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arXiv · 2506.23882

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

Pei Gao·Qiyu Yang
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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 $k+1$.

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Pei Gao, Qiyu Yang. 2025-06-30. An improved upper bound for the distribution of iterated Euler totient functions. https://arxiv.org/abs/2506.23882

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