Mathematische Annalen

, Volume 374, Issue 1–2, pp 517–551 | Cite as

Extreme biases in prime number races with many contestants

  • Kevin Ford
  • Adam J. Harper
  • Youness LamzouriEmail author


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 \(n \geqslant \varphi (q)^{\epsilon }\), 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 \(n = (\log q)^{1+o(1)}\). 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.



We would like to thank the anonymous referee for carefully reading the paper, and for thoughtful remarks and suggestions.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Mathematics Institute, Zeeman BuildingUniversity of WarwickCoventryEngland, UK
  3. 3.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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