An Improved Asymptotic on the Representations of Integers as Sums of Products

Abstract

In this paper, we improve the error terms of Chace’s results in the study by Chace (1994) on the number of ways of writing an integer $N$ as a sum of $k$ products of $l$ factors, valid for $k\ge 3$ and $l=2$ , 3. More precisely, for $l=2$ , 3, we improve the upper bound $N^{k-1-\left(2\left(k-2\right)/\left(k-1\right)\left(l+1\right)\right)+\epsilon }$ , $k\ge 3$ for the error term, to $N^{2-\left(2/2l+1\right)+\epsilon }$ when $k=3$ and $N^{k-1-\left(4\left(k-2\right)/\left(l+1\right)\left(k+l-2\right)\right)+\epsilon }$ when $k\ge 4$ .