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In [[mathematics]], in the field of [[number theory]], the '''Selberg 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 [[Atle Selberg]] in the 1940s.
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In terms of [[sieve theory]] the Selberg sieve is of ''combinatorial type'': that is, derives from a careful use of the [[inclusion-exclusion principle]]. Selberg replaced the values of the [[Möbius function]] which arise in this by a system of weights which are then optimised to fit the given problem.  The result gives an ''upper bound'' for the size of the sifted set.
 
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 product of the primes in ''P'' which are &le; ''z''. The object of the sieve is to estimate
 
:<math>S(A,P,z) = \left\vert A \setminus \bigcup_{p \mid 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{1}{f(d)} X + R_d . </math>
 
where ''f'' is a [[multiplicative function]] and ''X'' &nbsp; = &nbsp; |''A''|.  Let the function ''g'' be obtained from ''f'' by [[Möbius inversion formula|Möbius inversion]], that is
 
:<math> g(n) = \sum_{d \mid n} \mu(d) f(n/d) </math>
:<math> f(n) = \sum_{d \mid n} g(d) </math>
 
where &mu; is the [[Möbius function]].
Put
 
:<math> V(z) = \sum_{\begin{smallmatrix}d < z \\ d \mid P(z)\end{smallmatrix}} \frac{\mu^2(d)}{g(d)} . </math>
 
Then
 
:<math> S(A,P,z) \le \frac{X}{V(z)} + O\left({\sum_{\begin{smallmatrix} d_1,d_2 < z \\ d_1,d_2 \mid P(z)\end{smallmatrix}} \left\vert R_{[d_1,d_2]} \right\vert} \right) .</math>
 
It is often useful to estimate ''V''(''z'') by the bound
 
:<math> V(z) \ge \sum_{d \le z} \frac{1}{f(d)} . \, </math>
 
==Applications==
* The [[Brun–Titchmarsh theorem]] on the number of [[primes in arithmetic progression]];
* The number of ''n'' &le; ''x'' such that ''n'' is coprime to &phi;(''n'') is asymptotic to e<sup>&minus;&gamma;</sup> ''x'' / log log log (''x'') .
 
==References==
* {{cite book | first1=Alina Carmen | last1=Cojocaru | first2=M. Ram | last2=Murty | author2-link=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=113–134 | year=2005 | zbl=1121.11063 }}
* {{cite book | last1= Diamond | first1 = Harold G. | last2 = Halberstam | first2 = Heini | authorlink2 = Heini Halberstam | others = With William F. Galway | title = A Higher-Dimensional Sieve Method: with Procedures for Computing Sieve Functions |series= Cambridge Tracts in Mathematics |volume=177 | publisher =[[Cambridge University Press]] | location=[[Cambridge]] | year=2008 | isbn=978-0-521-89487-6 | zbl=1207.11099 }}
* {{cite book | first=George | last=Greaves | title=Sieves in number theory | publisher=[[Springer-Verlag]] | date=2001 | isbn=3-540-41647-1 | zbl=1003.11044 | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=43 | location=Berlin }}
* {{cite book | first1=Heini | last1=Halberstam | authorlink1 = Heini Halberstam | first2=H.E. | last2=Richert | authorlink2=Hans-Egon Richert | title=Sieve Methods | publisher=[[Academic Press]] | date=1974 | isbn=0-12-318250-6 | zbl=0298.10026 | series=London Mathematical Society Monographs | volume=4 }}
* {{cite book | first=Christopher | last=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| pages=7–12 | zbl=0327.10044 | series=Cambridge Tracts in Mathematics | volume=70 }}
* {{cite journal | first=Atle | last=Selberg | authorlink=Atle Selberg | title=On an elementary method in the theory of primes | journal=Norske Vid. Selsk. Forh. Trondheim | volume=19 | year=1947 | pages=64–67 | zbl=0041.01903 | issn=0368-6302 }}
 
[[Category:Sieve theory]]

Revision as of 18:52, 14 February 2014

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