标题： 等差数列上的除数之和 显示英文标题

标题： Sums of divisors on arithmetic progressions

摘要： 对于每个$s\in \mathbb R$和$n\in \mathbb N$，令$\sigma_s(n) = \sum_{d\mid n}d^s$。 在本文中，我们给出了$\sigma_s(an+b)$与$\sigma_s(cn+d)$的比较，其中$a$、$b$、$c$、$d$、$s$是固定的，向量$(a,b)$和$(c,d)$在$\mathbb Q$上线性无关，并且$n$遍历所有正整数。 例如，如果 $|s|\leq 1$， $a, b, c, d\in \mathbb N$固定并满足某些自然条件，那么 $$ \sigma_s(an+b) < \sigma_s(cn+d)\quad\text{ for all $n\leq M$} $$其中 $M$可能任意大，但事实上 $\sigma_s(an+b) - \sigma_s(cn+d)$有无限多个符号变化。 当 $|s|>1$时结果完全不同，可能出现以下三种情况： \begin{itemize} \item [(i)] $\sigma_s(an+b) < \sigma_s(cn+d)$对所有$n\in \mathbb N$； \item [(ii)] $\sigma_s(an+b) < \sigma_s(cn+d)$对所有$n\leq M$和$\sigma_s(an+b) > \sigma_s(cn+d)$对所有$n\geq M+1$; \item [(iii)] $\sigma_s(an+b) - \sigma_s(cn+d)$ 有无限多个符号变化。 \end{itemize}我们还给出了一些例子并提出了某些问题。 显示英文摘要

摘要： 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.