Primitive Root Conjecture in Arithmetic Progressions

Abstract

Let $x\ge 1$ be a large number, and let $1\le a<q$ 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 $p\in \{p=qn+a:n\ge 1\}$ with a fixed primitive root $u\ne \pm 1,v^2$ has the asymptotic formula $\ddot{I}{€}_u(x,q,a)=\hat{I}´(u,q,a)x/\log x+O(x/{\log }^{b}x),$ where $\hat{I}´(u,q,a)>0$ is the density, and $b=b(c)>1$ is a constant.

Sixteen Pages. Refined Error Term. Keywords: Prime Number; Primitive Root; Arithmetic Progression; Artin Primitive Root Conjecture. arXiv admin note: text overlap with arXiv:1609.01147