Prescribed Primitive Roots And The Least Primes

Let $q\ne \pm1,v^2$ be a fixed integer, and let $x\geq 1$ be a large number. The least prime number $p \geq3$ such that $q$ is a primitive root modulo $p$ is conjectured to be $p\ll (\log q)(\log \log q)^3),$ where $\gcd(p,q)=1$. This note proves the existence of small primes $p\ll(\log x)^c$, where $c>0$ is a constant, a close approximation to the conjectured upper bound.