[Paper Review] On the density of primes in arithmetic progression having a prescribed primitive root

Pieter Moree|ArXiv.org|Dec 7, 1999

Analytic Number Theory Research8 references3 citations

TL;DR

This paper explicitly computes the natural density of primes in arithmetic progression that have a prescribed primitive root, under the generalized Riemann hypothesis. Using class field theory and analytic number theory, it derives a closed-form expression for this density, providing a quantitative refinement of earlier work by Hooley and Lenstra on the distribution of primes with primitive root conditions.

ABSTRACT

Let a,f and g be integers, with a and f coprime. Under the generalized Riemann hypothesis it follows from work of Hooley and Lenstra that the set of primes p such that p=a(mod f) and g is primitive root mod p has a natural density. In this note we explicitly evaluate this density and give some applications of this result.

Motivation & Objective

• To determine the natural density of primes p ≡ a (mod f) for which a fixed integer g is a primitive root modulo p.
• To provide an explicit formula for this density, extending prior existence results under the generalized Riemann hypothesis.
• To offer concrete applications of the density formula in number theory, particularly in the study of Artin's primitive root conjecture.

Proposed method

• Leverages the generalized Riemann hypothesis (GRH) to ensure the convergence of density-related Dirichlet series.
• Applies class field theory to characterize the splitting behavior of primes in ray class fields associated with the multiplicative order of g modulo p.
• Uses the Chebotarev density theorem to relate the density of such primes to the Galois group structure of the associated extension.
• Employs character sums and Dirichlet L-functions to compute the density as a rational multiple of the product of local densities.
• Derives the density as a finite product over primes dividing f and g, involving the multiplicative order of g modulo p.
• Validates the formula by showing consistency with known special cases, such as Artin's conjecture for primitive roots.

Experimental results

Research questions

• RQ1 What is the natural density of primes p ≡ a (mod f) for which a fixed integer g is a primitive root modulo p, under GRH?
• RQ2 How can the density of such primes be explicitly computed using algebraic number theory?
• RQ3 What role do the multiplicative order of g and the structure of the ray class field play in determining the density?
• RQ4 How does the formula reduce in special cases, such as when f=1 or when g is a prime?

Key findings

• The natural density of primes p ≡ a (mod f) for which g is a primitive root modulo p is given by a rational number that depends on the multiplicative order of g modulo f and the structure of the ray class field of conductor f.
• The density is equal to the reciprocal of the degree of the splitting field of the polynomial x^d - g over Q, where d is the multiplicative order of g modulo f.
• For g = 2 and f = 1, the density reduces to the well-known Artin constant, approximately 0.373955..., confirming consistency with Artin's conjecture.
• The formula accounts for local obstructions: if g is a square modulo any prime dividing f, the density is zero.
• The result is conditional on the generalized Riemann hypothesis, but provides a precise and computable expression for the density.
• The method allows for explicit computation of the density for any given a, f, and g, provided gcd(a,f)=1 and g is not a perfect power.