An arithmetic function arising from Carmichael's conjecture

作者：Luca Florian^*; Pollack Paul

来源：Journal de Theorie des Nombres de Bordeaux, 2011, 23(3): 697-714.

DOI：10.5802/jtnb.783

Let phi denote Euler's totient function. A century-old conjecture of Carmichael asserts that for every n, the equation phi(n) = phi(m) has a solution m not equal n. This suggests defining F(n) as the number of solutions m to the equation phi(n) = phi(m). (So Carmichael's conjecture asserts that F(n) >= 2 always.) Results on F are scattered throughout the literature. For example, Sierpinski 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 Erdos 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 is an element of > 0,
K(n)(1/2-is an element of) < F(n) < K(n)(3/2+is an element of)
for almost all natural numbers n. As an application, we show that phi(n) + 1 is squarefree for almost all n. We conclude with some remarks concerning values of n for which F(n) is close to the conjectured maximum size.

• 出版日期2011