Larger sieve

In number theory, the larger sieve is a sifting device invented by P. X. Gallagher. The name denotes a heightening of the large sieve. In fact, combinatorial sieves like the Selberg sieve are strongest, when only a few residue classes are removed, while the term large sieve means that this sieve can take advantage of the removal of a large number of up to half of all residue classes. The larger sieve can exploit the deletion of an arbitrary number of classes.

Suppose that ${\displaystyle {\mathcal {S}}}$ is a set of prime powers, N an integer, ${\displaystyle {\mathcal {A}}}$ a set of integers in the interval [1, N], such that for ${\displaystyle q\in {\mathcal {S}}}$ there are at most ${\displaystyle g(q)}$ residue classes modulo ${\displaystyle q}$, which contain elements of ${\displaystyle {\mathcal {A}}}$.

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