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Statistics of Sieves and Square‐Free Numbers
Author(s) -
Grimmett Geoffrey
Publication year - 1991
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-43.1.1
Subject(s) - mathematics , square (algebra) , limit (mathematics) , combinatorics , number theory , interval (graph theory) , conjecture , representation (politics) , order (exchange) , function (biology) , prime number , statistics , discrete mathematics , mathematical analysis , geometry , finance , evolutionary biology , politics , political science , law , economics , biology
Let S = ( s 1 , s 2 ,…) be a collection of relatively prime numbers. The asymptotic properties of the process of sieving by S may be realized in terms of a stationary random process. In the case when S is the set of squares of the primes, one may make use of this representation to verify a conjecture of R. Hall: in a ‘typical’ interval of length k , the number S k of square‐free numbers has a probability mass function having order k −¼ in the limit as k → ∞.