On the \(r\)-rank Artin conjecture. II
type: Item
instance of: scholarly article
title: On the \(r\)-rank Artin conjecture. II
zbMATH Open document ID: 0926.11086
author: Leonardo Cangelmi
author: Francesco Pappalardi
published in: Journal of Number Theory
publication date: 1999-11-24T00:00:00Z
review text: 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.
reviewed by: George Greaves
Mathematics Subject Classification ID: 11R45
Mathematics Subject Classification ID: 11N69
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 1274815
zbMATH Keywords: primitive root
zbMATH Keywords: Artin conjecture
zbMATH Keywords: Riemann hypothesis
MaRDI profile type: Publication
full work available at URL: jnth.1998.2319
OpenAlex ID: W2023666083
Wikidata QID: Q123153779
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DOI: 10.1006/JNTH.1998.2319
description: scientific article; zbMATH DE number 1274815