Uniformly counting primes with a given primitive root and in an arithmetic progression

TL;DR

This paper derives explicit error bounds for counting primes with a specific primitive root in arithmetic progressions, assuming a form of the generalized Riemann Hypothesis, and applies it to a related Diophantine problem.

Contribution

It provides explicit error terms in prime counting with primitive roots in arithmetic progressions, extending previous asymptotic results by combining existing work with Hooley's method.

Findings

Explicit error bounds for prime counts with primitive roots

Application to a Diophantine problem involving such primes

Assumption of a suitable form of the generalized Riemann Hypothesis

Abstract

We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin's primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.