Two remarks on iterates of Euler's totient function

摘要

Let φ_κ denote the kth iterate of Euler's φ-function. We study two questions connected with these iterates. First, we determine the average order of φ_κ and 1/φ_κ; e. g., we show that for each κ ≥ 0, where γ is the Euler-Mascheroni constant. Second, for prime values of p, we study the number of distinct primes dividing ∏~∞_κ=_1φκ(p). These prime divisors are precisely the primes appearing in the Pratt tree for p, which has been the subject of recent work by Ford, Konyagin, and Luca. We show that for each ε > 0, the number of distinct primes appearing in the Pratt tree for p is >(log p)~(1/2-ε) for all but x~(o(1)) primes p ≤ x.