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.

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 $$\le x$$ and let $$P$$ be a set of primes. Let $$A_d$$ denote the set of elements of $$A$$ divisible by $$d$$ when $$d$$ is a product of distinct primes from $$P$$. Further let $$A_1$$ denote $$A$$ itself. Let $$z$$ be a positive real number and $$P(z)$$ denote the product of the primes in $$P$$ which are $$\le z$$. 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 |Ad| may be estimated by


 * $$ \left\vert A_d \right\vert = \frac{1}{f(d)} X + R_d . $$

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 &mu; is the Möbius function. Put


 * $$ V(z) = \sum_{\begin{smallmatrix}d < z \\ d \mid P(z)\end{smallmatrix}} \frac{1}{g(d)}. $$

Then


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

where $$[d_1,d_2]$$ denotes the least common multiple of $$d_1$$ and $$d_2$$. 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 &le; x such that n is coprime to &phi;(n) is asymptotic to e&minus;&gamma; x / log log log (x).