Some Applications of Large Sieve in Riemann Surfaces
نویسندگان
چکیده
1. Introduction. In [Ch] we gave some large sieve type inequalities involving elements of harmonic analysis in Riemann surfaces and compact Riemannian manifolds. In this paper we present some of their applications. Our results are related to the hyperbolic circle problem, which is a generalization of the classical circle problem. The latter can be formulated as counting the images of a point in the plane under integral translations belonging to a large circle. Similarly, in the hyperbolic version the integral translations are replaced by the elements of a Fuchsian group of the first kind, say Γ , and the problem is to find an asymptotic formula with a small error term for #{γ ∈ Γ : (γz, w) ≤ s} where is the hyperbolic distance. Only by notational convenience (
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