Simultaneous Primitive Roots over Finite Rings

TL;DR

This paper studies the distribution of prime numbers for which a randomly chosen element acts as a primitive root simultaneously over two related finite rings, revealing insights into their density as numbers grow large.

Contribution

It introduces the concept of simultaneous primitive roots over finite rings and analyzes their average density among primes as numbers tend to infinity.

Findings

Identifies the density of primes with simultaneous primitive roots over specified finite rings.

Provides asymptotic behavior of such primes as x approaches infinity.

Enhances understanding of primitive roots in the context of finite ring structures.

Abstract

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