Selberg sieve

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.

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 A be a set of positive integers ≤ x and let P be a set of primes. For each p in P, let A_p denote the set of elements of A divisible by p and extend this to let A_d the intersection of the A_p 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

$S(A,P,z) = \left\vert A \setminus \bigcup_{p \mid P(z)} A_p \right\vert .$

We assume that |A_d| may be estimated by

$\left\vert A_d \right\vert = \frac{1}{f(d)} X %2B R_d .$

where f is a multiplicative function and X = |A|. Let the function g be obtained from f by Möbius inversion, that is

$g(n) = \sum_{d \mid n} \mu(d) f(n/d)$

$f(n) = \sum_{d \mid n} g(d)$

where μ is the Möbius function. Put

$V(z) = \sum_{\begin{smallmatrix}d < z \\ d \mid P(z)\end{smallmatrix}} \frac{\mu^2(d)}{g(d)} .$

Then

$S(A,P,z) \le \frac{X}{V(z)} %2B O\left({\sum_{\begin{smallmatrix} d_1,d_2 < z \\ d_1,d_2 \mid P(z)\end{smallmatrix}} \left\vert R_{[d_1,d_2]} \right\vert} \right) .$

It is often useful to estimate V(z) by the bound

$V(z) \ge \sum_{d \le z} \frac{1}{f(d)} . \,$

Applications

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) .

References

Alina Carmen Cojocaru; M. Ram Murty (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.
George Greaves (2001). Sieves in number theory. Springer-Verlag. ISBN 3-540-41647-1.
Heini Halberstam; H.E. Richert (1974). Sieve Methods. Academic Press. ISBN 0-12-318250-6.
Christopher Hooley (1976). Applications of sieve methods to the theory of numbers. Cambridge University Press. pp. 7–12. ISBN 0-521-20915-3.
Atle Selberg (1947). "On an elementary method in the theory of primes". Norske Vid. Selsk. Forh. Trondheim 19: 64–67.