large sieve method
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The large sieve is a method (or family of methods and related ideas) in analytic number theory. It is a type of
sieve A sieve, fine mesh strainer, or sift, is a device for separating wanted elements from unwanted material or for controlling the particle size distribution of a sample, using a screen such as a woven mesh or net or perforated sheet materia ...
where up to half of all residue classes of numbers are removed, as opposed to small sieves such as the
Selberg sieve In number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s. Description In ...
wherein only a few residue classes are removed. The method has been further heightened by the
larger sieve In number theory, the larger sieve is a sieve invented by Patrick X. Gallagher. The name denotes a heightening of the large sieve. Combinatorial sieves like the Selberg sieve are strongest, when only a few residue classes are removed, while the te ...
which removes arbitrarily many residue classes.


Name

Its name comes from its original application: given a set S \subset \ 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 ''A''''p'' 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''.


History

The early history of the large sieve traces back to work of Yu. B. Linnik, in 1941, working on the problem of the
least quadratic non-residue In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that: :x^2\equiv q \pmod. Otherwise, ''q'' is called a quadratic no ...
. 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 Klaus Friedrich Roth (29 October 1925 – 10 November 2015) was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De Mo ...
and
Enrico Bombieri Enrico Bombieri (born 26 November 1940, Milan) is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory. Bombieri is currently Professor Emeritus in the School of Mathem ...
. It is also around that time that the connection with the duality principle became better understood. In the mid-1960s, the
Bombieri–Vinogradov theorem In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over a ...
was proved as a major application of large sieves using estimations of mean values of
Dirichlet character In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: :1)   \ch ...
s. In the late 1960s and early 1970s, many of the key ingredients and estimates were simplified by Patrick X. Gallagher.


Development

Large-sieve methods have been developed enough that they are applicable to small-sieve situations as well. Something is commonly seen as related to the large sieve not necessarily in terms of whether it is related to the kind of situation outlined above, but, rather, if it involves one of the two methods of proof traditionally used to yield a large-sieve result:


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 \widehat(a) of the characteristic function ''f''''p'' of the set ''S'' mod ''p'' are in average large. These coefficients can be lifted to values \widehat(a/p) of the Fourier transform \widehat of the characteristic function ''f'' of the set ''S'' (i.e., :\widehat(a/p)=\widehat(a)). By bounding derivatives, we can see that \widehat(x) 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 :, f, _2 = \sqrt, we get a contradiction with the Plancherel identity :, \widehat, _2=, f, _2 unless , ''S'', is small. (In practice, to optimise bounds, people nowadays modify the Plancherel identity into an equality rather than bound derivatives as above.)


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., :\sup_v , Av, _W/, v, _V, \, where ''A'' is an operator from a linear space ''V'' to a linear space ''W'') equals the norm of its adjoint i.e., :\sup_w , A^* w, ^*_V/, w, ^*_W). 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).


See also

*
Bombieri–Vinogradov theorem In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over a ...
*
Larger sieve In number theory, the larger sieve is a sieve invented by Patrick X. Gallagher. The name denotes a heightening of the large sieve. Combinatorial sieves like the Selberg sieve are strongest, when only a few residue classes are removed, while the te ...


References

* * * * * * * {{cite book , 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=62–73 Sieve theory