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'''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 primordial example of a sifted set is the set of [[prime number]]s up to some prescribed limit ''X''.  Correspondingly, the primordial 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 one of the major strands of number theory in the twentieth century, ways were found of avoiding some of the difficulties of a frontal attack with a naive idea of what sieving should be.
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One successful approach is to approximate a specific sifted set of numbers (e.g. the set of
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[[prime number]]s) by another, simpler set (e.g. the set of [[almost prime]] numbers), which is typically somewhat larger than the original set, and easier to analyze. More sophisticated sieves also do not work directly with sets ''per se'', but instead count them according to carefully chosen [[weight function]]s on these sets (options for giving some elements of these sets more "weight" than others). Furthermore, in some modern applications, sieves are used not to estimate the size of a sifted
set, but to produce a function that is large on the set and mostly small outside it, while being easier to analyze than
the [[Indicator function|characteristic function]] of the set.
 
== Types of sieving ==
 
Modern sieves include the [[Brun sieve]], the [[Selberg sieve]], and the [[large sieve]]. One of the original purposes of sieve theory was to try to prove conjectures in number theory such as the [[twin prime conjecture]]. While the original broad aims of sieve theory still are largely unachieved, there have been some partial successes, especially in combination with other number theoretic tools. Highlights include:
 
# ''[[Brun's theorem]]'', which asserts that the sum of the reciprocals of the twin primes converges (whereas the sum of the reciprocals of the primes themselves diverges);
# ''[[Chen's theorem]]'', which shows that there are infinitely many primes ''p'' such that ''p'' + 2 is either a prime or a [[semiprime]] (the product of two primes); a closely related theorem of [[Chen Jingrun]] asserts that every [[sufficiently large]] even number is the sum of a prime and another number which is either a prime or a semiprime. These can be considered to be near-misses to the [[twin prime conjecture]] and the [[Goldbach conjecture]] respectively.
# The ''[[fundamental lemma of sieve theory]]'', which (very roughly speaking) asserts that if one is sifting a set of ''N'' numbers, then one can accurately estimate the number of elements left in the sieve after <math>N^\epsilon</math> iterations provided that <math>\epsilon</math> is sufficiently small (fractions such as 1/10 are quite typical here). This lemma is usually too weak to sieve out primes (which generally require something like <math>N^{1/2}</math> iterations), but can be enough to obtain results regarding [[almost prime]]s.
# The ''[[Friedlander–Iwaniec theorem]]'', which asserts that there are infinitely many primes of the form <math>a^2 + b^4</math>.
 
== Techniques of sieve theory ==
 
The techniques of sieve theory can be quite powerful, but they seem to be limited by an obstacle known as the ''[[parity problem (sieve theory)|parity problem]]'', which roughly speaking asserts that sieve theory methods have extreme difficulty distinguishing between numbers with an odd number of prime factors and numbers with an even number of prime factors. This parity problem is still not very well understood.
 
Compared with other methods in number theory, sieve theory is comparatively ''elementary'', in the sense that it does not necessarily require sophisticated concepts from either [[algebraic number theory]] or [[analytic number theory]]. Nevertheless, the more advanced sieves can still get very intricate and delicate (especially when combined with other deep techniques in number theory), and entire textbooks have been devoted to this single subfield of number theory; a classic reference is {{harv|Halberstam|Richert|1974}} and a more modern text is {{harv|Iwaniec|Friedlander|2010}}.
 
The sieve methods discussed in this article are not closely related to the [[integer factorization]] sieve methods such as the [[quadratic sieve]] and the [[general number field sieve]]. Those factorization methods use the idea of the [[sieve of Eratosthenes]] to determine efficiently which members of a list of numbers can be completely factored into small primes.
 
==References==
{{No footnotes|date=July 2009}}
* {{citation | last1= Cojocaru | first1 = Alina Carmen | last2= Murty | first2= M. Ram | authorlink2 = M. Ram Murty | title= An introduction to sieve methods and their applications | publisher= [[Cambridge University Press]] | year= 2006 | isbn= 0-521-84816-4 | series= London Mathematical Society Student Texts | volume= 66 | mr= 2200366 }}
*{{citation | last=Motohashi | first=Yoichi |title=Lectures on Sieve Methods and Prime Number Theory |
series= Tata IFR LN | volume=72 | publisher=[[Springer-Verlag]] | date=1983 | isbn=3-540-12281-8}}
* {{citation | last=Greaves | first=George | 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}}
* {{cite book | first1=Heini | last1=Halberstam | authorlink1=Heini Halberstam | first2=Hans-Egon | last2=Richert | authorlink2=Hans-Egon Richert | title=Sieve Methods | publisher=[[Academic Press]] | date=1974 | isbn=0-12-318250-6}}
:* {{cite book | first1=Heini | last1=Halberstam | authorlink1=Heini Halberstam | first2=Hans-Egon | last2=Richert | authorlink2=Hans-Egon Richert | title=Sieve Methods | edition=2nd | publisher=Dover | year=2011 | isbn=0-486-47939-0 }}
* {{citation | last1=Iwaniec | first1=Henryk | authorlink1=Henryk Iwaniec | last2=Friedlander |first2= John | authorlink2=John Friedlander | title=Opera de cribro | publisher=[[American Mathematical Society|AMS]] | date=2010 | isbn=0-8218-4970-0}}
* {{citation | last=Hooley | first=Christopher | authorlink=Christopher Hooley | title=Applications of sieve methods to the theory of numbers | publisher=[[Cambridge University Press]] | date=1976 | isbn=0-521-20915-3}}
* {{citation | title=Introduction to Analytic and Probabilistic Number Theory | first=Gérald | last=Tenenbaum | series=Cambridge studies in advanced mathematics | volume=46 | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-41261-7 | pages=56–79 }}
 
==Further reading==
* {{cite book | last=Harman | first=Glyn | authorlink=Glyn Harman | title=Prime-detecting sieves | series=London Mathematical Society Monographs | volume=33 | publisher=Princeton University Press | year=2007 | isbn=978-0-691-12437-7 | zbl=1220.11118 }}
 
==External links==
* [http://www.dpmms.cam.ac.uk/~bjg23/expos.html Notes on sieve theory, Ben Green]
* {{SpringerEOM | title=Sieve method | id=12559 | last=Bredikhin | first=B.M. | author-link= }}
 
{{DEFAULTSORT:Sieve Theory}}
[[Category:Sieve theory| ]]

Revision as of 07:34, 10 February 2014

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