Distribution of primes in smooth moduli - Revision history

Teorth: /* Level 5 */

2013-10-11T18:56:59Z

Level 5

← Older revision		Revision as of 11:56, 11 October 2013
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	see [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 this comment].		see [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 this comment].

			A further averaging in the <math>l</math> parameter has led to a preliminary improvement of the main condition to

			:<math>48 \varpi + \frac{44}{3} \delta + \frac{38}{9} \sigma < 1</math>,

			see [http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247766 this comment].

	== Type II estimates ==		== Type II estimates ==

Teorth: /* Combinations */

2013-07-27T16:57:09Z

Combinations

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	typeI_4 := [ 236 * varpi/3 + 64 * delta/3 + 4*sigma < 1];		typeI_4 := [ 236 * varpi/3 + 64 * delta/3 + 4*sigma < 1];
	typeI_6 := [ 56 * varpi + 16delta + 4sigma < 1];		typeI_6 := [ 56 * varpi + 16delta + 4sigma < 1];
	typeI_5 := [ 160varpi/3 + 16delta + 34*sigma/9 < 1];		typeI_5 := [ 160varpi/3 + 16delta + 34sigma/9 < 1, 64\varpi+18\delta+2\sigma < 1];
	typeII_1 := [ 58 * varpi + 10 * delta < 1/2 ];		typeII_1 := [ 58 * varpi + 10 * delta < 1/2 ];
	typeII_1a := [48 * varpi + 7 * delta < 1/2 ];		typeII_1a := [48 * varpi + 7 * delta < 1/2 ];

Teorth: /* Level 5 */

2013-07-27T16:54:39Z

Level 5

← Older revision		Revision as of 09:54, 27 July 2013
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	Further improvement to these be obtainable by taking advantage of averaging in auxiliary parameters; in particular averaging over the parameter <math>d_1</math> has provisionally (subject to verification of some Deligne-level estimates) shown to establish <math>Type''''_I[\varpi,\delta,\sigma]</math> whenever		Further improvement to these be obtainable by taking advantage of averaging in auxiliary parameters; in particular averaging over the parameter <math>d_1</math> has provisionally (subject to verification of some Deligne-level estimates) shown to establish <math>Type''''_I[\varpi,\delta,\sigma]</math> whenever

	:<math>\frac{160}{3} \varpi + 16 \delta + \frac{34}{9} \sigma < 1</math>;		:<math>\frac{160}{3} \varpi + 16 \delta + \frac{34}{9} \sigma < 1</math>

			together with the secondary condition

			:<math>64\varpi+18\delta+2\sigma < 1</math>;

	see [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 this comment].		see [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 this comment].

Teorth: /* Combinations */

2013-07-20T17:20:57Z

Combinations

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	\| [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 details]		\| [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 details]
	\| Type I / combinatorial border		\| Type I / combinatorial border
			\|-
			\|Level 5
			\|Level 1c
			\|Level 4
			\|<math>\frac{600}{7} \varpi + \frac{180}{7} \delta < 1</math>
			\| [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 details]
			\| Type I / combinatorial border
	\|}		\|}

	For simplicity, only the constraint that is relevant for near-maximal values of <math>\varpi</math> is shown.		For simplicity, only the constraint that is relevant for near-maximal values of <math>\varpi</math> is shown.

	Here is some Maple code for finding the constraints coming from a certain set of inequalities (e.g. Type I level 6, Type II level 1c, and Type III level 4). To reduce the complexity of the output, one can introduce an artificial cutoff of, say, <math>\varpi > 1/200</math>, in the base constraints to restrict attention to the regime of large values of <math>\varpi</math>.		Here is some Maple code for finding the constraints coming from a certain set of inequalities (e.g. Type I level 5, Type II level 1c, and Type III level 4). To reduce the complexity of the output, one can introduce an artificial cutoff of, say, <math>\varpi > 1/200</math>, in the base constraints to restrict attention to the regime of large values of <math>\varpi</math>.

	with(SolveTools[Inequality]);		with(SolveTools[Inequality]);
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	typeI_4 := [ 236 * varpi/3 + 64 * delta/3 + 4*sigma < 1];		typeI_4 := [ 236 * varpi/3 + 64 * delta/3 + 4*sigma < 1];
	typeI_6 := [ 56 * varpi + 16delta + 4sigma < 1];		typeI_6 := [ 56 * varpi + 16delta + 4sigma < 1];
			typeI_5 := [ 160varpi/3 + 16delta + 34*sigma/9 < 1];
	typeII_1 := [ 58 * varpi + 10 * delta < 1/2 ];		typeII_1 := [ 58 * varpi + 10 * delta < 1/2 ];
	typeII_1a := [48 * varpi + 7 * delta < 1/2 ];		typeII_1a := [48 * varpi + 7 * delta < 1/2 ];
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	typeIII_3 := [ 3/2 * (1/2 + sigma) > (7/4) * (1/2 + 2varpi) + (3/8) delta ];		typeIII_3 := [ 3/2 * (1/2 + sigma) > (7/4) * (1/2 + 2varpi) + (3/8) delta ];
	typeIII_4 := [ 1/4 + (3/4) * (3/2) * (1/2 + sigma) > (7/4) * (1/2 + 2varpi) + (1/4) delta ];		typeIII_4 := [ 1/4 + (3/4) * (3/2) * (1/2 + sigma) > (7/4) * (1/2 + 2varpi) + (1/4) delta ];
	constraints := [ op(base), op(~~typeI_6~~), op(typeII_1c), op(typeIII_4) ];		constraints := [ op(base), op(typeI_5), op(typeII_1c), op(typeIII_4) ];
	LinearMultivariateSystem(constraints, [varpi,delta,sigma]);		LinearMultivariateSystem(constraints, [varpi,delta,sigma]);

Teorth: /* Type I estimates */

2013-07-20T17:13:30Z

Type I estimates

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	but this is inferior to the Level 3 estimates in practice. Details can be found [https://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236682 here].		but this is inferior to the Level 3 estimates in practice. Details can be found [https://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236682 here].

			=== Level 6 ===

			Even further improvement in the Type I sums may be possible by rebalancing the final Cauchy-Schwarz: instead of performing Cauchy-Schwarz in <math>n,q_1</math> (leaving <math>h,q_2</math> to be doubled), factor <math>q_2 = r_2 s_2</math> and Cauchy-Schwarz in <math>n,q_1,r_2</math> and only double <math>h,s_2</math>. The idea is to make the diagonal case <math>h s'_2 = h' s_2</math> do more of the work and the off-diagonal case <math>hs'_2 \neq h' s_2</math> do less of the work. This idea was first raised [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236673 here]. [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Preliminary computations] suggest that this allows one to take <math>56 \varpi + 16 \delta + 4 \sigma < 1</math> for the Type I sums in <math>x^\delta</math>-smooth case. In [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/ this post] it is shown that the same bound holds in the densely divisible case, thus <math>Type''_I[\varpi,\delta,\sigma]</math> holds whenever

			:<math>56 \varpi + 16 \delta + 4 \sigma < 1</math>.

	=== Level 5 ===		=== Level 5 ===

	~~Further improvement to the~~ (~~still sketchy) Level 4 estimate should be obtainable by taking advantage~~ of ~~averaging in auxiliary "h" parameters in~~ order to ~~reduce~~ the ~~contribution of the diagonal terms~~.		(The numbering here is out of order because the Level 5 estimates proved harder to implement than the Level 6 estimates.)

	~~=== Level 6 ===~~		Further improvement to these be obtainable by taking advantage of averaging in auxiliary parameters; in particular averaging over the parameter <math>d_1</math> has provisionally (subject to verification of some Deligne-level estimates) shown to establish <math>Type''''_I[\varpi,\delta,\sigma]</math> whenever

	~~Even further improvement in the Type I sums may be possible by rebalancing the final Cauchy-Schwarz~~: ~~instead of performing Cauchy-Schwarz in~~ <math>n,q_1</math> (leaving <math>h,q_2</math> to be doubled), factor <math>q_2 = r_2 s_2</math> and Cauchy-Schwarz in <math>n,q_1,r_2</math> and only double <math>h,s_2</math>. The idea is to make the diagonal case <math>h s'_2 = h' s_2</math> do more of the work and the off-diagonal case <math>hs'_2 \neq h' s_2</math> do less of the work. This idea was first raised [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236673 here]. [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Preliminary computations] suggest that this allows one to take <math>56 \varpi + 16 \delta + 4 \sigma < 1</math> for the Type I sums in <math>x^\delta</math>-smooth case. In [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/ this post] it is shown that the same bound holds in the densely divisible case, thus <math>Type''_I[\varpi,\delta]</math> holds whenever		:<math>\frac{160}{3} \varpi + 16 \delta + \frac{34}{9} \sigma < 1</math>;

	:~~<math>56 \varpi + 16 \delta + 4 \sigma < 1<~~/~~math>~~.		see [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 this comment].

	== Type II estimates ==		== Type II estimates ==

Teorth: /* Definitions */

2013-07-09T16:08:06Z

Definitions

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	A natural number <math>n</math> is said to be <em><math>y</math>-smooth</em> if all of its prime factors are less than or equal to <math>y</math>. We say that <math>n</math> is <em><math>y</math>-densely divisible</em> if, for every <math>1 \leq R \leq n</math>, one can find a factor of <math>n</math> in the interval <math>[y^{-1} R, R]</math>. Note that <math>y</math>-smooth numbers are automatically <math>y</math>-densely divisible, but the converse is not true in general. We say that <math>n</math> is <em>doubly <math>y</math>-densely divisible</em> if, for every <math>1 \leq R \leq n</math>, one can find a factor of <math>n</math> in the interval <math>[y^{-1} R, R]</math> which is itself <math>y</math>-densely divisible.		A natural number <math>n</math> is said to be <em><math>y</math>-smooth</em> if all of its prime factors are less than or equal to <math>y</math>. We say that <math>n</math> is <em><math>y</math>-densely divisible</em> if, for every <math>1 \leq R \leq n</math>, one can find a factor of <math>n</math> in the interval <math>[y^{-1} R, R]</math>. Note that <math>y</math>-smooth numbers are automatically <math>y</math>-densely divisible, but the converse is not true in general. We say that <math>n</math> is <em>doubly <math>y</math>-densely divisible</em> if, for every <math>1 \leq R \leq n</math>, one can find a factor of <math>n</math> in the interval <math>[y^{-1} R, R]</math> which is itself <math>y</math>-densely divisible.

	=== MPZ ===		We let <math>{\mathcal D}_y</math> denote the space of <math>y</math>-densely divisible numbers, and <math>{\mathcal D}_y^2</math> the space of doubly densely divisible numbers, thus

			:<math>{\mathcal S}_{[1,y]} \subset {\mathcal D}^2_y \subset {\mathcal D}_y</math>.

			=== MPZ and variants ===

	Let <math>0 < \varpi < 1/4</math> and <math>0 < \delta < \varpi + 1/4</math> be fixed. Let <math>\Lambda</math> denote the von Mangoldt function.		Let <math>0 < \varpi < 1/4</math> and <math>0 < \delta < \varpi + 1/4</math> be fixed. Let <math>\Lambda</math> denote the von Mangoldt function.
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	: <math>\displaystyle \sum_{q \in {\mathcal S}_I: q< x^{1/2+2\varpi}} \|\Delta(\Lambda 1_{[x,2x]}; a_q)\| \ll x \log^{-A} x</math>		: <math>\displaystyle \sum_{q \in {\mathcal S}_I: q< x^{1/2+2\varpi}} \|\Delta(\Lambda 1_{[x,2x]}; a_q)\| \ll x \log^{-A} x</math>

	for any fixed <math>A > 0</math>, any <math>I \subset [1,x^\delta]</math>, and any congruence class system <math> (\{a_q\})_{q \in {\mathcal S}_I}</math> ~~of controlled multiplicity~~.		for any fixed <math>A > 0</math>, any <math>I \subset [1,x^\delta]</math>, and any congruence class system <math> (\{a_q\})_{q \in {\mathcal S}_I}</math>.

	* We say that the estimate <math>MPZ'[\varpi,\delta]</math> holds if one has the estimate		* We say that the estimate <math>MPZ'[\varpi,\delta]</math> holds if one has the estimate
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	: <math>\displaystyle \sum_{q \in {\mathcal S}_I \cap {\mathcal D}_{x^\delta}: q< x^{1/2+2\varpi}} \|\Delta(\Lambda 1_{[x,2x]}; a_q)\| \ll x \log^{-A} x</math>		: <math>\displaystyle \sum_{q \in {\mathcal S}_I \cap {\mathcal D}_{x^\delta}: q< x^{1/2+2\varpi}} \|\Delta(\Lambda 1_{[x,2x]}; a_q)\| \ll x \log^{-A} x</math>

	for any fixed <math>A > 0</math>, any <math>I \subset {\mathbf R}</math>, and any congruence class system <math> (\{a_q\})_{q \in {\mathcal S}_I}</math> of controlled multiplicity.		for any fixed <math>A > 0</math>, any <math>I \subset {\mathbf R}</math>, and any congruence class system <math> (\{a_q\})_{q \in {\mathcal S}_I}</math>.


			* We say that the estimate <math>MPZ''[\varpi,\delta]</math> holds if one has the estimate

			: <math>\displaystyle \sum_{q \in {\mathcal S}_I \cap {\mathcal D}_{x^\delta}^2: q< x^{1/2+2\varpi}} \|\Delta(\Lambda 1_{[x,2x]}; a_q)\| \ll x \log^{-A} x</math>

			for any fixed <math>A > 0</math>, any <math>I \subset {\mathbf R}</math>, and any congruence class system <math> (\{a_q\})_{q \in {\mathcal S}_I}</math>.

			In early arguments an additional "controlled multiplicity" hypothesis was added to these assertions, but this hypothesis is no longer necessary.

	=== Type I, Type II, and Type III ===		=== Type I, Type II, and Type III ===
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	* We define <math>Type'_I[\varpi,\delta,\sigma]</math>, <math>Type'_{II}[\varpi,\delta]</math>, <math>Type_{III}[\varpi,\delta,\sigma]</math> analogously to <math>Type_I[\varpi,\delta,\sigma]</math>, <math>Type_{II}[\varpi,\delta]</math>, <math>Type_{III}[\varpi,\delta,\sigma]</math> but with the hypothesis <math>I \subset [1,x^\delta]</math> replaced with <math>I \subset \mathbf{R}</math>, and <math>{\mathcal S}_I</math> replaced with <math>{\mathcal S}_I \cap {\mathcal D}_{x^\delta}</math>. These estimates are slightly stronger than their unprimed counterparts.		* We define <math>Type'_I[\varpi,\delta,\sigma]</math>, <math>Type'_{II}[\varpi,\delta]</math>, <math>Type_{III}[\varpi,\delta,\sigma]</math> analogously to <math>Type_I[\varpi,\delta,\sigma]</math>, <math>Type_{II}[\varpi,\delta]</math>, <math>Type_{III}[\varpi,\delta,\sigma]</math> but with the hypothesis <math>I \subset [1,x^\delta]</math> replaced with <math>I \subset \mathbf{R}</math>, and <math>{\mathcal S}_I</math> replaced with <math>{\mathcal S}_I \cap {\mathcal D}_{x^\delta}</math>. These estimates are slightly stronger than their unprimed counterparts.

	There is also a "double-primed" variant <math>Type''_I[\varpi,\delta,\sigma], Type''_{II}[\varpi,\delta], Type''_{III}[\varpi,\delta,\sigma]</math> of these estimates, intermediate in strength between the primed and unprimed estimates, in which dense divisibility is replaced with "double dense divisibility" hypothesis.		* There is also a "double-primed" variant <math>Type''_I[\varpi,\delta,\sigma], Type''_{II}[\varpi,\delta], Type''_{III}[\varpi,\delta,\sigma]</math> of these estimates, intermediate in strength between the primed and unprimed estimates, in which dense divisibility is replaced with "double dense divisibility" hypothesis.

	== The combinatorial lemma ==		== The combinatorial lemma ==

Teorth: /* Level 4 */

2013-07-09T05:59:50Z

Level 4

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	<blockquote><b>Type III-4</b> We have <math>Type'_{III}[\varpi,\delta,\sigma]</math> (and hence <math>Type_{III}[\varpi,\delta,\sigma]</math>) whenever		<blockquote><b>Type III-4</b> We have <math>Type'_{III}[\varpi,\delta,\sigma]</math> (and hence <math>Type_{III}[\varpi,\delta,\sigma]</math> and <math>Type''_{III}[\varpi,\delta,\sigma]</math>) whenever

	:<math>\displaystyle \frac{1}{4} + \frac{3}{4} \frac{3}{2} (\frac{1}{2} + \sigma) > \frac{7}{4} (\frac{1}{2} + 2 \varpi) + \frac{1}{4} \delta </math>.		:<math>\displaystyle \frac{1}{4} + \frac{3}{4} \frac{3}{2} (\frac{1}{2} + \sigma) > \frac{7}{4} (\frac{1}{2} + 2 \varpi) + \frac{1}{4} \delta </math>.
	</blockquote>		</blockquote>

	This estimate is proven in [https://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 this comment]. It modifies the Level 3 argument by exploiting averaging in the <math>\alpha</math> parameter (this was suggested already by Fouvry, Kowalski, Michel, and Nelson).The constraint may also be written as a lower bound on <math>\sigma</math>:		This estimate is proven in [https://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 this comment] and then in [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/ this post]. It modifies the Level 3 argument by exploiting averaging in the <math>\alpha</math> parameter (this was suggested already by Fouvry, Kowalski, Michel, and Nelson).The constraint may also be written as a lower bound on <math>\sigma</math>:

	:<math>\displaystyle \sigma > \frac{1}{18} + \frac{28}{9} \varpi + \frac{2}{9} \delta</math>.		:<math>\displaystyle \sigma > \frac{1}{18} + \frac{28}{9} \varpi + \frac{2}{9} \delta</math>.

Teorth: /* Level 1c */

2013-07-09T05:58:28Z

Level 1c

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	=== Level 1c ===		=== Level 1c ===

	<blockquote><b>Type II-1c</b> We have <math>Type'_{II}[\varpi,\delta]</math> (and hence <math>Type_{II}[\varpi,\delta]</math>) whenever		<blockquote><b>Type II-1c</b> We have <math>Type'_{II}[\varpi,\delta]</math> (and hence <math>Type_{II}[\varpi,\delta]</math> and <math>Type''_{II}[\varpi,\delta]</math>) whenever
	:<math>\displaystyle 34\varpi + 7\delta < \frac{1}{2}</math>.		:<math>\displaystyle 34\varpi + 7\delta < \frac{1}{2}</math>.
	</blockquote>		</blockquote>

	This further refinement of the Level 1b estimate came from realising that R can in fact range in <math>[x^{-\delta-\varepsilon} N, x^{-\varepsilon} N]</math> if one strengthens the controlled multiplicity hypothesis slightly; see [https://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236482 this comment] for details.		This further refinement of the Level 1b estimate came from realising that R can in fact range in <math>[x^{-\delta-\varepsilon} N, x^{-\varepsilon} N]</math> if one strengthens the controlled multiplicity hypothesis slightly; see [https://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236482 this comment] or [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/ this post] for details.

	=== Level 2 ===		=== Level 2 ===

Teorth: /* Level 6 */

2013-07-09T05:57:37Z

Level 6

← Older revision		Revision as of 22:57, 8 July 2013
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	=== Level 6 ===		=== Level 6 ===

	Even further improvement in the Type I sums may be possible by rebalancing the final Cauchy-Schwarz: instead of performing Cauchy-Schwarz in <math>n,q_1</math> (leaving <math>h,q_2</math> to be doubled), factor <math>q_2 = r_2 s_2</math> and Cauchy-Schwarz in <math>n,q_1,r_2</math> and only double <math>h,s_2</math>. The idea is to make the diagonal case <math>h s'_2 = h' s_2</math> do more of the work and the off-diagonal case <math>hs'_2 \neq h' s_2</math> do less of the work. This idea was first raised [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236673 here]. [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Preliminary computations] suggest that this allows one to take <math>56 \varpi + 16 \delta + 4 \sigma < 1</math> for the Type I sums in <math>x^\delta</math>-smooth case~~, and then the doubly densely divisible case~~ [http://terrytao.wordpress.com/2013/06/30/~~bounded~~-~~gaps~~-~~between~~-primes-~~polymath8~~-a-~~progress~~-~~report~~/~~#comment-237306 here~~].		Even further improvement in the Type I sums may be possible by rebalancing the final Cauchy-Schwarz: instead of performing Cauchy-Schwarz in <math>n,q_1</math> (leaving <math>h,q_2</math> to be doubled), factor <math>q_2 = r_2 s_2</math> and Cauchy-Schwarz in <math>n,q_1,r_2</math> and only double <math>h,s_2</math>. The idea is to make the diagonal case <math>h s'_2 = h' s_2</math> do more of the work and the off-diagonal case <math>hs'_2 \neq h' s_2</math> do less of the work. This idea was first raised [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236673 here]. [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Preliminary computations] suggest that this allows one to take <math>56 \varpi + 16 \delta + 4 \sigma < 1</math> for the Type I sums in <math>x^\delta</math>-smooth case. In [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/ this post] it is shown that the same bound holds in the densely divisible case, thus <math>Type''_I[\varpi,\delta]</math> holds whenever

			:<math>56 \varpi + 16 \delta + 4 \sigma < 1</math>.

	== Type II estimates ==		== Type II estimates ==

Teorth: /* The combinatorial lemma */

2013-07-09T05:50:37Z

The combinatorial lemma

← Older revision		Revision as of 22:50, 8 July 2013
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	* If <math>Type_I[\varpi,\delta,\sigma]</math>, <math>Type_{II}[\varpi,\delta]</math>, and <math>Type_{III}[\varpi,\delta,\sigma]</math> all hold, then <math>MPZ[\varpi,\delta]</math> holds.		* If <math>Type_I[\varpi,\delta,\sigma]</math>, <math>Type_{II}[\varpi,\delta]</math>, and <math>Type_{III}[\varpi,\delta,\sigma]</math> all hold, then <math>MPZ[\varpi,\delta]</math> holds.
	* Similarly, if <math>Type'_I[\varpi,\delta,\sigma]</math>, <math>Type'_{II}[\varpi,\delta]</math>, and <math>Type'_{III}[\varpi,\delta,\sigma]</math> all hold, then <math>MPZ'[\varpi,\delta]</math> holds.		* Similarly, if <math>Type'_I[\varpi,\delta,\sigma]</math>, <math>Type'_{II}[\varpi,\delta]</math>, and <math>Type'_{III}[\varpi,\delta,\sigma]</math> all hold, then <math>MPZ'[\varpi,\delta]</math> holds.
			* Similarly, if <math>Type''_I[\varpi,\delta,\sigma]</math>, <math>Type''_{II}[\varpi,\delta]</math>, and <math>Type''_{III}[\varpi,\delta,\sigma]</math> all hold, then <math>MPZ''[\varpi,\delta]</math> holds.
	</blockquote>		</blockquote>