Vanishing of primitive root densities

I will explain why proving the correctness of the densities predicted by Artin’s primitive root conjecture and its elliptic analogies can currently be done if and only if the predicted density is equal to zero, and address the question what exactly can cause the vanishing of these densities. This is joint work with Francesco Campagna, Nathan Jones and Francesco Pappalardi. In the first half of the talk, I will pay attention to the interesting history of Artin’s primitive root problem.