[Paper Review] Prime geodesic theorem of Gallagher type

Muharem Avdispahić|arXiv (Cornell University)|Jan 9, 2017

Analytic Number Theory Research15 references5 citations

TL;DR

This paper improves the error term in the prime geodesic theorem for compact Riemann surfaces by reducing the exponent from $\frac{3}{4}$ to $\frac{7}{10}$ outside a set of finite logarithmic measure, using a Gallagher-type averaging method applied to the explicit formula for the Chebyshev function $\psi(x)$, leveraging zero-density estimates and $L^2$-bounds on the Selberg zeta function's non-trivial zeros.

ABSTRACT

We reduce the exponent in the error term of the prime geodesic theorem for compact Riemann surfaces from $\frac{3}{4}$ to $\frac{7}{10}$ outside a set of finite logarithmic measure.

Motivation & Objective

• To improve the error term in the prime geodesic theorem for compact Riemann surfaces beyond the classical $O(x^{3/4} \log x^{-1})$ bound.
• To extend the Gallagher-type averaging method—previously used in the context of the Riemann zeta function—to the Selberg zeta function on hyperbolic manifolds.
• To achieve an error term of $O(x^{7/10} (\log x)^{1/5} (\log \log x)^{1/5 + \varepsilon})$ outside a set of finite logarithmic measure.
• To refine the explicit formula for $\psi(x)$ by controlling the contribution of low-lying zeros and estimating the oscillatory sum over non-trivial zeros via $L^2$-norms.

Proposed method

• Applies Hejhal’s explicit formula for the integrated Chebyshev function $\psi_1(x)$, which includes a sum over non-trivial zeros $\rho$ of the Selberg zeta function.
• Uses the relation $\psi(x) \leq \frac{1}{h} \int_x^{x+h} \psi(t) dt$ for non-decreasing $\psi$, enabling pointwise control via averaging.
• Estimates the oscillatory sum over zeros with $\text{Re}(\rho) = \frac{1}{2}$ using $L^2$-bounds on the truncated sum, leveraging the mean square of $\sum \frac{t^{\rho+1}}{\rho(\rho+1)}$.
• Introduces a dyadic decomposition of the real line into intervals $[e^n, e^{n+1})$, and defines exceptional sets $E_n$ and $F_n$ where the zero sum exceeds a threshold, showing their logarithmic measure is finite.
• Applies zero-density estimates to control the number of zeros in dyadic intervals, using the bound $\sum_{Y < |\gamma| \leq T} \left| \frac{t^{\rho+1}}{\rho(\rho+1)} \right|^2 \frac{dt}{t^4} = O(1/Y)$.
• Optimizes the averaging parameter $h$ by balancing terms involving $h$, $x^2 / hT$, $x^{1/2} Y$, and $x^\alpha (\log x)^\beta (\log \log x)^\beta / h$, leading to $\alpha = \frac{7}{5}$, $\beta = \frac{2}{5}$.

Experimental results

Research questions

• RQ1 Can the error term in the prime geodesic theorem be improved beyond $O(x^{3/4})$ for compact Riemann surfaces?
• RQ2 Is it possible to achieve an exponent of $\frac{7}{10}$ in the error term using a Gallagher-type averaging method in the context of the Selberg zeta function?
• RQ3 What is the size of the exceptional set where such a strong error bound fails, and can it be shown to have finite logarithmic measure?
• RQ4 How do $L^2$-bounds on the zero sums and zero-density estimates interact to control the oscillatory contribution in the explicit formula?
• RQ5 Can the method be adapted to handle the transition between dyadic intervals where $x+h$ crosses into the next interval, without losing control of the error?

Key findings

• The error term in the prime geodesic theorem is reduced to $O\left(x^{7/10} (\log x)^{1/5} (\log \log x)^{1/5 + \varepsilon}\right)$ for $x \notin G$, where $G$ is a set of finite logarithmic measure.
• The main result establishes $\psi(x) = x + \sum_{\frac{7}{10} < \rho < 1} \frac{x^\rho}{\rho} + O\left(x^{7/10} (\log x)^{1/5} (\log \log x)^{1/5 + \varepsilon}\right)$ outside $G$, improving on prior bounds.
• The exceptional set $G = \bigcup E_n \cup \bigcup F_n$ has finite logarithmic measure, ensuring the bound holds almost everywhere in the logarithmic sense.
• The method successfully controls the contribution of low-lying zeros by splitting the sum and applying $L^2$-estimates, with the key bound $\int_T^{eT} \left| \sum \frac{t^{\rho+1}}{\rho(\rho+1)} \right|^2 \frac{dt}{t^4} = O(1/T)$.
• By choosing $h \approx x^{7/10} (\log x)^{1/5} (\log \log x)^{1/5}$, the competing error terms are balanced, yielding the optimal exponent.
• The result confirms that $\frac{7}{10}$ is achievable for the error exponent in the prime geodesic theorem for general compact Riemann surfaces, not just modular surfaces.