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

Abstract

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.