Densities in certain three-way prime number races

Abstract

Let $a_1$, $a_2$, and $a_3$ be distinct reduced residues modulo $q$ satisfying the congruences $a_1^2 \equiv a_2^2 \equiv a_3^2 \pmod q$. We conditionally derive an asymptotic formula, with an error term that has a power savings in $q$, for the logarithmic density of the set of real numbers $x$ for which $\pi(x;q,a_1) > \pi(x;q,a_2) > \pi(x;q,a_3)$. The relationship among the $a_i$ allows us to normalize the error terms for the $\pi(x;q,a_i)$ in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.