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. For each p in P, let Ap denote the set of elements of A divisible by p and extend this to let Ad the intersection of the Ap for p dividing 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 μ is the Möbius function. Put
It is often useful to estimate V(z) by the bound
- The Brun–Titchmarsh theorem on the number of primes in arithmetic progression;
- The number of n ≤ x such that n is coprime to φ(n) is asymptotic to e−γ x / log log log (x) .
- Cojocaru, Alina Carmen; Murty, M. Ram (2005). An introduction to sieve methods and their applications. London Mathematical Society Student Texts. 66. Cambridge University Press. pp. 113–134. ISBN 0-521-61275-6. Zbl 1121.11063.
- Diamond, Harold G.; Halberstam, Heini (2008). A Higher-Dimensional Sieve Method: with Procedures for Computing Sieve Functions. Cambridge Tracts in Mathematics. 177. With William F. Galway. Cambridge: Cambridge University Press. ISBN 978-0-521-89487-6. Zbl 1207.11099.
- Greaves, George (2001). Sieves in number theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 43. Berlin: Springer-Verlag. ISBN 3-540-41647-1. Zbl 1003.11044.
- Halberstam, Heini; Richert, H.E. (1974). Sieve Methods. London Mathematical Society Monographs. 4. Academic Press. ISBN 0-12-318250-6. Zbl 0298.10026.
- Hooley, Christopher (1976). Applications of sieve methods to the theory of numbers. Cambridge Tracts in Mathematics. 70. Cambridge University Press. pp. 7–12. ISBN 0-521-20915-3. Zbl 0327.10044.
- Selberg, Atle (1947). "On an elementary method in the theory of primes". Norske Vid. Selsk. Forh. Trondheim. 19: 64–67. ISSN 0368-6302. Zbl 0041.01903.