  An integer is a primitive root modulo a prime $p$ if it generates the whole
multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$. 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 $(\mathbb{Z}/n\mathbb{Z})^*$ 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 $\ell$-almost primes, i.e. integers with
at most $\ell$ prime factors, for which a given integer
$a\in\mathbb{Z}\backslash\{-1\}$, 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 $\ell$-almost primes.
