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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 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, the Turán 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 iterations provided that 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 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 . # Zhang's theorem that there are infinitely many pairs of primes within a bounded distance. The Maynard-Tao theorem generalizes Zhang's theorem to arbitrarily long sequences of primes. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sieve theory」の詳細全文を読む スポンサード リンク
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