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

Abstract

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

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