Functiones et Approximatio Commentarii Mathematici

On primes in arithmetic progression having a prescribed primitive root.II

Pieter Moree

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Abstract

Let $a$ and $f$ be coprime positive integers. Let $g$ be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes $p$ such that $p\equiv a({\rm mod~}f)$ and $g$ is a primitive root modulo $p$ has a natural density. In this note this density is explicitly evaluated with an Euler product as result. This extends a classical result of Hooley (1967) on Artin's primitive root conjecture. Various application are given, for example the integers $g$ and $f$ such that the set of primes $p$ such that $g$ is a primitive root modulo $p$ is equidistributed modulo $f$ is determined (on GRM).

Article information

Source
Funct. Approx. Comment. Math., Volume 39, Number 1 (2008), 133-144.

Dates
First available in Project Euclid: 19 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229696559

Digital Object Identifier
doi:10.7169/facm/1229696559

Mathematical Reviews number (MathSciNet)
MR2490093

Zentralblatt MATH identifier
1223.11118

Subjects
Primary: 11N69: Distribution of integers in special residue classes
Secondary: 11N13: Primes in progressions [See also 11B25] 11N36: Applications of sieve methods 11N56: Rate of growth of arithmetic functions 11R45: Density theorems

Keywords
Artin's primitive root conjecture arithmetic progression natural density

Citation

Moree, Pieter. On primes in arithmetic progression having a prescribed primitive root.II. Funct. Approx. Comment. Math. 39 (2008), no. 1, 133--144. doi:10.7169/facm/1229696559. https://projecteuclid.org/euclid.facm/1229696559


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