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

Abstract

Let $\varphi$ denote Euler's totient function. A classical result of Schoenberg asserts that $G(t):=\mathrm{dens}\{n \ge 1 : \varphi(n)/n \le t\}$ 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 $t = 1$. We provide, as $\varepsilon\to 0+$, an asymptotic expansion of $G(1-\varepsilon)$ according to negative powers of $\log(1/\varepsilon)$, together with an evaluation of the coefficients and an explicit bound for the remainder.