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

Then we have


 * $$|\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.

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

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

If the number of excluded residue classes modulo $$p$$ varies with $$p$$, then the larger sieve is often combined with the large sieve. The larger sieve is applied with the set $$\mathcal{S}$$ 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 $$\mathcal{S}$$.