An elementary proof of a power series identity for the weighted sum of all finite abelian p-groups

Using combinatorial techniques, we prove that the weighted sum of the inverse number of automorphisms of all finite abelian $p$-groups $\sum_G |G|^{-u} |$\text{Aut}$(G)|^{-1}$isequalto$\prod_{j=u+1}^\infty\left(1-1/p^j\right)^{-1}$,where$u$ is a non-negative integer. This result was originally obtained by H. Cohen and H. W. Lenstra, Jr. In this paper we give a new elementary proof of their result.