Primitive root biases for prime pairs I: Existence and non-totality of biases

We study the difference between the number of primitive roots modulo p and modulo p+k for prime pairs p,p+k. Assuming the Bateman-Horn conjecture, we prove the existence of strong sign biases for such pairs. More importantly, we prove that for a small positive proportion of prime pairs p,p+k, the dominant inequality is reversed.