Prime geodesic theorem for the modular surface

TL;DR

This paper improves the error term in the prime geodesic theorem for the modular surface under the generalized Lindelöf hypothesis, reducing the exponent to 5/8+ε outside a finite logarithmic measure set.

Contribution

It provides a refined error estimate for the prime geodesic theorem assuming the generalized Lindelöf hypothesis, enhancing previous bounds.

Findings

Error term exponent reduced to 5/8+ε

Improvement holds outside a finite logarithmic measure set

Conditional on the generalized Lindelöf hypothesis

Abstract

Under the generalized Lindel\"{o}f hypothesis, the exponent in the error term of the prime geodesic theorem for the modular surface is reduced to $\frac{5}{8}+\varepsilon $ outside a set of finite logarithmic measure.

Equations

$$Z_{\Gamma }(s)=\prod _{\{P_0\}}\underset{k=0}{\prod ^{\infty }}(1-N(P_0{)}^{-s-k})\text{, }\text{Re}(s)>1\text{,}$$

$$L\left(\frac{1}{2}+it,{\chi }_D\right)\ll {\left(1+\left|t\right|\right)}^A{\left|D\right|}^{\theta +\epsilon }$$

$${\pi }_{\Gamma }\left(x\right)={\int }_0^x\frac{dt}{\log t}+O\left(x^{\frac{5}{8}+\frac{\theta }{4}+\epsilon }\right)\left(x\to \infty ,x\notin B\right)\text{.}$$

$${\pi }_{\Gamma }\left(x\right)=li\left(x\right)+O\left(x^{\frac{2}{3}+\epsilon }\right)\left(x\to \infty ,x\notin B\right)\text{.}$$

$${\pi }_{\Gamma }\left(x\right)=li\left(x\right)+O\left(x^{\frac{5}{8}+\epsilon }\right)\left(x\to \infty ,x\notin B\right)\text{.}$$

$$S\left(u\right)=\sum _{\nu \in A}c\left(\nu \right)e^{2\pi i\nu u}$$

$${\int }_{-U}^U{\left|S\left(u\right)\right|}^{2}du\le {\left(\frac{\pi \eta }{\sin \pi \eta }\right)}^2{\int }_{-\infty }^{+\infty }{\left|\frac{U}{\eta }\sum _{t\le \nu \le t+\frac{\eta }{U}}c\left(\nu \right)\right|}^{2}dt\text{.}$$

$${\psi }_{\Gamma }\left(x\right)=x+\sum _{|\gamma |\le T}\frac{x^{\rho }}{\rho }+O\left(\frac{x}{T}{\left(\log x\right)}^2\right)\text{,}$$

$$\sum _{|\gamma |\le Y}{x}^{i\gamma }$$

$${\psi }_{\Gamma }\left(x\right)=x+\sum _{|\gamma |\le T}\frac{x^{\rho }}{\rho }+O\left(x^{\frac{1}{2}}{\left(\log x\right)}^4\right)\text{.}$$

$${\mu }^{\ast }{B}_n$$

$$2\pi \stackrel{\frac{1}{4\pi }}{\underset{-\frac{1}{4\pi }}{\int }}{\left|\sum _{|\gamma |\le T}\frac{e^{\left(n+\frac{1}{2}\right)i\gamma }}{\rho }{e}^{2\pi i\gamma u}\right|}^{2}du\text{.}$$

$$\stackrel{\frac{1}{4\pi }}{\underset{-\frac{1}{4\pi }}{\int }}{\left|\sum _{|\gamma |\le T}\frac{e^{\left(n+\frac{1}{2}\right)i\gamma }}{\rho }{e}^{2\pi i\gamma u}\right|}^{2}du\le {\left(\frac{\frac{1}{4}}{\sin \frac{1}{4}}\right)}^2\stackrel{+\infty }{\underset{-\infty }{\int }}{\left(\sum _{\begin{array}{c}t<\gamma \le t+1\\ |\gamma |\le Y\end{array}}^{+\infty }\frac{1}{\left|\rho \right|}\right)}^{2}dt\text{.}$$

$$\stackrel{+\infty }{\underset{-\infty }{\int }}{\left(\sum _{\begin{array}{c}t<\gamma \le t+1\\ |\gamma |\le Y\end{array}}^{+\infty }\frac{1}{\left|\rho \right|}\right)}^{2}dt=O\left(\underset{0}{\overset{Y}{\int }}dt\right)=O\left(Y\right)\text{.}$$

$$\left|\sum _{|\gamma |\le Y}\frac{x^{\rho }}{\rho }\right|\le x^{\frac{1}{2}+\epsilon }{Y}^{\frac{1}{2}}\text{.}$$

$$\sum _{Y<|\gamma |\le T}\frac{x^{i\gamma }}{\rho }=\frac{S\left(x,T\right)}{\frac{1}{2}+iT}-\frac{S\left(x,Y\right)}{\frac{1}{2}+iY}+i\underset{Y}{\overset{T}{\int }}\frac{S\left(x,u\right)}{{\left(\frac{1}{2}+iu\right)}^2}du\text{.}$$

$$\left|\sum _{Y<|\gamma |\le T}\frac{x^{\rho }}{\rho }\right|\ll \frac{x^{\frac{3}{4}+\frac{\theta }{2}+\epsilon }}{T^{\frac{1}{2}}}+\frac{x^{\frac{3}{4}+\frac{\theta }{2}+\epsilon }}{Y^{\frac{1}{2}}}+\underset{Y}{\overset{T}{\int }}\frac{x^{\frac{3}{4}+\frac{\theta }{2}+\epsilon }{u}^{\frac{1}{2}}}{u^2}du\ll \frac{x^{\frac{3}{4}+\frac{\theta }{2}+\epsilon }}{Y^{\frac{1}{2}}}\text{.}$$

$${\psi }_{\Gamma }\left(x\right)=x+O\left(x^{\frac{5}{8}+\frac{\theta }{4}+\epsilon }\right)\left(x\to \infty \text{, }x\notin B\right)\text{,}$$

Full text

1. Introduction

Let $\Gamma=PSL\left(2,\mathbb{Z}\right)$ be the modular group and $\mathcal{H}$ the upper half-plane equipped with the hyperbolic metric. The norms $N(P_{0})$ of primitive conjugacy classes $P_{0}$ in $\Gamma$ are sometimes called pseudo-primes. The length of the primitive closed geodesic on the modular surface $\Gamma\setminus\mathcal{H}$ joining two fixed points, which are the same for all representatives of $P_{0}$, equals $\log(N(P_{0}))$. The statement about the number $\pi_{\Gamma}(x)$ of classes $P_{0}$ such that $N(P_{0})\leq x$, for $x>0$, is known as the prime geodesic theorem, PGT.

The main tool in the proof of PGT is the Selberg zeta function, defined by

[TABLE]

and meromorphicaly continued to the whole complex plane.

The relationship between the prime geodesic theorem and the distribution of zeros of the Selberg zeta function resembles to a large extent the relationship between the prime number theorem and the zeros of the Riemann zeta.

However, the function $Z_{\Gamma}$ satisfies the Riemann hypothesis. It is an outstanding open problem whether the error term in the prime geodesic theorem is $O(x^{\frac{1}{2}+\varepsilon})$ as it would be the case in the prime number theorem once the Riemann hypothesis be proved.

The obstacles in establishing an analogue of von Koch’s theorem [13, p. 84] in this setting comes from the fact that $Z_{\Gamma}$ is a meromorphic function of order $2$, while the Riemann zeta is of order $1$.

In the case of Fuchsian groups $\Gamma\subset PSL\left(2,\mathbb{R}\right)$, the best estimate of the remainder term in PGT is still $O\left(\frac{x^{\frac{3}{4}}}{\log x}\right)$ obtained by Randol [18] (see also [7], [1] for different proofs). We note that its analogue $O\left(x^{\frac{3}{2}d_{0}}\left(\log x\right)^{-1}\right)$ is valid also for strictly hyperbolic manifolds of higher dimensions, where $d_{0}=\frac{d-1}{2}$ and $d\geq 3$ is the dimension of a manifold [5, Theorem 1].

The attempts to reduce the exponent $\frac{3}{4}$ in PGT were successful only in special cases. The chronological list of improvements for the modular group $\Gamma=PSL(2,\mathbb{Z})$ includes $\frac{35}{48}+\varepsilon$ (Iwaniec [15]), $\frac{7}{10}+\varepsilon$ (Luo and Sarnak [17]), $\frac{71}{102}+\varepsilon$ (Cai [8]) and the present $\frac{25}{36}+\varepsilon$ (Soundararajan and Young [19]).

Iwaniec [14] remarked that the generalized Lindelöf hypothesis for Dirichlet $L$-functions would imply $\frac{2}{3}+\varepsilon$.

We proved [2] that $\frac{2}{3}+\varepsilon$ is valid outside a set of finite logarithmic measure. In the present note, we relate the error term in the Gallagherian $PGT$ on $PSL(2,\mathbb{Z})$ to the subconvexity bound for Dirichlet $L$- functions. This enables us to replace $\frac{2}{3}+\varepsilon$ by $\frac{5}{8}+\varepsilon$ under the generalized Lindelöf hypothesis. More precisely, the main result of this paper is the following theorem.

Theorem**.**

Let $\Gamma=PSL(2,\mathbb{Z})$ be the modular group, $\varepsilon>0$ arbitrarily small and $\theta$ be such that

[TABLE]

for some fixed $A>0$, where $D$ is a fundamental discriminant. There exists a set $B$ of finite logarithmic measure such that

[TABLE]

Inserting the Conrey-Iwaniec [9] value $\theta=\frac{1}{6}$ into Theorem, we obtain

Corollary 1**.**

[TABLE]

Any improvement of $\theta$ immediately results in the obvious improvement of the error term in PGT. Taking into account that the Lindelöf hypothesis allows $\theta=0$, we get

Corollary 2**.**

Under the Lindelöf hypothesis,

[TABLE]

Remark 1**.**

The obtained exponent for strictly hyperbolic Fuchsian groups is $\frac{7}{10}+\varepsilon$ outside a set of finite logarithmic measure [3] and coincides with the above mentioned Luo-Sarnak unconditional result for $\Gamma=PSL(2,\mathbb{Z})$. In the case of a cocompact Kleinian group or a noncompact congruence group for some imaginary quadratic number field, the respective Gallagherian bound is $\frac{13}{9}+\varepsilon$ [4].

2. Preliminaries.

The motivation for Theorem comes from several sources, including Gallagher [11], Iwaniec [15] and Balkanova and Frolenkov [6].

Recall that $\pi_{\Gamma}\left(x\right)=\mathit{li}\left(x\right)+O\left(x^{\frac{5}{8}+\frac{\theta}{4}+\varepsilon}\right)$ is equivalent to $\psi_{\Gamma}\left(x\right)=x+O\left(x^{\frac{5}{8}+\frac{\theta}{4}+\varepsilon}\right)$, where $\psi_{\Gamma}\left(x\right)=\underset{N\left(P_{0}\right)^{k}\leq x}{\sum}\log N\left(P_{0}\right)$ is the $\Gamma$ analogue of the classical Chebyshev function $\psi$.

Under the Riemann hypothesis, Gallagher improved von Koch’s remainder term in the prime number theorem from $\psi(x)=x+O\left(x^{\frac{1}{2}}(\log x)^{2}\right)$ to $\psi(x)=x+O\left(x^{\frac{1}{2}}(\log\log x)^{2}\right)$ outside a set of finite logarithmic measure.

Following Koyama [16], we shall apply the next lemma [10] due to Gallagher to our setting.

Lemma A**.**

Let $A$ be a discrete subset of $\mathbb{R}$ and $\eta\in(0,1)$. For any sequence $c(\nu)\in\mathbb{C}$, $\nu\in A$, let the series

[TABLE]

be absolutely convergent. Then

[TABLE]

Iwaniec [15] established the following explicit formula with an error term for $\psi_{\Gamma}$ on $\Gamma=PSL(2,\mathbb{Z})$.

Lemma B**.**

For $1\leq T\leq\frac{x^{\frac{1}{2}}}{\left(\log x\right)^{2}}$, one has

[TABLE]

where $\rho=\frac{1}{2}+i\gamma$ denote zeros of $Z_{\Gamma}$.

Recently, O. Balkanova and D. Frolenkov have proved the following estimate.

Lemma C**.**

[TABLE]

where $\rho=\frac{1}{2}+i\gamma$ are the zeros of $Z_{\Gamma}$, $\theta$ is the subconvexity exponent for Dirichlet $L-$functions, and $\kappa\left(x\right)$ is the distance from $\sqrt{x}+\frac{1}{\sqrt{x}}$ to the nearest integer.

3. Proof of Theorem.

Inserting $T=\frac{x^{\frac{1}{2}}}{\left(\log x\right)^{2}}$ into Lemma B, we obtain

[TABLE]

We would like to bound the expression $\underset{\left|\gamma\right|\leq T}{\sum}\frac{x^{\rho}}{\rho}$, where $Y\in\left(0,T\right)$ is a parameter to be determined later on.

Let $n=\left\lfloor\log x\right\rfloor$ and $B_{n}=\left\{x\in\left[e^{n},e^{n+1}\right):\left|\underset{\left|\gamma\right|\leq T}{{\sum}}\frac{x^{i\gamma}}{\rho}\right|>x^{\varepsilon}Y^{\frac{1}{2}}\right\}$. Looking at the logarithmic measure of $B_{n}$, we get

[TABLE]

After substitution $x=e^{n}\cdot e^{2\pi\left(u+\frac{1}{4\pi}\right)}$, the last integral becomes

[TABLE]

Applying Lemma A, with $\eta=U=\frac{1}{4\pi}$ and $c_{\gamma}=\frac{e^{\left(n+\frac{1}{2}\right)i\gamma}}{\rho}$ for $\left|\gamma\right|\leq T$, $c_{\gamma}=0$ otherwise, we get

[TABLE]

Note that $\sum_{\begin{subarray}{c}t<\gamma\leq t+1\end{subarray}}\frac{1}{\left|\rho\right|}=O\left(1\right)$ since $\#\left\{\gamma:t<\left|\gamma\right|\leq t+1\right\}=O\left(t\right)$ by the Weyl law.

Thus,

[TABLE]

The relations (3), (3) and (4) imply $\mu^{\ast}B_{n}\ll\frac{Y}{e^{2n\varepsilon}Y}=\frac{1}{e^{2n\varepsilon}}$. Hence, the set $B=\cup B_{n}$ has a finite logarithmic measure.

For $x\notin B$, we have $\left|\underset{\left|\gamma\right|\leq Y}{{\sum}}\frac{x^{i\gamma}}{\rho}\right|\leq x^{\varepsilon}Y^{\frac{1}{2}}$, i.e.

[TABLE]

Now, we rely on Lemma C to estimate $\left|\underset{Y<\left|\gamma\right|\leq T}{{\sum}}\frac{x^{\rho}}{\rho}\right|$. Let us put $S\left(x,T\right)=\underset{\left|\gamma\right|\leq T}{{\sum}}x^{i\gamma}$. By Abel’s partial summation, we have

[TABLE]

Multiplying the last relation by $x^{\frac{1}{2}}$ and recalling that Lemma C yields $\underset{\left|\gamma\right|\leq Y}{\sum}x^{i\gamma}\ll x^{\frac{1}{4}+\frac{\theta}{2}+\varepsilon}Y^{\frac{1}{2}}$ for $Y<T=\frac{x^{\frac{1}{2}}}{\left(\log x\right)^{2}}$, we get

[TABLE]

Combining (5) and (6), we see that the optimal choice for the parameter $Y$ is $Y\approx x^{\frac{1}{4}+\frac{\theta}{2}}$. Then, $\underset{\left|\gamma\right|\leq T}{{\sum}}\frac{x^{\rho}}{\rho}=O\left(x^{\frac{1}{2}+\varepsilon}Y^{\frac{1}{2}}\right)=O\left(x^{\frac{5}{8}+\frac{\theta}{4}+\varepsilon}\right)$ for $x\notin B$.

The relation (1) becomes

[TABLE]

as asserted.