Dickson-Hardy-Littlewood theorems - Revision history

Teorth: /* Converting MPZ to DHL */

2013-07-28T02:53:26Z

Converting MPZ to DHL

← Older revision		Revision as of 19:53, 27 July 2013
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	There is a variant of <math>MPZ'</math> which we call <math>MPZ''</math> in which dense divisibility is replaced by the stronger condition of double dense divisibility; see [Distribution of primes in smooth moduli] for details. <math>MPZ''[\varpi,\delta]</math> is weaker than <math>MPZ'[\varpi,\delta]</math> but stronger than <math>MPZ[\varpi,\delta]</math>. It turns out (details [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 here]) that one can also deduce <math>DHL[k_0,2]</math> from <math>MPZ''[\varpi,\delta]</math> with an almost identical numerology to the previous section, except that <math>\tilde \theta</math> is increased to		There is a variant of <math>MPZ'</math> which we call <math>MPZ''</math> in which dense divisibility is replaced by the stronger condition of double dense divisibility; see [Distribution of primes in smooth moduli] for details. <math>MPZ''[\varpi,\delta]</math> is weaker than <math>MPZ'[\varpi,\delta]</math> but stronger than <math>MPZ[\varpi,\delta]</math>. It turns out (details [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 here]) that one can also deduce <math>DHL[k_0,2]</math> from <math>MPZ''[\varpi,\delta]</math> with an almost identical numerology to the previous section, except that <math>\tilde \theta</math> is increased to

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

	So the Maple code is now changed slightly to the following:		So the Maple code is now changed slightly to the following:
Line 148:		Line 148:
	A := [INSERT VALUE HERE];		A := [INSERT VALUE HERE];
	theta := deltap / (1/4 + varpi);		theta := deltap / (1/4 + varpi);
	thetat := min( ((deltap - delta) + varpi) / (1/4 + varpi), 1);		thetat := min( ((deltap - delta/2) + varpi) / (1/4 + varpi), 1);
	deltat := delta / (1/4 + varpi);		deltat := delta / (1/4 + varpi);
	j := BesselJZeros(k0-2,1);		j := BesselJZeros(k0-2,1);
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	eps2 := 2*(kappa1+kappa2+kappa3);		eps2 := 2*(kappa1+kappa2+kappa3);
	# we win if eps2 < eps		# we win if eps2 < eps

			More generally, to deduce <math>DHL[k_0,2]</math> from <math>MPZ^{(k)}[\varpi,\delta]</math>, we need to use

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

Teorth: /* Converting EH to DHL */

2013-07-19T21:14:31Z

Converting EH to DHL

← Older revision		Revision as of 14:14, 19 July 2013
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	:<math>2\theta > \frac{j_{k_0-2}^2}{k_0(k_0-1)}</math>		:<math>2\theta > \frac{j_{k_0-2}^2}{k_0(k_0-1)}</math>

	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.		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. Note that even if the full Elliott-Halberstam conjecture was obtained, one could not reduce <math>k_0</math> to below 6 by this criterion (and similarly for the variants of this implication given below), so <math>DHL[6,2]</math> is the limit of this method.

	== Converting MPZ to DHL ==		== Converting MPZ to DHL ==

Charles R Greathouse IV: fix typo

2013-07-16T23:00:23Z

fix typo

← Older revision		Revision as of 16:00, 16 July 2013
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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 precise definition of the <math>MPZ</math>, <math>MPZ'</math> and <math>MPZ''</math> estimates can be found at the page on [[distribution of primes in smooth moduli]].		The Elliott-Halberstam estimates are the simplest to use, but unfortunately no estimate of the form <math>EH[\theta]</math> for any <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 <math>MPZ</math>, <math>MPZ'</math> and <math>MPZ''</math> estimates can be found at the page on [[distribution of primes in smooth moduli]].

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

Teorth at 16:05, 9 July 2013

2013-07-09T16:05:12Z

← Older revision		Revision as of 09:05, 9 July 2013
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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 precise definition of the MPZ and MPZ' estimates can be found at the page on [[distribution of primes in smooth moduli]].		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 <math>MPZ</math>, <math>MPZ'</math> and <math>MPZ''</math> estimates can be found at the page on [[distribution of primes in smooth moduli]].

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

Teorth: /* Converting MPZ to DHL */

2013-07-09T16:04:37Z

Converting MPZ to DHL

← Older revision		Revision as of 09:04, 9 July 2013
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	# we win if eps2 < eps		# we win if eps2 < eps

	== Converting MPZ'' to DHL ==		== Converting <math>MPZ''</math> to DHL ==

	There is a variant of MPZ' which we call MPZ'' in which dense divisibility is replaced by the stronger condition of double dense divisibility; see [Distribution of primes in smooth moduli] for details. <math>MPZ''[\varpi,\delta]</math> is weaker than <math>MPZ'[\varpi,\delta]</math> but stronger than <math>MPZ[\varpi,\delta]</math>. It turns out (details [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 here]) that one can also deduce <math>DHL[k_0,2]</math> from <math>MPZ''[\varpi,\delta]</math> with an almost identical numerology to the previous section, except that <math>\tilde \theta</math> is increased to		There is a variant of <math>MPZ'</math> which we call <math>MPZ''</math> in which dense divisibility is replaced by the stronger condition of double dense divisibility; see [Distribution of primes in smooth moduli] for details. <math>MPZ''[\varpi,\delta]</math> is weaker than <math>MPZ'[\varpi,\delta]</math> but stronger than <math>MPZ[\varpi,\delta]</math>. It turns out (details [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 here]) that one can also deduce <math>DHL[k_0,2]</math> from <math>MPZ''[\varpi,\delta]</math> with an almost identical numerology to the previous section, except that <math>\tilde \theta</math> is increased to

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

	So the Maple code is now changed slightly to the following:		So the Maple code is now changed slightly to the following:
Line 148:		Line 148:
	A := [INSERT VALUE HERE];		A := [INSERT VALUE HERE];
	theta := deltap / (1/4 + varpi);		theta := deltap / (1/4 + varpi);
	thetat := ((deltap - delta) + varpi) / (1/4 + varpi);		thetat := min( ((deltap - delta) + varpi) / (1/4 + varpi), 1);
	deltat := delta / (1/4 + varpi);		deltat := delta / (1/4 + varpi);
	j := BesselJZeros(k0-2,1);		j := BesselJZeros(k0-2,1);

Teorth: /* Converting MPZ' to DHL */

2013-07-09T05:46:16Z

Converting MPZ' to DHL

@@ Line 120: / Line 120: @@
   theta := deltap / (1/4 + varpi);
   thetat := ((deltap - delta)/2 + varpi) / (1/4 + varpi);
   deltat := delta / (1/4 + varpi);
   j := BesselJZeros(k0-2,1);

Pace: Added a 2 to an exponent on a j.

2013-07-01T20:25:03Z

Added a 2 to an exponent on a j.

← Older revision		Revision as of 13:25, 1 July 2013
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	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		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>		:<math> (1+4\varpi) (1-2\kappa_1 - 2\kappa_2 - 2\kappa_3) > \frac{j^{2}_{k_0-2}}{k_0(k_0-1)}</math>

	where		where

Pace: /* Converting MPZ to DHL */ Added the exponent 2 to some of the j's

2013-07-01T20:24:14Z

Converting MPZ to DHL: Added the exponent 2 to some of the j's

← Older revision		Revision as of 13:24, 1 July 2013
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	Using the optimal Bessel weight, this condition was improved to		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>;		:<math> (1+4\varpi) > \frac{j^{2}_{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].		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].
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	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		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>		:<math> (1+4\varpi) (1-\kappa') > \frac{j^{2}_{k_0-2}}{k_0(k_0-1)} (1 + \kappa)</math>

	where		where

Teorth: /* Converting MPZ' to DHL */

2013-06-26T07:04:49Z

Converting MPZ' to DHL

← Older revision		Revision as of 00:04, 26 June 2013
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	:<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 }		:<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>		\exp( A + (k_0-1) \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> \alpha := \frac{j_{k_0-2}^2}{4(k_0-1)}</math>

Teorth: /* Converting MPZ' to DHL */

2013-06-26T07:03:57Z

Converting MPZ' to DHL

← Older revision		Revision as of 00:03, 26 June 2013
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	:<math> \alpha := \frac{j_{k_0-2}^2}{4(k_0-1)}</math>		:<math> \alpha := \frac{j_{k_0-2}^2}{4(k_0-1)}</math>
	:<math> \theta := \delta' ~~/ (~~1/4 + \varpi)</math>		:<math> \theta := \frac{\delta'}{1/4 + \varpi}</math>
	:<math> \tilde \theta := \frac{(\delta' - \delta)/2 + \varpi}{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>		:<math> \tilde \delta := \frac{\delta}{1/4 + \varpi}</math>