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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 integer]]s which satisfy a set of conditions which are expressed by [[Congruence relation#Modular arithmetic|congruence]]s.  It was developed by [[Viggo Brun]] in 1915.
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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 &le; ''x'' and let ''P'' be a set of primes. For each ''p'' in ''P'', let ''A''<sub>''p''</sub> denote the set of elements of ''A'' divisible by ''p'' and extend this to let ''A''<sub>''d''</sub> the intersection of the ''A''<sub>''p''</sub> for ''p'' dividing ''d'', when ''d'' is a product of distinct primes from ''P''.  Further let A<sub>1</sub> denote ''A'' itself. Let ''z'' be a positive real number and ''P''(''z'') denote the primes in ''P'' &le; ''z''. The object of the sieve is to estimate
 
:<math>S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert . </math>
 
We assume that |''A''<sub>''d''</sub>| may be estimated by
 
:<math> \left\vert A_d \right\vert = \frac{w(d)}{d} X + R_d</math>
 
where ''w'' is a [[multiplicative function]] and ''X'' &nbsp; = &nbsp; |''A''|.  Let
 
:<math> W(z) = \prod_{p \in P(z)} \left(1 - \frac{w(p)}{p} \right) . </math>
 
===Brun's pure sieve===
This formulation is from [http://books.google.ro/books?id=1swo9Yf3d2YC&printsec=frontcover#v=onepage&q&f=false Cojocaru & Murty], Theorem 6.1.2. With the notation as above, assume that
*|''R''<sub>''d''</sub>| &le; ''w''(''d'') for any squarefree ''d'' composed of primes in ''P'' ;
* ''w''(''p'') < ''C'' for all ''p'' in ''P'' ;
* <math> \sum_{p \in P_z} \frac{w(p)}{p} < D \log\log z + E; </math>
 
where ''C'', ''D'', ''E'' are constants.
 
Then
 
:<math> 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) .</math>
 
In particular, if log ''z'' &lt; ''c'' log ''x'' / log log ''x'' for a suitably small ''c'', then
 
:<math> S(A,P,z) = X \cdot W(z) (1+o(1)) . \, </math>
 
==Applications==
* [[Brun's theorem]]: the sum of the reciprocals of the [[twin prime]]s 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==
* {{cite journal | author=Viggo Brun | authorlink=Viggo Brun | title=Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare | journal=Archiv for Mathematik og Naturvidenskab | volume=B34 | issue=8 | year=1915 }}
* {{cite journal | author=Viggo Brun | title=La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+..., où les dénominateurs sont nombres premiers jumeaux est convergente ou finie | journal=[[Bulletin des Sciences Mathématiques]] | year=1919 | volume=43 | pages=100–104,124–128 }}
* {{cite book | author=Alina Carmen Cojocaru | coauthors=M. Ram Murty | title=An introduction to sieve methods and their applications | series=London Mathematical Society Student Texts | volume=66 | publisher=[[Cambridge University Press]] | isbn=0-521-61275-6 | pages=80–112 | year=2005 }}
* {{cite book | author=George Greaves | title=Sieves in number theory | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge) | volume=43 | publisher=[[Springer-Verlag]] | date=2001 | isbn=3-540-41647-1 | pages=71–101}}
* {{cite book | author=[[Heini Halberstam]] | coauthors=H.E. Richert | title=Sieve Methods | publisher=[[Academic Press]] | date=1974 | isbn=0-12-318250-6}}
* {{cite book | author= Christopher Hooley | authorlink=Christopher Hooley | title=Applications of sieve methods to the theory of numbers | publisher=Cambridge University Press | date=1976 | isbn=0-521-20915-3}}.
 
[[Category:Sieve theory]]
 
[[es:Criba de Brun]]

Revision as of 19:17, 7 February 2014

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