The method has been further heightened by the larger sieve which removes arbitrarily many residue classes.
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?
The early history of the large sieve traces back to work of Yu.
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.
In the mid-1960s, the Bombieri–Vinogradov theorem was proved as a major application of large sieves using estimations of mean values of Dirichlet characters.
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: 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.
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.