Sums of divisors on arithmetic progressions

Abstract

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 $|s|\le 1$, $a,b,c,d\in \mathbb{N}$ are fixed and satisfy certain natural conditions, then $${\sigma }_s(an+b)<{\sigma }_s(cn+d)\text{ for all n≤M}$$ 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 $n\le M$ and ${\sigma }_s(an+b)>{\sigma }_s(cn+d)$ for all $n\ge M+1$; \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.