An effective bound for the Huber constant for cofinite Fuchsian groups

Let $Γ$ be a cofinite Fuchsian group acting on hyperbolic two-space $\HH.$ Let $M=Γ\setminus \HH $ be the corresponding quotient space. For $γ,$ a closed geodesic of $M$, let $l(γ)$ denote its length. The prime geodesic counting function $π_{M}(u)$ is defined as the number of $Γ$-inconjugate, primitive, closed geodesics $γ$ such that $e^{l(γ)} \leq u.$ The \emph{prime geodesic theorem} implies: $π_{M}(u)=\sum_{0 \leq λ_{M,j} \leq 1/4} \text{li}(u^{s_{M,j}}) + O_{M}(\frac{u^{3/4}}{\log{u}}),$ where $0=λ_{M,0} < λ_{M,1} <...$ are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth functions on $M$ and $s_{M,j} = \frac{1}{2}+\sqrt{\frac{1}{4} - λ_{M,j}}.$ Let $C_{M}$ be smallest implied constant so that $|π_{M}(u)-\sum_{0 \leq λ_{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 $C_{M} \leq 16,607,349,020,658 \approx \exp(30.44086643)$.

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