Artin's conjecture on primitive roots

Statement

Infinitude version

Suppose $a$ is an integer that is not equal to $-1$ and is not a perfect square, i.e., $a$ is not the square of an integer. Then, there exist infinitely many primes $p$ such that $a$ is a primitive root modulo $p$.

Density version

Fill this in later

Relation with other conjectures and known facts

Name of conjecture/fact !! Statement !! Conditional to ...
Hooley's theorem	Artin's conjecture holds for all $a$	(special cases of) generalized Riemann hypothesis
Gupta-Ram Murty theorem	Artin's conjecture holds for infinitely many $a$	Unconditional
Heath-Brown theorem on Artin's conjecture	Artin's conjecture holds for all but two exceptional values of $a$. However, no explicit information about the explicit values of $a$	Unconditional