[Turkmath:9133] namık kemal erdogan, sen de haklısın!...

[yilmaz akyildiz]

On Wed, Jun 19, 2013 at 2:08 PM, Namık Kemal ERDOĞAN <
nkerdoga at anadolu.edu.tr> wrote:

Burada matematikten başka her şey konuşuluyor.


haklısın namık kemal erdoğan meslektaş.
gel biraz da matematikten haber verelim:
benim son zamanlarda dikkatimi çeken ve içinde türk matematikçilerimizin de
bulunduğu iki mühim netice var:

1. [AAT12] Charles Akemann, Joel Anderson, and Betul Tanbay. The
Kadison-Singer problem
for the direct sum of matrix algebras. Positivity, 16(1):53-66, 2012.
makalesi mühim bir hipotezin çözüldüğü şu makalenin bel kemiğidir:
http://arxiv.org/abs/1306.3969

2. Yalçın  Yıldırım ın baş aktörlerden olduğu asal sayılar hususundaki o
müthiş teorem: (benim anladığım kadarı ile...)

aralarında 70,000,000 dan daha küçük fark olan asal sayı çiftlerinin sayısı
sonsuzdur.

iki netice de az-buz değil. ..

       New post on *What's new*
<http://terrytao.wordpress.com/author/teorth/>  A truncated elementary
Selberg sieve of
Pintz<http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/>
by
Terence Tao <http://terrytao.wordpress.com/author/teorth/>

This post is a continuation of the previous post on sieve
theory<http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/>,
which is an ongoing part of the Polymath8
project<http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes>.
As the previous post was getting somewhat full, we are rolling the thread
over to the current post.

In this post we will record a new truncation of the elementary Selberg
sieve discussed in this previous post (and also analysed in the context of
bounded prime gaps by
Graham-Goldston-Pintz-Yildirim<http://www.ams.org/mathscinet-getitem?mr=2515812>and
Motohashi-Pintz <http://www.ams.org/mathscinet-getitem?mr=2414788>) that
was recently worked out by Janos Pintz, who has kindly given permission to
share this new idea with the Polymath8 project. This new sieve decouples
the [image: {\delta}] parameter that was present in our previous analysis
of Zhang's argument into two parameters, a quantity [image: {\delta}] that
used to measure smoothness in the modulus, but now measures a weaker notion
of ``dense divsibility" which is what is really needed in the
Elliott-Halberstam type estimates, and a second quantity [image:
{\delta]which still measures smoothness but is allowed to be
substantially larger
than [image: {\delta}]. Through this decoupling, it appears that the [image:
{\kappa}] type losses in the sieve theoretic part of the argument can be
almost completely eliminated (they basically decay exponential in [image:
{\delta] and have only mild dependence on [image: {\delta}], whereas the
Elliott-Halberstam analyhsis is sensitive only to [image: {\delta}],
allowing one to set [image: {\delta}] far smaller than previously by
keeping [image: {\delta] large). This should lead to noticeable gains in
the [image: {k_0}] quantity in our analysis.

To describe this new truncation we need to review some notation. As in all
previous posts (in particular, the first post in this
series<http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/>),
we have an asymptotic parameter [image: {x}] going off to infinity, and all
quantities here are implicitly understood to be allowed to depend on [image:
{x}] (or to range in a set that depends on [image: {x}]) unless they are
explicitly declared to be *fixed*. We use the usual asymptotic notation [image:
{O(), o(), \ll}] relative to this parameter [image: {x}]. To be able to
ignore local factors (such as the singular series [image: {{\mathfrak G}}]),
we also use the ``[image: {W}]-trick" (as discussed in the first post in
this series<http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/>):
we introduce a parameter [image: {w}] that grows very slowly with [image:
{x}], and set [image: {W := \prod_{p<w} p}].

For any fixed natural number [image: {k_0}], define an *admissible [image:
{k_0}]-tuple* to be a fixed tuple [image: {{\mathcal H}}] of [image:
{k_0}]distinct integers which avoids at least one residue class modulo
[image:
{p}] for each prime [image: {p}]. Our objective is to obtain the following
conjecture [image: {DHL[k_0,2]}] for as small a value of the parameter [image:
{k_0}] as possible:

*Conjecture 1* ([image: {DHL[k_0,2]}]) Let [image: {{\mathcal H}}] be a
fixed admissible [image: {k_0}]-tuple. Then there exist infinitely many
translates [image: {n+{\mathcal H}}] of [image: {{\mathcal H}}] that
contain at least two primes.

 The twin prime conjecture asserts that [image: {DHL[k_0,2]}] holds for [image:
{k_0}] as small as [image: {2}], but currently we are only able to
establish this result for [image: {k_0 \geq 6329}] (see this
comment<http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693>).
However, with the new truncated sieve of Pintz described in this post, we
expect to be able to lower this threshold [image: {k_0 \geq 6329}]somewhat.

In previous posts, we deduced [image: {DHL[k_0,2]}] from a technical
variant [image: {MPZ[\varpi,\delta]}] of the Elliot-Halberstam conjecture
for certain choices of parameters [image: {0 < \varpi < 1/4}], [image: {0 <
\delta < 1/4+\varpi}]. We will use the following formulation of [image:
{MPZ[\varpi,\delta]}]:

*Conjecture 2* ([image: {MPZ[\varpi,\delta]}]) Let [image: {{\mathcal
H}}]be a fixed [image:
{k_0}]-tuple (not necessarily admissible) for some fixed [image: {k_0 \geq
2}], and let [image: {b\ (W)}] be a primitive residue class. Then

[image: \displaystyle \sum_{q \in {\mathcal S}_I: q< x^{1/2+2\varpi}}
\sum_{a \in C(q)} |\Delta_{b,W}(\Lambda; q,a)| = O( x \log^{-A} x) \ \ \ \
\ (1)]

for any fixed [image: {A>0}], where [image: {I = (w,x^{\delta})}], [image:
{{\mathcal S}_I}] are the square-free integers whose prime factors lie
in [image:
{I}], and [image: {\Delta_{b,W}(\Lambda;q,a)}] is the quantity

[image: \displaystyle \Delta_{b,W}(\Lambda;q,a) := | \sum_{x \leq n \leq
2x: n=b\ (W); n = a\ (q)} \Lambda(n) \ \ \ \ \ (2)]

[image: \displaystyle - \frac{1}{\phi(q)} \sum_{x \leq n \leq 2x: n = b\
(W)} \Lambda(n)|.]

and [image: {C(q)}] is the set of congruence classes

[image: \displaystyle C(q) := \{ a \in ({\bf Z}/q{\bf Z})^\times: P(a) = 0
\}]

and [image: {P}] is the polynomial

[image: \displaystyle P(a) := \prod_{h \in {\mathcal H}} (a+h).]

 The conjecture [image: {MPZ[\varpi,\delta]}] is currently known to hold
whenever [image: {87 \varpi + 17 \delta < \frac{1}{4}}] (see this
comment<http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670>and
this
confirmation<http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742>).
Actually, we can prove a stronger result than [image:
{MPZ[\varpi,\delta]}]in this regime in a couple ways. Firstly, the
congruence classes [image:
{C(q)}] can be replaced by a more general systetm of congruence classes
obeying a certain controlled multiplicity axiom; see this
post<http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/>.
Secondly, and more importantly for this post, the requirement that the
modulus [image: {q}] lies in [image: {{\mathcal S}_I}] can be relaxed; see
below.

To connect the two conjectures, the previously best known implication was
the folowing (see Theorem 2 from this
post<http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/>):


*Theorem 3* Let [image: {0 < \varpi < 1/4}], [image: {0 < \delta < 1/4 +
\varpi}] and [image: {k_0 \geq 2}] be such that we have the inequality

[image: \displaystyle (1 +4 \varpi) (1-\kappa]

where [image: {j_{k_0-2} = j_{k_0-2,1}}] is the first positive zero of the
Bessel function [image: {J_{k_0-2}}], and [image: {\kappa,\kappa] are the
quantities

[image: \displaystyle \kappa := \sum_{1 \leq n < \frac{1+4\varpi}{4\delta}}
\frac{3^n+1}{2} \frac{k_0^n}{n!} (\int_{4\delta/(1+4\varpi)}^1
(1-t)^{k_0/2}\ \frac{dt}{t})^n]

and

[image: \displaystyle \kappa]

[image: \displaystyle (\int_{4\delta/(1+4\varpi)}^1 (1-t)^{(k_0-1)/2}\
\frac{dt}{t})^n.]

Then [image: {MPZ[\varpi,\delta]}] implies [image: {DHL[k_0,2]}].

 Actually there have been some slight improvements to the quantities [image:
{\kappa,\kappa]; see the comments to this previous
post<http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/>.
However, the main error [image: {\kappa}] remains roughly of the order [image:
{\delta^{-1} \exp( - 2 k_0\delta )}], which limits one from taking [image:
{\delta}] too small.

To improve beyond this, the first basic observation is that the smoothness
condition [image: {q \in {\mathcal S}_I}], which implies that all prime
divisors of [image: {q}] are less than [image: {x^\delta}], can be relaxed
in the proof of [image: {MPZ[\varpi,\delta]}]. Indeed, if one inspects the
proof of this proposition (described in these
three<http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/>
previous<http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/>
posts<http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/>),
one sees that the key property of [image: {q}] needed is not so much the
smoothness, but a weaker condition which we will call (for lack of a better
term) *dense divisibility*:

*Definition 4* Let [image: {y > 1}]. A positive integer [image: {q}] is
said to be *[image: {y}]-densely divisible* if for every [image: {1 \leq R
\leq q}], one can find a factor of [image: {q}] in the interval [image:
{[y^{-1} R, R]}]. We let [image: {{\mathcal D}_y}] denote the set of
positive integers that are [image: {y}]-densely divisible.

 Certainly every integer which is [image: {y}]-smooth (i.e. has all prime
factors at most [image: {y}] is also [image: {y}]-densely divisible, as can
be seen from the greedy algorithm; but the property of being [image:
{y}]-densely
divisible is strictly weaker than [image: {y}]-smoothness, which is a
property we will exploit shortly.

We now define [image: {MPZ] to be the same statement as [image:
{MPZ[\varpi,\delta]}], but with the condition [image: {q \in {\mathcal
S}_I}] replaced by the weaker condition [image: {q \in {\mathcal
S}_{[w,+\infty)} \cap {\mathcal D}_{x^\delta}}]. The arguments in previous
posts then also establish [image: {MPZ] whenever [image: {87 \varpi + 17
\delta < \frac{1}{4}}].

The main result of this post is then the following implication, essentially
due to Pintz:

*Theorem 5* Let [image: {0 < \varpi < 1/4}], [image: {0 < \delta \leq
\delta], [image: {A \geq 0}], and [image: {k_0 \geq 2}] be such that

[image: \displaystyle (1 +4 \varpi) (1-2\kappa_1 - 2\kappa_2 - 2\kappa_3) >
\frac{j^2_{k_0-2}}{k_0(k_0-1)}]

where

[image: \displaystyle \kappa_1 := \int_{\theta}^1 (1-t)^{(k_0-1)/2}\
\frac{dt}{t}]

[image: \displaystyle \kappa_2 := (k_0-1) \int_{\theta}^1 (1-t)^{k_0-1}\
\frac{dt}{t}]

[image: \displaystyle \kappa_3 := e^A \frac{G_{k_0-1,\tilde
\theta}(0,0)}{G_{k_0-1}(0,0)} \sum_{0 \leq J \leq 1/\tilde \delta}
\frac{(k_0-1)^J}{J!} (\int_{\tilde \delta}^\theta e^{-At} \frac{dt}{t})^J]

and

[image: \displaystyle \theta := \frac{\delta]

[image: \displaystyle \tilde \theta := \frac{\delta]

[image: \displaystyle \tilde \delta := \frac{\delta}{1/4+\varpi}]

[image: \displaystyle G_{k_0-1}(0,0) := \int_0^1 f(t)^2
\frac{t^{k_0-2}}{(k_0-2)!}\ dt]

[image: \displaystyle G_{k_0-1,\tilde \theta}(0,0) := \int_0^{\tilde
\theta} f(t)^2 \frac{t^{k_0-2}}{(k_0-2)!}\ dt]

and

[image: \displaystyle f(t) := t^{1-k_0/2} J_{k_0-2}( \sqrt{t / j_{k_0-2}}
).]

Then [image: {MPZ] implies [image: {DHL[k_0,2]}].

 This theorem has rather messy constants, but we can isolate some special
cases which are a bit easier to compute with. Setting [image: {\delta], we
see that [image: {\kappa_3}] vanishes (and the argument below will show
that we only need [image: {MPZ[\varpi,\delta]}] rather than [image: {MPZ]),
and we obtain the following slight improvement of Theorem
3<#13f5a3a2fe43b667_impl>:


*Theorem 6* Let [image: {0 < \varpi < 1/4}], [image: {0 < \delta < 1/4 +
\varpi}] and [image: {k_0 \geq 2}] be such that we have the inequality

[image: \displaystyle (1 +4 \varpi) (1-2\kappa_1-2\kappa_2) >
\frac{j^2_{k_0-2}}{k_0(k_0-1)} \ \ \ \ \ (4)]

where

[image: \displaystyle \kappa_1 := \int_{4\delta/(1+4\varpi)}^1
(1-t)^{(k_0-1)/2}\ \frac{dt}{t}]

[image: \displaystyle \kappa_2 := (k_0-1) \int_{4\delta/(1+4\varpi)}^1
(1-t)^{k_0-1}\ \frac{dt}{t}.]

Then [image: {MPZ[\varpi,\delta]}] implies [image: {DHL[k_0,2]}].

 This is a little better than Theorem 3 <#13f5a3a2fe43b667_impl>, because
the error [image: {2\kappa_1+2\kappa_2}] has size about [image: {\frac{1}{2
k_0 \delta} \exp( - 2 k_0 \delta) + \frac{1}{2 \delta} \exp(-4 k_0 \delta)}],
which compares favorably with the error in Theorem
3<#13f5a3a2fe43b667_impl>which is about [image:
{\frac{1}{\delta} \exp(-2 k_0 \delta)}]. This should already give a
``cheap" improvement to our current threshold [image: {k_0 \geq 6329}],
though it will fall short of what one would get if one fully optimised over
all parameters in the above theorem.

Returning to the full strength of Theorem 5 <#13f5a3a2fe43b667_pintz>, let
us obtain a crude upper bound for [image: {\kappa_3}] that is a little
simpler to understand. Extending the [image: {J}] summation to infinity and
using the Taylor series for the exponential, we have

[image: \displaystyle \kappa_3 \leq \frac{G_{k_0-1,\tilde
\theta}(0,0)}{G_{k_0-1}(0,0)} \exp( A + (k_0-1) \int_{\tilde \delta}^\theta
e^{-At} \frac{dt}{t} ).]

We can crudely bound

[image: \displaystyle \int_{\tilde \delta}^\theta e^{-At} \frac{dt}{t} \leq
\frac{1}{A \tilde \delta}]

and then optimise in [image: {A}] to obtain

[image: \displaystyle \kappa_3 \leq \frac{G_{k_0-1,\tilde
\theta}(0,0)}{G_{k_0-1}(0,0)} \exp( 2 (k_0-1)^{1/2} \tilde \delta^{-1/2} ).]

Because of the [image: {t^{k_0-2}}] factor in the integrand for [image:
{G_{k_0-1}}] and [image: {G_{k_0-1,\tilde \theta}}], we expect the
ratio [image:
{\frac{G_{k_0-1,\tilde \theta}(0,0)}{G_{k_0-1}(0,0)}}] to be of the order
of [image: {\tilde \theta^{k_0-1}}], although one will need some
theoretical or numerical estimates on Bessel functions to make this
heuristic more precise. Setting [image: {\tilde \theta}] to be something
like [image: {1/2}], we get a good bound here as long as [image: {\tilde
\delta \gg 1/k_0}], which at current values of [image: {\delta, k_0}] is a
mild condition.

Pintz's argument uses the elementary Selberg sieve, discussed in this
previous post<http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/>,
but with a more efficient estimation of the quantity [image: {\beta}], in
particular avoiding the truncation to moduli [image: {d}] between [image:
{x^{-\delta} R}] and [image: {R}] which was the main source of inefficiency
in that previous post. The basic idea is to ``linearise" the effect of the
truncation of the sieve, so that this contribution can be dealt with by the
union bound (basically, bounding the contribution of each large prime one
at a time). This mostly avoids the more complicated combinatorial analysis
that arose in the analytic Selberg sieve, as seen in this previous
post<http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/>.


Read more of this
post<http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#more-6849>
  *Terence Tao <http://terrytao.wordpress.com/author/teorth/>* | 18 June,
2013 at 6:18 pm | Tags: Janos
Pintz<http://terrytao.wordpress.com/?tag=janos-pintz>,
polymath8 <http://terrytao.wordpress.com/?tag=polymath8>, Selberg
sieve<http://terrytao.wordpress.com/?tag=selberg-sieve>| Categories:
math.NT <http://terrytao.wordpress.com/?cat=1440053>,
polymath<http://terrytao.wordpress.com/?cat=123554>| URL:
http://wp.me/p3qzP-1Mt

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