Larger sieve

In number theory, the larger sieve is a sieve invented by Patrick X. Gallagher. The name denotes a heightening of the large sieve. 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.

Statement

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 ${\displaystyle {\mathcal {A}}}$.

Then we have

${\displaystyle |{\mathcal {A}}|\leq {\frac {\sum _{q\in {\mathcal {S}}}\Lambda (q)-\log N}{\sum _{q\in {\mathcal {S}}}{\frac {\Lambda (q)}{g(q)}}-\log N}},}$

provided the denominator on the right is positive.^[1]

Applications

A typical application is the following result, for which the large sieve fails (specifically for ${\displaystyle \theta >{\frac {1}{2}}}$), due to Gallagher:^[2]

The number of integers ${\displaystyle n\leq N}$, such that the order of ${\displaystyle n}$ modulo ${\displaystyle p}$ is ${\displaystyle \leq N^{\theta }}$ for all primes ${\displaystyle p\leq N^{\theta +\epsilon }}$ is ${\displaystyle {\mathcal {O}}(N^{\theta })}$.

If the number of excluded residue classes modulo ${\displaystyle p}$ varies with ${\displaystyle p}$, then the larger sieve is often combined with the large sieve. The larger sieve is applied with the set ${\displaystyle {\mathcal {S}}}$ above defined to be the set of primes for which many residue classes are removed, while the large sieve is used to obtain information using the primes outside ${\displaystyle {\mathcal {S}}}$.^[3]