On a generalization of Artin's conjecture among almost primes (Q6634761)

In 1927, Artin conjectured that an integer \(a \ne -1\) that is not a square is a primitive root for infinitely many primes; this was proved by \textit{C. Hooley} [J. Reine Angew. Math. 225, 209--220 (1967; Zbl 0221.10048)] in 1967 assuming GRH. Call \(a\) a generalized primitive root if it generates a subgroup of \((\mathbb Z/n\mathbb Z)^\times\) of maximal size. \textit{S. Li} and \textit{C. Pomerance} [J. Reine Angew. Math. 556, 205--224 (2003; Zbl 1022.11049)] showed, assuming GRH, that the set of integers for which a given integer is a generalized primitive root does not have an asymptotic density among all integers.\N\NCall an integer an \(\ell\)-almost prime if it has at most \(\ell\) prime factors. In this article it is shown, assuming GRH, that the set of \(\ell\)-almost primes for which an integer \(a \ne -1\) that is not a square is a generalized primitive root has an asymptotic density among all \(\ell\)-almost primes.