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 ${\displaystyle A}$ be a set of positive integers ${\displaystyle \leq x}$ and let ${\displaystyle P}$ be a set of primes. Let ${\displaystyle A_{d}}$ denote the set of elements of ${\displaystyle A}$ divisible by ${\displaystyle d}$ when ${\displaystyle d}$ is a product of distinct primes from ${\displaystyle P}$. Further let ${\displaystyle A_{1}}$ denote ${\displaystyle A}$ itself. Let ${\displaystyle z}$ be a positive real number and ${\displaystyle P(z)}$ denote the product of the primes in ${\displaystyle P}$ which are ${\displaystyle \leq z}$. The object of the sieve is to estimate

${\displaystyle S(A,P,z)=\left\vert A\setminus \bigcup _{p\mid P(z)}A_{p}\right\vert .}$

We assume that |A_d| may be estimated by

${\displaystyle \left\vert A_{d}\right\vert ={\frac {1}{f(d)}}X+R_{d}.}$

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

${\displaystyle g(n)=\sum _{d\mid n}\mu (d)f(n/d)}$

${\displaystyle f(n)=\sum _{d\mid n}g(d)}$

where μ is the Möbius function. Put

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

Then

${\displaystyle S(A,P,z)\leq {\frac {X}{V(z)}}+O\left({\sum _{\begin{smallmatrix}d_{1},d_{2}<z\\d_{1},d_{2}\mid P(z)\end{smallmatrix}}\left\vert R_{[d_{1},d_{2}]}\right\vert }\right)}$

where ${\displaystyle [d_{1},d_{2}]}$ denotes the least common multiple of ${\displaystyle d_{1}}$ and ${\displaystyle d_{2}}$. It is often useful to estimate ${\displaystyle V(z)}$ by the bound

${\displaystyle V(z)\geq \sum _{d\leq z}{\frac {1}{f(d)}}.\,}$

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

• Cojocaru, Alina Carmen; Murty, M. Ram (2005). An introduction to sieve methods and their applications. London Mathematical Society Student Texts. Vol. 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. Vol. 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. Vol. 43. Berlin: Springer-Verlag. ISBN 3-540-41647-1. Zbl 1003.11044.
• Halberstam, Heini; Richert, H.E. (1974). Sieve Methods. London Mathematical Society Monographs. Vol. 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. Vol. 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.