An effective bound for the Huber constant for cofinite Fuchsian groups

Source abstract

Let Γ \Gamma be a cofinite Fuchsian group acting on hyperbolic two-space H \mathbb {H} . Let M = Γ ∖ H M=\Gamma \setminus \mathbb {H} be the corresponding quotient space. For γ \gamma , a closed geodesic of M M , let l ( γ ) l(\gamma ) denote its length. The prime geodesic counting function π M ( u ) \pi _{M}(u) is defined as the number of Γ \Gamma -inconjugate, primitive, closed geodesics γ \gamma such that e l ( γ ) ≤ u . e^{l(\gamma )} \leq u. The prime geodesic theorem states that: $$π M ( u ) = ∑ 0 ≤ λ M , j ≤ 1 / 4 li ⁡ ( u s M , j ) + O M ( u 3 / 4 log ⁡ u ) , \pi _M(u) = \sum _{0 \leq \lambda _{M,j} \leq 1/4} \operatorname {li}(u^{s_{M,j}}) + O_M \left (\frac {u^{3/4}}{\log u}\right ),$$ where 0 = λ M , 0 > λ M , 1 > ⋯ 0=\lambda _{M,0} > \lambda _{M,1} > \cdots are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth functions on M M and s M , j = 1 2 + 1 4 − λ M , j s_{M,j} = \frac {1}{2}+\sqrt {\frac {1}{4} - \lambda _{M,j} } . Let C M C_{M} be the smallest implied constant so that $$| π M ( u ) − ∑ 0 ≤ λ M , j ≤ 1 / 4 li ⁡ ( u s M , j ) | ≤ C M u 3 / 4 log ⁡ u for all u > 1. \left |\pi _{M}(u)-\sum _{0 \leq \lambda _{M,j} \leq 1/4} \operatorname {li}(u^{s_{M,j}})\right | \leq C_{M}\frac {u^{3/4}}{\log {u}} \quad \text {for all $u > 1.$}$$ We call the (absolute) constant C M C_{M} the Huber constant. The objective of this paper is to give an effectively computable upper bound of C M C_{M} for an arbitrary cofinite Fuchsian group. As a corollary we bound the Huber constant for P S L ( 2 , Z ) PSL(2,\mathbb {Z}) , showing that C M ≤ 16,607,349,020,658 ≈ exp ⁡ ( 30.44086643 ) C_{M} \leq 16{,}607{,}349{,}020{,}658 \approx \exp (30.44086643) .