The Circle Problem in the Hyperbolic Plane

This paper deal with the analogue of the classical circle problem in the hyperbolic plane; that is, we count the number NΓ(s, z) of translates of a base point z by a Fuchsian group Γ, which lie in a geodesic ball of radius s about z. If Σ(s, z) is the theoretical best approximation to N(s, z) (which has πes/vol(Γ) as its leading term), we set d(s, z) = N(s, z) − Σ(s, z), the best known upper bound for which is O(e2s/3). We get omega results for d(s, z), which in the co-compact case are d(s, z) = Ω(es/2β(s)), where β(s) → ∞ as s → ∞. We also show that the normalized remainder term e(s, z) = d(s, z)/es/2 has finite mean, which is zero unless Γ is noncompact and has null forms. Further we carry out a numerical investigation of the Fermat groups and the results are consistent with an upper bound e(s, z) = O(eϵS) for all ϵ > 0. The problem in hyperbolic n-space is also investigated.