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.
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.
be a set of positive integers
denote the set of elements of
is a product of distinct primes from
be a positive real number and
denote the product of the primes in
The object of the sieve is to estimate We assume that |Ad| may be estimated by where f is a multiplicative function and X = |A|.
Let the function g be obtained from f by Möbius inversion, that is where μ is the Möbius function.
denotes the least common multiple of