On the natural density of integers $n$ for which $\sigma(kn+r_1) >\sigma(kn+r_2)$

Abstract

For any positive integer $n$, let $\sigma(n)=\sum_{d\mid n} d$. In 2020, M. Kobayashi and T. Trudgian showed that the natural density of positive integers n with $\sigma(kn+r_1) \geq \sigma(kn+r_2)$ is between 0.053 and 0.055. In this paper, we extend their result. For integers $k>r_1>r_2\geq 0,$ we provide an estimate on the natural density of positive integers $n$ for which $\sigma(kn+r_1) > \sigma(kn+r_2)$. We also calculate some special cases with certain $k,r_1$ and $r_2$. We also compute explicit bounds for specific $k,r_1,r_2$ to illustrate the variation of density.