Larger Sieve
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In
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, the larger sieve is a
sieve A sieve (), fine mesh strainer, or sift is a tool used for separating wanted elements from unwanted material or for controlling the particle size distribution of a sample, using a screen such as a woven mesh or net or perforated sheet m ...
invented by
Patrick X. Gallagher Patrick Ximenes Gallagher (January 2, 1935 – March 30, 2019) was an American mathematician who pioneered large sieve, large sieve theory and invented the larger sieve. Biography Early life Patrick Ximenes Gallagher was born on January 2, 1935, ...
. The name denotes a heightening of the
large sieve The large sieve is a method (or family of methods and related ideas) in analytic number theory. It is a type of sieve where up to half of all residue classes of numbers are removed, as opposed to small sieves such as the Selberg sieve wherein on ...
. Combinatorial sieves like the
Selberg sieve In number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s. Description In ...
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 is a set of prime powers, ''N'' an integer, \mathcal a set of integers in the interval , ''N'' such that for q\in \mathcal there are at most g(q) residue classes modulo q, which contain elements of \mathcal. Then we have :, \mathcal, \leq \frac, 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), 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^ is \mathcal(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 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.Croot, Elsholtz, 2004


Notes


References

* * {{cite journal , first1=Ernie , last1=Croot , first2 = Christian , last2 = Elsholtz , author-link=Ernie Croot , title=On variants of the larger sieve , journal= Acta Mathematica Hungarica , volume=103 , year=2004 , issue=3 , pages=243–254 , doi=10.1023/B:AMHU.0000028411.04500.e2 , doi-access=free Sieve theory