The Goldston–Pintz–Yıldırım sieve (also called GPY sieve or GPY method) is a sieve method and variant of the Selberg sieve with generalized, multidimensional sieve weights.
The sieve led to a series of important breakthroughs in analytic number theory.
It is named after the mathematicians Dan Goldston, János Pintz and Cem Yıldırım.
[1] 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.
The sieve was then modified by Yitang Zhang in order to prove a finite bound on the smallest gap between two consecutive primes that is attained infinitely often.
[2] Later the sieve was again modified by James Maynard (who lowered the bound to
[3]) and by Terence Tao.
be admissible and consider the following sifting function where
is a weight function we derive later.
this sifting function counts the primes of the form
minus some threshold
has not so nice analytic properties one chooses rather the following sifting function Since
Next we have to choose the weight function
so that we can detect prime k-tuples.
A candidate for the weight function is the generalized von Mangoldt function which has the following property: if
This functions also detects factors which are proper prime powers, but this can be removed in applications with a negligible error.
is a prime k-tuple, then the function will not vanish.
The (classical) von Mangoldt function can be approximated with the truncated von Mangoldt function where
now no longer stands for the length of
Analogously we approximate
with For technical purposes we rather want to approximate tuples with primes in multiple components than solely prime tuples and introduce another parameter
or less distinct prime factors.
This leads to the final form Without this additional parameter
but by introducing this parameter one gets the more looser restriction
-dimensional sieve problem.
[4] The GPY sieve has the following form with Consider
In their paper, Goldston, Pintz and Yıldırım proved in two propositions that under suitable conditions two asymptotic formulas of the form and hold, where
are two singular series whose description we omit here.
Finally one can apply these results to
to derive the theorem by Goldston, Pintz and Yıldırım on infinitely many prime tuples whose distances are arbitrarily smaller than the average distance.