On the density of primes in arithmetic progression having a prescribed primitive root

TL;DR

This paper explicitly calculates the density of primes in a given arithmetic progression for which a specified integer is a primitive root, under the assumption of the generalized Riemann hypothesis, and explores its applications.

Contribution

It provides an explicit formula for the density of such primes, extending previous theoretical results with concrete evaluations and applications.

Findings

Explicit density formula derived under GRH

Application to distribution of primes with specific primitive roots

Enhanced understanding of prime distribution in arithmetic progressions

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.