# Brun sieve

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

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 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

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}} where C, D, E are constants.

Then

$S(A,P,z)=X\cdot W(z)\cdot \left({1+O\left((\log z)^{-b\log b}\right)}\right)+O\left(z^{b\log \log z}\right)$ where b is any positive integer. In particular, if log z < c log x / log log x for a suitably small c, then

$S(A,P,z)=X\cdot W(z)(1+o(1)).\,$ ## 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The last two results were superseded by Chen's theorem, and the second by Goldbach's weak conjecture (C = 3).