On the Normal Behavior of the Iterates Of some Arithmetic Functions

Abstract

Let ϕ₁(n) = ϕ(n) where ϕ is Euler’s function, let ϕ₂(n) = ϕ(ϕ(n)), etc. We prove several theorems about the normal order of ϕ_k(n) and state some open problems. In particular, we show that the normal order of ϕ _k (n)/ϕ _{k +1}(n)is k e ^γ log log log n where γ is Euler’s constant. We also show that there is some positive constant c such that for all n, but for a set of asymptotic density 0, there is some k with ϕ _k (n) divisible by every prime up to (log n)^c. With k(n) the first subscript k with ϕ _k (n) = 1, we show, conditional on a certain form of the Elliott-Halberstam conjecture, that there is some positive constant α such that k(n) has normal order α log n. Let s(n) = σ(n) − n where σ is the sum of the divisors function, let s ₂ (n) = S(S(n)), etc. We prove that S ₂(n)/s(n) = s(n)/n + o(1) on a set of asymptotic density 1 and conjecture the same is true for S _k+1 (n)/ _{S k} (n) for any fixed k.

Let ϕ₁(n) = ϕ(n) where ϕ is Euler’s function, let ϕ₂(n) = ϕ(ϕ(n)), etc. We prove several theorems about the normal order of ϕ_k(n) and state some open problems. In particular, we show that the normal order of ϕ _k (n)/ϕ _{k +1}(n)is k e ^γ log log log n where γ is Euler’s constant. We also show that there is some positive constant c such that for all n, but for a set of asymptotic density 0, there is some k with ϕ _k (n) divisible by every prime up to (log n)^c. With k(n) the first subscript k with ϕ _k (n) = 1, we show, conditional on a certain form of the Elliott-Halberstam conjecture, that there is some positive constant α such that k(n) has normal order α log n. Let s(n) = σ(n) − n where σ is the sum of the divisors function, let s ₂ (n) = S(S(n)), etc. We prove that S ₂(n)/s(n) = s(n)/n + o(1) on a set of asymptotic density 1 and conjecture the same is true for S _k+1 (n)/ _{S k} (n) for any fixed k.

Supported in part by an NSERC grant

Supported in part by an NSF grant