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Extreme biases in prime number races with many contestants

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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 \(n \geqslant \varphi (q)^{\epsilon }\), 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.

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Notes

  1. We remark that the case \(k=1\) is not excluded from the non-uniformity result for technical reasons, but because it is genuinely different. Harper and Lamzouri [8] showed that \(\delta ^\flat _1(q;a_1, \dots , a_n) \sim 1/n\) provided \(n = o(\varphi (q)^{1/32})\), when \(q \rightarrow \infty \). Notice that Theorem 1.3, below, deals with \(\delta ^\flat _{2k}(q;a_1, \dots , a_n)\) and \(\delta ^\sharp _{2k}(q;a_1, \dots , a_n)\), so the case of \(\delta ^\flat _1(q;a_1, \dots , a_n)\) is again excluded.

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL, 61801, USA

    Kevin Ford

  2. Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, England, UK

    Adam J. Harper

  3. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J1P3, Canada

    Youness Lamzouri

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  1. Kevin Ford
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  2. Adam J. Harper
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  3. Youness Lamzouri
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Correspondence to Youness Lamzouri.

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Communicated by Kannan Soundararajan.

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Kevin Ford was supported by National Science Foundation Grant DMS-1501982. Adam Harper was supported, for part of this research, by a research fellowship at Jesus College, Cambridge. Youness Lamzouri is partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. In addition, part of this work was carried out at MSRI, Berkeley during the Spring semester of 2017, supported in part by NSF Grant DMS 1440140; and part of the writing up was supported by funding (for Ford and Harper) from the Simons Foundation and the Centre de Recherches Mathématiques, Montréal, through the Simons-CRM scholar-in-residence program.

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Ford, K., Harper, A.J. & Lamzouri, Y. Extreme biases in prime number races with many contestants. Math. Ann. 374, 517–551 (2019). https://doi.org/10.1007/s00208-019-01810-x

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  • DOI: https://doi.org/10.1007/s00208-019-01810-x

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