TY - JOUR
T1 - Extreme biases in prime number races with many contestants
AU - Ford, Kevin
AU - Harper, Adam J.
AU - Lamzouri, Youness
N1 - Funding Information:
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.
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2019/6/1
Y1 - 2019/6/1
N2 - 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.
AB - 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.
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U2 - 10.1007/s00208-019-01810-x
DO - 10.1007/s00208-019-01810-x
M3 - Article
AN - SCOPUS:85062705057
VL - 374
SP - 517
EP - 551
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 1-2
ER -