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

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 $k+1$.