  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.
