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.)
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