Large sieve
Encyclopedia
In mathematics
, the large sieve is a method (or family of methods and related ideas) in analytic number theory
.
Its name comes from its original application: given a set
such that the elements of S are forbidden to lie in a set Ap ⊂ Z/p Z modulo every prime p, how large can S be? Here Ap is thought of as being large, i.e., at least as large as a constant times p; if this is not the case, we speak of a small sieve. (The term "sieve" is seen as alluding to, say, sifting ore for gold: we "sift out" the integers falling in one of the forbidden congruence classes modulo p, and ask ourselves how much is left at the end.)
Large-sieve methods have been developed enough that they are applicable to small-sieve situations as well. By now, something is seen as related to the large sieve not necessarily in terms of whether it related to the kind situation outlined above, but, rather, if it involves one of the two methods of proof traditionally used to yield a large-sieve result:
It is also possible to derive the large sieve from majorants in the style of Selberg (see Selberg, Collected Works, vol II, Lectures on sieves).
, in 1941, working on the problem of the least quadratic non-residue. Subsequently Alfréd Rényi
worked on it, using probability methods. It was only two decades later, after quite a number of contributions by others, that the large sieve was formulated in a way that was more definitive. This happened in the early 1960s, in independent work of Klaus Roth
and Enrico Bombieri
. It is also around that time that the connection with the duality principle became better understood.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the large sieve is a method (or family of methods and related ideas) in analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
.
Its name comes from its original application: given a set
such that the elements of S are forbidden to lie in a set Ap ⊂ Z/p Z modulo every prime p, how large can S be? Here Ap is thought of as being large, i.e., at least as large as a constant times p; if this is not the case, we speak of a small sieve. (The term "sieve" is seen as alluding to, say, sifting ore for gold: we "sift out" the integers falling in one of the forbidden congruence classes modulo p, and ask ourselves how much is left at the end.)Large-sieve methods have been developed enough that they are applicable to small-sieve situations as well. By now, something is seen as related to the large sieve not necessarily in terms of whether it related to the kind situation outlined above, but, rather, if it involves one of the two methods of proof traditionally used to yield a large-sieve result:
- An approximate Plancherel inequality. If a set 'S' is ill-distributed modulo p (by virtue, for example, of being excluded from the congruence classes Ap) then the Fourier coefficients
of the characteristic function fp of the set S mod p are in average large. These coefficients can be lifted to values 
of the Fourier transform 
of the characteristic function f of the set S (i.e., 
). By bounding derivatives, we can see that 
must be large, on average, for all x near rational numbers of the form a/p. Large here means "a relatively large constant times |S|". Since 
, we get a contradiction with the Plancherel identity 
unless |S| is small. (In practice, to optimise bounds, people nowadays modify the Plancherel identity into an equality rather than bound derivatives as above.)
- The duality principle. One can prove a strong large-sieve result easily by noting the following basic fact from functional analysis: the norm of a linear operator (i.e.,
, where A is an operator from a linear space V to a linear space W) equals the norm of its adjoint (i.e., 
). This principle itself has come to acquire the name "large sieve" in some of the mathematical literature.
It is also possible to derive the large sieve from majorants in the style of Selberg (see Selberg, Collected Works, vol II, Lectures on sieves).
History
The early history of the large sieve traces back to work of Yu. B. LinnikYuri Linnik
Yuri Vladimirovich Linnik was a Soviet mathematician active in number theory, probability theory and mathematical statistics.Linnik was born in Bila Tserkva, in present-day Ukraine. He went to St Petersburg University where his supervisor was Vladimir Tartakovski, and later worked at that...
, in 1941, working on the problem of the least quadratic non-residue. Subsequently Alfréd Rényi
Alfréd Rényi
Alfréd Rényi was a Hungarian mathematician who made contributions in combinatorics, graph theory, number theory but mostly in probability theory.-Life:...
worked on it, using probability methods. It was only two decades later, after quite a number of contributions by others, that the large sieve was formulated in a way that was more definitive. This happened in the early 1960s, in independent work of Klaus Roth
Klaus Roth
Klaus Friedrich Roth is a British mathematician known for work on diophantine approximation, the large sieve, and irregularities of distribution. He was born in Breslau, Prussia, but raised and educated in the UK. He graduated from Peterhouse, Cambridge in 1945...
and Enrico Bombieri
Enrico Bombieri
Enrico Bombieri is a mathematician who has been working at the Institute for Advanced Study in Princeton, New Jersey. Bombieri's research in number theory, algebraic geometry, and mathematical analysis have earned him many international prizes --- a Fields Medal in 1974 and the Balzan Prize in 1980...
. It is also around that time that the connection with the duality principle became better understood.