Simple Proof of the Primitive Root Conjecture

Abstract

Let \(u\neq \pm 1,v^2\) be a fixed integer, let \(p\geq 2\) be a prime, and let ${\text{ord}}_p(u)\mid p-1$ be the multiplicative order of $u\text{ mod }p$. Define a prime counting function by $\ddot{I}€(u,x)=\#\{p\le x:{\text{ord}}_p(u)=p-1\}$. In 1967 Hooley proved a conditional asymptotic formula $\ddot{I}€(u,x)=\hat{I}´(u)x(\log x{)}^{-1}+O(\log \log x(\log x{)}^{-2}$ for the primitive root conjecture. This note proves an unconditional asymptotic formula $\ddot{I}€(u,x)=\hat{I}´(u)x(\log x{)}^{-1}+O(x(\log x{)}^{-2}$ of the same result, where $\hat{I}´(u)>0$ is the density constant.

Seventeen Pages. Refined analysis of the error term. Keywords: Repeated Decimal; Primitive root; Distribution of Prime; Artin Primitive Root Conjecture