Editing Sieve theory (section)

== Types of sieving ==

Modern sieves include the [[Brun sieve]], the [[Selberg sieve]], the [[Turán sieve]], the [[large sieve]], the [[larger sieve]] and the [[Goldston–Pintz–Yıldırım 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 shows that the sum of the reciprocals of the twin primes converges (whereas the sum of the reciprocals of all primes 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 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^\varepsilon</math> iterations provided that <math>\varepsilon</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 primes.
# The ''[[Friedlander–Iwaniec theorem]]'', which asserts that there are infinitely many primes of the form <math>a^2 + b^4</math>.
# [[Yitang Zhang|Zhang]]'s theorem {{harv|Zhang|2014}}, which shows that there are infinitely many [[prime gap|pairs of primes within a bounded distance]]. The Maynard–Tao theorem {{harv|Maynard|2015}} generalizes Zhang's theorem to arbitrarily long sequences of primes.