Orderings of weakly correlated random variables, and prime number races with many contestants

Orderings of weakly correlated random variables, and prime number races with many contestants 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 $$n, q\rightarrow \infty $$ 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 $$\log q$$ log q . 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Probability Theory and Related Fields Springer Journals

Orderings of weakly correlated random variables, and prime number races with many contestants

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Mathematics; Probability Theory and Stochastic Processes; Theoretical, Mathematical and Computational Physics; Quantitative Finance; Mathematical and Computational Biology; Statistics for Business/Economics/Mathematical Finance/Insurance; Operations Research/Decision Theory
ISSN
0178-8051
eISSN
1432-2064
D.O.I.
10.1007/s00440-017-0800-2
Publisher site
See Article on Publisher Site

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 $$n, q\rightarrow \infty $$ 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 $$\log q$$ log q . 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.

Journal

Probability Theory and Related FieldsSpringer Journals

Published: Sep 18, 2017

References

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