Orderings of weakly correlated random variables, and prime number races with many contestants

TL;DR

This paper studies prime number races with many contestants under standard conjectures, revealing persistent biases and providing asymptotic formulas for leader probabilities using harmonic analysis and probabilistic methods.

Contribution

It introduces new results on prime races with many classes where biases persist and develops advanced analytical tools for orderings of weakly correlated variables.

Findings

Prime races with many classes exhibit persistent biases.

Asymptotic formulas for leader probabilities when competitors grow as a power of q.

Development of harmonic analysis and probabilistic techniques for correlation control.

Abstract

We investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI. Among our results we exhibit, for the first time, prime races modulo $q$ with $n$ competitor classes where the biases do not dissolve when $n, q\to \infty$. We also study the leaders in the prime number race, obtaining asymptotic formulae for logarithmic densities when the number of competitors can be as large as a power of $q$, whereas previous methods could only allow a power of $\log q$. The proofs use harmonic analysis related to the Hardy--Littlewood circle method to control the average size of correlations in prime number races. They also use various probabilistic tools, including an exchangeable pairs version of Stein's method, normal comparison tools, and conditioning arguments. In the process we derive some general results about orderings of weakly correlated random variables, which may be of independent interest.