What is the best upper bound for "how often" a number n is a primitive root modulo a prime p?

Question

Let $n$ be a non-square positive number. The Artin Conjecture states that there are infinitely primes $p$ for which $n$ is a primitive root.

Question: Given a number $n$, what is the best upper bound on the proportion of primes for which $n$ is a primitive root.