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
[edit]
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
![{\displaystyle \theta (n)={\begin{cases}\log(n)&{\text{if }}n\in \mathbb {P} \\0&{\text{else.}}\end{cases}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/11864fb526514e09516ac39b0b3a123c444ee978)
- Notice that
.
For an
we also define
,
![{\displaystyle P_{\mathcal {H}}(n):=(n+h_{1})(n+h_{2})\cdots (n+h_{k})}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/59ccf02e4db640568d5e7e64eee0c8af22edd87f)
is the amount of distinct residue classes of
modulo
. For example
and
because
and
.
If
for all
, then we call
admissible.
Let
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,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/c9453d8f59f76b43e0fcf0ee841ff32d128ba7cd)
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
![{\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}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/74d057386e311417f2d78c3bbf8eeedc1be52277)
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
[edit]
A candidate for the weight function is the generalized von Mangoldt function
![{\displaystyle \Lambda _{k}(n)=\sum \limits _{d\mid n}\mu (d)\left(\log \left({\frac {n}{d}}\right)\right)^{k},}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/7fd5f14b48db5a99a94fc8b99b04f5429505963c)
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
![{\displaystyle \Lambda _{k}(n;{\mathcal {H}})={\frac {1}{k!}}\Lambda _{k}(P_{\mathcal {H}}(n))}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/225dac6a72a700ec131d599c5272d817eaad632c)
will not vanish. The factor
is just for computational purposes. The (classical) von Mangoldt function can be approximated with the truncated von Mangoldt function
![{\displaystyle \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),}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/01a3c583f9b8d8df7d458caee18fbd6086b50bd7)
where
now no longer stands for the length of
but for the truncation position. Analogously we approximate
with
![{\displaystyle \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}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/dcd137166d85f4b82fb3e7dda7c1935fc377ce00)
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
![{\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 }}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/980e516762b45eda2069e5b3afe9b367fadc48b1)
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
[edit]
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}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/60799481656076740b4ed97f870d2d4a94389ec6)
with
.[1]: 827–829
Proof of the main theorem by Goldston, Pintz and Yıldırım
[edit]
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
![{\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}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/93f7e94091e8d09068f68dde001748c9d8928c6c)
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}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/80a7acb5951276d9b578ee5e8d3ba55ead864b58)
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