# Turan sieve

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In mathematics, in the field of 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

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 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

${\displaystyle S(A,P,z)=\left\vert A\setminus \bigcup _{p\in P(z)}A_{p}\right\vert .}$

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

${\displaystyle \left\vert A_{p}\right\vert ={\frac {1}{f(p)}}X+R_{p}}$

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

${\displaystyle \left\vert A_{pq}\right\vert ={\frac {1}{f(p)f(q)}}X+R_{p,q}}$

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

${\displaystyle U(z)=\sum _{p\mid P(z)}f(p).}$

Then

${\displaystyle S(A,P,z)\leq {\frac {X}{U(z)}}+{\frac {2}{U(z)}}\sum _{p\mid P(z)}\left\vert R_{p}\right\vert +{\frac {1}{U(z)^{2}}}\sum _{p,q\mid P(z)}\left\vert R_{p,q}\right\vert .}$
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