Smooth Values of the Iterates of the Euler’s Phi-function

Let φ(n) be the Euler-phi function, define φ0(n) = n and φk+1(n) = φ(φk(n)) for all k ≥ 0. We will determine an asymptotic formula for the set of integers n less than x for which φk(n) is y-smooth, conditionally on a weak form of the Elliott-Halberstam conjecture. 1.Introduction Integers without large prime factors, usually called smooth numbers, play a central role in several topics of number theory. From multiplicative questions to analytic methods, they have various and wide applications, and understanding their behavior will have important consequences for number theoretic algorithms, which are an important tool in cryptography. Let φ(n) be the Euler-phi function, define φ0(n) = n and φk+1(n) = φ(φk(n)) for all k ≥ 0. There are several interesting results on the behavior of the functions φk (Erdös, Granville, Pomerance and Spiro [5]). It is known that the understanding of the multiplicative structure of the phi-function and its iterates is in some sense equivalent to studying the behavior of the integers of the form p−1 where p is prime. It is also believed that the distribution of the prime factors of such an integer behaves like that of a random integer, in the following sense: Define Ψ(x, y) = ∣∣{n ≤ x : p|n =⇒ p ≤ y}∣∣ and π(x, y) = ∣∣{p ≤ x : q|p− 1 =⇒ q ≤ y}∣∣. Conjecture 1. Fix U ≥ 1. If x ≤ y ≤ x then π(x, y) π(x) ∼ Ψ(x, y) x as x →∞. Assuming this conjecture one can deduce the behavior of the function π(x, y) from the known asymptotic formula Ψ(x, y) ∼ xρ(u) as x →∞ with x = y AMS subject classification: 11N37, 11B37, 34K05, 45J05. Typeset by AMS-TEX 1