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

TL;DR

This paper refines the asymptotic understanding of the sum involving the counting function of non-isomorphic abelian groups, providing a more precise formula than previously known.

Contribution

It presents a sharper asymptotic formula for the sum of the counting function of abelian groups evaluated at n + a(n), improving on Ivić's 1991 result.

Findings

Derived a more accurate asymptotic formula for the sum involving a(n) and n.

Enhanced the understanding of the distribution of non-isomorphic abelian groups.

Extended previous asymptotic results with improved precision.

Abstract

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