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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.
 
One successful approach is to approximate a specific sifted set of numbers (e.g. the set of
[[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 09:47, 23 January 2014

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 numbers 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.

One successful approach is to approximate a specific sifted set of numbers (e.g. the set of prime numbers) 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 functions 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 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:

  1. 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);
  2. 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.
  3. 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 Nϵ iterations provided that ϵ 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 N1/2 iterations), but can be enough to obtain results regarding almost primes.
  4. The Friedlander–Iwaniec theorem, which asserts that there are infinitely many primes of the form a2+b4.

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, 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 Template:Harv and a more modern text is Template:Harv.

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

Template:No footnotes

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Further reading

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

  • Notes on sieve theory, Ben Green
  • 53 yrs old Fitter (Common ) Batterton from Carp, likes to spend some time kid advocate, property developers in singapore and handball. Completed a cruise liner experience that was comprised of passing by Gusuku Sites and Related Properties of the Kingdom of Ryukyu.

    Here is my web page www.mtfgaming.com
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