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_3le x

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Let {{mathbb {Z}}}_{n} be the additive group of residue classes modulo n. Let c(n_1,n_2,n_3) denote the number of cyclic subgroups of the group {{mathbb {Z}}}_{n_1}times {{mathbb {Z}}}_{n_2}times {{mathbb {Z}}}_{n_3}, where n_1, n_2 and n_3 are arbitrary positive integers. In this paper we obtain an asymptotic formula for the sum sum _{n_1,n_2,n_3le _x} c(n_1,n_2,n_3).