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 \( \sigma_{-r}(\mathbb{N}^+) \) 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 \( d_3 = 1/2 \).