Extreme biases in prime number races with many contestants

TL;DR

This paper demonstrates that for many residue classes modulo q, prime number races can be extremely biased when the number of contestants n grows faster than log q, under standard conjectures, revealing a phase transition in bias behavior.

Contribution

It improves the range for bias existence in prime number races and establishes the transition point from unbiased to biased races as n approaches log^{1+o(1)} q.

Findings

Biases in prime number races can be constructed for many classes when n/ log q → ∞.

The transition from unbiased to biased races occurs around n = log^{1+o(1)} q.

A new Gaussian approximation theorem is developed for analyzing these biases.

Abstract

We continue to investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI. We show that provided $n/\log q \rightarrow \infty$ as $q \rightarrow \infty$, we can find $n$ competitor classes modulo $q$ so that the corresponding $n$-way prime number race is extremely biased. This improves on the previous range $n \geq \varphi(q)^{\epsilon}$, and (together with an existing result of Harper and Lamzouri) establishes that the transition from all $n$-way races being asymptotically unbiased, to biased races existing, occurs when $n = \log^{1+o(1)}q$. The proofs involve finding biases in certain auxiliary races that are easier to analyse than a full $n$-way race. An important ingredient is a quantitative, moderate deviation, multi-dimensional Gaussian approximation theorem, which we prove using a Lindeberg type method.