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

TL;DR

This paper investigates the properties of functions derived from Euler's totient and Dedekind's functions, establishing their maximal and average behaviors, and proving density theorems for these composite functions.

Contribution

It introduces new results on the maximal and average orders of specific compositions of Euler's and Dedekind's functions, along with density theorems for these functions.

Findings

Maximal order of I(n) determined

Average orders of I(n) and K(n) computed

Density theorems for I(n) and K(n) proved

Abstract

Let $I(n) = \frac{\psi(\phi(n))}{\phi(\psi(n))}$ and $K(n) = \frac{\psi(\phi(n))}{\phi(\phi(n))}$, where $\phi(n)$ is Euler's function and $\psi(n)$ 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 $K(n)$.