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Lithuanian Mathematical Journal

, Volume 39, Issue 1, pp 86–101 | Cite as

A sieve result for Farey fractions

  • Vilius Stakénas
Article

Abstract

This paper is concerned with the sieve problem for Farey fractions (i.e., rational numbers with denominators less thanx) lying in an interval (λ1, λ2). An asymptotic formula for the sifting function is derived under the assumption that (λ1, λ2)x→∞ asx→∞. Two applications of this result are made. In the first one, the value distribution of the vector η(m/n)=(ξ(m), ξ(n)) is considered; here, fork=p 1 p 2...p s ,p 1p 2>-..., ξk)_is defined by ξ(k)=(logp 1/logk, logp 2/logk,..., logp s /logk, 0, ...); allp i are prime numbers. It is shown that the limit distribution is π×π, where π is the Poisson-Dirichlet distribution. The asymptotical behavior of finite-dimensional distributions of ξ(k) for natural numbers was studied by Billingsley, Knuth, Trabb Pardo, Vershik, and others; the result of weak convergence to the Poisson-Dirichlet distribution appears in Donnelly and Grimmett. The second application is concerned with the density of sets {m/n: f(m/n)=a}, wheref is a function with the almost squareful kernel.

Key words

Farey fractions Poisson-Dirichlet distribution 

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Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • Vilius Stakénas

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