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The asymptotics of random sieves

Part of: Multiplicative number theory

Published online by Cambridge University Press:  26 February 2010

G. R. Grimmett  and
R. R. Hall
G. R. Grimmett
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW
R. R. Hall
Affiliation:
Department of Mathematics, University of York, Heslington, York. YO1 5DD
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Extract

Let J = (s1, s2, … ) be a collection of relatively prime integers, and suppose that π(n) = |J∩{1,2,…, n}| is a regularly varying function with index a satisfying 0 < α < l. We investigate the “stationary random sieve” generated by J, proving that the number of integers less than k which escape the action of the sieve has a probability mass function with approximate order k-α/2 in the limit as k → ∞. This result may be used to deduce certain asymptotic properties of the set of integers which are divisible by no s є J, in that it gives new information about the usual deterministic (that is, non-random) sieve. This work extends previous results valid when si=pi2, the square of the ith prime.

MSC classification

Secondary: 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Mathematika , Volume 38 , Issue 2 , December 1991 , pp. 285 - 302
Copyright
Copyright © University College London 1991

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