AbstractFor any real number s, let σsbe the generalized divisor function, i.e., the arithmetic function defined by σs(n) := ∑d|nds, for all positive integers n. We prove that for any r > 1 the topological closure of σ−r(N+) is the union of a finite number of pairwise disjoint closed intervals I1, . . . , Iℓ. Moreover, for k = 1, . . . , ℓ, we show that the set of positive integers n such that σ−r(n) ∈ Ikhas a positive rational asymptotic density dk. In fact, we provide a method to give exact closed form expressions for I1, . . . , Iℓand d1, . . . , dℓ, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results ℓ = 3, I1= [1, π2/9], I2= [10/9, π2/8], I3= [5/4, π2/6], d1= 1/3, d2 = 1/6, and d3= 1/2.