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

[math]\displaystyle{ S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert . }[/math]

We assume that |A_d| may be estimated by

[math]\displaystyle{ \left\vert A_d \right\vert = \frac{1}{f(d)} X + R_d . }[/math]

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

[math]\displaystyle{ g(n) = \sum_{d \mid n} \mu(d) f(n/d) }[/math]

[math]\displaystyle{ f(n) = \sum_{d \mid n} g(d) }[/math]

where μ is the Möbius function. Put

[math]\displaystyle{ V(z) = \sum_{\begin{smallmatrix}d \lt z \\ d \mid P(z)\end{smallmatrix}} \frac{\mu^2(d)}{g(d)} . }[/math]

Then

[math]\displaystyle{ S(A,P,z) \le \frac{X}{V(z)} + O\left({\sum_{\begin{smallmatrix} d_1,d_2 \lt z \\ d_1,d_2 \mid P(z)\end{smallmatrix}} \left\vert R_{[d_1,d_2]} \right\vert} \right) . }[/math]

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

[math]\displaystyle{ V(z) \ge \sum_{d \le z} \frac{1}{f(d)} . \, }[/math]

Applications

• The Brun-Titchmarsh theorem on the number of primes in an 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. 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.