Counting integers with a smooth totient

TL;DR

This paper corrects a previous proof regarding the distribution of integers with a smooth totient and establishes a potentially optimal upper bound for their count.

Contribution

It fixes a gap in the proof of an upper bound for integers with smooth Euler totients and derives a stronger, possibly best-possible result.

Findings

Established a corrected upper bound for integers with smooth totients.

Derived a potentially optimal bound improving previous results.

Provided a more accurate understanding of the distribution of integers with smooth Euler functions.

Abstract

We fix a gap in our proof of an upper bound for the number of positive integers $n\le x$ for which the Euler function $\varphi(n)$ has all prime factors at most $y$. While doing this we obtain a stronger, likely best-possible result.