Artin's conjecture for primitive roots
type: Item
instance of: scholarly article
title: Artin's conjecture for primitive roots
zbMATH Open document ID: 0656.10044
DOI: 10.1007/BF03023749
author: M. Ram Murty
published in: The Mathematical Intelligencer
publication date: 1988-01-01T00:00:00Z
review text: This paper is a nice expository article discussing work by \textit{C. Hooley} [J. Reine Angew. Math. 225, 209-220 (1967; Zbl 0221.10048)], \textit{R. Gupta} and the author [Invent. Math. 78, 127-130 (1984; Zbl 0549.10037)], and by the reviewer [Q. J. Math., Oxf. II. Ser. 37, 27-38 (1986; Zbl 0586.10025)]. These authors are concerned with Artin's conjecture that if k is any integer other than -1 or a square, then there are infinitely many primes for which k is a primitive root. Hooley proved the conjecture, and even an asymptotic formula for the density of such primes, subject to the Riemann hypothesis for certain Dedekind zeta- functions. The other authors proved unconditional results, showing that the conjecture does at least hold for most values of k, in an appropriate sense. In this article the underlying ideas behind the various proofs are explained, and the corresonding question for elliptic curves is discussed.
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11N69
Mathematics Subject Classification ID: 11R45
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 4071031
zbMATH Keywords: expository article
zbMATH Keywords: Artin's conjecture
zbMATH Keywords: primitive root
zbMATH Keywords: asymptotic formula
zbMATH Keywords: density
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: Dedekind zeta-functions
zbMATH Keywords: elliptic curves
Wikidata QID: Q123190270
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4071031