Simultaneous Primitive Roots over Finite Rings

Abstract

This note investigates the average density of prime numbers $p\in [x,2x]$ with respect to a random simultaneous primitive root $g\le p^{1/2+\epsilon }$ over the finite rings $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}/p^2\mathbb{Z}$ as $x\to \infty$.

Twenty Pages. Keywords: Prime number; Primitive root; Simultaneous primitive root; Hensel lemma; Artin Primitive Root Conjecture