On the Density of Ranges of Generalized Divisor Functions

Abstract

The range of the divisor function $\ddot{I}{ƒ}_{-1}$ is dense in the interval $[1,\infty )$. However, the range of the function $\ddot{I}{ƒ}_{-2}$ is not dense in the interval $\left[1,\frac{\ddot{I}{€}^2}{6}\right)$. We begin by generalizing the divisor functions to a class of functions $\ddot{I}{ƒ}_t$ for all real $t$. We then define a constant $\hat{I}\cdot \approx 1.8877909$ and show that if $r\in (1,\infty )$, then the range of the function $\ddot{I}{ƒ}_{-r}$ is dense in the interval $[1,\hat{I}\P (r))$ if and only if $r\le \hat{I}\cdot$. We end with an open problem.

10 pages, 0 figures