Subgroups of cyclic groups and values of the Riemann zeta function

Abstract

Let k be a positive integer, x a large real number, and let $$C_n$$ be the cyclic group of order n. For $$k\le n\le x$$ we determine the mean average order of the subgroups of $$C_n$$ generated by k distinct elements and we give asymptotic results of related averaging functions of the orders of subgroups of cyclic groups. The average order is expressed in terms of Jordan’s totient functions and Stirling numbers of the second kind. We have the following consequence. Let k and x be as above. For $$k\le n\le x$$ , the mean average proportion of $$C_n$$ generated by k distinct elements approaches $$\zeta (k+2)/\zeta (k+1)$$ as x grows, where $$\zeta (s)$$ is the Riemann zeta function.