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For any positive integer $n$, let $\sigma(n)=\sum_{d\mid n} d$. In 1936, P.Erd\"os proved that the natural density of the set of positive integers $n$ forwhich $\sigma(n+1) \geq \sigma(n)$ is $\frac{1}{2}$. In 2020, M. Kobayashi andT. Trudgian showed that the natural density of positive integers $n$ with$\sigma(2n+1) \geq \sigma(2n)$ is between 0.053 and 0.055. In this paper, wegeneralize M. Kobayashi and T. Trudgian's result \cite{KT20} and give a newestimate on the natural density of positive integers $n$ for which$\sigma(kn+r_1) \geq \sigma(kn+r_2)$. We also calculate some special cases withcertain $k,r_1$ and $r_2$.

The estimate on the natural density of integers $n$ for which $\sigma(kn+r_1) \geq \sigma(kn+r_2)$