Concept

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 The last two results were superseded by Chen's theorem, and the second by Goldbach's weak conjecture ().

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https://en.wikipedia.org/wiki/Brun_sieve

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We show that the prime divisors of a random polynomial in F-q[t] are typically "Poisson distributed". This result is analogous to the result in Z of Granville [1]. Along the way, we use a sieve developed by Granville and Soundararajan [2] to give a simple ...

2009

Related concepts (4)

Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the prototypical example of a sieve is the sieve of Eratosthenes, or the more general Legendre sieve. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the accumulation of error terms.

In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B2 . Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods. The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes. Let denote the number of primes p ≤ x for which p + 2 is also prime (i.

Viggo Brun (13 October 1885 – 15 August 1978) was a Norwegian professor, mathematician and number theorist. In 1915, he introduced a new method, based on Legendre's version of the sieve of Eratosthenes, now known as the Brun sieve, which addresses additive problems such as Goldbach's conjecture and the twin prime conjecture. He used it to prove that there exist infinitely many integers n such that n and n+2 have at most nine prime factors, and that all large even integers are the sum of two numbers with at most nine prime factors.