On the \(r\)-rank Artin conjecture. II
Let \(\Gamma\) be a finitely generated multiplicative subgroup of the rationals of rank \(r\), and let \(N_\Gamma(x)\) denote the number of primes \(p\) for which the reduction \(\bmod p\) of \(\Gamma\) consists of all the non-zero residue classes \(\bmod p\). Assuming an appropriate Riemann hypothesis for Kummer extensions, the author's result implies an asymptotic formula of the shape \(N_\Gamma(x) \sim \delta_r x/ \log x\). In the case when \(\Gamma\) is generated by positive integers \(a_1, \ldots,a_r\) the second-named author derived such a result in [Math.\ Comput. 66, 853--868 (1997; Zbl 0883.11041)]. The current result also improves on the error term in this formula, in which the case \(r=1\) corresponds to Artin's well-known conjecture on primitive roots treated under a similar hypothesis by \textit{C. Hooley} [J. Reine Angew. Math. 226, 207--220 (1967; Zbl 0221.10048)]. An explicit formula is provided for the density \(\delta_r\) which permits its effective computation. In this way some support is obtained for conjectures concerning which groups \(\Gamma\) of rank \(r\) give the largest \(\delta_r\). An error in the first paper of this series is corrected.
scientific article; zbMATH DE number 1274815