  The range of the divisor function $\sigma_{-1}$ is dense in the interval
$[1,\infty)$. However, the range of the function $\sigma_{-2}$ is not dense in
the interval $\displaystyle{\left[1,\frac{\pi^2}{6}\right)}$. We begin by
generalizing the divisor functions to a class of functions $\sigma_{t}$ for all
real $t$. We then define a constant $\eta\approx 1.8877909$ and show that if
$r\in(1,\infty)$, then the range of the function $\sigma_{-r}$ is dense in the
interval $[1,\zeta(r))$ if and only if $r\leq\eta$. We end with an open
problem.
