On the enumeration of finite Abelian and solvable groups

Let <f>a(n)</f> be the number of nonisomorphic abelian groups of order <f>n</f>. We obtain a short interval result for the local density of <f>a(n)</f>. More generally, we get short interval version of results of Ivic´ on the local density of prime independent multiplicative functions. Also we prove a short interval version of the theorem of Erdo¨s and Szekeres on the summatory function of <f>a(n)</f> and the theorem of Greenberg and Newman on the enumeration of a certain type of finite solvable groups. [Copyright &y& Elsevier]