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

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 $n⩽x$ such that ${\phi }_{k+1}(n)⩾cn$. Comparing with the upper bound which was obtained from Pollack’s asymptotic formula of the summation of ${\phi }_{k+1}(n)$ for $n⩽x$, we have successfully increased the denominator exponent of the main term of the upper bound from $k$ to $k+1$.