On the Density of Ranges of Generalized Divisor Functions With Restricted Domains

We begin by defining functions σt,k, which are generalized divisor functions with restricted domains. For each positive integer k, we show that, for ζ(r) r > 1, the range of σ−r,k is a subset of the interval1, ζ((k + 1)r). After some work, we define constants ηk which satisfy the following: If k ∈ N and r > 1, then ζ(r) the range of the function σ−r,k is dense in 1, if and only if r ≤ ηk.ζ((k + 1)r)We end with an open problem.