An effective bound for the Huber constant for cofinite Fuchsian groups

Abstract

Let Γ be a cofinite Fuchsian group acting on hyperbolic two-space H{double-struck}. Let M = Γ\H{double-struck} be the corresponding quotient space. For γ, a closed geodesic of M, let l(γ) denote its length. The prime geodesic counting function πM(u) is defined as the number of Γ-inconjugate, primitive, closed geodesics γ such that el(γ) ≤ u. The prime geodesic theorem states that: where 0 = λM,0 < λM,1 < ... are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth functions on M and sM,j =. Let CM be the smallest implied constant so that We call the (absolute) constant CM the Huber constant. The objective of this paper is to give an effectively computable upper bound of CM for an arbitrary cofinite Fuchsian group. As a corollary we bound the Huber constant for PSL(2,Z{double-struck}), showing that CM ≤ 16,607,349,020,658 ≈ exp(30.44086643). © 2010 American Mathematical Society.