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 study the number of sign changes in the difference \(\sigma _s(an+b)-\sigma _s(cn+d)\) 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. We also give several examples and propose some problems.