On the Closure of the Image of the Generalized Divisor Function

For any real number s, let σ_s be the generalized divisor function, i.e., the arithmetic function defined by σ_s(n) := ∑_d|n d^s, 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, . . . , I_ℓ. Moreover, for k = 1, . . . , ℓ, we show that the set of positive integers n such that σ−_r(n) ∈ I_k has a positive rational asymptotic density d_k. In fact, we provide a method to give exact closed form expressions for I₁, . . . , I_ℓ and d₁, . . . , d_ℓ, 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.