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
) and by
Terence Tao.
Goldston–Pintz–Yıldırım sieve
Notation
Fix a
and the following notation:
*
is the set of prime numbers and
the characteristic function of that set,
*
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 ...
,
*
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
)
*
is a set of distinct nonnegative integers
.
*
is another characteristic function of the primes defined as
::
: Notice that
.
For an
we also define
*
,
*
*
is the amount of distinct residue classes of
modulo
. For example
and
because
and
.
If