An arithmetic function arising from Carmichael’s conjecture

Let φ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every n, the equation φ(n) = φ(m) has a solution m 6= n. This suggests defining F (n) as the number of solutions m to the equation φ(n) = φ(m). (So Carmichael’s conjecture asserts that F (n) ≥ 2 always.) Results on F are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of F contains every natural number k ≥ 2. Also, the maximal order of F has been investigated by Erdős and Pomerance. In this paper we study the normal behavior of F . Let K(x) := (log x)(log log x)(log log log x). We prove that for every fixed > 0, K(n)1/2− < F (n) < K(n)3/2+ Manuscrit reçu le 15 mai 2010.