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.

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