On the distribution of $ϕ(ψ(n))$ and $ψ(ψ(n))$

Abstract

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