On a conjecture of Rodier on primitive roots
type: Item
instance of: scholarly article
title: On a conjecture of Rodier on primitive roots
zbMATH Open document ID: 0888.11035
DOI: 10.1007/BF02940827
author: Pieter Moree
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1998-03-25T00:00:00Z
review text: Let \(N\geq3\) be an integer. A sequence \(a_1,a_2,\dots\) of integers is said to be weakly uniformly distributed \(\bmod\;N\) (WUD \(\bmod\;N\)) if it contains infinitely many elements prime to \(N\) and the condition \((r,N)=(s,N)=1\) implies asymptotic equality of \(\#\{j\leq t:\;a_j\equiv r\pmod N\}\) and \(\#\{j\leq t:\;a_j\equiv s\pmod N\}\) for \(t\) tending to infinity. Let also for squarefree \(g\neq 0,\pm1\) \(P_g\) be the sequence of all primes having \(g\) for its primitive root. The author shows, assuming GRH, that this sequence is WUD \(\bmod N\) if and only if either \(g\equiv 1\pmod 4\) and \(N\) is a power of 2 or \(g\equiv 2 \pmod 4\) and \(N=4\). Details are given only in the case \(g=2\), the general case being treated in a preprint (MPI Bonn, preprint 5/1997). The proof in this case relies on the methods used by \textit{C. Hooley} [J. Reine Angew. Math. 225, 209-220 (1967; Zbl 0221.10048)], \textit{H. W. Lenstra, jun.} [Invent. Math. 42, 201-224 (1977; Zbl 0362.12012)] and \textit{G. Cooke} and \textit{P. J. Weinberger} [Commun. Algebra 3, 481-524 (1975; Zbl 0315.12001)].
Mathematics Subject Classification ID: 11N69
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 1114170
zbMATH Keywords: primitive roots
zbMATH Keywords: weak uniform distribution
zbMATH Keywords: conjecture of Rodier
zbMATH Keywords: sequence of primes
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
Wikidata QID: Q123312433
cites work: On the construction of division chains in algebraic number rings,with applications to SL2
cites work: On Artin's conjecture.
cites work: Q4092051
cites work: On Artin's conjecture and Euclid's algorithm in global fields
cites work: Uniform distribution of sequences of integers in residue classes
cites work: Three-manifold subgroup growth, homology of coverings and simplicial volume
cites work: Asymptotic estimation of the minimum distance of the duals of BCH codes and Dickson polynomials
cites work: Artin L-series and his conjectures
description: scientific article; zbMATH DE number 1114170