Sur une généralisation de la conjecture d'Artin parmi les presque-premiers

An integer is a primitive root modulo a prime $p$ if it generates the whole multiplicative group $(\mathbb{Z}\mathrm{/}p\mathbb{Z}{)}^{\ast }$. In 1927 Artin conjectured that an integer $a$ which is not $-1$ or a square is a primitive root for infintely many primes, and that the set of those primes has a positive asymptotic density among all primes. This conjectured was proved, under the generalized Riemann hypothesis (GRH), in 1967 by Hooley. More generally, an integer is called a generalized primitive root modulo $n$ if it generates a subgroup of $(\mathbb{Z}\mathrm{/}n\mathbb{Z}{)}^{\ast }$ of maximal size. Li and Pomerance showed, under GRH, that the set of integers for which a given integer is a generalized primitive root doesn’t have an asymptotic density among all integers. We study here the set of the $\mathrm{\ell }$-almost primes, i.e. integers with at most $\mathrm{\ell }$ prime factors, for which a given integer $a\in \mathbb{Z}\mathrm{\backslash }\{-1\}$, which is not a square, is a generalized primitive root, and we prove, under GRH, that this set has an asymptotic density among all the $\mathrm{\ell }$-almost primes.