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 $\sigma_s(n) := \sum_{d \, \mid \, n} d^s$, for all positive integers $n$. We prove that for any $r > 1$ the topological closure of $\sigma_{-r}(\mathbf{N}^+)$ is the union of a finite number of pairwise disjoint closed intervals $I_1, \ldots, I_\ell$. Moreover, for $k=1,\ldots,\ell$, 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 $I_1, \ldots, I_\ell$ and $d_1, \ldots, d_\ell$, assuming to know $r$ with sufficient precision. As an example, we show that for $r = 2$ it results $\ell = 3$, $I_1 = [1, \pi^2/9]$, $I_2 = [10/9, \pi^2/8]$, $I_3 = [5/4, \pi^2 / 6]$, $d_1 = 1/3$, $d_2 = 1/6$, and $d_3 = 1/2$.