An improved upper bound for the distribution of iterated Euler totient functions

TL;DR

This paper improves the upper bound on the count of positive integers up to x for which the (k+1)-th iterate of Euler's totient function exceeds a linear threshold, refining previous asymptotic estimates.

Contribution

It provides a tighter upper bound for the distribution of iterated Euler totient functions, enhancing the understanding of their asymptotic behavior.

Findings

Increased the denominator exponent from k to k+1 in the upper bound.

Refined the asymptotic estimate for the distribution of iterated totient functions.

Improved the understanding of the density of integers with large iterated totient values.

Abstract

Let $\phi(n)$ be the Euler totient function and $\phi_k(n)$ its $k$-fold iterate. In this note, we improve the upper bound for the number of positive $n\leqslant x$ such that $\phi_{k+1}(n)\geqslant cn$. Comparing with the upper bound which was obtained from Pollack's asymptotic formula of the summation of $\phi_{k+1}(n)$ for $n\leqslant x$, we have successfully increased the denominator exponent of the main term of the upper bound from $k$ to $k+1$.