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The asymptotics of random sieves
Author(s) -
Grimmett G. R.,
Hall R. R.
Publication year - 1991
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300006628
Subject(s) - mathematics , sieve (category theory) , combinatorics , limit (mathematics) , square (algebra) , number theory , prime number , order (exchange) , function (biology) , prime (order theory) , asymptotic formula , discrete mathematics , mathematical analysis , geometry , finance , evolutionary biology , economics , biology
Let J = (s 1 , s 2 , …) 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 s i =p i 2 , the square of the i th prime.