# Dickson-Hardy-Littlewood theorems

### From Polymath1Wiki

(New page: For any integer <math>k_0 \geq 2</math>, let <math>DHL[k_0,2]</math> denote the assertion that given any admissible <math>k_0</math>-tuple <math>{\mathcal H}</math>, that infinitely many t...) |
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* Motohashi-Pintz-Zhang estimates <math>MPZ'[\varpi,\delta]</math> for densely divisible moduli for some <math>0 < \varpi < 1/4</math> and <math>0 < \delta < 1/4+\varpi</math>. | * Motohashi-Pintz-Zhang estimates <math>MPZ'[\varpi,\delta]</math> for densely divisible moduli for some <math>0 < \varpi < 1/4</math> and <math>0 < \delta < 1/4+\varpi</math>. | ||

- | The Elliott-Halberstam estimates are the simplest to use, but unfortunately no estimate of the form <math>EH[\theta]</math> for nay <math>\theta > 1/2</math> is known unconditionally at present. Zhang was the first to establish a result of the form <math>MPZ[\varpi,\theta]</math>, which is weaker than <math>EH[1/2+2\varpi]</math>, for some <math>\varpi,\theta>0</math>. More recently, we have switched to using <math>MPZ'[\varpi,\theta]</math>, an estimate of intermediate strength between <math>MPZ[\varpi,\delta]</math> and <math>EH[1/2+2\varpi]</math>, as the conversion of this estimate to a <math>DHL[k_0,2]</math> result is more efficient in the <math>\delta</math> parameter. | + | The Elliott-Halberstam estimates are the simplest to use, but unfortunately no estimate of the form <math>EH[\theta]</math> for nay <math>\theta > 1/2</math> is known unconditionally at present. Zhang was the first to establish a result of the form <math>MPZ[\varpi,\theta]</math>, which is weaker than <math>EH[1/2+2\varpi+]</math>, for some <math>\varpi,\theta>0</math>. More recently, we have switched to using <math>MPZ'[\varpi,\theta]</math>, an estimate of intermediate strength between <math>MPZ[\varpi,\delta]</math> and <math>EH[1/2+2\varpi+]</math>, as the conversion of this estimate to a <math>DHL[k_0,2]</math> result is more efficient in the <math>\delta</math> parameter. The precise definition of the MPZ and MPZ' estimates can be found at the page on [[distribution of primes in smooth moduli]]. |

== Converting EH to DHL == | == Converting EH to DHL == | ||

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Some further optimisation of this condition was performed in the paper of Goldston, Pintz, and Yildirim by working with general polynomial weights rather than monomial weights. In [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf this paper of Farkas, Pintz, and Revesz], the optimal weight was found (coming from a Bessel function), and the optimised condition | Some further optimisation of this condition was performed in the paper of Goldston, Pintz, and Yildirim by working with general polynomial weights rather than monomial weights. In [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf this paper of Farkas, Pintz, and Revesz], the optimal weight was found (coming from a Bessel function), and the optimised condition | ||

- | :<math>2\theta > \frac{j_{k_0- | + | :<math>2\theta > \frac{j_{k_0-2}^2}{k_0(k_0-1)}</math> |

- | was obtained, where <math>j_{k_0- | + | was obtained, where <math>j_{k_0-2}=j_{k_0-2,1}</math> is the first positive zero of the Bessel function <math>J_{k_0-2}</math>. See for instance [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ this post] for details. |

+ | == Converting MPZ to DHL == | ||

+ | The observation that <math>DHL[k_0,2]</math> could be deduced from <math>MPZ[\varpi,\delta]</math> if <math>k_0</math> was sufficiently large depending on <math>\varpi,\delta</math> was first made in the literature [http://www.ams.org/mathscinet-getitem?mr=2414788 by Motohashi and Pintz]. In the [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf paper of Zhang], an explicit implication was established: <math>MPZ[\varpi,\varpi]</math> implies <math>DHL[k_0,2]</math> whenever there exists an integer <math>l_0>0</math> such that | ||

+ | :<math> (1+4\varpi) (1-\kappa_2) > (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0}) (1 + \kappa_1)</math> | ||

+ | where | ||

+ | :<math> \kappa_1 := \delta_1( 1 + \delta_2^2 + k_0 \log(1+\frac{1}{4\varpi}) \binom{k_0+2l_0}{k_0}</math> | ||

- | = | + | :<math> \kappa_2 := \delta_1 (1+4\varpi) ( 1 + \delta_2^2 + k_0 \log(1+\frac{1}{4\varpi}) \binom{k_0+2l_0+1}{k_0-1}</math> |

+ | :<math> \delta_1 := (1+4\varpi)^{-k_0}</math> | ||

+ | |||

+ | :<math> \delta_2 := \sum_{0 \leq j < 1+\frac{1}{4\varpi}} \frac{ \log(1+\frac{1}{4\varpi}) k_0)^j}{j!}.</math> | ||

+ | |||

+ | The value of <math>\delta_2</math> was lowered to <math>\prod_{0 \leq j < 1+\frac{1}{4\varpi}} (1 + k_0 \log(1+\frac{1}{j})</math> in [http://terrytao.files.wordpress.com/2013/05/bounds.pdf these notes]. Subsequently, the values of <math>\kappa_1,\kappa_2</math> were improved to | ||

+ | |||

+ | :<math> \kappa_1 := (\delta_1 + \sum_{j=1}^{1/4\varpi} \delta_1^j \delta_{2,j}^2 + \delta_1 k_0 \log(1+\frac{1}{4\varpi})) \binom{k_0+2l_0}{k_0}</math> | ||

+ | |||

+ | :<math> \kappa_2 := (\delta_1 (1+4\varpi) + \sum_{j=1}^{1/4\varpi} \delta_1^j (1+4\varpi)^j \delta_{2,j}^2 + \delta_1 (1+4\varpi) k_0 \log(1+\frac{1}{4\varpi}) \binom{k_0+2l_0+1}{k_0-1}</math> | ||

+ | |||

+ | where | ||

+ | |||

+ | :<math> \delta_{2,j} := \prod_{i=1}^j (1 + k_0 \log(1+\frac{1}{i}) )</math>; | ||

+ | |||

+ | again, see [http://terrytao.files.wordpress.com/2013/05/bounds.pdf these notes]. As before, <math>l_0</math> can be taken to be non-integer. | ||

+ | |||

+ | The constraint was then [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ improved further in this post] to deduce <math>DHL[k_0,2]</math> from <math>MPZ[\varpi,\delta]</math> whenever | ||

+ | |||

+ | :<math> (1+4\varpi) > (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0}) (1 + \kappa)</math> | ||

+ | |||

+ | where | ||

+ | |||

+ | :<math> \kappa = \sum_{1 \leq n \leq \frac{1+4\varpi}{2\delta}} (1 - \frac{2n\delta}{1+4\varpi})^{k_0/2+l_0} \prod_{j=1}^n (1+3k_0 \log(1+\frac{1}{j}))</math>. | ||

+ | |||

+ | Using the optimal Bessel weight, this condition was improved to | ||

+ | |||

+ | :<math> (1+4\varpi) > \frac{j_{k_0-2}}{k_0(k_0-1)} (1 + \kappa)</math>; | ||

+ | |||

+ | again, see [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ this post]. | ||

+ | |||

+ | A variant of this criterion was developed using the elementary Selberg sieve in [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ this post], but never used. A subsequent refined criterion was established in [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ this post], namely that | ||

+ | |||

+ | :<math> (1+4\varpi) (1-\kappa') > \frac{j_{k_0-2}}{k_0(k_0-1)} (1 + \kappa)</math> | ||

+ | |||

+ | where | ||

+ | |||

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

+ | |||

+ | :<math> \kappa' := \sum_{2 \leq n < \frac{1+4\varpi}{2\delta}} \frac{3^n-1}{2} \frac{(k_0-1)^n}{n!} (\int_{4\delta/(1+\varpi)}^1 (1-t)^{(k_0-1)/2} \frac{dt}{t})^n.</math> | ||

+ | |||

+ | A slight refinement in [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234845 this comment] allows the condition <math>n \geq 2</math> in the definition of <math>\kappa'</math> to be raised to <math>n \geq 3</math>. | ||

+ | |||

+ | An [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz argument of Pintz] yields the following improved values of <math>\kappa,\kappa'</math> in the above criterion: | ||

+ | |||

+ | :<math> \kappa := 0 </math> | ||

+ | |||

+ | :<math> \kappa' := 2 \kappa_1 + 2 \kappa_2</math> | ||

+ | |||

+ | :<math> \kappa_1 := \int_{4\delta/(1+4\varpi)}^1 (1-t)^{(k_0-1)/2} \frac{dt}{t}</math> | ||

+ | |||

+ | :<math> \kappa_2 := (k_0-1) \int_{4\delta/(1+4\varpi)}^1 (1-t)^{k_0-1} \frac{dt}{t}</math>. | ||

== Converting MPZ' to DHL == | == Converting MPZ' to DHL == | ||

+ | |||

+ | An efficient [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz argument of Pintz], based on the elementary Selberg sieve, allows one to deduce <math>DHL[k_0,2]</math> from <math>MPZ'[\varpi,\delta]</math> with almost no loss with respect to the <math>\delta</math> parameter. As currently optimised, the criterion takes the form | ||

+ | |||

+ | :<math> (1+4\varpi) (1-2\kappa_1 - 2\kappa_2 - 2\kappa_3) > \frac{j_{k_0-2}}{k_0(k_0-1)}</math> | ||

+ | |||

+ | where | ||

+ | |||

+ | :<math> \kappa_1 := \int_{\theta}^1 (1-t)^{(k_0-1)/2} \frac{dt}{t}</math> | ||

+ | |||

+ | :<math> \kappa_2 := (k_0-1) \int_{\theta}^1 (1-t)^{k_0-1} \frac{dt}{t}</math> | ||

+ | |||

+ | :<math> \kappa_3 := \tilde \theta \frac{J_{k_0-2}(\sqrt{\tilde \theta} j_{k_0-2})^2 - J_{k_0-3}(\sqrt{\tilde \theta} j_{k_0-2}) J_{k_0-1}(\sqrt{\tilde \theta} j_{k_0-2})}{ J_{k_0-3,j_{k_0-2}}^2 } | ||

+ | \exp( A + (k0-1) \int_{\tilde \delta}^\theta e^{-(A+2\alpha)t} \frac{dt}{t} )</math> | ||

+ | |||

+ | :<math> \alpha := \frac{j_{k_0-2}^2}{4(k_0-1)}</math> | ||

+ | :<math> \theta := \delta' / (1/4 + \varpi)</math> | ||

+ | :<math> \tilde \theta := \frac{(\delta' - \delta)/2 + \varpi}{1/4 + \varpi}</math> | ||

+ | :<math> \tilde \delta := \frac{\delta}{1/4 + \varpi}</math> | ||

+ | |||

+ | and <math>A>0</math> and <math>\delta \leq \delta' \leq \frac{1}{4} + \varpi</math> are parameters one is free to optimise over. | ||

+ | |||

+ | Here is some simple Maple code to verify the above criterion for given choices of <math>k_0,\varpi,\delta,\delta',A</math>: | ||

+ | |||

+ | k0 := [INSERT VALUE HERE]; | ||

+ | varpi := [INSERT VALUE HERE]; | ||

+ | delta := [INSERT VALUE HERE]; | ||

+ | deltap := [INSERT VALUE HERE]; | ||

+ | A := [INSERT VALUE HERE]; | ||

+ | theta := deltap / (1/4 + varpi); | ||

+ | thetat := ((deltap - delta)/2 + varpi) / (1/4 + varpi); | ||

+ | deltat := delta / (1/4 + varpi); | ||

+ | j := BesselJZeros(k0-2,1); | ||

+ | eps := 1 - j^2 / (k0 * (k0-1) * (1+4*varpi)); | ||

+ | kappa1 := int( (1-t)^((k0-1)/2)/t, t = theta..1, numeric); | ||

+ | kappa2 := (k0-1) * int( (1-t)^(k0-1)/t, t=theta..1, numeric); | ||

+ | alpha := j^2 / (4 * (k0-1)); | ||

+ | e := exp( A + (k0-1) * int( exp(-(A+2*alpha)*t)/t, t=deltat..theta, numeric ) ); | ||

+ | gd := (j^2/2) * BesselJ(k0-3,j)^2; | ||

+ | tn := sqrt(thetat)*j; | ||

+ | gn := (tn^2/2) * (BesselJ(k0-2,tn)^2 - BesselJ(k0-3,tn)*BesselJ(k0-1,tn)); | ||

+ | kappa3 := (gn/gd) * e; | ||

+ | eps2 := 2*(kappa1+kappa2+kappa3); | ||

+ | # we win if eps2 < eps |

## Revision as of 06:53, 26 June 2013

For any integer , let *D**H**L*[*k*_{0},2] denote the assertion that given any admissible *k*_{0}-tuple , that infinitely many translates of contain at least two primes. Thus for instance *D**H**L*[2,2] would imply the twin prime conjecture. The acronym DHL stands for "Dickson-Hardy-Littlewood", and originates from this paper of Pintz.

It is known how to deduce results *D**H**L*[*k*_{0},2] from three classes of estimates:

- Elliott-Halberstam estimates
*E**H*[θ] for some 1 / 2 < θ < 1. - Motohashi-Pintz-Zhang estimates for some and .
- Motohashi-Pintz-Zhang estimates for densely divisible moduli for some and .

The Elliott-Halberstam estimates are the simplest to use, but unfortunately no estimate of the form *E**H*[θ] for nay θ > 1 / 2 is known unconditionally at present. Zhang was the first to establish a result of the form , which is weaker than , for some . More recently, we have switched to using , an estimate of intermediate strength between and , as the conversion of this estimate to a *D**H**L*[*k*_{0},2] result is more efficient in the δ parameter. The precise definition of the MPZ and MPZ' estimates can be found at the page on distribution of primes in smooth moduli.

## Converting EH to DHL

In the breakthrough paper of Goldston, Pintz, and Yildirim, it was shown that *E**H*[θ] implied *D**H**L*[*k*_{0},2] whenever

for some positive integer *l*_{0}. Actually (as noted here), there is nothing preventing the argument for working for non-integer *l*_{0} > 0 as well, so we can optimise this condition as

- .

Some further optimisation of this condition was performed in the paper of Goldston, Pintz, and Yildirim by working with general polynomial weights rather than monomial weights. In this paper of Farkas, Pintz, and Revesz, the optimal weight was found (coming from a Bessel function), and the optimised condition

was obtained, where is the first positive zero of the Bessel function . See for instance this post for details.

## Converting MPZ to DHL

The observation that *D**H**L*[*k*_{0},2] could be deduced from if *k*_{0} was sufficiently large depending on was first made in the literature by Motohashi and Pintz. In the paper of Zhang, an explicit implication was established: implies *D**H**L*[*k*_{0},2] whenever there exists an integer *l*_{0} > 0 such that

where

The value of δ_{2} was lowered to in these notes. Subsequently, the values of κ_{1},κ_{2} were improved to

where

- ;

again, see these notes. As before, *l*_{0} can be taken to be non-integer.

The constraint was then improved further in this post to deduce *D**H**L*[*k*_{0},2] from whenever

where

- .

Using the optimal Bessel weight, this condition was improved to

- ;

again, see this post.

A variant of this criterion was developed using the elementary Selberg sieve in this post, but never used. A subsequent refined criterion was established in this post, namely that

where

A slight refinement in this comment allows the condition in the definition of κ' to be raised to .

An argument of Pintz yields the following improved values of κ,κ' in the above criterion:

- κ: = 0

- κ': = 2κ
_{1}+ 2κ_{2}

- .

## Converting MPZ' to DHL

An efficient argument of Pintz, based on the elementary Selberg sieve, allows one to deduce *D**H**L*[*k*_{0},2] from with almost no loss with respect to the δ parameter. As currently optimised, the criterion takes the form

where

and *A* > 0 and are parameters one is free to optimise over.

Here is some simple Maple code to verify the above criterion for given choices of :

k0 := [INSERT VALUE HERE]; varpi := [INSERT VALUE HERE]; delta := [INSERT VALUE HERE]; deltap := [INSERT VALUE HERE]; A := [INSERT VALUE HERE]; theta := deltap / (1/4 + varpi); thetat := ((deltap - delta)/2 + varpi) / (1/4 + varpi); deltat := delta / (1/4 + varpi); j := BesselJZeros(k0-2,1); eps := 1 - j^2 / (k0 * (k0-1) * (1+4*varpi)); kappa1 := int( (1-t)^((k0-1)/2)/t, t = theta..1, numeric); kappa2 := (k0-1) * int( (1-t)^(k0-1)/t, t=theta..1, numeric); alpha := j^2 / (4 * (k0-1)); e := exp( A + (k0-1) * int( exp(-(A+2*alpha)*t)/t, t=deltat..theta, numeric ) ); gd := (j^2/2) * BesselJ(k0-3,j)^2; tn := sqrt(thetat)*j; gn := (tn^2/2) * (BesselJ(k0-2,tn)^2 - BesselJ(k0-3,tn)*BesselJ(k0-1,tn)); kappa3 := (gn/gd) * e; eps2 := 2*(kappa1+kappa2+kappa3); # we win if eps2 < eps