Ω-results for the hyperbolic lattice point problem

2015/12/13 by Dimitrios Chatzakos, Chatzakos, Dimitrios · 2 citations
Mathematics · #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1512.04137

openalex publication_date 2015/12/13 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

For Γ a cocompact or cofinite Fuchsian group, we study the lattice point problem on the Riemann surface Γ\backslashℍ. The main asymptotic for the counting of the orbit Γz inside a circle of radius r centered at z grows like c e^r. Phillips and Rudnick studied Ω-results for the error term and mean results in r for the normalized error term. We investigate the normalized error term in the natural parameter X=2 \cosh r and prove Ω_±-results for the orbit Γw and circle centered at z, even for z ≠ w.

Cited by

• The hyperbolic lattice counting problem in large dimensions
• The hyperbolic circle problem over Heegner points

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• The hyperbolic lattice point problem in conjugacy classes
• Local average in hyperbolic lattice point counting
• The hyperbolic lattice counting problem in large dimensions
• Local average of the hyperbolic circle problem for Fuchsian groups