On the average number of cyclic subgroups of the groups $${\mathbb {Z}}_{n_1} \times {\mathbb {Z}}_{n_2}\times {\mathbb {Z}}_{n_3}$$ with $$n_1,n_2,n_3\le x$$

Abstract

AbstractLet $${{\mathbb {Z}}}_{n}$$Zn be the additive group of residue classes modulo n. Let $$c(n_1,n_2,n_3)$$c(n1,n2,n3) denote the number of cyclic subgroups of the group $${{\mathbb {Z}}}_{n_1}\times {{\mathbb {Z}}}_{n_2}\times {{\mathbb {Z}}}_{n_3}$$Zn1×Zn2×Zn3, where $$n_1, n_2$$n1,n2 and $$n_3$$n3 are arbitrary positive integers. In this paper we obtain an asymptotic formula for the sum $$\sum _{n_1,n_2,n_3\le _x} c(n_1,n_2,n_3).$$∑n1,n2,n3≤xc(n1,n2,n3).