The correction factor in Artin's primitive root conjecture.
type: Item
instance of: scholarly article
title: The correction factor in Artin's primitive root conjecture.
zbMATH Open document ID: 1043.11078
author: Peter Stevenhagen
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2004-03-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/228800
review text: Emil Artin proposed in 1927 a conjectural density for the set of primes \(p\) for which a given integer \(g\) is a primitive root modulo \(p\). However, in 1957 computer calculations by D. H. and E. Lehmer showed unexpected deviations for certain small \(g\). Artin, when informed of this, reconsidered his original statement and came up with a correction. The Lehmers, unfortunately, do not seem to have mentioned this in any of their papers. The correction is now sometimes attributed to Heilbronn. It is therefore not without historical interest that the author and H. W. Lenstra unearthed the correspondence between the Lehmers and Artin. This correspondence makes for amusing reading, as it shows an interesting clash of mathematical background and outlook. Some of the highlights of this correspondence are reprinted and discussed in the first part of this paper. Assuming the Generalized Riemann Hypothesis it was proved in 1967 by \textit{C. Hooley} [J. Reine Angew. Math. 225, 209--220 (1967; Zbl 0221.10048)] that the density for Artin's problem exists and equals a rational number times the so-called Artin constant \(A=\prod_{p}(1-{1\over p(p-1)})\approx 0.3739\), where \(p\) runs over the primes. On averaging these densities over \(g\), the average turns out to be \(A\). Thus Artin's constant can be considered to be the density for a generic number \(g\). Many variations of Artin's problem follow the same reasoning. Unfortunately, analytic number theory gives the densities involved in terms of sums involving the Möbius function. It can then be (and usually is) a challenge to derive the actual density in Euler product form. Indeed, already Hooley spoke of a `rather harder' calculation [ibid., p. 219]. \textit{H. W. Lenstra}, the author and the reviewer (in preparation) have found a way to infer the Euler product in a much more direct way. Also, for the first time the correction term is written as a product of local terms. The form of the density is then the generic positi…
reviewed by: Pieter Moree
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 2058875
zbMATH Keywords: primitive roots
zbMATH Keywords: character
zbMATH Keywords: Frobenius element
MaRDI profile type: Publication
OpenAlex ID: W2064771743
Wikidata QID: Q123265808
cites work: Q3955331
cites work: Q5797106
cites work: Q5812293
cites work: On Artin's conjecture.
cites work: Q5337291
cites work: On Artin's conjecture and Euclid's algorithm in global fields
cites work: Character sums for primitive root densities
cites work: Chebotarëv and his density theorem
DOI: 10.5802/JTNB.408
description: scientific article; zbMATH DE number 2058875