On the Density of Ranges of Generalized Divisor Functions

The range of the divisor function ${\sigma }_{-1}$ is dense in the interval $[1,\mathrm{\infty })$. However, the range of the function ${\sigma }_{-2}$ is not dense in the interval ${\displaystyle [1,\frac{{\pi }^2}{6})}$. 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 $r\in (1,\mathrm{\infty })$, then the range of the function ${\sigma }_{-r}$ is dense in the interval $[1,\zeta (r))$ if and only if $r\le \eta$. We end with an open problem.