The second moment for counting prime geodesics

A brighter light has freshly been shed upon the second moment of the Prime Geodesic Theorem. We work with such moments in the two and three dimensional hyperbolic spaces. Letting Ег(Х) be the error term arising from counting prime geodesics associated to Γ = PSL2(Z[i]), the bound Ег(Х) X3/2+e is proved in a square mean sense. Our second moment bound is the pure counterpart of the work of Balog et al. for Γ = PSL2(Z), and the main innovation entails the delicate analysis of sums of Kloosterman sums. We also infer pointwise bounds from the standpoint of the second moment. Finally, we announce the pointwise bound Er(X) x67=42+? for Γ = PSL2(Z[i]) by an application of the Weyl-type subconvexity.