Sums of divisors on arithmetic progressions

For each s is an element of R and n is an element of N, let sigma(s)(n)= Sigma(d vertical bar 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 Q, and n runs over all positive integers. We also give several examples and propose some problems.