$PGT$ on $PSL(2,\mathbb{Z})$. A short proof

TL;DR

This paper provides a concise proof of a refined error term bound in the prime geodesic theorem for the modular surface, utilizing the Iwaniec explicit formula.

Contribution

It offers a shorter proof of a more precise error bound in the prime geodesic theorem for PSL(2,Z) based on the Iwaniec explicit formula.

Findings

Achieved a 2/3 exponent bound in the error term

Provided a simplified proof approach

Enhanced understanding of prime geodesic distribution

Abstract

Taking the Iwaniec explicit formula as a starting point, we give a short proof of a more precise $\frac{2}{3}$ bound for the exponent in the error term of the Gallagher-type prime geodesic theorem for the modular surface.