Simple Proof of the Primitive Root Conjecture

Let $u\mathrm{\ne }\pm 1,v^2$ be a fixed integer, let $p\ge 2$ 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 $\pi (u,x)=\mathrm{\#}\{p\le x:{\text{ord}}_p(u)=p-1\}$. In 1967 Hooley proved a conditional asymptotic formula $\pi (u,x)=\delta (u)x(\log x{)}^{-1}+O(\log \log x(\log x{)}^{-2}$ for the primitive root conjecture. This note proves an unconditional asymptotic formula $\pi (u,x)=\delta (u)x(\log x{)}^{-1}+O(x(\log x{)}^{-2}$ of the same result, where $\delta (u)>0$ is the density constant.