Counting congruence subroups

Abstract

Let $\Gamma$ denote the modular group $SL(2,\Bbb Z)$ and $C_n(\Gamma)$ the number of congruence subgroups of $\Gamma$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log C_n(\Gamma)}{(\log n)^2/\log\log n} = \frac{3-2\sqrt{2}}{4}.$ We also present a very general conjecture giving an asymptotic estimate for $C_n(\Gamma)$ 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.