Cangelmi, Leonardo; Marcovecchio, Raffaele Optimal groups for the \(r\)-rank Artin conjecture. (English) Zbl 1476.11005 Funct. Approximatio, Comment. Math. 60, No. 1, 77-86 (2019). This article completes and refines a previous work on the same topic, by the first-named author and F. Pappalardi [J. Number Theory 75, 120–132 (1999; Zbl 0926.11086)], which left pending the search for subgroups \(\Gamma\) of \(\mathbb{Q}^*\) with maximal density \(\delta_\Gamma\), also called optimal groups.We refer to the density of the primes \(p\) for which the reduction modulo \(p\) of a finitely generated subgroup \(\Gamma\) of \(\mathbb{Q}^*\) contains a primitive root modulo \(p\), that Cangelmi and Pappalardi were able to describe through an explicit formula based on the the \(r\)-rank Artin constant \(A_r\) and simplified as \(\delta_\Gamma = A_r b_\Gamma c_\Gamma\).Here, by maximizing the product \(b_\Gamma c_\Gamma\) (especially the factor \(c_\Gamma\), via similar sums over subgroups of \(\mathbb{Q}^* / \mathbb{Q}^{*2}\)), the authors characterize the optimal free group, the optimal torsion group, and the optimal positive group among all the subgroups \(\Gamma\) of \(\mathbb{Q}^*\) with any given rank \(r\).Noteworthy is the induction as key-method of proving throughout the paper. Reviewer: Enzo Bonacci (Latina) MSC: 11A07 Congruences; primitive roots; residue systems 11R45 Density theorems 20K15 Torsion-free groups, finite rank 11N13 Primes in congruence classes 11N69 Distribution of integers in special residue classes Keywords:Artin’s conjecture; finitely generated subgroups of \(\mathbb{Q}^*\); density theorems; character sums; primitive roots Citations:Zbl 1353.11104; Zbl 0926.11086 PDFBibTeX XMLCite \textit{L. Cangelmi} and \textit{R. Marcovecchio}, Funct. Approximatio, Comment. Math. 60, No. 1, 77--86 (2019; Zbl 1476.11005) Full Text: DOI Euclid