Artin's primitive root conjecture for function fields revisited
type: Item
instance of: scholarly article
title: Artin's primitive root conjecture for function fields revisited
zbMATH Open document ID: 1486.11140
author: M. Ram Murty
author: Seo Young Kim
published in: Finite Fields and their Applications
publication date: 2021-02-18T00:00:00Z
review text: Artin's primitive root conjecture claims that for each integer \(g > 0\), either \(g\) is a perfect square or there exist infinitely many primes \(p\) such that \(g\) is a primitive root modulo \(p\). The corresponding property for function fields was proved by \textit{H. Bilharz} [Math. Ann. 114, 476--492 (1937; Zbl 0016.34301)] assuming the Riemann hypothesis for function fields, which was later proved by André Weil. In this article the authors prove Bilharz's theorem without using the Riemann hypothesis for function fields; instead they apply an idea used by Hadamard and de la Vallée Poussin for proving the prime number theorem: it is sufficient to obtain a sufficiently large zero-free region of the congruence zeta functions. The proof follows the strategy already used by \textit{E. Jensen} and \textit{M. R. Murty} [in: Number theory. Basel: Birkhäuser. 167--181 (2000; Zbl 0987.11074)].
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 7312701
zbMATH Keywords: Artin's conjecture
zbMATH Keywords: function fields
zbMATH Keywords: finite fields
zbMATH Keywords: congruence zeta function
Wikidata QID: Q122937360
MaRDI profile type: Publication
full work available at URL: j.ffa.2020.101713
OpenAlex ID: W3042559833
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DOI: 10.1016/J.FFA.2020.101713
description: scientific article; zbMATH DE number 7312701