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Brun sieve

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In the field of number theory, the Brun sieve (also called Brun's pure 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 Viggo Brun in 1915.

Contents

Description

In terms of sieve theory the Brun sieve is of combinatorial type; that is, it derives from a careful use of the inclusion-exclusion principle.

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 primes in P ≤ z. The object of the sieve is to estimate

S ( A , P , z ) = | A ∖ ⋃ p ∈ P ( z ) A p | .

We assume that |Ad| may be estimated by

| A d | = w ( d ) d X + R d

where w is a multiplicative function and X   =   |A|. Let

W ( z ) = ∏ p ∈ P ( z ) ( 1 − w ( p ) p ) .

Brun's pure sieve

This formulation is from Cojocaru & Murty, Theorem 6.1.2. With the notation as above, assume that

  • |Rd| ≤ w(d) for any squarefree d composed of primes in P ;
  • w(p) < C for all p in P ;
  • ∑ p ∈ P z w ( p ) p < D log ⁡ log ⁡ z + E ;

    • where C, D, E are constants.

    Then

    S ( A , P , z ) = X ⋅ W ( z ) ⋅ ( 1 + O ( ( log ⁡ z ) − b log ⁡ b ) ) + O ( z b log ⁡ log ⁡ z )

    where b is any positive integer. In particular, if log z < c log x / log log x for a suitably small c, then

    S ( A , P , z ) = X ⋅ W ( z ) ( 1 + o ( 1 ) ) .

    Applications

  • Brun's theorem: the sum of the reciprocals of the twin primes converges;
  • Schnirelmann's theorem: every even number is a sum of at most C primes (where C can be taken to be 6);
  • There are infinitely many pairs of integers differing by 2, where each of the member of the pair is the product of at most 9 primes;
  • Every even number is the sum of two numbers each of which is the product of at most 9 primes.

    • The last two results were superseded by Chen's theorem, and the second by Goldbach's weak conjecture (C = 3).

    References

    Brun sieve Wikipedia
    (Text) CC BY-SA