[Turkmath:9046] The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang

yilmaz akyildiz yilmaz.akyildiz at gmail.com
3 Haz 2013 Pzt 20:01:38 EEST


       <http://terrytao.wordpress.com/author/teorth/>  The prime tuples
conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim,
Motohashi-Pintz, and
Zhang<http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/>
by
Terence Tao <http://terrytao.wordpress.com/author/teorth/>

Suppose one is given a [image: {k_0}]-tuple [image: {{\mathcal H} =
(h_1,\ldots,h_{k_0})}] of [image: {k_0}] distinct integers for some [image:
{k_0 \geq 1}], arranged in increasing order. When is it possible to find
infinitely many translates [image: {n + {\mathcal H}
=(n+h_1,\ldots,n+h_{k_0})}] of [image: {{\mathcal H}}] which consists
entirely of primes? The case [image: {k_0=1}] is just Euclid's
theorem<http://en.wikipedia.org/wiki/Euclid%27s_theorem>on the
infinitude of primes, but the case [image:
{k_0=2}] is already open in general, with the [image: {{\mathcal H} =
(0,2)}] case being the notorious twin prime
conjecture<http://en.wikipedia.org/wiki/Twin_prime>.


On the other hand, there are some tuples [image: {{\mathcal H}}] for which
one can easily answer the above question in the negative. For instance, the
only translate of [image: {(0,1)}] that consists entirely of primes is [image:
{(2,3)}], basically because each translate of [image: {(0,1)}] must contain
an even number, and the only even prime is [image: {2}]. More generally, if
there is a prime [image: {p}] such that [image: {{\mathcal H}}] meets each
of the [image: {p}] residue classes [image: {0 \hbox{ mod } p, 1 \hbox{ mod
} p, \ldots, p-1 \hbox{ mod } p}], then every translate of [image:
{{\mathcal H}}] contains at least one multiple of [image: {p}]; since [image:
{p}] is the only multiple of [image: {p}] that is prime, this shows that
there are only finitely many translates of [image: {{\mathcal H}}] that
consist entirely of primes.

To avoid this obstruction, let us call a [image: {k_0}]-tuple [image:
{{\mathcal H}}] *admissible* if it avoids at least one residue class [image:
{\hbox{ mod } p}] for each prime [image: {p}]. It is easy to check for
admissibility in practice, since a [image: {k_0}]-tuple is automatically
admissible in every prime [image: {p}] larger than [image: {k_0}], so one
only needs to check a finite number of primes in order to decide on the
admissibility of a given tuple. For instance, [image: {(0,2)}] or [image:
{(0,2,6)}] are admissible, but [image: {(0,2,4)}] is not (because it covers
all the residue classes modulo [image: {3}]). We then have the famous
Hardy-Littlewood
prime tuples conjecture <http://en.wikipedia.org/wiki/Prime_k-tuple>:

*Conjecture 1 (Prime tuples conjecture, qualitative form)* If [image:
{{\mathcal H}}] is an admissible [image: {k_0}]-tuple, then there exists
infinitely many translates of [image: {{\mathcal H}}] that consist entirely
of primes.

 This conjecture is extremely difficult (containing the twin prime
conjecture, for instance, as a special case), and in fact there is
*no*explicitly known example of an admissible [image:
{k_0}]-tuple with [image: {k_0 \geq 2}] for which we can verify this
conjecture (although, thanks to the recent work of
Zhang<http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf>,
we know that [image: {(0,d)}] is admissible for some [image: {0 < d <
70,000,000}], even if we can't yet say what the precise value of [image:
{d}] is).

Actually, Hardy and Littlewood conjectured a more precise version of
Conjecture 1 <#13f0af938d02c8f9_tuples>. Given an admissible [image:
{k_0}]-tuple
[image: {{\mathcal H}}], and for each prime [image: {p}], let [image:
{\nu_p = \nu_p({\mathcal H}) := |{\mathcal H} \hbox{ mod } p|}] denote the
number of residue classes modulo [image: {p}] that [image: {{\mathcal
H}}]meets; thus we have [image:
{1 \leq \nu_p \leq p-1}] for all [image: {p}] by admissibility, and
also [image:
{\nu_p = k}] for all [image: {p>k_0}]. We then define the *singular
series* [image:
{{\mathfrak H} = {\mathfrak G}({\mathcal H})}] associated to [image:
{{\mathfrak H}}] by the formula

[image: \displaystyle {\mathfrak G} := \prod_{p \in {\mathcal P}}
\frac{1-\frac{\nu_p}{p}}{(1-\frac{1}{p})^{k_0}}]

where [image: {{\mathcal P} = \{2,3,5,\ldots\}}] is the set of primes; by
the previous discussion we see that the infinite product in [image:
{{\mathfrak G}}] converges to a finite non-zero number.

We will also need some asymptotic notation (in the spirit of "cheap
nonstandard analysis<http://terrytao.wordpress.com/2012/04/02/a-cheap-version-of-nonstandard-analysis/>").
We will need a parameter [image: {x}] that one should think of going to
infinity. Some mathematical objects (such as [image: {{\mathcal H}}]
and [image:
{k}]) will be independent of [image: {x}] and referred to as *fixed*; but
unless otherwise specified we allow all mathematical objects under
consideration to depend on [image: {x}]. If [image: {X}] and [image:
{Y}]are two such quantities, we say that [image:
{X = O(Y)}] if one has [image: {|X| \leq CY}] for some fixed [image: {C}],
and [image: {X = o(Y)}] if one has [image: {|X| \leq c(x) Y}] for some
function [image: {c(x)}] of [image: {x}] (and of any fixed parameters
present) that goes to zero as [image: {x \rightarrow \infty}] (for each
choice of fixed parameters).

*Conjecture 2 (Prime tuples conjecture, quantitative form)* Let [image:
{k_0 \geq 1}] be a fixed natural number, and let [image: {{\mathcal H}}] be
a fixed admissible [image: {k_0}]-tuple. Then the number of natural
numbers [image:
{n < x}] such that [image: {n+{\mathcal H}}] consists entirely of
primes is [image:
{({\mathfrak G} + o(1)) \frac{x}{\log^{k_0} x}}].

 Thus, for instance, if Conjecture 2
<#13f0af938d02c8f9_tuples-quant>holds, then the number of twin primes
less than [image:
{x}] should equal [image: {(2 \Pi_2 + o(1)) \frac{x}{\log^2 x}}], where [image:
{\Pi_2}] is the twin prime
constant<http://mathworld.wolfram.com/TwinPrimesConstant.html>

[image: \displaystyle \Pi_2 := \prod_{p \in {\mathcal P}: p>2} (1 -
\frac{1}{(p-1)^2}) = 0.6601618\ldots.]

As this conjecture is stronger than Conjecture 1 <#13f0af938d02c8f9_tuples>,
it is of course open. However there are a number of partial results on this
conjecture. For instance, this conjecture is known to be true if one
introduces some additional averaging in [image: {{\mathcal H}}]; see for
instance this previous
post<http://terrytao.wordpress.com/2008/11/18/marker-lectures-ii-linear-equations-in-primes/>.
>From the methods of sieve theory <http://en.wikipedia.org/wiki/Sieve_theory>,
one can obtain an *upper bound* of [image: {(C_{k_0} {\mathfrak G} + o(1))
\frac{x}{\log^{k_0} x}}] for the number of [image: {n < x}] with [image: {n
+ {\mathcal H}}] all prime, where [image: {C_{k_0}}] depends only on [image:
{k_0}]. Sieve theory can also give analogues of Conjecture
2<#13f0af938d02c8f9_tuples-quant>if the primes are replaced by a
suitable notion of almost
prime <http://en.wikipedia.org/wiki/Almost_prime> (or more precisely, by a
weight function concentrated on almost primes).

Another type of partial result towards Conjectures 1<#13f0af938d02c8f9_tuples>,
2 <#13f0af938d02c8f9_tuples-quant> come from the results of
Goldston-Pintz-Yildirim <http://www.ams.org/mathscinet-getitem?mr=2552109>,
Motohashi-Pintz <http://www.ams.org/mathscinet-getitem?mr=2414788>, and of
Zhang <http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf>.
For each [image: {k_0>2}], let [image: {P[k_0]}] denote the following
assertion:

*Conjecture 3* ([image: {P[k_0]}]) Let [image: {{\mathcal H}}] be a fixed
admissible [image: {k_0}]-tuple. Then there are infinitely many
translates [image:
{n+{\mathcal H}}] of [image: {{\mathcal H}}] which contain *at least
two*primes.

 This conjecture gets harder as [image: {k_0}] gets smaller. Note for
instance that [image: {P[2]}] would imply all the [image: {k_0=2}] cases of
Conjecture 1 <#13f0af938d02c8f9_tuples>, including the twin prime
conjecture. More generally, if one knew [image: {P[k_0]}] for some [image:
{k_0}], then one would immediately conclude that there are an infinite
number of pairs of consecutive primes of separation at most [image:
{H(k_0)}], where [image: {H(k_0)}] is the minimal diameter [image:
{h_{k_0}-h_1}] amongst all admissible [image: {k_0}]-tuples [image:
{{\mathcal H}}]. Values of [image: {H(k_0)}] for small [image: {k_0}] can
be found at this link <http://www.opertech.com/primes/k-tuples.html>
(with [image:
{H(k_0)}] denoted [image: {w}] in that page). For large [image: {k_0}], the
best upper bounds on [image: {H(k_0)}] have been found by using
admissible [image:
{k_0}]-tuples [image: {{\mathcal H}}] of the form

[image: \displaystyle {\mathcal H} = \{ - p_{m+\lfloor k_0/2\rfloor - 1},
\ldots, - p_{m+1}, -1, +1, p_{m+1}, \ldots, p_{m+\lfloor (k_0+1)/2\rfloor -
1} \}]

where [image: {p_n}] denotes the [image: {n^{th}}] prime and [image:
{m}]is a parameter to be optimised over (in practice it is an order of
magnitude or two smaller than [image: {k_0}]); see this blog post for
details<http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/>.
The upshot is that one can bound [image: {H(k_0)}] for large [image:
{k_0}]by a quantity slightly smaller than [image:
{k_0 \log k_0}] (and the large sieve inequality shows that this is sharp up
to a factor of two, see e.g. this previous
post<http://terrytao.wordpress.com/2011/12/31/montgomerys-uncertainty-principle/>for
more discussion).

In a key breakthrough, Goldston, Pintz, and
Yildirim<http://www.ams.org/mathscinet-getitem?mr=2552109>were able to
establish the following conditional result a few years ago:

*Theorem 4 (Goldston-Pintz-Yildirim)* Suppose that the Elliott-Halberstam
conjecture <http://en.wikipedia.org/wiki/Elliott-Halberstam_conjecture> [image:
{EH[\theta]}] is true for some [image: {1/2 < \theta < 1}]. Then [image:
{P[k_0]}] is true for some finite [image: {k_0}]. In particular, this
establishes an infinite number of pairs of consecutive primes of
separation [image:
{O(1)}].

 The dependence of constants between [image: {k_0}] and [image:
{\theta}]given by the Goldston-Pintz-Yildirim argument is basically of
the form [image:
{k_0 \sim (\theta-1/2)^2}].

Unfortunately, the Elliott-Halberstam conjecture (which we will state
properly below) is only known for [image: {\theta<1/2}], an important
result known as the Bombieri-Vinogradov
theorem<http://en.wikipedia.org/wiki/Bombieri%E2%80%93Vinogradov_theorem>.
If one uses the Bombieri-Vinogradov theorem instead of the
Elliott-Halberstam conjecture, Goldston, Pintz, and Yildirim were still
able to show the highly non-trivial result that there were infinitely many
pairs [image: {p_{n+1},p_n}] of consecutive primes with [image:
{(p_{n+1}-p_n) / \log p_n \rightarrow 0}] (actually they showed more than
this; see e.g. this survey of
Soundarajan<http://www.ams.org/mathscinet-getitem?mr=2265008>for
details).

Actually, the full strength of the Elliott-Halberstam conjecture is not
needed for these results. There is a technical specialisation of the
Elliott-Halberstam conjecture which does not presently have a commonly
accepted name; I will call it the *Motohashi-Pintz-Zhang conjecture* [image:
{MPZ[\varpi]}] in this post, where [image: {0 < \varpi < 1/4}] is a
parameter. We will define this conjecture more precisely later, but let us
remark for now that [image: {MPZ[\varpi]}] is a consequence of [image:
{EH[\frac{1}{2}+2\varpi]}].

We then have the following two theorems. Firstly, we have the following
strengthening of Theorem 4 <#13f0af938d02c8f9_gpy>:

*Theorem 5 (Motohashi-Pintz-Zhang)* Suppose that [image: {MPZ[\varpi]}] is
true for some [image: {0 < \varpi < 1/4}]. Then [image: {P[k_0]}] is true
for some [image: {k_0}].

 A version of this result (with a slightly different formulation of [image:
{MPZ[\varpi]}]) appears in this paper of Motohashi and
Pintz<http://www.ams.org/mathscinet-getitem?mr=2414788>,
and in the paper of
Zhang<http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf>,
Theorem 5 <#13f0af938d02c8f9_mpz> is proven for the concrete values [image:
{\varpi = 1/1168}] and [image: {k_0 = 3,500,000}]. We will supply a
self-contained proof of Theorem 5 <#13f0af938d02c8f9_mpz> below the fold,
the constants upon those in Zhang's paper (in particular, for [image:
{\varpi = 1/1168}], we can take [image: {k_0}] as low as [image: {866,805}]).
As with Theorem 4 <#13f0af938d02c8f9_gpy>, we have an inverse quadratic
relationship [image: {k_0 \sim \varpi^{-2}}].

In his paper, Zhang obtained for the first time an unconditional
advance on [image:
{MPZ[\varpi]}]:

*Theorem 6 (Zhang)* [image: {MPZ[\varpi]}] is true for all [image: {0 <
\varpi \leq 1/1168}].

 This is a deep result, building upon the work of
Fouvry-Iwaniec<http://www.ams.org/mathscinet-getitem?mr=610700>,
Friedlander-Iwaniec <http://www.ams.org/mathscinet-getitem?mr=786351> and
Bombieri <http://www.ams.org/mathscinet-getitem?mr=834613>-Friedlander<http://www.ams.org/mathscinet-getitem?mr=891581>
-Iwaniec <http://www.ams.org/mathscinet-getitem?mr=976723> which
established results of a similar nature to [image: {MPZ[\varpi]}] but
simpler in some key respects. We will not discuss this result further here,
except to say that they rely on the (higher-dimensional case of the) Weil
conjectures <http://en.wikipedia.org/wiki/Weil_conjectures>, which were
famously proven by Deligne using methods from l-adic
cohomology<http://en.wikipedia.org/wiki/L-adic_cohomology>.
Also, it was believed among at least some experts that the methods of
Bombieri, Fouvry, Friedlander, and Iwaniec were not quite strong enough to
obtain results of the form [image: {MPZ[\varpi]}], making Theorem
6<#13f0af938d02c8f9_zh-thm>a particularly impressive achievement.

Combining Theorem 6 <#13f0af938d02c8f9_zh-thm> with Theorem
5<#13f0af938d02c8f9_mpz>we obtain [image:
{P[k_0]}] for some finite [image: {k_0}]; Zhang obtains this for [image:
{k_0 = 3,500,000}] but as detailed below, this can be lowered to [image:
{k_0 = 866,805}]. This in turn gives infinitely many pairs of consecutive
primes of separation at most [image: {H(k_0)}]. Zhang gives a simple
argument that bounds [image: {H(3,500,000)}] by [image: {70,000,000}],
giving his famous result that there are infinitely many pairs of primes of
separation at most [image: {70,000,000}]; by being a bit more careful (as
discussed in this
post<http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/>)
one can lower the upper bound on [image: {H(3,500,000)}] to [image:
{57,554,086}], and if one instead uses the newer value [image: {k_0 =
866,605}] for [image: {k_0}] one can instead use the bound [image:
{H(866,605) \leq 13,008,612}]. (Many thanks to Scott Morrison for these
numerics.)

In this post we would like to give a self-contained proof of both
Theorem 4<#13f0af938d02c8f9_gpy>and Theorem
6 <#13f0af938d02c8f9_zh-thm>, which are both sieve-theoretic results that
are mainly elementary in nature. (But, as stated earlier, we will not
discuss the deepest new result in Zhang's paper, namely Theorem
6<#13f0af938d02c8f9_zh-thm>.)
Our presentation will deviate a little bit from the traditional
sieve-theoretic approach in a few places. Firstly, there is a portion of
the argument that is traditionally handled using contour integration and
properties of the Riemann zeta function; we will present a ``cheaper"
approach (which Ben Green and I used in our papers, e.g. in this
one<http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/>)
using Fourier analysis, with the only property used about the zeta
function [image:
{\zeta(s)}] being the elementary fact that blows up like [image:
{\frac{1}{s-1}}] as one approaches [image: {1}] from the right. To deal
with the contribution of small primes (which is the source of the singular
series [image: {{\mathfrak G}}]), it will be convenient to use the ``[image:
{W}]-trick" (introduced in this paper of mine with
Ben<http://www.ams.org/mathscinet-getitem?mr=2415379>),
passing to a single residue class mod [image: {W}] (where [image: {W}] is
the product of all the small primes) to end up in a situation in which all
small primes have been ``turned off" which leads to better pseudorandomness
properties (for instance, once one eliminates all multiples of small
primes, almost all pairs of remaining numbers will be coprime).

Read more of this
post<http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#more-6728>
  *Terence Tao <http://terrytao.wordpress.com/author/teorth/>* | 3 June,
2013 at 8:57 am | Tags: Cem
Yildirim<http://terrytao.wordpress.com/?tag=cem-yildirim>,
Dan Goldston <http://terrytao.wordpress.com/?tag=dan-goldston>,
Elliot-Halberstam
conjecture <http://terrytao.wordpress.com/?tag=elliot-halberstam-conjecture>,
Janos Pintz <http://terrytao.wordpress.com/?tag=janos-pintz>, prime
gaps<http://terrytao.wordpress.com/?tag=prime-gaps>,
sieve theory <http://terrytao.wordpress.com/?tag=sieve-theory>, Yitong
Zhang<http://terrytao.wordpress.com/?tag=yitong-zhang>,
Yoichi Motohashi <http://terrytao.wordpress.com/?tag=yoichi-motohashi> |
Categories: expository <http://terrytao.wordpress.com/?cat=162892>,
math.NT<http://terrytao.wordpress.com/?cat=1440053>| URL:
http://wp.me/p3qzP-1Kw

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