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The '''Goldston–Pintz–Yıldırım sieve''' (also called '''GPY sieve''' or '''GPY method''') is a [[sieve theory|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]].<ref name="GPY2005">{{cite journal|first1=Daniel A.|last1=Goldston|first2=János|last2=Pintz|first3=Cem Y.|last3=Yıldırım|title=Primes in Tuples I|journal=Annals of Mathematics|volume=170|number=2|year=2009|pages=819–862|doi=10.4007/annals.2009.170.819|doi-access=free}}</ref> 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 [[prime number|primes]] that is attained infinitely often.<ref>{{cite journal|first1=Yitang|last1=Zhang|title=Bounded gaps between primes|journal=Annals of Mathematics|volume=179|year=2014|pages=1121–1174|doi=10.4007/annals.2014.179.3.7|doi-access=free}}</ref>
Later the sieve was again modified by [[James Maynard (mathematician)|James Maynard]] (who lowered the bound to <math>600</math><ref>{{cite journal|first1=James|last1=Maynard|title=Small gaps between primes|journal=Annals of Mathematics|pages=383–413|volume=181|year=2015|number=1|doi=10.4007/annals.2015.181.1.7|arxiv=1311.4600}}</ref>) and by [[Terence Tao]].

== Goldston–Pintz–Yıldırım sieve ==
=== Notation ===
Fix a <math>k\in \N</math> and the following notation:
*<math>\mathbb{P}</math> is the set of prime numbers and <math>1_{\mathbb{P}}(n)</math> the characteristic function of that set,
*<math>\Lambda(n)</math> is the [[von Mangoldt function]],
*<math>\omega(n)</math> is the small [[prime omega function]] (which counts the distinct prime factors of <math>n</math>)
*<math>\mathcal{H}=\{h_1,\dots,h_k\}</math> is a set of distinct nonnegative integers <math>h_i\in\Z_+\cup \{0\}</math>.
*<math>\theta(n)</math> is another characteristic function of the primes defined as
::<math>\theta(n)=\begin{cases} \log(n) & \text{if }n\in \mathbb{P}\\ 0 & \text{else.}\end{cases}</math>
: Notice that <math>\theta(n)=\log((n-1)1_{\mathbb{P}}(n)+1)</math>.
For an <math>\mathcal{H}</math> we also define
*<math>\mathcal{H}(n):=(n+h_1,\dots,n+h_k)</math>,
*<math>P_{\mathcal{H}}(n):=(n+h_1)(n+h_2)\cdots (n+h_k)</math>
*<math>\nu_p(\mathcal{H})</math> is the amount of distinct residue classes of <math>\mathcal{H}</math> modulo <math>p</math>. For example <math>\nu_3(\{0,2,4\})=3</math> and <math>\nu_3(\{0,2\})=2</math> because <math>\{0,2,4\}\stackrel{\pmod{3}}{=}\{0,1,2\}</math> and <math>\{0,2\}\stackrel{\pmod{3}}{=}\{0,2\}</math>.
If <math>\nu_p(\mathcal{H})<k</math> for all <math>p\in \mathbb{P}</math>, then we call <math>\mathcal{H}</math> admissible.

=== Construction ===
Let <math>\mathcal{H}=\{h_1,\dots,h_k\}</math> be admissible and consider the following sifting function
:<math>\mathcal{S}(N,c;\mathcal{H}):=\sum\limits_{n=N+1}^{2N}\left(\sum\limits_{h_i\in \mathcal {H}}1_{\mathbb{P}}(n+h_i)-c\right)w(n)^2,\quad w(n)\in \R,\quad c>0,</math>
where <math>w(n)</math> is a weight function we derive later.

For each <math>n\in [N+1,2N]</math> this sifting function counts the primes of the form <math>n+h_i</math> minus some threshold <math>c</math>, so if <math>\mathcal{S}>0</math> then there exist some <math>n</math> such that at least <math>\lfloor c \rfloor +1</math> are prime numbers in <math>\mathcal{H}(n)</math>.

Since <math>1_{\mathbb{P}}(n)</math> has not so nice analytic properties one chooses rather the following sifting function
:<math>\mathcal{S}(N;\mathcal{H}):=\sum\limits_{n=N+1}^{2N}\left(\sum\limits_{h_i\in \mathcal{H}}\theta(n+h_i)-\log(3N)\right)w(n)^2.</math>
Since <math>\log(N)<\theta(n+h_i)<\log(2N)</math> and <math>c=\log(3n)</math>, we have <math>\mathcal{S}>0</math> only if there are at least two prime numbers <math>n+h_i</math> and <math>n+h_j</math>. Next we have to choose the weight function <math>w(n)</math> so that we can detect [[prime k-tuple]]s.

==== Derivation of the weights ====
A candidate for the weight function is the [[Von Mangoldt function#Generalized von Mangoldt function|generalized von Mangoldt function]]
:<math>\Lambda_k(n)=\sum\limits_{d\mid n}\mu(d)\left(\log\left(\frac{n}{d}\right)\right)^k,</math>
which has the following property: if <math>\omega(n)>k</math>, then <math>\Lambda_k(n)=0</math>. This functions also detects factors which are proper prime powers, but this can be removed in applications with a negligible error.<ref name="GPY2005" />{{rp|826}}

So if <math>\mathcal{H}(n)</math> is a prime k-tuple, then the function
:<math>\Lambda_k(n;\mathcal{H})=\frac{1}{k!}\Lambda_k(P_{\mathcal{H}}(n))</math>
will not vanish. The factor <math>1/k!</math> is just for computational purposes. The (classical) von Mangoldt function can be approximated with the ''truncated von Mangoldt function''
:<math>\Lambda(n)\approx \Lambda_R(n):=\sum\limits_{\begin{array}{c} d\mid n\\ d\leq R \end{array}}\mu(d)\log\left(\frac{R}{d}\right),</math>
where <math>R</math> now no longer stands for the length of <math>\mathcal{H}</math> but for the truncation position. Analogously we approximate <math>\Lambda_k(n;\mathcal{H})</math> with
:<math>\Lambda_R(n;\mathcal{H})=\frac{1}{k!}\sum\limits_{\begin{array}{c} d\mid P_{\mathcal{H}}(n)\\ d\leq R \end{array}}\mu(d)\left(\log\left(\frac{R}{d}\right)\right)^k</math>

For technical purposes we rather want to approximate tuples with primes in multiple components than solely prime tuples and introduce another parameter <math>0\leq \ell \leq k</math> so we can choose to have <math>k+\ell</math> or less distinct prime factors. This leads to the final form
:<math>\Lambda_R(n;\mathcal{H},\ell)=\frac{1}{(k+\ell)!}\sum\limits_{\begin{array}{c} d\mid P_{\mathcal{H}}(n)\\ d\leq R \end{array}}\mu(d)\left(\log\left(\frac{R}{d}\right)\right)^{k+\ell}</math>
Without this additional parameter <math>\ell</math> one has for a distinct <math>d=d_1d_2\cdots d_k</math> the restriction <math>d_1\leq R, d_2\leq R, \dots ,d_k\leq R</math> but by introducing this parameter one gets the more looser restriction <math>d_1d_2\dots d_k\leq R</math>.<ref name="GPY2005" />{{rp|827}}
So one has a <math>k+\ell</math>-dimensional sieve for a <math>k</math>-dimensional sieve problem.<ref>{{cite journal|title=Small gaps between primes or almost primes|first1=Daniel A.|last1=Goldston|first2=János|last2=Pintz|first3=Cem Y.|last3=Yıldırım|first4=Sidney W.|last4=Graham|year=2009|journal=Transactions of the American Mathematical Society |volume=361|number=10|arxiv=math/0506067|page=7}}</ref>

=== Goldston–Pintz–Yıldırım sieve ===
The GPY sieve has the following form
:<math>\mathcal{S}(N;\mathcal{H},\ell):=\sum\limits_{n=N+1}^{2N}\left(\sum\limits_{h_i\in \mathcal{H}}\theta(n+h_i)-\log(3N)\right)\Lambda_R(n;\mathcal{H},\ell)^2,\qquad |\mathcal{H}|=k</math>
with
:<math>\Lambda_R(n;\mathcal{H},\ell)=\frac{1}{(k+\ell)!}\sum\limits_{\begin{array}{c} d\mid P_{\mathcal{H}}(n)\\ d\leq R \end{array}}\mu(d)\left(\log\left(\frac{R}{d}\right)\right)^{k+\ell},\quad 0\leq \ell\leq k</math>.<ref name="GPY2005" />{{rp|827-829}}

=== Proof of the main theorem by Goldston, Pintz and Yıldırım ===
Consider <math>(\mathcal{H}_1,\ell_1, k_1)</math> and <math>(\mathcal{H}_2,\ell_2, k_2)</math> and <math>1\leq h_0\leq R</math>  and define <math>M:=k_1+k_2+\ell_1+\ell_2</math>. In their paper, Goldston, Pintz and Yıldırım proved in two propositions that under suitable conditions two asymptotic formulas of the form
:<math>\sum\limits_{n\leq N}\Lambda_R(n;\mathcal{H}_1,\ell_1)\Lambda_R(n;\mathcal{H}_2,\ell_2) = C_1\left(\mathcal{S}(\mathcal{H}^{i})+o_M(1)\right)N</math>
and
:<math>\sum\limits_{n\leq N}\Lambda_R(n;\mathcal{H}_1,\ell_1)\Lambda_R(n;\mathcal{H}_2,\ell_2)\theta(n+h_0)
= C_2\left(\mathcal{S}(\mathcal{H}^j)+o_M(1)\right)N</math>
hold, where <math>C_1,C_2</math> are two constants, <math>\mathcal{S}(\mathcal{H}^{i})</math> and <math>\mathcal{S}(\mathcal{H}^{j})</math> are two singular series whose description we omit here.

Finally one can apply these results to <math>\mathcal{S}</math> to derive the theorem by Goldston, Pintz and Yıldırım on infinitely many prime tuples whose distances are arbitrarily smaller than the average distance.<ref name="GPY2005" />{{rp|827-829}}

== References ==
{{Reflist}}

[[Category:Sieve theory]]