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 terms of sieve theory the Selberg sieve is of combinatorial type: that is, derives from a careful use of the inclusion–exclusion principle. Selberg replaced the values of the Möbius function which arise in this by a system of weights which are then optimised to fit the given problem. The result gives an upper bound for the size of the sifted set.
Let be a set of positive integers and let be a set of primes. Let denote the set of elements of divisible by when is a product of distinct primes from . Further let denote itself. Let be a positive real number and denote the product of the primes in which are . The object of the sieve is to estimate
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