An effective bound for the Huber constant for cofinite Fuchsian groups

Abstract

Let $\Gamma$ be a cofinite Fuchsian group acting on hyperbolic two-space $\HH.$ Let $M=Γ\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 $e^{l(\gamma )}\le u.$ The \emph{prime geodesic theorem} implies: $${\pi }_M(u)=\sum _{0\le {\lambda }_{M,j}\le 1/4}\text{li}(u^{s_{M,j}})+O_M(\frac{u^{3/4}}{\log u}),$$ 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 $$|{\pi }_M(u)-\sum _{0\le {\lambda }_{M,j}\le 1/4}\text{li}(u^{s_{M,j}})|\le C_M\frac{u^{3/4}}{\log u}\text{for all u>1.}$$ 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 $C_M\le 16,607,349,020,658\approx \exp (30.44086643)$.

To appear in Mathematics of Computation

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