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→ ∞ as q→ ∞, 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⩾ φ(q) ϵ, 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 q) 1+o(1). 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.
ASJC Scopus subject areas