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.

Description[edit]

In terms of sieve theory the Brun sieve is of combinatorial type: that is, 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 Pz. The object of the sieve is to estimate

S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert .

We assume that |Ad| may be estimated by

 \left\vert A_d \right\vert = \frac{w(d)}{d} X + R_d

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

 W(z) = \prod_{p \in P(z)} \left(1 - \frac{w(p)}{p} \right) .

Brun's pure sieve[edit]

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 ;
  •  \sum_{p \in P_z} \frac{w(p)}{p} < D \log\log z + E;

where C, D, E are constants.

Then

 S(A,P,z) = X \cdot W(z) \cdot \left({1 + O\left((\log z)^{-b \log b}\right)}\right) + O\left(z^{b \log\log z}\right) .

In particular, if log z < c log x / log log x for a suitably small c, then

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

Applications[edit]

  • 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The last two results were superseded by Chen's theorem.

References[edit]