Sums of divisors on arithmetic progressions

For each $s\in \mathbb R$ and $n\in \mathbb N$, let $\sigma_s(n) = \sum_{d\mid n}d^s$. In this article, we give a comparison between $\sigma_s(an+b)$ and $\sigma_s(cn+d)$ where $a$, $b$, $c$, $d$, $s$ are fixed, the vectors $(a,b)$ and $(c,d)$ are linearly independent over $\mathbb Q$, and $n$ runs over all positive integers. For example, if $|s|\leq 1$, $a, b, c, d\in \mathbb N$ are fixed and satisfy certain natural conditions, then $$\sigma_s(an+b) < \sigma_s(cn+d)\quad\text{ for all $n\leq M$}$$ where $M$ may be arbitrarily large, but in fact $\sigma_s(an+b) - \sigma_s(cn+d)$ has infinitely many sign changes. The results are entirely different when $|s|>1$, where the following three cases may occur: \begin{itemize} \item[(i)] $\sigma_s(an+b) < \sigma_s(cn+d)$ for all $n\in \mathbb N$; \item[(ii)] $\sigma_s(an+b) < \sigma_s(cn+d)$ for all $n\leq M$ and $\sigma_s(an+b) > \sigma_s(cn+d)$ for all $n\geq M+1$; \item[(iii)] $\sigma_s(an+b) - \sigma_s(cn+d)$ has infinitely many sign changes. \end{itemize} We also give several examples and propose some problems.