Zeta functions related to the group SL2(ℤ p )

An explicit formula is given for the number of subgroups of indexpn in the principle congruence subgroups of SL2(ℤp) (for odd primesp), and for the zeta function associated with the group. Asymptotically this number iscnpn, wherec is a constant depending on the congruence subgroup. Also, the zeta function of thei-th congruence subgroup coincides with the partial zeta function of the 3-generated subgroups of thei+1-th congruence subgroup, and for each indexpn the ratio between 2-generated subgroups and 3-generated subgroups tends top - 1:1, asn tends to infinity.