On the Composition of the Euler Function and the Dedekind Arithmetic Function

Abstract

Let $I(n)=\frac{\ddot{I}ˆ(\ddot{I}•(n))}{\ddot{I}•(\ddot{I}ˆ(n))}$ and $K(n)=\frac{\ddot{I}ˆ(\ddot{I}•(n))}{\ddot{I}•(\ddot{I}•(n))}$, where $\ddot{I}•(n)$ is Euler's function and $\ddot{I}ˆ(n)$ is Dedekind's arithmetic function. We obtain the maximal order of $I(n)$, as well as the average orders of $I(n)$ and $K(n)$. Additionally, we prove a density theorem for both $I(n)$ and $K(n)$.