On Koyama's refinement of the prime geodesic theorem

TL;DR

This paper presents a new proof for the prime geodesic theorem on compact Riemann surfaces, improving the error term without restrictive assumptions and extending the Gallagher-Koyama approach for better bounds.

Contribution

It introduces a novel proof technique that refines the error term in the prime geodesic theorem without excluding any finite measure set, advancing previous results.

Findings

Improved error term in the prime geodesic theorem.

Elimination of the assumption excluding a finite logarithmic measure set.

Extension of Gallagher-Koyama approach for stronger results.

Abstract

We give a new proof of the best presently known error term in the prime geodesic theorem for compact Riemann surfaces, without the assumption of excluding a set of finite logarithmic measure. Stronger implications of the Gallagher-Koyama approach are derived yielding to a further reduction of the error term outside a set of finite logarithmic measure.

Equations

$${\psi }_{\Gamma }\left(x\right)=\sum _{\begin{array}{c}P,k\\ N{\left(P\right)}^k\le x\end{array}}\log N\left(P\right)$$

$${\psi }_{\Gamma }\left(x\right)=x+\sum _{\frac{3}{4}<\rho <1}\frac{x^{\rho }}{\rho }+O\left(x^{\frac{3}{4}}{\left(\log \log x\right)}^{\frac{1}{4}+\epsilon }\right)\left(x\to \infty ,x\notin E\right)\text{.}$$

$${\psi }_{\Gamma }\left(x\right)=x+\sum _{\frac{3}{4}<\rho <1}\frac{x^{\rho }}{\rho }+O\left(x^{\frac{3}{4}}\right)\left(x\to \infty \right)\text{.}$$

$${\psi }_{\Gamma }\left(x\right)=x+\sum _{\frac{3}{4}-\epsilon <\rho <1}\frac{x^{\rho }}{\rho }+O\left(\frac{x^{\frac{3}{4}}}{(\log x{)}^{\alpha }}\right)\left(x\to \infty ,x\notin H\right)\text{,}$$

$$\begin{array}{r}{\psi }_{1,\Gamma }\left(x\right)={\alpha }_{0}x+{\beta }_{0}x\log x+{\alpha }_1+{\beta }_1\log x\\ +F\left(\frac{1}{x}\right)+\frac{x^2}{2}+\sum _{\begin{array}{c}\rho \\ |\gamma |<T\end{array}}\frac{x^{\rho +1}}{\rho \left(\rho +1\right)}+O\left(\frac{x^2\log x}{T}\right)\left(x\to \infty \right)\text{.}\end{array}$$

$$\begin{gathered} {\psi }_{2,\Gamma }\left(x\right)={\alpha }_0^{\prime }{x}^2+{\beta }_0^{\prime }{x}^2\log x+{\alpha }_1^{\prime }x+{\beta }_{1}x\log x+\frac{x^3}{6}+{\beta }_2 \\ +\left(2g-2\right)\stackrel{\infty }{\sum _{k=2}}\frac{2k+1}{k\left(k-1\right)}\frac{x^{2-k}}{\left(2-k\right)}+\sum _{\frac{1}{2}<\rho <1}\frac{x^{\rho +2}}{\rho \left(\rho +1\right)\left(\rho +2\right)}+\sum _{\text{Re}\rho =\frac{1}{2}}\frac{x^{\rho +2}}{\rho \left(\rho +1\right)\left(\rho +2\right)}\text{.} \end{gathered}$$

$${\Delta }_2^{+}f\left(x\right)$$

$${\Delta }_2^{-}f\left(x\right)$$

$$\frac{1}{h^2}{\Delta }_2^{-}{\psi }_{2,\Gamma }\left(x\right)\le {\psi }_{\Gamma }\left(x\right)\le \frac{1}{h^2}{\Delta }_2^{+}{\psi }_{2,\Gamma }\left(x\right)\text{.}$$

$$\frac{1}{h^2}{\Delta }_2^{+}\frac{x^{\rho +2}}{\rho \left(\rho +1\right)\left(\rho +2\right)}=O\left(\min \left(\frac{x^{\frac{1}{2}}}{\left|\rho \right|},\frac{x^{\frac{5}{2}}}{h^2{\left|\rho \right|}^3}\right)\right)\text{.}$$

$$\frac{1}{h^{2}}\Delta_{2}^{+}\underset{\text{Re}\left(\rho\right)=\frac{1}{2}}{\sum}\frac{x^{\rho+2}}{\rho\left(\rho+1\right)\left(\rho+2\right)}=O\left(x^{\frac{1}{2}}\underset{{\scriptsize\begin{array}[]{c}\text{Re}\left(\rho\right)=\frac{1}{2}\\ \left|\rho\right|<M\end{array}}}{\sum}\frac{1}{\left|\rho\right|}\right)+O\left(\frac{x^{\frac{5}{2}}}{h^{2}}\underset{{\scriptsize\begin{array}[]{c}\text{Re}\left(\rho\right)=\frac{1}{2}\\ \left|\rho\right|\geq M\end{array}}}{\sum}\frac{1}{\left|\rho\right|^{3}}\right)\\ =O\left(x^{\frac{1}{2}}M\right)+O\left(\frac{x^{\frac{5}{2}}}{h^{2}M}\right)\text{ \ for }M>2\text{.}$$

$${\psi }_{\Gamma }\left(x\right)=x+\sum _{\frac{3}{4}<\rho <1}\frac{x^{\rho }}{\rho }+O\left(x^{\frac{3}{4}}\right)\text{.}$$

$$\begin{gathered} {\psi }_{\Gamma }\left(x\right)\le \frac{1}{h}{\int }_x^{x+h}{\psi }_{\Gamma }\left(t\right)dt \\ =x+\sum _{\frac{1}{2}<\rho <1}\frac{x^{\rho }}{\rho }+O\left(\log x\right)+O\left(h\right)+O\left(\frac{x^2\log x}{hT}\right)+\frac{1}{h}\left|\sum _{\begin{array}{c}\text{Re}(\rho )=\frac{1}{2}\\ |\gamma |\le T\end{array}}\frac{{\left(x+h\right)}^{\rho +1}-x^{\rho +1}}{\rho \left(\rho +1\right)}\right|\text{.} \end{gathered}$$

$$\sum _{\begin{array}{c}\text{Re}(\rho )=\frac{1}{2}\\ |\gamma |\le T\end{array}}\frac{{\left(x+h\right)}^{\rho +1}-x^{\rho +1}}{\rho \left(\rho +1\right)}=\sum _{\begin{array}{c}\text{Re}(\rho )=\frac{1}{2}\\ |\gamma |\le (\log T{)}^{\beta }\end{array}}\frac{{\left(x+h\right)}^{\rho +1}-x^{\rho +1}}{\rho \left(\rho +1\right)}+\sum _{\begin{array}{c}\text{Re}(\rho )=\frac{1}{2}\\ (\log T{)}^{\beta }<|\gamma |\le T\end{array}}\frac{{\left(x+h\right)}^{\rho +1}-x^{\rho +1}}{\rho \left(\rho +1\right)}\text{.}$$

$$\frac{1}{h}\left|\sum _{\begin{array}{c}\text{Re}\left(\rho \right)=\frac{1}{2}\\ |\gamma |\le (\log T{)}^{\beta }\end{array}}\frac{{\left(x+h\right)}^{\rho +1}-x^{\rho +1}}{\rho \left(\rho +1\right)}\right|=O\left(x^{\frac{1}{2}}\sum _{\begin{array}{c}\text{Re}\left(\rho \right)=\frac{1}{2}\\ |\gamma |\le (\log T{)}^{\beta }\end{array}}\frac{1}{\left|\rho \right|}\right)=O\left(x^{\frac{1}{2}}{\left(\log T\right)}^{\beta }\right)\text{.}$$

$$\sum _{\begin{array}{c}\text{Re}\left(\rho \right)=\frac{1}{2}\\ (\log T{)}^{\beta }<|\gamma |\le T\end{array}}\frac{{\left(x+h\right)}^{\rho +1}}{\rho \left(\rho +1\right)}-\sum _{\begin{array}{c}\text{Re}\left(\rho \right)=\frac{1}{2}\\ (\log T{)}^{\beta }<|\gamma |\le T\end{array}}\frac{x^{\rho +1}}{\rho \left(\rho +1\right)}={\sigma }_{\beta ,T}\left(x+h\right)-{\sigma }_{\beta ,T}\left(x\right)\text{.}$$

$$D_{Y}^{T}=\left\{x\in\left[T,eT\right):\left|\underset{{\scriptsize\begin{array}[]{c}\text{Re}\left(\rho\right)=\frac{1}{2}\\ Y<\left|\gamma\right|\leq T\end{array}}}{\sum}\frac{x^{\rho+1}}{\rho\left(\rho+1\right)}\right|>\frac{x^{\frac{3}{2}}}{(\log x)^{2\alpha}}\right\}\text{, }Y<T\text{.}$$

$$Y^{-1}\gg \frac{1}{(\log eT{)}^{4\alpha }}\underset{D_Y^T}{\int }\frac{dx}{x}=\frac{1}{(1+\log T{)}^{4\alpha }}{\mu }^{\times }{D}_Y^T\text{.}$$

$${\mu }^{\times }{D}_Y^T\ll \frac{(1+\log T{)}^{4\alpha }}{Y}\text{.}$$

$${\mu }^{\times }E$$

$${\mu }^{\times }F$$

$${\mu }^{\times }G$$

$${\sigma }_{\beta ,T}\left(x\right)=O\left(\frac{x^{\frac{3}{2}}}{(\log x{)}^{2\alpha }}\right)\text{.}$$

$${\sigma }_{\beta ,T}\left(x+h\right)=O\left(\frac{{\left(x+h\right)}^{\frac{3}{2}}}{(\log \left(x+h\right){)}^{2\alpha }}\right)=O\left(\frac{x^{\frac{3}{2}}}{(\log x{)}^{2\alpha }}\right)\text{.}$$

$${\sigma }_{\beta ,T}\left(x+h\right)=\sum _{\begin{array}{c}\text{Re}\left(\rho \right)=\frac{1}{2}\\ n^{\beta }<|\gamma |\le e^{n+1}\end{array}}\frac{{\left(x+h\right)}^{\rho +1}}{\rho \left(\rho +1\right)}-\sum _{\begin{array}{c}\text{Re}\left(\rho \right)=\frac{1}{2}\\ e^n<|\gamma |\le e^{n+1}\end{array}}\frac{{\left(x+h\right)}^{\rho +1}}{\rho \left(\rho +1\right)}\text{.}$$

$$\sum _{\begin{array}{c}\text{Re}\left(\rho \right)=\frac{1}{2}\\ e^n<|\gamma |\le e^{n+1}\end{array}}\frac{{\left(x+h\right)}^{\rho +1}}{\rho \left(\rho +1\right)}=O\left(\frac{x^{\frac{3}{2}}}{(\log x{)}^{2\alpha }}\right)\text{,}$$

$${\psi }_{\Gamma }\left(x\right)\le x+\sum _{\frac{1}{2}<\rho <1}\frac{x^{\rho }}{\rho }+O\left(\log x\right)+O\left(h\right)+O\left(\frac{x\log x}{h}\right)+O\left(\frac{x^{\frac{3}{2}}}{h(\log x{)}^{2\alpha }}\right)\text{.}$$

$${\psi }_{\Gamma }\left(x\right)\le x+\sum _{\frac{1}{2}<\rho <1}\frac{x^{\rho }}{\rho }+O\left(\frac{x^{\frac{3}{4}}}{(\log x{)}^{\alpha }}\right)\text{.}$$

Full text

1. Introduction

Let $\Gamma\subset PSL\left(2,\mathbb{R}\right)$ be a strictly hyperbolic Fuchsian group acting on the upper half-plane $\mathcal{H}$ equipped with the hyperbolic metric. The quotient space $\Gamma\setminus\mathcal{H}$ can be identified with a compact Riemann surface $\mathcal{F}$ of a genus $g\geq 2$. The object of our attention is the asymptotic behaviour of the summatory von Mangoldt function

[TABLE]

where the sum is taken over primitive hyperbolic conjugacy classes $P$ in $\Gamma$ (prime geodesics on $\mathcal{F}$), $N\left(P\right)=\exp\left(\text{length}\left(P\right)\right)$ is the norm of a class $P$ and $k$ runs through positive integers.

In the recent paper [6], Shin-ya Koyama studied the existence of a subset $E$ in $\mathbb{R}_{\geq 2}$ with finite logarithmic measure such that

[TABLE]

Here and in the sequel, $\rho$ denotes zeros of the Selberg zeta function $Z_{\Gamma}$. It is known that the complex zeros of $Z_{\Gamma}$ are of the form $\rho=\frac{1}{2}\pm\mathrm{i}\gamma$ and that $Z_{\Gamma}$ has finitely many real zeros, all laying in the interval $\left[0,1\right]$.

Koyama was motivated by Gallagher’s [4] approach to the prime number theorem under Riemann hypothesis.

We give a new proof of the following sharper result (cf. [7], [3]).

Theorem 1**.**

[TABLE]

We observe that the analogue is also valid for higher dimensional hyperbolic manifolds with cusps. Applying the Gallagher-Koyama method, we further reduce the error term outside a set of finite logarithmic measure.

Theorem 2**.**

For $\alpha>0$, there exists a set $H$ of finite logarithmic measure such that

[TABLE]

where $\varepsilon>0$ is arbitrarily small.

2. From Hejhal to Randol

Proof of Theorem 1. We shall take the same starting point as in [6], i.e. Hejhal’s explicit formula with an error term for the function $\psi_{1,\Gamma}\left(x\right)=\underset{1}{\overset{x}{\int}}\psi_{\Gamma}\left(x\right)dx$ (cf. [5, Theorem 6.16. on p. 110]):

[TABLE]

Recall that $F\left(x\right)=\left(2g-2\right)\overset{\infty}{\underset{k=2}{\sum}}\frac{2k+1}{k\left(k-1\right)}x^{1-k}$.

The novelty of our approach consists in integrating (1) at this point and then temporarily getting rid of Hejhal’s error term. Indeed, the integration of (1) firstly yields the explicit formula with an error term for $\psi_{2,\Gamma}\left(x\right)=\underset{1}{\overset{x}{\int}}\psi_{1,\Gamma}\left(x\right)dx$. Now, letting $T\rightarrow\infty$ in the obtained formula, we end up with

[TABLE]

As usually, to derive the asymptotics of $\psi_{\Gamma}\left(x\right)$ from the asymptotics of $\psi_{2,\Gamma}\left(x\right)$, one introduces the second-difference operators:

[TABLE]

where $h>0$ is to be determined later.

Since $\psi_{\Gamma}$ is a non-decreasing function, we have

[TABLE]

We apply $\Delta_{2}^{+}$ to all summands in the explicit formula for $\psi_{2,\Gamma}\left(x\right)$. E.g., $\Delta_{2}^{+}\left(\frac{x^{3}}{6}\right)=xh^{2}+h^{3}$, what gives us $\frac{1}{h^{2}}\Delta_{2}^{+}\left(\frac{x^{3}}{6}\right)=x+h$, etc.

Applying $\frac{1}{h^{2}}\Delta_{2}^{+}$ to the sum $\underset{\frac{1}{2}<\rho<1}{\sum}\frac{x^{\rho+2}}{\rho\left(\rho+1\right)\left(\rho+2\right)}$, we end up with $\underset{\frac{1}{2}<\rho<1}{\sum}\frac{x^{\rho}}{\rho}+O\left(h\right)$.

When dealing with the absolutely convergent series $\underset{\text{Re}\left(\rho\right)=\frac{1}{2}}{\sum}\frac{x^{\rho+2}}{\rho\left(\rho+1\right)\left(\rho+2\right)}$, we take into account that

[TABLE]

Thus,

[TABLE]

We are left to optimize the terms $O\left(h\right)$, $O\left(x^{\frac{1}{2}}M\right)$, $O\left(\frac{x^{\frac{5}{2}}}{h^{2}M}\right)$. This is achieved by choosing $h=x^{\frac{3}{4}}$, $M=x^{\frac{1}{4}}$. All other ingredients are dominated by $O\left(x^{\frac{3}{4}}\right)$.

The same procedure works in case of $\Delta_{2}^{-}\psi_{2,\Gamma}\left(x\right)$, i.e., for estimating $\psi_{2,\Gamma}\left(x\right)$ from below.

So,

[TABLE]

Remark 1**.**

The error term $O\left(x^{\frac{3}{4}}\right)$ in Theorem 1. yields $O\left(x^{\frac{3}{4}}/\log x\right)$ in the prime geodesic theorem. Concerning the explicit formula for $\psi_{1,\Gamma}$, one can consult [2], where a better estimate for the logarithmic derivative of the Selberg zeta function is established.

Remark 2**.**

The full analogue is valid for higher dimensional hyperbolic manifolds with cusps. Namely, the error term in the prime geodesic theorem in that setting reads $O\left(x^{\frac{3}{2}d_{0}}\left(\log x\right)^{-1}\right)$, where $d_{0}=\frac{d-1}{2}$ and $d$ is the dimension of a manifold [1, Theorem 1].

3. An application of the Gallagher-Koyama method

Proof of Theorem 2. In estimating $\psi_{\Gamma}\left(x\right)$, we shall use the explicit formula (1) and the relation $\frac{1}{h}\Delta_{1}^{-}$ $\psi_{1,\Gamma}\left(x\right)\leq$ $\psi_{\Gamma}\left(x\right)\leq\frac{1}{h}\Delta_{1}^{+}$ $\psi_{1,\Gamma}\left(x\right)$, where $0<h<\frac{x}{2}$ is to be determined later on. Here, $\Delta_{1}^{+}f\left(x\right)=f\left(x+h\right)-f\left(x\right)$ and $\Delta_{1}^{-}f\left(x\right)=f\left(x\right)-f\left(x-h\right)$.

Let $\beta>4\alpha+1$. According to (1) and the relation above, we have

[TABLE]

Now,

[TABLE]

For the first sum on the right hand side, we have

[TABLE]

The second sum is to be split into

[TABLE]

Let

[TABLE]

By Koyama’s argument [6, p. 80],

[TABLE]

Hence,

[TABLE]

For $x\in\left[e^{n},e^{n+1}\right)$, let $T=e^{n}$. The error term in (2) becomes $O\left(x\log x/h\right)$. Let $Y$ take values $Y_{1}=\left(\log T\right)^{\beta}=n^{\beta}$, $Y_{2}=\left(n-1\right)^{\beta}$, $Y_{3}=e^{n-1}$. Denote $E_{n}=D_{Y_{1}}^{T}$, $F_{n}=D_{Y_{2}}^{T}$, $G_{n}=D_{Y_{3}}^{T}$ and $E=\cup E_{n}$, $F=\cup F_{n}$, $G=\cup G_{n}$, respectively. We have

[TABLE]

Put $H=E\cup F\cup G$. Obviously, $\mu^{\times}H<\infty$. We take $x,x+h\in\mathbb{R}_{\geq 2}\setminus H$.

For $x\in\left[e^{n},e^{n+1}\right)\setminus E_{n}$, $T=e^{n}$, we get

[TABLE]

Case I. If $x+h\in\left[e^{n},e^{n+1}\right)\setminus H$, then we also have

[TABLE]

Case II. If $x+h\in\left[e^{n+1},e^{n+2}\right)\setminus H$, we shall express the sum $\sigma_{\beta,T}\left(x+h\right)$ in the form

[TABLE]

The first sum is $O\left(\frac{x^{\frac{3}{2}}}{(\log x)^{2\alpha}}\right)$ because $x+h\notin F_{n+1}=D_{Y_{2}}^{eT}$. The second sum

[TABLE]

since $x+h\notin G_{n+1}=D_{Y_{3}}^{eT}$.

So, in both cases, the relation (2) becomes

[TABLE]

The optimal bound is achieved by $h=\frac{x^{\frac{3}{4}}}{(\log x)^{\alpha}}$. Thus,

[TABLE]

The opposite inequality is derived from $\psi_{\Gamma}\left(x\right)\geq\frac{1}{h}\Delta_{1}^{-}\psi_{1,\Gamma}\left(x\right)$ by the same procedure. If $\varepsilon>0$ is arbitrarily small, then $\sum_{\frac{1}{2}<\rho<\frac{3}{4}-\varepsilon}\frac{x^{\rho}}{\rho}$ is obviously dominated by the error term. This completes the proof.