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Orderings of weakly correlated random variables, and prime number races with many contestants
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Harper, Adam and Lamzouri, Youness (2018) Orderings of weakly correlated random variables, and prime number races with many contestants. Probability Theory and Related Fields, 170 (3-4). pp. 961-1010. doi:10.1007/s00440-017-0800-2 ISSN 0178-8051.
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WRAP-orderings-weakly-correlated-random-variables-contestants-Harper-2017.pdf - Accepted Version - Requires a PDF viewer. Download (492Kb) | Preview |
Official URL: http://dx.doi.org/10.1007/s00440-017-0800-2
Abstract
We investigate the race between prime numbers in many residue classes modulo q, assuming the standard conjectures GRH and LI. Among our results we exhibit, for the first time, n-way prime number races modulo q where the biases do not dissolve when n,q→∞n,q→∞ . We also study the leaders in the prime number race, obtaining asymptotic formulae for logarithmic densities when the number of competitors can be as large as a power of q, whereas previous methods could only allow a power of logqlogq . The proofs use harmonic analysis related to the Hardy–Littlewood circle method to control the average size of correlations in prime number races. They also use various probabilistic tools, including an exchangeable pairs version of Stein’s method, normal comparison tools, and conditioning arguments. In the process we derive some general results about orderings of weakly correlated random variables, which may be of independent interest.
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, Random variables | ||||||||
Journal or Publication Title: | Probability Theory and Related Fields | ||||||||
Publisher: | Springer | ||||||||
ISSN: | 0178-8051 | ||||||||
Official Date: | April 2018 | ||||||||
Dates: |
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Volume: | 170 | ||||||||
Number: | 3-4 | ||||||||
Page Range: | pp. 961-1010 | ||||||||
DOI: | 10.1007/s00440-017-0800-2 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 29 September 2017 | ||||||||
Date of first compliant Open Access: | 18 September 2018 | ||||||||
Funder: | Jesus College (University of Cambridge), Université de Montréal. Centre de recherches mathématiques, Natural Sciences and Engineering Research Council Canada |
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