On the average number of automorphisms of the group ℤm×ℤn

Let [Formula: see text] be the additive group of the residue classes modulo [Formula: see text]. For positive integers [Formula: see text] and [Formula: see text], let [Formula: see text] denote the number of the automorphisms of the group [Formula: see text]. We obtain an asymptotic formula for the sum [Formula: see text], which indicates that the average value of [Formula: see text] is [Formula: see text] up to multiplying by a constant.