Sums of Kloosterman sums in the prime geodesic theorem

TL;DR

This paper introduces a novel approach to analyze sums of Kloosterman sums, leading to a new proof of the error term estimate in the prime geodesic theorem, enhancing understanding of spectral exponential sums.

Contribution

The paper develops a new method for studying Kloosterman sum sums, providing an alternative proof of the prime geodesic theorem's error term estimate.

Findings

New proof of the error term in the prime geodesic theorem

Enhanced understanding of spectral exponential sums

Development of a novel method for Kloosterman sum analysis

Abstract

We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.