On refinements of rank one Gallagherian prime geodesic theorems

TL;DR

This paper advances the understanding of prime geodesic theorems in rank one locally symmetric spaces by deriving new explicit formulas and improving error bounds, aligning with known results in special cases and surpassing previous bounds in certain settings.

Contribution

It introduces new explicit formulas for functions related to prime geodesic counting and refines error estimates, extending Gallagherian techniques to improve bounds in various geometric contexts.

Findings

Improved error bounds for prime geodesic theorems in compact, even-dimensional, rank one spaces.

New explicit formulas for functions al_{j}(x) with applications to asymptotic analysis.

Enhanced bounds that match or surpass known estimates in special cases like Riemann surfaces and manifolds with cusps.

Abstract

In his recent research, the author improved the error term in the prime geodesic theorem for compact, even-dimensional, rank one locally symmetric spaces. It turned out that the obtained estimate $O(x^{2\rho-\frac{\rho}{n}}(\log x)^{-1})$ coincides with the best known results for compact Riemann surfaces, three manifolds, and manifolds with cusps, where $n$ stands for the dimension of the space, and $\rho$ is the half-sum of positive roots. The above bound was then reduced to $O(x^{2\rho-\rho\frac{2\cdot(2n)+1}{2n\cdot(2n)+1}}(\log x)^{\frac{n-1}{2n\cdot(2n)+1}-1}(\log\log x)^{\frac{n-1}{2n\cdot(2n)+1}+\varepsilon})$ in the Gallagherian sense, with $\varepsilon$ $>$ $0$, and the key role played by the counting function $\psi_{2n}(x)$. The purpose of this research is to prove that the latter $O$-term can be further reduced. To do so, we derive new explicit formulas for the functions $\psi_{j}(x)$, $j$ $\geq$ $n$, and conditional formula for $\psi_{n-1}(x)$. Applying the Gallagher-Koyama techniques, we deduce the asymptotics for $\psi_{0}(x)$, and the Gallagherian prime geodesic theorems. The obtained error terms $O(x^{2\rho-\rho\frac{2j+1}{2nj+1}}(\log x)^{\frac{n-1}{2nj+1}-1}(\log\log x)^{\frac{n-1}{2nj+1}+\varepsilon})$, $n-1$ $\leq$ $j$ $<$ $2n$, improve the $O$-term given above, with the optimal unconditional (conditional) size achieved for $j$ $=$ $n$ ($j$ $=$ $n-1$). If $j$ $=$ $n$ $\geq$ $4$, our new bound coincides with the best known estimate in the manifolds with cusps case. If $j$ $=$ $n-1$, the $O$-term fully agrees with the results in the Riemann surface case ($n$ $=$ $2$, $\rho$ $=$ $\frac{1}{2}(n-1)$ $=$ $\frac{1}{2}$), and the three manifolds case ($n$ $=$ $2$, $\rho$ $=$ $1$). Finally, for $j$ $=$ $n-1$, $n$ $\geq$ $4$, our result improves the best known bound in the manifolds with cusps case.