left[\tau_{k_{2}}\prod_{i\in
J}\tau_{d_{i}}\right]_{g_{2},|J|+1}$ $\displaystyle=$ $\displaystyle
16\sum_{I\amalg
J=\\{2,\cdots,n\\}}\sum_{g_{1}+g_{2}=g}\sum_{L=d_{1}-2}^{3g-5+n-|d|+d_{1}}\sum_{k_{1}+k_{2}=L}$
$\displaystyle a_{L-d_{1}+2}\left[\tau_{k_{1}}\prod_{i\in
I}\tau_{d_{i}}\right]_{g_{1},|I|+1}\times\left[\tau_{k_{2}}\prod_{i\in
J}\tau_{d_{i}}\right]_{g_{2},|J|+1}.$

For convenience, here we denote $a_{i}=0$ if $i<0$ and
$[\tau_{d_{1}}\cdots\tau_{d_{n}}]_{g,n}=0$ if some index $d_{i}<0$.

Mirzakhani showed that

######  Lemma 37 ([Mir13, Lemma 3.1]).

The sequence $\\{a_{n}\\}_{n=0}^{\infty}$ is increasing. Moreover,

$\lim_{n\to\infty}a_{n}=1\ \ \text{and}\ \
a_{n+1}-a_{n}\asymp\frac{1}{2^{2n}}.$

So clearly we have

$\left[\tau_{d_{1}+1}\cdots\tau_{d_{n}}\right]_{g,n}^{A,j}<\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}^{A,j},$
$\left[\tau_{d_{1}+1}\cdots\tau_{d_{n}}\right]_{g,n}^{B}<\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}^{B},$
$\left[\tau_{d_{1}+1}\cdots\tau_{d_{n}}\right]_{g,n}^{C}<\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}^{C}.$

These directly yield to

$\left[\tau_{d_{1}+1}\cdots\tau_{d_{n}}\right]_{g,n}<\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}\leq
V_{g,n}$

which implies

(162)
$\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}\leq\left[\tau_{0}\tau_{d_{2}}\cdots\tau_{d_{n}}\right]_{g,n}\leq
V_{g,n}.$

Our main result on intersection numbers in this section is as follows. Theorem
5 is as a direct consequence.

######  Theorem 38.

There exists a universal constant $c>0$ such that for any
$d=(d_{1},\cdots,d_{n})$, assume that $I\neq\emptyset\subset\\{1,\cdots,n\\}$
is the subset consisting of all $i$’s with
$h_{i}=c\cdot\frac{d_{i}^{2}}{2g-2+n}>1$, then we have

$\frac{\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}}{V_{g,n}}\leq\prod_{i\in
I}\left(\frac{1}{h_{i}}\right)^{\frac{1}{2}\log_{4}(h_{i})}.$

First assuming Theorem 38, we prove Theorem 5.

###### Proof of Theorem 5.

Let $I\subset\\{1,\cdots,n\\}$ be the subset in Theorem 38. If $I=\emptyset$,
we have $\frac{2g-2+n}{\max\limits_{1\leq i\leq n}\\{d_{i}^{2}\\}}\geq c$.
Then the conclusion clearly holds by choosing $c^{\prime}=\frac{1}{c}$ because
$\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}\leq V_{g,n}$. If
$I\neq\emptyset$, let $i_{0}\in[1,n]$ with $d_{i_{0}}^{2}=\max\limits_{1\leq
i\leq n}\\{d_{i}^{2}\\}$. It is clear that the function
$f(x)=\left(\frac{1}{x}\right)^{\frac{1}{2}\log_{4}x-1}$ is uniformly bounded
on $[1,\infty)$. Then it follows by Theorem 38 that

(163)
$\displaystyle\frac{\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}}{V_{g,n}}\leq\left(\frac{1}{h_{i_{0}}}\right)^{\frac{1}{2}\log_{4}(h_{i_{0}})}\leq\frac{c^{\prime\prime}}{h_{i_{0}}}$

for some universal constant $c^{\prime\prime}>0$. Then the conclusion follows
by choosing $c^{\prime}=\frac{c^{\prime\prime}}{c}$. ∎

Now we return to prove Theorem 38.

###### Proof of Theorem 38.

For convenience, we denote $a_{i}=0$ if $i<0$ and
$[\tau_{d_{1}}\cdots\tau_{d_{n}}]_{g,n}=0$ if some index $d_{i}<0$. For any
$d^{\prime}=(d^{\prime}_{1},\cdots,d^{\prime}_{n+1})$, firstly by (159) we
have

$\displaystyle\left[\tau_{d^{\prime}_{1}}\tau_{d^{\prime}_{2}}\cdots\tau_{d^{\prime}_{n+1}}\right]_{g,n+1}^{A,j}-\left[\tau_{d^{\prime}_{1}+1}\tau_{d^{\prime}_{2}}\cdots\tau_{d^{\prime}_{n+1}}\right]_{g,n+1}^{A,j}$
$\displaystyle=$ $\displaystyle
8\sum_{L=d^{\prime}_{1}+d^{\prime}_{j}-1}^{3g-3+n-\sum_{i\neq
1,j}d^{\prime}_{i}}(2d^{\prime}_{j}+1)(a_{L-d^{\prime}_{1}-d^{\prime}_{2}+1}-a_{L-d^{\prime}_{1}-d^{\prime}_{2}})\left[\tau_{L}\prod_{i\neq
1,j}\tau_{d^{\prime}_{i}}\right]_{g,n}$ $\displaystyle\geq$ $\displaystyle
8(2d^{\prime}_{j}+1)a_{0}\left[\tau_{d^{\prime}_{1}+d^{\prime}_{j}-1}\prod_{i\neq
1,j}\tau_{d^{\prime}_{i}}\right]_{g,n}.$

So

(164)
$\displaystyle\left[\tau_{0}\tau_{d^{\prime}_{2}}\cdots\tau_{d^{\prime}_{n+1}}\right]_{g,n+1}^{A,j}=\left[\tau_{d^{\prime}_{1}}\cdots\tau_{d^{\prime}_{n+1}}\right]_{g,n+1}^{A,j}$
$\displaystyle+\sum_{i=0}^{d^{\prime}_{1}-1}\left(\left[\tau_{i}\tau_{d^{\prime}_{2}}\cdots\tau_{d^{\prime}_{n+1}}\right]_{g,n+1}^{A,j}-\left[\tau_{i+1}\tau_{d^{\prime}_{2}}\cdots\tau_{d^{\prime}_{n+1}}\right]_{g,n+1}^{A,j}\right)$
$\displaystyle\geq\sum_{i=0}^{d^{\prime}_{1}-1}8(2d^{\prime}_{j}+1)a_{0}\left[\tau_{i+d^{\prime}_{j}-1}\prod_{i\neq
1,j}\tau_{d^{\prime}_{i}}\right]_{g,n}$ $\displaystyle\geq
8d^{\prime}_{1}(2d^{\prime}_{j}+1)a_{0}\left[\tau_{d^{\prime}_{1}+d^{\prime}_{j}-2}\prod_{i\neq
1,j}\tau_{d^{\prime}_{i}}\right]_{g,n}.$

And then for any $d^{\prime}_{1}\geq 0$ and $2\leq j\leq n+1$,

(165)
$\left[\tau_{0}\tau_{d^{\prime}_{2}}\cdots\tau_{d^{\prime}_{n+1}}\right]_{g,n+1}\geq
8d^{\prime}_{1}(2d^{\prime}_{j}+1)a_{0}\left[\tau_{d^{\prime}_{1}+d^{\prime}_{j}-2}\prod_{i\neq
1,j}\tau_{d^{\prime}_{i}}\right]_{g,n}.$

For $d=(d_{1},\cdots,d_{n})$, taking
$d^{\prime}_{1}=\lfloor\frac{d_{1}+2}{2}\rfloor$,
$d^{\prime}_{j}=\lceil\frac{d_{1}+2}{2}\rceil$ and $j=2$ we have

(166) $\displaystyle\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}$
$\displaystyle\leq$
$\displaystyle\frac{\left[\tau_{0}\tau_{\lceil\frac{d_{1}+2}{2}\rceil}\tau_{d_{2}}\cdots\tau_{d_{n}}\right]_{g,n+1}}{8a_{0}\lfloor\frac{d_{1}+2}{2}\rfloor(2\lceil\frac{d_{1}+2}{2}\rceil+1)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2d^{2}_{1}}\left[\tau_{0}\tau_{\lceil\frac{d_{1}+2}{2}\rceil}\tau_{d_{2}}\cdots\tau_{d_{n}}\right]_{g,n+1}$

where the fact $a_{0}=\frac{1}{2}$ is used in the last inequality. Now we
apply this argument for $p_{1}\geq 0$ times and each time $d_{1}$ is reduced
by no more than $\frac{1}{2}$. So combine with (162), we get

$\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}\leq\prod_{k=1}^{p_{1}}\frac{1}{2(d_{1}/2^{k-1})^{2}}\left[\tau_{0}\cdots\tau_{0}\tau_{d_{2}}\cdots\tau_{d_{n}}\right]_{g,n+p_{1}}.$

By induction, for any $p=(p_{1},\cdots,p_{n})$,

(167)
$\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}\leq\prod_{i=1}^{n}\left(\prod_{k_{i}=1}^{p_{i}}\frac{1}{2(d_{i}/2^{k_{i}-1})^{2}}\right)V_{g,n+|p|}.$

By Theorem 10 we know that

$V_{g,n+|p|}\leq(b(2g-2+n+|p|))^{|p|}V_{g,n}$

for a universal constant $b=\frac{120}{10-\pi^{2}}\approx 920.276$. So

(168)
$\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}\leq\prod_{i=1}^{n}\left(\prod_{k_{i}=1}^{p_{i}}\frac{b(2g-2+n+|p|)}{2(d_{i}/2^{k_{i}-1})^{2}}\right)V_{g,n}.$

Let $h_{i}=\frac{2}{3b}\frac{d_{i}^{2}}{2g-2+n}$. Assume
$I\subset\\{1,\cdots,n\\}$ is the subset consisting of all $i$’s with
$h_{i}>1$. For any $i\in[1,n]$, take

$p_{i}=\begin{cases}0&\text{if}\ h_{i}\leq 1\\\ 1+[\log_{4}(h_{i})]&\text{if}\
h_{i}>1\end{cases}.$

Recall that the function $\log_{4}(x)$ is concave on $(0,\infty)$. In
particular,

$\frac{1}{|I|}\log_{4}\left(\prod_{i\in
I}d_{i}\right)\leq\log_{4}\left(\frac{\sum\limits_{i\in I}d_{i}}{|I|}\right).$

Then we have

(169) $\displaystyle|p|$ $\displaystyle=\sum\limits_{i\in
I}p_{i}\leq|I|+\log_{4}\left(\left(\frac{2}{3b(2g-2+n)}\right)^{|I|}\prod_{i\in
I}d_{i}^{2}\right)$
$\displaystyle\leq|I|+|I|\log_{4}\left(\frac{2}{3b(2g-2+n)}\left(\frac{\sum\limits_{i\in
I}d_{i}}{|I|}\right)^{2}\right)$
$\displaystyle\leq|I|+|I|\log_{4}\left(\frac{2(3g-3+n)^{2}}{3b(2g-2+n)}\frac{1}{|I|^{2}}\right)$
$\displaystyle\leq|I|+|I|\frac{3g-3+n}{b}\frac{1}{|I|^{2}}$ $\displaystyle\leq
2(2g-2+n).$

And so for $i\in I$,

(170) $\displaystyle\prod_{k_{i}=1}^{p_{i}}$
$\displaystyle\frac{b(2g-2+n+|p|)}{2(d_{i}/2^{k_{i}-1})^{2}}\leq\prod_{k=1}^{p_{i}}\frac{3b(2g-2+n)}{2(d_{i}/2^{k-1})^{2}}$
$\displaystyle=\prod_{k=1}^{p_{i}}\frac{4^{k-1}}{h_{i}}=\frac{4^{\frac{1}{2}(1+[\log_{4}(h_{i})])[\log_{4}(h_{i})]}}{h_{i}^{1+[\log_{4}(h_{i})]}}$
$\displaystyle\leq(h_{i})^{-\frac{1}{2}\log_{4}(h_{i})}.$

Then the conclusion follows by (168), (169) and (170). ∎

###  6.2. Weil-Petersson volumes

When $\sum x_{i}^{2}=o(2g-2+n)$, for the asymptotic behavior of
$V_{g,n}(x_{1},\cdots,x_{n})$, the bounds in Theorem 12 are effective. In this
subsection we apply Theorem 38 to prove Theorem 6, which is a more effective
upper bound for $V_{g,n}(x_{1},\cdots,x_{n})$ when some of $x_{i}$’s are
larger than $\sqrt{2g-2+n}$. We first prove the following result.

######  Theorem 39.

For any $k\geq 1$, there exists a constant $c(k)>0$ independent of $g,n$ and
$x_{i}$’s such that

$\frac{V_{g,n}(x_{1},\cdots,x_{n})}{V_{g,n}}\leq\left(\prod_{i\in
I}c(k)\frac{2g-2+n}{x_{i}^{2}}\right)^{k}\prod_{i=1}^{n}\frac{\sinh(x_{i}/2)}{x_{i}/2}$

where we assume that $I\neq\emptyset\subset\\{1,\cdots,n\\}$ is the subset
consisting of all $i$’s satisfying that $x_{i}^{2}>c(k)(2g-2+n)$.

###### Proof.

As in Theorem 38 we denote $h_{i}=c\frac{d_{i}^{2}}{2g-2+n}$. Then similar as
in the proof of Theorem 5 we have that for any fixed $k\geq 1$,
$\left(\frac{1}{h_{i}}\right)^{\frac{1}{2}\log_{4}(h_{i})}\prec_{k}\left(\frac{2g-2+n}{(2d_{i}+2k+1)^{2}}\right)^{k}$
whenever $h_{i}>1$, and
$1\prec_{k}\left(\frac{2g-2+n}{(2d_{i}+2k+1)^{2}}\right)^{k}$ whenever
$h_{i}\leq 1$. So combine with Theorem 38 we have that for any subset
$I^{\prime}\subset\\{1,\cdots,n\\}$,

(171)
$\frac{\left[\tau_{d_{1}}\cdots\tau_{d_{n}}\right]_{g,n}}{V_{g,n}}\leq\prod_{i\in
I^{\prime}}\left(c^{\prime}(k)\frac{2g-2+n}{(2d_{i}+2k+1)^{2}}\right)^{k}$

for some $c^{\prime}(k)>0$. Then it follows by Theorem 9 that

(172)
$\displaystyle\frac{V_{g,n}(2x_{1},\cdots,2x_{n})}{V_{g,n}}\leq\sum_{|d|\leq
3g-3+n}\prod_{i\in
I^{\prime}}\left(c^{\prime}(k)\frac{2g-2+n}{(2d_{i}+2k+1)^{2}}\right)^{k}\prod_{i=1}^{n}\frac{x_{i}^{2d_{i}}}{(2d_{i}+1)!}$
$\displaystyle\leq\left(c^{\prime}(k)(2g-2+n)\right)^{k|I^{\prime}|}\cdot\left(\prod_{i\in
I^{\prime}}\sum_{d_{i}=0}^{\infty}\frac{x_{i}^{2d_{i}}}{(2d_{i}+2k+1)!}\right)\cdot\left(\prod_{i\notin
I^{\prime}}\sum_{d_{i}=0}^{\infty}\frac{x_{i}^{2d_{i}}}{(2d_{i}+1)!}\right).$

By Taylor’s expansion,

$\sum_{d=0}^{\infty}\frac{x^{2d}}{(2d+1)!}=\frac{\sinh(x)}{x}$

and hence

$\sum_{d=0}^{\infty}\frac{x^{2d}}{(2d+2k+1)!}\leq\frac{1}{x^{2k}}\sum_{d=0}^{\infty}\frac{x^{2d}}{(2d+1)!}=\frac{\sinh(x)}{x^{2k+1}}.$

Then again by (172),

(173) $\frac{V_{g,n}(2x_{1},\cdots,2x_{n})}{V_{g,n}}\leq\left(\prod_{i\in
I^{\prime}}c^{\prime}(k)\frac{2g-2+n}{x_{i}^{2}}\right)^{k}\prod_{i=1}^{n}\frac{\sinh(x_{i})}{x_{i}}$

for any subset $I^{\prime}\subset\\{1,\cdots,n\\}$. Now let $I^{\prime}=I$,
then the conclusion follows by choosing $c(k)=4c^{\prime}(k)$. The proof is
complete. ∎

Now we complete the proof of Theorem 6.

###### Proof of Theorem 6.

Recall that the upper bound in Theorem 12 says that

(174)
$\frac{V_{g,n}(x_{1},\cdots,x_{n})}{V_{g,n}}\leq\prod_{i=1}^{n}\frac{\sinh(x_{i}/2)}{x_{i}/2}.$

Then the conclusion directly follows by (174) and Theorem 39. ∎

##  7\. Proof of Theorem 4

In this section we prove Theorem 4 confirming [LW21, Conjecture $1.2$].

Similar as [NWX20, WX22], we first make the following notations. Let
$X_{g}\in\mathcal{M}_{g}$ be a closed hyperbolic surface of genus $g$ and
$\mathcal{P}^{\prime}(X_{g})$ be the set of all unoriented primitive closed
geodesics on $X_{g}$. For any $L>0$, we define

  1. (1)

$N(X_{g},L):=\\#\\{\gamma\in\mathcal{P}^{\prime}(X_{g});\
\ell_{\gamma}(X_{g})\leq L\\}$,

  2. (2)

$N^{s}(X_{g},L):=\\#\\{\gamma\in\mathcal{P}^{\prime}(X_{g})\text{ is simple};\
\ell_{\gamma}(X_{g})\leq L\\}$,

  3. (3)

$N^{ns}(X_{g},L):=\\#\\{\gamma\in\mathcal{P}^{\prime}(X_{g})\text{ is non-
simple};\ \ell_{\gamma}(X_{g})\leq L\\}$,

  4. (4)

$N_{nsep}^{s}(X_{g},L):=\\#\\{\gamma\in\mathcal{P}^{\prime}(X_{g})\text{ is
simple and non-separating};\ \ell_{\gamma}(X_{g})\leq L\\}$,

  5. (5)

$N_{sep,k}^{s}(X_{g},L):=\\#\\{\gamma\in\mathcal{P}^{\prime}(X_{g})\text{ and
$X_{g}\setminus\gamma\cong S_{k,1}\cup S_{g-k,1}$};\ \ell_{\gamma}(X_{g})\leq
L\\}$ where $1\leq k\leq\left[\frac{g}{2}\right]$.

Figure 1. Examples: $\alpha$ is simple and non-separating; $\beta$ is simple
and separating; $\gamma$ is non-simple.

It is clear that

$N(X_{g},L)=\frac{\pi_{X_{g}}(e^{L})}{2}$

and

$\displaystyle N(X_{g},L)=N^{s}(X_{g},L)+N^{ns}(X_{g},L)$
$\displaystyle=N_{nsep}^{s}(X_{g},L)+\sum_{k=1}^{\left[\frac{g}{2}\right]}N_{sep,k}^{s}(X_{g},L)+N^{ns}(X_{g},L).$

We split the proof of Theorem 4 into two parts.

###  7.1. Geodesics of length significantly less than $\sqrt{g}$

In this subsection we confirm the first part of [LW21, Conjecture $1.2$]. That
is, we show that as $g\to\infty$, on most surfaces in $\mathcal{M}_{g}$ most
geodesics of length significantly less than $\sqrt{g}$ are simple and non-
separating. More precisely,

######  Theorem 40.

Assume that $L=L(g)$ satisfies

$L(g)>12\ln g\text{ and }\lim\limits_{g\to\infty}\frac{L^{2}(g)}{g}=0.$

Then for any $\delta=\delta(g)>0$ satisfying
$\lim\limits_{g\to\infty}\delta(g)=0$ and
$\lim\limits_{g\to\infty}\frac{1}{\delta}\left(\frac{L^{2}}{g}+\frac{1}{g^{1-\epsilon_{0}}}\right)=0$
for some $\epsilon_{0}\in(0,1)$, we have

$\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\left|1-\frac{N_{nsep}^{s}(X_{g},L)}{N(X_{g},L)}\right|<\delta\right)=1.$

We first make the following lemma.

######  Lemma 41.

Assume that $L=L(g)$ satisfies

$\lim\limits_{g\to\infty}L(g)=\infty\text{ and
}\lim\limits_{g\to\infty}\frac{L^{2}(g)}{g}=0.$

Then

$\mathbb{E}_{\rm WP}^{g}\left[N_{nsep}^{s}(X_{g},L)\right]=\frac{\mathop{\rm
Li}(e^{L})}{2}\cdot\left(1+O\left(\frac{L^{2}}{g}\right)\right)$

where the implied constant is universal.

###### Proof.

First by Mirzakhani’s integration formula, i.e., Theorem 8, we have

$\mathbb{E}_{\rm
WP}^{g}\left[N_{nsep}^{s}(X_{g},L)\right]=\frac{1}{V_{g}}\frac{1}{2}\int_{0}^{L}V_{g-1,2}(x,x)xdx.$

Since $L=L(g)=o(\sqrt{g})$, by Theorem 12 we have that for all $x\in[0,L]$,

$V_{g-1,2}(x,x)=V_{g-1,2}\cdot\left(\frac{\sinh\left(\frac{x}{2}\right)}{\frac{x}{2}}\right)^{2}\cdot\left(1+O\left(\frac{x^{2}}{g}\right)\right)$

where the implied constant is universal. By Theorem 10 we also know that

$\frac{V_{g}}{V_{g-1,2}}=1+O\left(\frac{1}{g}\right).$

Thus,

$\displaystyle\mathbb{E}_{\rm
WP}^{g}\left[N_{nsep}^{s}(X_{g},L)\right]=\int_{0}^{L}\left(\sinh\left(\frac{x}{2}\right)\right)^{2}\cdot\frac{2}{x}\cdot\left(1+O\left(\frac{x^{2}}{g}\right)\right)$
$\displaystyle\cdot\left(1+O\left(\frac{1}{g}\right)\right)dx=\frac{\mathop{\rm
Li}(e^{L})}{2}\cdot\left(1+O\left(\frac{L^{2}}{g}\right)\right)$

where we apply $\lim\limits_{g\to\infty}L=\lim\limits_{g\to\infty}L(g)=\infty$
in the last equality.

The proof is complete. ∎

Now we are ready to prove Theorem 40.

###### Proof of Theorem 40.

Recall that $N(X_{g},L)=\frac{\pi_{X_{g}}(e^{L})}{2}$. Since $L>12\ln g$, it
follows by Theorem 3 that for any $\epsilon\in(0,1)$,

$\mathbb{E}_{\rm WP}^{g}\left[N(X_{g},L)\right]=\frac{\mathbb{E}_{\rm
WP}^{g}\left[\pi_{X_{g}}(e^{L})\right]}{2}=\frac{\mathop{\rm
Li}(e^{L})}{2}\cdot\left(1+O_{\epsilon}\left(\frac{1}{g^{1-\epsilon}}\right)\right)$

which together with Lemma 41 imply that

$\mathbb{E}_{\rm
WP}^{g}\left[\left|N(X_{g},L)-N_{nsep}^{s}(X_{g},L)\right|\right]=\mathop{\rm
Li}(e^{L})\cdot
O_{\epsilon}\left(\frac{L^{2}}{g}+\frac{1}{g^{1-\epsilon}}\right).$

Now we choose $\epsilon=\epsilon_{0}$. Then by Markov’s inequality we have

(175) $\displaystyle\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\left|N(X_{g},L)-N_{nsep}^{s}(X_{g},L)\right|\geq\frac{\delta\cdot\mathop{\rm
Li}(e^{L})}{4}\right)$
$\displaystyle=O_{\epsilon_{0}}\left(\frac{1}{\delta}\left(\frac{L^{2}}{g}+\frac{1}{g^{1-\epsilon_{0}}}\right)\right)\to
0\text{ as }g\to\infty.$

Let $\mathcal{A}_{g}$ be in Theorem 28. Then we have

(176) $\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{A}_{g}\right)=1$

and for any $X_{g}\in\mathcal{A}_{g}$ and large enough $g>0$,

(177) $\displaystyle N(X_{g},L)\geq\frac{1}{2}\cdot\mathop{\rm
Li}(e^{L})\cdot\left(1+O\left(\frac{g}{e^{\frac{L}{4}}}\right)\right)\geq\frac{\mathop{\rm
Li}(e^{L})}{4}$

where we apply $L>12\ln g$ in the last inequality. Thus, it follows by (175)
and (177) that for large enough $g>0$,

$\displaystyle\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\left|1-\frac{N_{nsep}^{s}(X_{g},L)}{N(X_{g},L)}\right|\geq\delta\right)\leq\mathop{\rm
Prob}\nolimits_{\rm WP}^{g}\left(X_{g}\notin\mathcal{A}_{g}\right)$
$\displaystyle+\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\left|N(X_{g},L)-N_{nsep}^{s}(X_{g},L)\right|\geq\frac{\delta\cdot\mathop{\rm
Li}(e^{L})}{4}\right)$ $\displaystyle=\left(1-\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{A}_{g}\right)\right)+O_{\epsilon_{0}}\left(\frac{1}{\delta}\left(\frac{L^{2}}{g}+\frac{1}{g^{1-\epsilon_{0}}}\right)\right).$

By letting $g\to\infty$, then the conclusion follows by (176) and our
assumption on $\delta$. The proof is complete. ∎

###  7.2. Geodesics of length significantly greater than $\sqrt{g}$

In this subsection we confirm the second part of [LW21, Conjecture $1.2$].
That is, we show that as $g\to\infty$, on most surfaces in $\mathcal{M}_{g}$
most geodesics of length significantly greater than $\sqrt{g}$ are non-simple.
More precisely,

######  Theorem 42.

Assume that $L=L(g)$ satisfies

$\lim\limits_{g\to\infty}\frac{L^{2}(g)}{g}=\infty.$

Then for any $k>0\in\mathbb{Z}$, we have

$\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
\left|1-\frac{N^{ns}(X_{g},L)}{N(X_{g},L)}\right|<\left(\frac{g}{L^{2}}\right)^{k}\right)=1.$

We first begin with the following easy consequence of Theorem 6.

######  Lemma 43.

For any $x>0$ and $k>0\in\mathbb{Z}$, we have that for all $g\geq 2$ and
$1\leq i\leq\left[\frac{g}{2}\right]$,

$\frac{V_{g-1,2}(x,x)}{V_{g-1,2}}\prec_{k}\left(\frac{g}{1+x^{2}}\right)^{k+1}\cdot\left(\frac{\sinh\left(\frac{x}{2}\right)}{\frac{x}{2}}\right)^{2}$

and

$\frac{V_{g-i,1}(x)\cdot V_{i,1}(x)}{V_{g-i,1}\cdot
V_{i,1}}\prec_{k}\left(\frac{g}{1+x^{2}}\right)^{k+1}\cdot\left(\frac{\sinh\left(\frac{x}{2}\right)}{\frac{x}{2}}\right)^{2}.$

###### Proof.

By Theorem 6 we have

$\displaystyle\frac{V_{g-1,2}(x,x)}{V_{g-1,2}}$
$\displaystyle\leq\left(\min\left\\{c(k+1)\frac{2g-2}{x^{2}},1\right\\}\right)^{2k+2}\cdot\left(\frac{\sinh\left(\frac{x}{2}\right)}{\frac{x}{2}}\right)^{2}$
$\displaystyle\leq\left(\min\left\\{c(k+1)\frac{2g-2}{x^{2}},1\right\\}\right)^{k+1}\cdot\left(\frac{\sinh\left(\frac{x}{2}\right)}{\frac{x}{2}}\right)^{2}$
$\displaystyle\prec_{k}\left(\frac{g}{1+x^{2}}\right)^{k+1}\cdot\left(\frac{\sinh\left(\frac{x}{2}\right)}{\frac{x}{2}}\right)^{2}.$

Now we prove the second inequality. Recall that the upper bound in Theorem 12
implies that
$\frac{V_{i,1}(x)}{V_{i,1}}\leq\frac{\sinh\left(\frac{x}{2}\right)}{\frac{x}{2}}.$
Similarly, then it follows by Theorem 6 that for all $1\leq
i\leq\left[\frac{g}{2}\right]$,

$\displaystyle\frac{V_{g-i,1}(x)\cdot V_{i,1}(x)}{V_{g-i,1}\cdot V_{i,1}}$
$\displaystyle\leq\frac{V_{g-i,1}(x)}{V_{g-i,1}}\cdot\left(\frac{\sinh\left(\frac{x}{2}\right)}{\frac{x}{2}}\right)$
$\displaystyle\leq\left(\min\left\\{c(k+1)\frac{2g-2i-1}{x^{2}},1\right\\}\right)^{k+1}\cdot\left(\frac{\sinh\left(\frac{x}{2}\right)}{\frac{x}{2}}\right)^{2}$
$\displaystyle\prec_{k}\left(\frac{g}{1+x^{2}}\right)^{k+1}\cdot\left(\frac{\sinh\left(\frac{x}{2}\right)}{\frac{x}{2}}\right)^{2}.$

The proof is complete. ∎

######  Lemma 44.

Assume that $L=L(g)$ satisfies

$\lim\limits_{g\to\infty}\frac{L^{2}(g)}{g}=\infty.$

Then for any $k>0\in\mathbb{Z}$,

$\mathbb{E}_{\rm WP}^{g}\left[N_{nsep}^{s}(X_{g},L)\right]\prec_{k}\mathop{\rm
Li}(e^{L})\cdot\left(\frac{g}{L^{2}}\right)^{k+1}.$

###### Proof.

It follows by Mirzakhani’s integration formula, i.e., Theorem 8, and Lemma 43
that

$\displaystyle\mathbb{E}_{\rm WP}^{g}\left[N_{nsep}^{s}(X_{g},L)\right]$
$\displaystyle=\frac{1}{V_{g}}\frac{1}{2}\int_{0}^{L}V_{g-1,2}(x,x)xdx$
$\displaystyle\prec_{k}\frac{V_{g-1,2}}{V_{g}}\cdot\int_{0}^{L}\left(\sinh\left(\frac{x}{2}\right)\right)^{2}\cdot\frac{1}{x}\cdot\left(\frac{g}{1+x^{2}}\right)^{k+1}dx.$

By Theorem 10 we also know that

$\frac{V_{g}}{V_{g-1,2}}=1+O\left(\frac{1}{g}\right).$

Thus,

$\displaystyle\mathbb{E}_{\rm WP}^{g}\left[N_{nsep}^{s}(X_{g},L)\right]$
$\displaystyle\prec_{k}\left(1+O\left(\frac{1}{g}\right)\right)\cdot\int_{0}^{L}\left(\sinh\left(\frac{x}{2}\right)\right)^{2}\frac{1}{x}\left(\frac{g}{1+x^{2}}\right)^{k+1}dx$
$\displaystyle\asymp\mathop{\rm
Li}(e^{L})\cdot\left(\frac{g}{L^{2}}\right)^{k+1}$

where we apply $\lim\limits_{g\to\infty}\frac{L^{2}(g)}{g}=\infty$ in the last
equation. The proof is complete. ∎

######  Lemma 45.

Assume that $L=L(g)$ satisfies

$\lim\limits_{g\to\infty}\frac{L^{2}(g)}{g}=\infty.$

Then for any $k>0\in\mathbb{Z}$,

$\sum_{i=1}^{\left[\frac{g}{2}\right]}\mathbb{E}_{\rm
WP}^{g}\left[N_{sep,i}^{s}(X_{g},L)\right]\prec_{k}\frac{\mathop{\rm
Li}(e^{L})}{g}\cdot\left(\frac{g}{L^{2}}\right)^{k+1}.$

###### Proof.

It follows by Theorem 8 and Lemma 43 that

$\displaystyle\sum_{i=1}^{\left[\frac{g}{2}\right]}\mathbb{E}_{\rm
WP}^{g}\left[N_{sep,i}^{s}(X_{g},L)\right]\leq\frac{1}{V_{g}}\sum_{i=1}^{\left[\frac{g}{2}\right]}\int_{0}^{L}V_{g-i,1}(x)V_{i,1}(x)xdx$
$\displaystyle\prec_{k}\left(\sum_{i=1}^{\left[\frac{g}{2}\right]}\frac{V_{g-i,1}V_{i,1}}{V_{g}}\right)\cdot\int_{0}^{L}\left(\sinh\left(\frac{x}{2}\right)\right)^{2}\cdot\frac{1}{x}\cdot\left(\frac{g}{1+x^{2}}\right)^{k+1}dx.$

By Theorem 11 we also know that

$\sum_{i=1}^{\left[\frac{g}{2}\right]}\frac{V_{g-i,1}V_{i,1}}{V_{g}}\asymp\frac{1}{g}.$

Thus,

$\displaystyle\sum_{i=1}^{\left[\frac{g}{2}\right]}\mathbb{E}_{\rm
WP}^{g}\left[N_{sep,i}^{s}(X_{g},L)\right]$
$\displaystyle\prec_{k}\frac{1}{g}\cdot\int_{0}^{L}\left(\sinh\left(\frac{x}{2}\right)\right)^{2}\frac{1}{x}\left(\frac{g}{1+x^{2}}\right)^{k+1}dx$
$\displaystyle\asymp\frac{\mathop{\rm
Li}(e^{L})}{g}\cdot\left(\frac{g}{L^{2}}\right)^{k+1}$

where we apply $\lim\limits_{g\to\infty}\frac{L^{2}(g)}{g}=\infty$ in the last
equation. The proof is complete. ∎

Now we are ready to prove Theorem 42.

###### Proof of Theorem 42.

First by Lemma 44 and Lemma 45 we know that for any $k>0\in\mathbb{Z}$,

$\displaystyle\mathbb{E}_{\rm WP}^{g}\left[N^{s}(X_{g},L)\right]$
$\displaystyle=\mathbb{E}_{\rm
WP}^{g}\left[N_{nsep}^{s}(X_{g},L)\right]+\sum_{i=1}^{\left[\frac{g}{2}\right]}\mathbb{E}_{\rm
WP}^{g}\left[N_{sep,i}^{s}(X_{g},L)\right]$ $\displaystyle\prec_{k}\mathop{\rm
Li}(e^{L})\cdot\left(\frac{g}{L^{2}}\right)^{k+1}.$

Then by Markov’s inequality we have

(178) $\displaystyle\mathop{\rm Prob}\nolimits_{\rm WP}^{g}$
$\displaystyle\left(X_{g}\in\mathcal{M}_{g};\
N^{s}(X_{g},L)\geq\frac{1}{4}\left(\frac{g}{L^{2}}\right)^{k}\cdot\mathop{\rm
Li}(e^{L})\right)$ $\displaystyle\prec_{k}\frac{g}{L^{2}}\to 0\text{ as
}g\to\infty.$

Recall that $N(X_{g},L)=\frac{\pi_{X_{g}}(e^{L})}{2}$. Let $\mathcal{A}_{g}$
be in Theorem 28. Then we have

(179) $\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{A}_{g}\right)=1$

and for any $X_{g}\in\mathcal{A}_{g}$ and large enough $g>0$,

(180) $\displaystyle N(X_{g},L)\geq\frac{1}{2}\cdot\mathop{\rm
Li}(e^{L})\cdot\left(1+O\left(\frac{g}{e^{\frac{L}{4}}}\right)\right)\geq\frac{\mathop{\rm
Li}(e^{L})}{4}$

where we apply $\lim\limits_{g\to\infty}\frac{L^{2}(g)}{g}=\infty$ in the last
inequality. Since $N^{s}(X_{g},L)=N(X_{g},L)-N^{ns}(X_{g},L)$, it follows by
(178) and (180) that for large enough $g>0$,

$\displaystyle\mathop{\rm Prob}\nolimits_{\rm WP}^{g}$
$\displaystyle\left(X_{g}\in\mathcal{M}_{g};\
\left|1-\frac{N^{ns}(X_{g},L)}{N(X_{g},L)}\right|\geq\left(\frac{g}{L^{2}}\right)^{k}\right)\leq\mathop{\rm
Prob}\nolimits_{\rm WP}^{g}\left(X_{g}\notin\mathcal{A}_{g}\right)$
$\displaystyle+\mathop{\rm Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{M}_{g};\
N^{s}(X_{g},L)\geq\frac{1}{4}\left(\frac{g}{L^{2}}\right)^{k}\cdot\mathop{\rm
Li}(e^{L})\right)$ $\displaystyle\prec_{k}\left(\left(1-\mathop{\rm
Prob}\nolimits_{\rm
WP}^{g}\left(X_{g}\in\mathcal{A}_{g}\right)\right)+\frac{g}{L^{2}}\right).$

By letting $g\to\infty$, then the conclusion follows by (179) and our
assumption on $L=L(g)$. The proof is complete. ∎

###### Proof of Theorem 4.

It clearly follows from Theorem 40 and 42. ∎

## References

  * [Agg21] Amol Aggarwal, _Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials_ , Invent. Math. 226 (2021), no. 3, 897–1010.
  * [AM22] Nalini Anantharaman and Laura Monk, _A high-genus asymptotic expansion of Weil-Petersson volume polynomials_ , J. Math. Phys. 63 (2022), no. 4, Paper No. 043502, 26.
  * [Ber16] Nicolas Bergeron, _The spectrum of hyperbolic surfaces_ , Universitext, Springer, Cham; EDP Sciences, Les Ulis, 2016, Appendix C by Valentin Blomer and Farrell Brumley, Translated from the 2011 French original by Brumley [2857626].
  * [BMM16] Werner Ballmann, Henrik Matthiesen, and Sugata Mondal, _Small eigenvalues of closed surfaces_ , J. Differential Geom. 103 (2016), no. 1, 1–13.
  * [BMM17] by same author, _On the analytic systole of Riemannian surfaces of finite type_ , Geom. Funct. Anal. 27 (2017), no. 5, 1070–1105.
  * [BP22] Maxime Fortier Bourque and Bram Petri, _Kissing numbers of closed hyperbolic manifolds_ , Amer. J. Math. 144 (2022), no. 4, 1067–1086.
  * [Bus77] Peter Buser, _Riemannsche Flächen mit Eigenwerten in $(0,$ $1/4)$_, Comment. Math. Helv. 52 (1977), no. 1, 25–34.
  * [Bus10] by same author, _Geometry and spectra of compact Riemann surfaces_ , Modern Birkhäuser Classics, Birkhäuser Boston, Ltd., Boston, MA, 2010, Reprint of the 1992 edition.
  * [Cai02] Yingchun Cai, _Prime geodesic theorem_ , J. Théor. Nombres Bordeaux 14 (2002), no. 1, 59–72.
  * [Cha84] Isaac Chavel, _Eigenvalues in Riemannian geometry_ , Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984, Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk.
  * [DGZZ20] Vincent Delecroix, Élise Goujard, Peter Zograf, and Anton Zorich, _Uniform lower bound for intersection numbers of $\psi$-classes_, SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), 086, 13 pages.
  * [GLMST21] Clifford Gilmore, Etienne Le Masson, Tuomas Sahlsten, and Joe Thomas, _Short geodesic loops and $L^{p}$ norms of eigenfunctions on large genus random surfaces_, Geom. Funct. Anal. 31 (2021), no. 1, 62–110.
  * [GPY11] Larry Guth, Hugo Parlier, and Robert Young, _Pants decompositions of random surfaces_ , Geom. Funct. Anal. 21 (2011), no. 5, 1069–1090.
  * [Hej76] Dennis A. Hejhal, _The Selberg trace formula for ${\rm PSL}(2,R)$. Vol. I_, Lecture Notes in Mathematics, Vol. 548, Springer-Verlag, Berlin-New York, 1976.
  * [Hid21] Will Hide, _Spectral gap for Weil-Petersson random surfaces with cusps_ , arXiv e-prints (2021), arXiv:2107.14555.
  * [HM21] Will Hide and Michael Magee, _Near optimal spectral gaps for hyperbolic surfaces_ , arXiv e-prints (2021), arXiv:2107.05292.
  * [Hub61] Heinz Huber, _Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II_ , Math. Ann. 142 (1960/61), 385–398.
  * [HW22] Yuxin He and Yunhui Wu, _On second eigenvalues of closed hyperbolic surfaces for large genus_ , arXiv e-prints (2022), arXiv:2207.12919.
  * [Iwa84] Henryk Iwaniec, _Prime geodesic theorem_ , J. Reine Angew. Math. 349 (1984), 136–159.
  * [LRS95] W. Luo, Z. Rudnick, and P. Sarnak, _On Selberg’s eigenvalue conjecture_ , Geom. Funct. Anal. 5 (1995), no. 2, 387–401.
  * [LS20] Etienne Le Masson and Tuomas Sahlsten, _Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus_ , arXiv e-prints (2020), arXiv:2006.14935.
  * [LW21] Michael Lipnowski and Alex Wright, _Towards optimal spectral gaps in large genus_ , arXiv e-prints (2021), arXiv:2103.07496.
  * [LX14] Kefeng Liu and Hao Xu, _A remark on Mirzakhani’s asymptotic formulae_ , Asian J. Math. 18 (2014), no. 1, 29–52.
  * [Mir07a] Maryam Mirzakhani, _Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces_ , Invent. Math. 167 (2007), no. 1, 179–222.
  * [Mir07b] by same author, _Weil-Petersson volumes and intersection theory on the moduli space of curves_ , J. Amer. Math. Soc. 20 (2007), no. 1, 1–23.
  * [Mir08] by same author, _Growth of the number of simple closed geodesics on hyperbolic surfaces_ , Ann. of Math. (2) 168 (2008), no. 1, 97–125.
  * [Mir13] by same author, _Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus_ , J. Differential Geom. 94 (2013), no. 2, 267–300.
  * [MNP22] Michael Magee, Frédéric Naud, and Doron Puder, _A random cover of a compact hyperbolic surface has relative spectral gap $\frac{3}{16}-\epsilon$_, Geom. Funct. Anal. 32 (2022), no. 3, 595–661.
  * [Mon22] Laura Monk, _Benjamini-Schramm convergence and spectra of random hyperbolic surfaces of high genus_ , Anal. PDE 15 (2022), no. 3, 727–752.
  * [MP19] Maryam Mirzakhani and Bram Petri, _Lengths of closed geodesics on random surfaces of large genus_ , Comment. Math. Helv. 94 (2019), no. 4, 869–889.
  * [MZ15] Maryam Mirzakhani and Peter Zograf, _Towards large genus asymptotics of intersection numbers on moduli spaces of curves_ , Geom. Funct. Anal. 25 (2015), no. 4, 1258–1289.
  * [NWX20] Xin Nie, Yunhui Wu, and Yuhao Xue, _Large genus asymptotics for lengths of separating closed geodesics on random surfaces_ , arXiv e-prints (2020), arXiv:2009.07538.
  * [OR09] Jean-Pierre Otal and Eulalio Rosas, _Pour toute surface hyperbolique de genre $g,\ \lambda_{2g-2}>1/4$_, Duke Math. J. 150 (2009), no. 1, 101–115.
  * [PWX21] Hugo Parlier, Yunhui Wu, and Yuhao Xue, _The simple separating systole for hyperbolic surfaces of large genus_ , Journal of the Institute of Mathematics of Jussieu (2021), to appear.
  * [Ran77] Burton Randol, _On the asymptotic distribution of closed geodesics on compact Riemann surfaces_ , Trans. Amer. Math. Soc. 233 (1977), 241–247.
  * [Rud22] Zeév Rudnick, _GOE statistics on the moduli space of surfaces of large genus_ , arXiv e-prints (2022), arXiv:2202.06379.
  * [Sar80] Peter Clive Sarnak, _PRIME GEODESIC THEOREMS_ , ProQuest LLC, Ann Arbor, MI, 1980, Thesis (Ph.D.)–Stanford University. MR 2630950
  * [Sel56] A. Selberg, _Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series_ , J. Indian Math. Soc. (N.S.) 20 (1956), 47–87.
  * [SW22] Yang Shen and Yunhui Wu, _Arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps_ , arXiv e-prints (2022), arXiv:2203.15681.
  * [SY13] K. Soundararajan and Matthew P. Young, _The prime geodesic theorem_ , J. Reine Angew. Math. 676 (2013), 105–120.
  * [Tho20] Joe Thomas, _Delocalisation of eigenfunctions on large genus random surfaces_ , Israel Journal of Mathematics _, to appear_ (2020).
  * [Wol82] Scott Wolpert, _The Fenchel-Nielsen deformation_ , Ann. of Math. (2) 115 (1982), no. 3, 501–528.
  * [Wri20] Alex Wright, _A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces_ , Bull. Amer. Math. Soc. (N.S.) 57 (2020), no. 3, 359–408.
  * [WX22] Yunhui Wu and Yuhao Xue, _Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-\epsilon$_, Geom. Funct. Anal. 32 (2022), no. 2, 340–410.