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

Xin-qi Luo, Chen-kai Ren

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 $σ(kn+r_1) \geq σ(kn+r_…