HOME

TheInfoList



OR:

The Goldston–Pintz–Yıldırım sieve (also called GPY sieve or GPY method) is a sieve method and variant of the
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 ...
with generalized, multidimensional sieve weights. The sieve led to a series of important breakthroughs in
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...
. It is named after the mathematicians Dan Goldston, János Pintz and Cem Yıldırım. They used it in 2005 to show that there are infinitely many prime tuples whose distances are arbitrarily smaller than the average distance that follows from the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by p ...
. The sieve was then modified by
Yitang Zhang Yitang Zhang (; born February 5, 1955) is a Chinese-American mathematician primarily working on number theory and a professor of mathematics at the University of California, Santa Barbara since 2015. Previously working at the University of New ...
in order to prove a finite bound on the smallest gap between two consecutive primes that is attained infinitely often. Later the sieve was again modified by James Maynard (who lowered the bound to 600) and by Terence Tao.


Goldston–Pintz–Yıldırım sieve


Notation

Fix a k\in \N and the following notation: *\mathbb is the set of prime numbers and 1_(n) the characteristic function of that set, *\Lambda(n) is the
von Mangoldt function In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive. Definition The von Mang ...
, *\omega(n) is the small
prime omega function In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n. The number of ''distinct'' prime factors is assigned to \omega(n) (little omega), while \Omega(n) (big omega) counts the '' ...
(which counts the distinct prime factors of n) *\mathcal=\ is a set of distinct nonnegative integers h_i\in\Z_+\cup \. *\theta(n) is another characteristic function of the primes defined as ::\theta(n)=\begin \log(n) & \textn\in \mathbb\\ 0 & \text\end : Notice that \theta(n)=\log((n-1)1_(n)+1). For an \mathcal we also define *\mathcal(n):=(n+h_1,\dots,n+h_k), *P_(n):=(n+h_1)(n+h_2)\cdots (n+h_k) *\nu_p(\mathcal) is the amount of distinct residue classes of \mathcal modulo p. For example \nu_3(\)=3 and \nu_3(\)=2 because \\stackrel\ and \\stackrel\. If \nu_p(\mathcal) for all p\in \mathbb, then we call \mathcal admissible.


Construction

Let \mathcal=\ be admissible and consider the following sifting function :\mathcal(N,c;\mathcal):=\sum\limits_^\left(\sum\limits_1_(n+h_i)-c\right)w(n)^2,\quad w(n)\in \R,\quad c>0, where w(n) is a weight function we derive later. For each n\in +1,2N/math> this sifting function counts the primes of the form n+h_i minus some threshold c, so if \mathcal>0 then there exist some n such that at least \lfloor c \rfloor +1 are prime numbers in \mathcal(n). Since 1_(n) has not so nice analytic properties one chooses rather the following sifting function :\mathcal(N;\mathcal):=\sum\limits_^\left(\sum\limits_\theta(n+h_i)-\log(3N)\right)w(n)^2. Since \log(N)<\theta(n+h_i)<\log(2N) and c=\log(3n), we have \mathcal>0 only if there are at least two prime numbers n+h_i and n+h_j. Next we have to choose the weight function w(n) so that we can detect prime k-tuples.


Derivation of the weights

A candidate for the weight function is the generalized von Mangoldt function :\Lambda_k(n)=\sum\limits_\mu(d)\left(\log\left(\frac\right)\right)^k, which has the following property: if \omega(n)>k, then \Lambda_k(n)=0. This functions also detects factors which are proper prime powers, but this can be removed in applications with a negligible error. So if \mathcal(n) is a prime k-tuple, then the function :\Lambda_k(n;\mathcal)=\frac\Lambda_k(P_(n)) will not vanish. The factor 1/k! is just for computational purposes. The (classical) von Mangoldt function can be approximated with the ''truncated von Mangoldt function'' :\Lambda(n)\approx \Lambda_R(n):=\sum\limits_\mu(d)\log\left(\frac\right), where R now no longer stands for the length of \mathcal but for the truncation position. Analogously we approximate \Lambda_k(n;\mathcal) with :\Lambda_R(n;\mathcal)=\frac\sum\limits_\mu(d)\left(\log\left(\frac\right)\right)^k For technical purposes we rather want to approximate tuples with primes in multiple components than solely prime tuples and introduce another parameter 0\leq \ell \leq k so we can choose to have k+\ell or less distinct prime factors. This leads to the final form :\Lambda_R(n;\mathcal,\ell)=\frac\sum\limits_\mu(d)\left(\log\left(\frac\right)\right)^ Without this additional parameter \ell one has for a distinct d=d_1d_2\cdots d_k the restriction d_1\leq R, d_2\leq R, \dots ,d_k\leq R but by introducing this parameter one gets the more looser restriction d_1d_2\dots d_k\leq R. So one has a k+\ell-dimensional sieve for a k-dimensional sieve problem.


Goldston–Pintz–Yıldırım sieve

The GPY sieve has the following form :\mathcal(N;\mathcal,\ell):=\sum\limits_^\left(\sum\limits_\theta(n+h_i)-\log(3N)\right)\Lambda_R(n;\mathcal,\ell)^2,\qquad , \mathcal, =k with :\Lambda_R(n;\mathcal,\ell)=\frac\sum\limits_\mu(d)\left(\log\left(\frac\right)\right)^,\quad 0\leq \ell\leq k.


Proof of the main theorem by Goldston, Pintz and Yıldırım

Consider (\mathcal_1,\ell_1, k_1) and (\mathcal_2,\ell_2, k_2) and 1\leq h_0\leq R and define M:=k_1+k_2+\ell_1+\ell_2. In their paper, Goldston, Pintz and Yıldırım proved in two propositions that under suitable conditions two asymptotic formulas of the form :\sum\limits_\Lambda_R(n;\mathcal_1,\ell_1)\Lambda_R(n;\mathcal_2,\ell_2) = C_1\left(\mathcal(\mathcal^)+o_M(1)\right)N and :\sum\limits_\Lambda_R(n;\mathcal_1,\ell_1)\Lambda_R(n;\mathcal_2,\ell_2)\theta(n+h_0) = C_2\left(\mathcal(\mathcal^j)+o_M(1)\right)N hold, where C_1,C_2 are two constants, \mathcal(\mathcal^) and \mathcal(\mathcal^) are two singular series whose description we omit here. Finally one can apply these results to \mathcal to derive the theorem by Goldston, Pintz and Yıldırım on infinitely many prime tuples whose distances are arbitrarily smaller than the average distance.


References

{{Reflist Sieve theory