On the weighted average number of subgroups of ℤm×ℤn with mn≤x.

Let ℤ m be the additive group of residue classes modulo m. For any positive integers m and n , let s (m , n) and c (m , n) denote the total number of subgroups and cyclic subgroups of the group ℤ m × ℤ n , respectively. Define D ̃ s (x) = ∑ m n ≤ x s (m , n) log x m n and D ̃ c (x) = ∑ m n ≤ x c (m , n) log x m n. In this paper, we study the asymptotic behavior of the functions D ̃ s (x) and D ̃ c (x). [ABSTRACT FROM AUTHOR]