Counting congruence subroups

Let $\mathrm{\Gamma }$ denote the modular group $SL(2,\mathbb{Z})$ and $C_n(\mathrm{\Gamma })$ the number of congruence subgroups of $\mathrm{\Gamma }$ 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 $C_n(\mathrm{\Gamma })$ 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.