The large sieve is a method (or family of methods and related ideas) in analytic number theory.
Contents
Its name comes from its original application: given a set
Development
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:
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
By bounding derivatives, we can see that
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.)
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. Linnik, 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.