An effective bound for the Huber constant for cofinite Fuchsian groups

Abstract:

Let $\Gamma$ be a cofinite Fuchsian group acting on hyperbolic two-space $\mathbb {H}$. Let $M=\Gamma \setminus \mathbb {H}$ 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 prime geodesic theorem states that: \[ \pi _M(u) = \sum _{0 \leq \lambda _{M,j} \leq 1/4} \operatorname {li}(u^{s_{M,j}}) + O_M \left (\frac {u^{3/4}}{\log u}\right ), \] where $0=\lambda _{M,0} < \lambda _{M,1} < \cdots$ 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 the smallest implied constant so that \[ \left |\pi _{M}(u)-\sum _{0 \leq \lambda _{M,j} \leq 1/4} \operatorname {li}(u^{s_{M,j}})\right | \leq C_{M}\frac {u^{3/4}}{\log {u}} \quad \text {for all $u > 1.$} \] 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 bound the Huber constant for $PSL(2,\mathbb {Z})$, showing that $C_{M} \leq 16{,}607{,}349{,}020{,}658 \approx \exp (30.44086643)$.