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.
In the late 1960s and early 1970s, many of the key ingredients and estimates were simplified by Patrick X.
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.