Ranges of Unitary Divisor Functions

Abstract

For any real $t$, the unitary divisor function $\ddot{I}{ƒ}_t^{\ast }$ is the multiplicative arithmetic function defined by $\ddot{I}{ƒ}_t^{\ast }(p^{\hat{I}}\pm )=1+p^{\hat{I}\pm t}$ for all primes $p$ and positive integers $\hat{I}\pm$. Let $\overline{\ddot{I}{ƒ}_t^{\ast }(\mathbb{N})}$ denote the topological closure of the range $\ddot{I}{ƒ}_t^{\ast }$. We calculate an explicit constant $\hat{I}{\cdot }^{\ast }\approx 1.9742550$ and show that $\overline{\ddot{I}{ƒ}_{-r}^{\ast }(\mathbb{N})}$ is connected if and only if $r\in (0,\hat{I}{\cdot }^{\ast }]$. We end with an open problem.

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