Extreme biases in prime number races with many contestants

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.