Densities in certain three-way prime number races

TL;DR

This paper derives an asymptotic formula for the density of real numbers where three prime counting functions in a specific residue class order, revealing independence among their distributions under certain quadratic residue conditions.

Contribution

It provides a new asymptotic formula with power savings for the density in three-way prime number races under quadratic residue conditions, using elementary probability tools.

Findings

Asymptotic formula with power savings for the density

Mutual independence of distributions under quadratic residue conditions

Elementary probability methods applied to 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.