Primitive Root Conjecture in Arithmetic Progressions

Abstract

Let $x\geq 1$ be a large number, and let $1 \leq a <q $ be integers such that $\gcd(a,q)=1$ and $q=O(\log^c)$ with $c>0$ constant. This note proves that the counting function for the number of primes $p \in \{p=qn+a: n \geq1 \}$ with a fixed primitive root $u\ne \pm 1, v^2$ has the asymptotic formula $\pi_u(x,q,a)=\delta(u,q,a)x/ \log x +O(x/\log^b x),$ where $\delta(u,q,a)>0$ is the density, and $b=b(c)>1$ is a constant.