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Extreme biases in prime number races with many contestants
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Ford, Kevin, Harper, Adam and Lamzouri, Youness (2020) Extreme biases in prime number races with many contestants. Mathematische Annalen, 374 . pp. 517-551. doi:10.1007/s00208-019-01810-x ISSN 0025-5831.
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WRAP-extreme-biases-prime-number-races-contestants-Harper-2019.pdf - Accepted Version - Requires a PDF viewer. Download (803Kb) | Preview |
Official URL: https://doi.org/10.1007/s00208-019-01810-x
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/logq→∞ 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=(logq)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.
Item Type: | Journal Article | |||||||||||||||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||||||||||||||
Library of Congress Subject Headings (LCSH): | Numbers, Prime, Probabilities | |||||||||||||||||||||
Journal or Publication Title: | Mathematische Annalen | |||||||||||||||||||||
Publisher: | Springer Verlag | |||||||||||||||||||||
ISSN: | 0025-5831 | |||||||||||||||||||||
Official Date: | 1 June 2020 | |||||||||||||||||||||
Dates: |
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Volume: | 374 | |||||||||||||||||||||
Page Range: | pp. 517-551 | |||||||||||||||||||||
DOI: | 10.1007/s00208-019-01810-x | |||||||||||||||||||||
Status: | Peer Reviewed | |||||||||||||||||||||
Publication Status: | Published | |||||||||||||||||||||
Reuse Statement (publisher, data, author rights): | This is a post-peer-review, pre-copyedit version of an article published in Mathematische Annalen. The final authenticated version is available online at: https://doi.org/10.1007/s00208-019-01810-x | |||||||||||||||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||||||||
Date of first compliant deposit: | 12 February 2019 | |||||||||||||||||||||
Date of first compliant Open Access: | 21 December 2020 | |||||||||||||||||||||
RIOXX Funder/Project Grant: |
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