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Turán sieve

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Pál Turán

In number theory, the Turán sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Pál Turán in 1934.

Description

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In terms of sieve theory the Turán sieve is of combinatorial type: deriving from a rudimentary form of the inclusion–exclusion principle. The result gives an upper bound for the size of the sifted set.

Let A be a set of positive integers ≤ x and let P be a set of primes. For each p in P, let Ap denote the set of elements of A divisible by p and extend this to let Ad be the intersection of the Ap for p dividing d, when d is a product of distinct primes from P. Further let A1 denote A itself. Let z be a positive real number and P(z) denote the product of the primes in P which are ≤ z. The object of the sieve is to estimate

We assume that |Ad| may be estimated, when d is a prime p by

and when d is a product of two distinct primes d = p q by

where X   =   |A| and f is a function with the property that 0 ≤ f(d) ≤ 1. Put

Then

Applications

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References

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  • Alina Carmen Cojocaru; M. Ram Murty. An introduction to sieve methods and their applications. London Mathematical Society Student Texts. Vol. 66. Cambridge University Press. pp. 47–62. ISBN 0-521-61275-6.
  • Greaves, George (2001). Sieves in number theory. Springer-Verlag. ISBN 3-540-41647-1.
  • Halberstam, Heini; Richert, H.-E. (1974). Sieve Methods. London Mathematical Society Monographs. Vol. 4. Academic Press. ISBN 0-12-318250-6. MR 0424730. Zbl 0298.10026.
  • Christopher Hooley (1976). Applications of sieve methods to the theory of numbers. Cambridge University Press. p. 21. ISBN 0-521-20915-3.