On the number of Abelian groups of a given order and on certain related multiplicative functions

AbstractLet a(n) denote the number of nonisomorphic Abelian groups with n elements. Several problems concerning a(n) and a class of related multiplicative functions are discussed. These include precise characterizations of local densities, distribution of values (where the Tauberian theorem of Hardy-Ramanujan is used), sharp formulas for the sums Σn≤xF(a(n)) for a wide class of functions F, and a comparison of values taken on the average by a(n) and some other common arithmetic functions