The Prime Geodesic Theorem and Bounds for Character Sums

TL;DR

This paper improves the error bounds in the prime geodesic theorem for the modular surface to an exponent of 2/3 unconditionally and further to 5/8 conditionally, using advanced automorphic and L-function techniques.

Contribution

It refines existing bounds for the prime geodesic theorem, achieving the first unconditional exponent of 2/3 and breaking the barrier to 5/8 under the generalized Lindelöf hypothesis.

Findings

Unconditional exponent improved to 2/3.

Conditional exponent improved to 5/8.

Breaks previous barriers in error term bounds.

Abstract

We establish the prime geodesic theorem for the modular surface with exponent $\frac{2}{3}+\varepsilon$, improving upon the long-standing exponent $\frac{25}{36}+\varepsilon$ of Soundararajan-Young (2013). This was previously known conditionally on the generalised Lindel\"{o}f hypothesis for quadratic Dirichlet $L$-functions. Our argument goes through a well-trodden trail via the automorphic machinery, and refines the techniques of Iwaniec (1984) and Cai (2002) to a maximum extent. A key ingredient is an asymptotic for bilinear forms with a counting function in Kloosterman sums via hybrid Weyl-strength subconvex bounds for quadratic Dirichlet $L$-functions due to Young (2017), zero density estimates due to Heath-Brown (1995) near the edge of the critical strip, and an asymptotic for averages of Zagier $L$-series due to Balkanova-Frolenkov-Risager (2022). Furthermore, we strengthen our exponent to $\frac{5}{8}+\varepsilon$ conditionally on the generalised Lindel\"{o}f hypothesis for quadratic Dirichlet $L$-functions, which breaks the existing barrier.