On the Density of Ranges of Generalized Divisor Functions

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

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