Bounds for a spectral exponential sum

TL;DR

This paper develops a refined method to bound spectral exponential sums by accounting for oscillations, leading to improved bounds, proof of a conjecture in certain ranges, and a new proof of an error estimate in the prime geodesic theorem.

Contribution

The paper introduces a novel approach that considers oscillatory behavior in mean value evaluations, improving bounds and confirming a conjecture in specific parameter ranges.

Findings

Improved upper bounds for spectral exponential sums.

Proof of Petridis and Risager's conjecture in certain ranges.

New proof of the Soundararajan-Young error estimate in the prime geodesic theorem.

Abstract

We prove new upper bounds for a spectral exponential sum by refining the process by which one evaluates mean values of $L$-functions multiplied by an oscillating function. In particular, we introduce a method which is capable of taking into consideration the oscillatory behaviour of the function. This gives an improvement of the result of Luo and Sarnak when $T\geq X^{1/6+2\theta/3}$. Furthermore, this proves the conjecture of Petridis and Risager in some ranges. Finally, this allows obtaining a new proof of the Soundararajan-Young error estimate in the prime geodesic theorem.