Sur le comportement local de la répartition de l'indicatrice d'Euler

Gérald Tenenbaum (1) , Vincent Toulmonde (1)

1 IECN - Institut Élie Cartan de Nancy

Let phi denote Euler's totient function. A classical result of Schoenberg asserts that G(t):=dens{n>= 1:varphi(n)/n <= t} is well-defined for every t in [0,1] and recent results of the second author show that the local behaviour of G around any given t may essentially be described in terms of the variations around t=1. As epsilon tends to 0+, we provide an asymptotic expansion of G(1-epsilon) according to negative powers of log(1/epsilon), together with an evaluation of the coefficients and an explicit bound for the remainder.

Domaines: Théorie des nombres [math.NT]

Gérald Tenenbaum, Vincent Toulmonde. Sur le comportement local de la répartition de l'indicatrice d'Euler. Functiones et Approximatio Commentarii Mathematici, 2006, 35, pp.321-338. ⟨hal-00091289⟩