Goldston-Pintz-Yıldırım sieve

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.

Goldston-Pintz-Yıldırım sieve

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Notation

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Fix a   and the following notation:

  •   is the set of prime numbers and   the characteristic function of that set,
  •   is the von Mangoldt function,
  •   is the small prime omega function (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   for all  , then we call   admissible.

Construction

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Let   be admissible and consider the following sifting function

 

where   is a weight function we derive later.

For each   this sifting function counts the primes of the form   minus some threshold  , so if   then there exist some   such that at least   are prime numbers in  .

Since   has not so nice analytic properties one chooses rather the following sifting function

 

Since   and  , we have   only if there are at least two prime numbers   and  . Next we have to choose the weight function   so that we can detect prime k-tuples.

Derivation of the weights

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A candidate for the weight function is the generalized von Mangoldt function

 

which has the following property: if  , then  . This functions also detects factors which are proper prime powers, but this can be removed in applications with a negligible error.[1]: 826 

So if   is a prime k-tuple, then the function

 

will not vanish. The factor   is just for computational purposes. The (classical) von Mangoldt function can be approximated with the truncated von Mangoldt function

 

where   now no longer stands for the length of   but for the truncation position. 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   so we can choose to have   or less distinct prime factors. This leads to the final form

 

Without this additional parameter   one has for a distinct   the restriction   but by introducing this parameter one gets the more looser restriction  .[1]: 827  So one has a  -dimensional sieve for a  -dimensional sieve problem.[4]

Goldston-Pintz-Yıldırım sieve

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The GPY sieve has the following form

 

with

 .[1]: 827–829 

Proof of the main theorem by Goldston, Pintz and Yıldırım

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Consider   and   and   and define  . 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 constants,   and   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.[1]: 827–829 

References

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  1. ^ a b c d e Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. (2009). "Primes in Tuples I". Annals of Mathematics. 170 (2): 819–862. doi:10.4007/annals.2009.170.819.
  2. ^ Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179: 1121–1174. doi:10.4007/annals.2014.179.3.7.
  3. ^ Maynard, James (2015). "Small gaps between primes". Annals of Mathematics. 181 (1): 383–413. arXiv:1311.4600. doi:10.4007/annals.2015.181.1.7.
  4. ^ Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y.; Graham, Sidney W. (2009). "Small gaps between primes or almost primes". Transactions of the American Mathematical Society. 361 (10): 7. arXiv:math/0506067.