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

Loading next page...
 
/lp/springer_journal/orderings-of-weakly-correlated-random-variables-and-prime-number-races-01nrbIpCPz
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

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches

$49/month

Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588

$360/year

billed annually
Start Free Trial

14-day Free Trial