ISRAEL JOURNAL OF MATHEMATICS 109 (1999), 157-172 ZETA FUNCTIONS RELATED TO THE GROUP SL2(Zp) BY ISHAI ILANI* Dolev, D.N. Modiin, 71935, Israel e-mail: ilani@netmedia.net.il ABSTRACT An explicit formula is given for the number of subgroups of index pn in the principle congruence subgroups of SL2(Zp) (for odd primes p), and for the zeta function associated with the group. Asymptotically this number is cnp n, where c is a constant depending on the congruence subgroup. Also, the zeta function of the i-th congruence subgroup coincides with the partial zeta function of the 3-generated subgroups of the i-t-l-th congruence subgroup, and for each index pn the ratio between 2-generated subgroups and 3-generated subgroups tends to p - 1: 1, as n tends to infinity. 1. Introduction Let G be a finitely generated group; let an -~ an(G) be the number of subgroups of G of index n. Interest in the function G -+ (an(G)}~_ 1 and the related zeta function (c(s) -- En°°=l ann -s has grown in the last few years and some interesting results were obtained. Two of the most interesting results are: THEOREM A ([dS]): If G is a finitely generated, (topologically), compact, p-adic analytic group, then 0C3 G,p(s) := Z a;°; n~O is a rational fimction in the variable p-S. * This work is part of the author's Ph.D. thesis carried out at the Hebrew University of Jerusalem under the supervision of Prof. A. Lubotzky. I wish to thank Prof. Lubotzky for his continual interest and encouragement without which this paper would not have been published. Received February 4, 1997 157