Cesaro asymptotics for the orders of SL_k(Z_n)$ and GL_k(Z_n) as n -> infinity

Abstract

Given an integer k>0, our main result states that the sequence of orders of the groups SL_k(\Z_n) (respectively, of the groups GL_k(Z_n)) is Cesaro equivalent as n -> infinity to the sequence C_1(k) n^{k^2-1} (respectively, C_2(k)n^{k^2}), where the coefficients C_1(k) and C_2(k) depend only on k; we give explicit formulas for C_1(k) and C_2(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function is Cesaro equivalent to n * 6/pi^2. We present some experimental facts related to the main result.