Asymmetries in the Shanks-Renyi Prime Number Race

Abstract

It has been well-observed that an inequality of the type $\pi(x;q,a) > \pi(x;q,b)$ is more likely to hold if $a$ is a non-square modulo $q$ and $b$ is a square modulo $q$ (the so-called ``Chebyshev Bias''). For instance, each of $\pi(x;8,3)$, $\pi(x;8,5)$, and $\pi(x;8,7)$ tends to be somewhat larger than $\pi(x;8,1)$. However, it has come to light that the tendencies of these three $\pi(x;8,a)$ to dominate $\pi(x;8,1)$ have different strengths. A related phenomenon is that the six possible inequalities of the form $\pi(x;8,a_1) > \pi(x;8,a_2) > \pi(x;8,a_3)$ with $\{a_1,a_2,a_3\}=\{3,5,7\}$ are not all equally likely---some orderings are preferred over others. In this paper we discuss these phenomena, focusing on the moduli $q=8$ and $q=12$, and we explain why the observed asymmetries (as opposed to other possible asymmetries) occur.