Namespaces
Variants
Actions

Selberg sieve

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


2020 Mathematics Subject Classification: Primary: 11N35 [MSN][ZBL]

Selberg method

A special, and at the same time fairly universal, sieve method created by A. Selberg [1]. The Selberg sieve enables one to obtain a good upper bound of the sifting function $S(A;P,z)$, which denotes the number of elements of a set $A$ of integers that are not divisible by prime numbers $p < z$ and that belong to a certain set $P$ of prime numbers.

Let $P(z) = \prod_{p<z\,;\,p \in P} p$. The Selberg method is based on the obvious inequality $$ \begin{equation}\label{e:1} S(A;P,z) \le \sum_{a \in A} \left({ \sum_{d | a\,;\,d | P(z)} \lambda_d }\right)^2 \end{equation} $$ which holds for $\lambda_1 = 1$ and arbitrary real numbers $\lambda_d$ ($d \ge 2$). Selberg's idea consists of the following: Set $\lambda_d = 0$ for $d \ge z$, and minimize the right-hand side of \ref{e:1} by a suitable choice of the remaining numbers $\lambda_d$ ($2 \le d < z$).

When combined with other sieve methods, the Selberg sieve enables one to obtain lower bounds that are particularly powerful when used with weight functions.

References

[1] A. Selberg, "On an elementary method in the theory of primes" Norsk. Vid. Selsk. Forh. , 19 : 18 (1947) pp. 64–67 Zbl 0041.01903
[2] K. Prachar, "Primzahlverteilung", Die Grundlehren der Mathematischen Wissenschaften 91, Springer (1957) Zbl 0080.25901
[3] H. Halberstam, H.-E. Richert, "Sieve methods", London Mathematical Society Monographs 4, Academic Press (1974) Zbl 0298.10026
How to Cite This Entry:
Selberg sieve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Selberg_sieve&oldid=34721
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article