On the distribution of $\varphi (\psi (n))$ and $\psi (\psi (n))$

Abstract

Let $\psi (n)$ and $\varphi (n)$ denote Dedekind's arithmetic function and Euler's totient function, respectively. We study the distribution of the compositions $\varphi (\psi (n))$ and $\psi (\psi (n))$. In particular, we obtain quantitative upper bounds for the exceptional set associated with $\varphi (\psi (n))$, thereby refining a density result of Sándor. We also prove that, for every fixed $c>0$, the set of integers $n\le x$ satisfying $\psi (\psi (n))\le cn$ has asymptotic density zero. Our method adapts sieve ideas used by Dixit and Bhattacharjee to compositions involving Dedekind's arithmetic function.

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