An effective bound for the Huber constant for cofinite Fuchsian groups

Let Γ be a cofinite Fuchsian group acting on hyperbolic two-space H. Let M = Γ\H 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: πM (u) = ∑ 0≤λM,j≤1/4 li(uM,j ) +OM ( u3/4 log u ) , 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 = 1 2 + √ 1 4 − λM,j . Let CM be the smallest implied constant so that ∣∣∣∣∣ πM (u)− ∑ 0≤λM,j≤1/4 li(uM,j ) ∣∣∣∣∣ ≤ CM u3/4 log u for all u > 1. 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), showing that CM ≤ 16,607,349,020,658 ≈ exp(30.44086643). Introduction Let Γ be a cofinite Fuchsian group, and let M = Γ \ H be the corresponding hyperbolic orbifold. Let C(M) denote the set of closed geodesics of M, and let P(M) denote the set of prime (or primitive) closed geodesics (see [Bus92, page 245]). For each γ ∈ C(M) there exists a unique prime geodesic γ0 and a unique exponent m ≥ 1 so that γ = γ 0 . Let l(γ) denote the length of γ. Associated to γ is a unique hyperbolic conjugacy class {Pγ}Γ with norm Nγ ≡ N(Pγ) = e. The prime geodesic counting function πM (u) is defined to be the number of Γ-inconjugate, primitive, hyperbolic elements γ ∈ Γ such that eγ < u. Received by the editor September 25, 2009 and, in revised form, March 2, 2010. 2010 Mathematics Subject Classification. Primary 11F72; Secondary 30F35. The second named author acknowledges support from grants from the NSF and PSC-CUNY.. The third named author acknowledges support from the DFG Graduate School Berlin Mathematical School and from the DFG Research Training Group Arithmetic and Geometry. c ©2010 American Mathematical Society Reverts to public domain 28 years from publication