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 N_Γ(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 πe^s/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(e^2s/3). We get omega results for d(s, z), which in the co-compact case are d(s, z) = Ω(e^s/2β(s)), where β(s) → ∞ as s → ∞. We also show that the normalized remainder term e(s, z) = d(s, z)/e^s/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(e^ε(lunate)S) for all ε(lunate) > 0. The problem in hyperbolic n-space is also investigated.