An effective bound for the Huber constant for cofinite Fuchsian groups

TL;DR

This paper establishes an explicit upper bound for the Huber constant associated with cofinite Fuchsian groups, providing a computable estimate and applying it to the modular group with a specific numerical bound.

Contribution

It offers the first effectively computable upper bound for the Huber constant for any cofinite Fuchsian group, including the modular group.

Findings

Derived an explicit upper bound for the Huber constant.

Applied the bound to the modular group PSL(2,Z).

Obtained a numerical estimate C_M ≤ 16,607,349,020,658.

Abstract

Let $\Gamma$ be a cofinite Fuchsian group acting on hyperbolic two-space $\HH.$ Let $M=\Gamma \setminus \HH $ be the corresponding quotient space. For $\gamma,$ a closed geodesic of $M$, let $l(\gamma)$ denote its length. The prime geodesic counting function $\pi_{M}(u)$ is defined as the number of $\Gamma$-inconjugate, primitive, closed geodesics $\gamma $ such that $e^{l(\gamma)} \leq u.$ The \emph{prime geodesic theorem} implies: $$\pi_{M}(u)=\sum_{0 \leq \lambda_{M,j} \leq 1/4} \text{li}(u^{s_{M,j}}) + O_{M}(\frac{u^{3/4}}{\log{u}}), $$ where $0=\lambda_{M,0} < \lambda_{M,1} <...$ are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth functions on $M$ and $s_{M,j} = \frac{1}{2}+\sqrt{\frac{1}{4} - \lambda_{M,j}}. $ Let $C_{M}$ be smallest implied constant so that $$|\pi_{M}(u)-\sum_{0 \leq \lambda_{M,j} \leq 1/4} \text{li}(u^{s_{M,j}})|\leq C_{M}\frac{u^{3/4}}{\log{u}} \quad \text{\text{for all} $u > 1.$}$$ We call the (absolute) constant $C_{M}$ the Huber constant. The objective of this paper is to give an effectively computable upper bound of $C_{M}$ for an arbitrary cofinite Fuchsian group. As a corollary we estimate the Huber constant for $\PSL(2,\ZZ),$ we obtain $C_{M} \leq 16,607,349,020,658 \approx \exp(30.44086643)$.