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

We begin by defining functions $\sigma_{t,k}$, which are generalized divisor functions with restricted domains. For each positive integer $k$, we show that, for $r>1$, the range of $\sigma_{-r,k}$ is a subset of the interval $\displaystyle{\left[1,\frac{\zeta(r)}{\zeta((k+1)r)}\right)}$. After some work, we define constants $\eta_k$ which satisfy the following: If $k\in\mathbb{N}$ and $r>1$, then the range of the function $\sigma_{-r,k}$ is dense in $\displaystyle{\left[1,\frac{\zeta(r)}{\zeta((k+1)r)}\right)}$ if and only if $r\leq\eta_k$. We end with an open problem.