Counting congruence subroups

2004/06/12 by D. Goldfeld, Dorian Goldfeld, Alexander Lubotzky +5
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Finite Group Theory Research #math.GR

paper · pdf · doi:10.48550/arxiv.math/0406249

30 pages

arxiv created 2004/06/12 · arxiv updated 2009/12/01

Abstract

Let Γ denote the modular group SL(2,\Bbb Z) and C_n(Γ) the number of congruence subgroups of Γ of index at most n. We prove that lim_{n→ ∞} (log C_n(Γ))/((log n)²/loglog n) = (3-2√(2))/(4). We also present a very general conjecture giving an asymptotic estimate for C_n(Γ) for general arithmetic groups. The lower bound of the conjecture is proved modulo the generalized Riemann hypothesis for Artin-Hecke L-functions, and in many cases is also proved unconditionally.

Citations

• Highly Composite Numbers

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