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arXiv · 2608.03749

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

Aimin Guo
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Abstract

Let $\psi(n)$ and $\phi(n)$ denote Dedekind's arithmetic function and Euler's totient function, respectively. We study the distribution of the compositions $\phi(\psi(n))$ and $\psi(\psi(n))$. In particular, we obtain quantitative upper bounds for the exceptional set associated with $\phi(\psi(n))$, thereby refining a density result of S\'andor. We also prove that, for every fixed $c>0$, the set of integers $n\leq x$ satisfying $\psi(\psi(n))\leq cn$ 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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Aimin Guo. 2026-08-04. On the distribution of $\phi(\psi(n))$ and $\psi(\psi(n))$. https://arxiv.org/abs/2608.03749

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