Extreme biases in prime number races with many contestants

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.