Selberg sieve

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Atle Selberg

In mathematics, in the field of 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.

Let A be a set of positive integers ≤ x and let P be a set of primes. Let Ad denote the set of elements of A divisible by d when d is a product of distinct primes from P. Further let A1 denote A itself. Let z be a positive real number and P(z) denote the product of the primes in P which are ≤ z. 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. Put


where [d1,d2] denotes the least common multiple of d1 and d2. It is often useful to estimate V(z) by the bound



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