On some sums involving the counting function of non-isomorphic abelian groups

Abstract

Let $a(n)$ denote the number of non-isomorphic abelian groups with $n$ elements. In 1991 Ivić proved an asymptotic formula of the sum ${\sum }_{n\le x}a(n+a(n)).$ In this paper, we will prove a sharper asymptotic formula for this sum.