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 and later generalized to the fundamental lemma of sieve theory by others.

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 finite set of positive integers. Let $$P$$ be some set of prime numbers. For each prime $$p$$ in $$P$$, let $$A_p$$ denote the set of elements of $$A$$ that are divisible by $$p$$. This notation can be extended to other integers $$d$$ that are products of distinct primes in $$P$$. In this case, define $$A_d$$ to be the intersection of the sets $$A_p$$ for the prime factors $$p$$ of $$d$$. Finally, define $$A_1$$ to be $$A$$ itself. Let $$z$$ be an arbitrary positive real number. The object of the sieve is to estimate: $$S(A,P,z) = \biggl\vert A \setminus \bigcup_{p \in P\atop p \le z} A_p \biggr\vert, $$

where the notation $$|X|$$ denotes the cardinality of a set $$X$$, which in this case is just its number of elements. Suppose in addition that $$|A_d|$$ may be estimated by $$ \left\vert A_d \right\vert = \frac{w(d)}{d} |A| + R_d,$$ where $$w$$ is some multiplicative function, and $$R_d$$ is some error function. Let $$ W(z) = \prod_{p \in P\atop p \le z} \left(1 - \frac{w(p)}{p} \right). $$

Brun's pure sieve
This formulation is from Cojocaru & Murty, Theorem 6.1.2. With the notation as above, suppose that
 * $$| R_d | \leq w(d) $$ for any squarefree $$d$$ composed of primes in $$P$$;
 * $$w(p) < C $$ for all $$p$$ in $$P$$;
 * There exist constants $$C, D, E$$ such that, for any positive real number $$z$$, $$ \sum_{p \in P\atop p \le z} \frac{w(p)}{p} < D \log\log z + E.$$

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 $$X$$ is the cardinal of $$A$$, $$b$$ is any positive integer and the $$O$$ invokes big O notation. In particular, letting $$x$$ denote the maximum element in $$A$$, 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 ($$C=3$$).