Groups of rational numbers and remarks on the analytic part of Hooley's proof

Vinogradov in 1971 proved a lower bound for the number of primes up to $x$ for which a given integer $a$ is a primitive root subject to a zero-density assumption for the Dedekind zeta function of $\mathbf{Q}(a^{1/\ell})$ with a certain uniformity with respect to the prime $\ell$. Some more research has followed this approach as, for example, van der Waall in 1975. We shall examine the analogue of this result for the extension of Artin Conjecture to finitely generated groups of rational numbers. Further we shall review another possible weakening of the Generalized Rieman Hypothesis for groups of sufficiently large rank.