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

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).