On the closure of the image of the generalized divisor function

For any real number s, let sigma_s be the generalized divisor function, i.e., the arithmetic function defined by sigma_s(n) := sum_{d | n} d^s, for all positive integers n. We prove that for any r > 1 the topological closure of sigma_{-r}(N) is the union of a finite number of pairwise disjoint closed intervals I_1, ..., I_l. Moreover, for k=1,...,l, we show that the set of positive integers n such that sigma_{-r}(n) in I_k has a positive rational asymptotic density d_k. In fact, we provide a method to give exact closed form expressions for I_1,..., I_l and d_1, ..., d_l, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results l = 3, I_1 = [1, pi^2/9], I_2 = [10/9, pi^2/8], I_3 = [5/4, pi^2 / 6], d_1 = 1/3, d_2 = 1/6, and d_3 = 1/2.