Notation

Fix a ${\displaystyle k\in \mathbb {N} }$ and the following notation:

• ${\displaystyle \mathbb {P} }$ is the set of prime numbers and ${\displaystyle 1_{\mathbb {P} }(n)}$ the characteristic function of that set,
• ${\displaystyle \Lambda (n)}$ is the von Mangoldt function,
• ${\displaystyle \omega (n)}$ is the small prime omega function (which counts the distinct prime factors of ${\displaystyle n}$)
• ${\displaystyle {\mathcal {H}}=\{h_{1},\dots ,h_{k}\}}$ is a set of distinct nonnegative integers ${\displaystyle h_{i}\in \mathbb {Z} _{+}\cup \{0\}}$.
• ${\displaystyle \theta (n)}$ is another characteristic function of the primes defined as

${\displaystyle \theta (n)={\begin{cases}\log(n)&{\text{if }}n\in \mathbb {P} \\0&{\text{else.}}\end{cases}}}$

Notice that ${\displaystyle \theta (n)=\log((n-1)1_{\mathbb {P} }(n)+1)}$.

For an ${\displaystyle {\mathcal {H}}}$ we also define

• ${\displaystyle {\mathcal {H}}(n):=(n+h_{1},\dots ,n+h_{k})}$,
• ${\displaystyle P_{\mathcal {H}}(n):=(n+h_{1})(n+h_{2})\cdots (n+h_{k})}$
• ${\displaystyle \nu _{p}({\mathcal {H}})}$ is the amount of distinct residue classes of ${\displaystyle {\mathcal {H}}}$ modulo ${\displaystyle p}$. For example ${\displaystyle \nu _{3}(\{0,2,4\})=3}$ and ${\displaystyle \nu _{3}(\{0,2\})=2}$ because ${\displaystyle \{0,2,4\}{\stackrel {\pmod {3}}{=}}\{0,1,2\}}$ and ${\displaystyle \{0,2\}{\stackrel {\pmod {3}}{=}}\{0,2\}}$.

If ${\displaystyle \nu _{p}({\mathcal {H}})<k}$ for all ${\displaystyle p\in \mathbb {P} }$, then we call ${\displaystyle {\mathcal {H}}}$ admissible.

Construction

Let ${\displaystyle {\mathcal {H}}=\{h_{1},\dots ,h_{k}\}}$ be admissible and consider the following sifting function

${\displaystyle {\mathcal {S}}(N,c;{\mathcal {H}}):=\sum \limits _{n=N+1}^{2N}\left(\sum \limits _{h_{i}\in {\mathcal {H}}}1_{\mathbb {P} }(n+h_{i})-c\right)w(n)^{2},\quad w(n)\in \mathbb {R} ,\quad c>0.}$

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

Since ${\displaystyle 1_{\mathbb {P} }(n)}$ has not so nice analytic properties one chooses rather the following sifting function

${\displaystyle {\mathcal {S}}(N;{\mathcal {H}}):=\sum \limits _{n=N+1}^{2N}\left(\sum \limits _{h_{i}\in {\mathcal {H}}}\theta (n+h_{i})-\log(3N)\right)w(n)^{2}.}$

Since ${\displaystyle \log(N)<\theta (n+h_{i})<\log(2N)}$ and ${\displaystyle c=\log(3n)}$, we have ${\displaystyle {\mathcal {S}}>0}$ only if there are at least two prime numbers ${\displaystyle n+h_{i}}$ and ${\displaystyle n+h_{j}}$. Next we have to choose the weight function ${\displaystyle w(n)}$ so that we can detect prime k-tuples.

Goldston-Pintz-Yıldırım sieve

The GPY sieve has the following form

${\displaystyle {\mathcal {S}}(N;{\mathcal {H}},\ell ):=\sum \limits _{n=N+1}^{2N}\left(\sum \limits _{h_{i}\in {\mathcal {H}}}\theta (n+h_{i})-\log(3N)\right)\Lambda _{R}(n;{\mathcal {H}},\ell )^{2},\qquad |{\mathcal {H}}|=k}$

with

${\displaystyle \Lambda _{R}(n;{\mathcal {H}},\ell )={\frac {1}{(k+\ell )!}}\sum \limits _{\begin{array}{c}d\mid P_{\mathcal {H}}(n)\\d\leq R\end{array}}\mu (d)\left(\log \left({\frac {R}{d}}\right)\right)^{k+\ell },\quad 0\leq \ell \leq k}$.^{[1]: 827–829}

Proof of the main theorem by Goldston, Pintz and Yıldırım

Consider ${\displaystyle ({\mathcal {H}}_{1},\ell _{1},k_{1})}$ and ${\displaystyle ({\mathcal {H}}_{2},\ell _{2},k_{2})}$ and ${\displaystyle 1\leq h_{0}\leq R}$ and define ${\displaystyle M:=k_{1}+k_{2}+\ell _{1}+\ell _{2}}$. In their paper, Goldston, Pintz and Yıldırım proved in two propositions that under suitable conditions two asymptotic formulas of the form

${\displaystyle \sum \limits _{n\leq N}\Lambda _{R}(n;{\mathcal {H}}_{1},\ell _{1})\Lambda _{R}(n;{\mathcal {H}}_{2},\ell _{2})=C_{1}\left({\mathcal {S}}({\mathcal {H}}^{i})+o_{M}(1)\right)N}$

and

${\displaystyle \sum \limits _{n\leq N}\Lambda _{R}(n;{\mathcal {H}}_{1},\ell _{1})\Lambda _{R}(n;{\mathcal {H}}_{2},\ell _{2})\theta (n+h_{0})=C_{2}\left({\mathcal {S}}({\mathcal {H}}^{j})+o_{M}(1)\right)N}$

hold, where ${\displaystyle C_{1},C_{2}}$ are two constants, ${\displaystyle {\mathcal {S}}({\mathcal {H}}^{i})}$ and ${\displaystyle {\mathcal {S}}({\mathcal {H}}^{j})}$ are two singular series whose description we omit here.

Finally one can apply these results to ${\displaystyle {\mathcal {S}}}$ 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}