# Sieve theory

**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 .

## 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

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

## Further reading

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

## External links

As of 21Oct2014, both of these links are broken.

- Notes on sieve theory, Ben Green
- {{#invoke:citation/CS1|citation

|CitationClass=citation }}