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Concept
Brun sieve
Summary
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.
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 be a finite set of positive integers.
Let be some set of prime numbers.
For each prime in , let denote the set of elements of that are divisible by .
This notation can be extended to other integers that are products of distinct primes in . In this case, define to be the intersection of the sets for the prime factors of .
Finally, define to be itself.
Let be an arbitrary positive real number.
The object of the sieve is to estimate:
where the notation denotes the cardinality of a set , which in this case is just its number of elements.
Suppose in addition that may be estimated by
where is some multiplicative function, and is some error function.
Let
This formulation is from Cojocaru & Murty, Theorem 6.1.2. With the notation as above, suppose that
for any squarefree composed of primes in ;
for all in ;
There exist constants such that, for any positive real number ,
Then
where is the cardinal of , is any positive integer and the invokes big O notation.
In particular, letting denote the maximum element in , if for a suitably small , then
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 primes (where 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The last two results were superseded by Chen's theorem, and the second by Goldbach's weak conjecture ().
Official source
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