Distribution of primes in smooth moduli

2013-06-26T10:25:11Z

DLyons: Typo 1/12.

A key input to Zhang's proof that bounded gaps occur infinitely often is a distribution result on primes in smooth moduli, which we have called <math>MPZ[\varpi,\delta]</math> (and later strengthened to <math>MPZ'[\varpi,\delta]</math>. These estimates are obtained as a combination of three other estimates, which we will call <math>Type_I[\varpi,\delta,\sigma]</math>, <math>Type_{II}[\varpi,\delta]</math>, and <math>Type_{III}[\varpi,\delta,\sigma]</math>.

== Definitions ==

=== Asymptotic notation ===

<math>x</math> is a parameter going off to infinity, and all quantities may depend on <math>x</math> unless explicitly declared to be "fixed". The asymptotic notation <math>O(), o(), \ll</math> is then defined relative to this parameter. A quantity <math>q</math> is said to be of polynomial size if one has <math>q = O(x^{O(1)})</math>, and bounded if <math>q=O(1)</math>. We also write <math>X \lessapprox Y</math> for <math>X \ll x^{o(1)} Y</math>, and <math>\displaystyle X \sim Y</math> for <math>X \ll Y \ll X</math>.

=== Coefficient sequences ===

We need a fixed quantity <math>A_0>0</math>.

A coefficient sequence is a finitely supported sequence <math>\alpha: {\mathbf N} \rightarrow {\mathbf R}</math> that obeys the bounds

:<math>\displaystyle |\alpha(n)| \ll \tau^{O(1)}(n) \log^{O(1)}(x)</math>

* If <math>\alpha</math> is a coefficient sequence and <math>a\ (q) = a \hbox{ mod } q</math> is a primitive residue class, the (signed) discrepancy <math>\Delta(\alpha; a\ (q))</math> of <math>\alpha</math> in the sequence is defined to be the quantity

: <math>\displaystyle \Delta(\alpha; a \ (q)) := \sum_{n: n = a\ (q)} \alpha(n) - \frac{1}{\phi(q)} \sum_{n: (n,q)=1} \alpha(n).</math>

* A coefficient sequence <math>\alpha</math> is said to be at scale <math>N</math> for some <math>N \geq 1</math> if it is supported on an interval of the form <math>[(1-O(\log^{-A_0} x)) N, (1+O(\log^{-A_0} x)) N]</math>.

* A coefficient sequence <math>\alpha</math> at scale <math>N</math> is said to obey the Siegel-Walfisz theorem if one has

: <math> \displaystyle | \Delta(\alpha 1_{(\cdot,q)=1}; a\ (r)) | \ll \tau(qr)^{O(1)} N \log^{-A} x</math>

for any <math>q,r \geq 1</math>, any fixed <math>A</math>, and any primitive residue class <math>a\ (r)</math>.

* A coefficient sequence <math>\alpha</math> at scale <math>N</math> is said to be smooth if it takes the form <math>\alpha(n) = \psi(n/N)</math> for some smooth function <math>\psi: {\mathbf R} \rightarrow {\mathbf C}</math> supported on <math>[1-O(\log^{-A_0} x), 1+O(\log^{-A_0} x)]</math> obeying the derivative bounds

:<math>\displaystyle \psi^{(j)}(t) = O( \log^{j A_0} x ) </math>

for all fixed <math>j \geq 0</math> (note that the implied constant in the <math>O()</math> notation may depend on <math>j</math>).

=== Congruence class systems ===

Let <math>I \subset {\mathbf R}</math>, and let <math>{\mathcal S}_I</math> denote the square-free numbers whose prime factors lie in <math>I</math>.

* A singleton congruence class system on <math>I</math> is a collection <math>{\mathcal C} = (\{a_q\})_{q \in {\mathcal S}_I}</math> of primitive residue classes <math>a_q \in ({\mathbf Z}/q{\mathbf Z})^\times</math> for each </math>q \in {\mathcal S}_I</math>, obeying the Chinese remainder theorem property

:<math>\displaystyle a_{qr}\ (qr) = (a_q\ (q)) \cap (a_r\ (r))</math>

whenever <math>q,r \in {\mathcal S}_I</math> are coprime. We say that such a system <math>{\mathcal C}</math> has controlled multiplicity if the quantity

:<math>\displaystyle \tau_{\mathcal C}(n) := |\{ q \in {\mathcal S}_I: n = a_q\ (q) \}|</math>

obeys the estimate

:<math>\displaystyle \sum_{C^{-1} x \leq n \leq Cx: n = a\ (r)} \tau_{\mathcal C}(n)^2 \ll \frac{x}{r} \tau(r)^{O(1)} \log^{O(1)} x + x^{o(1)}. </math>

for any fixed <math>C > 1</math> and any congruence class <math>a\ (r)</math> with <math>r \in {\mathcal S}_I</math>. Here <math>\tau</math> is the divisor function.

=== Smooth and densely divisible numbers ===

A natural number <math>n</math> is said to be <math>y</math>-smooth if all of its prime factors are less than or equal to <math>y</math>. We say that <math>n</math> is <math>y</math>-densely divisible 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.

=== MPZ ===

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.

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

: <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.

* 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}: 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.

=== Type I, Type II, and Type III ===

Let <math>0 < \varpi < 1/4</math>, <math>0 < \delta < 1/4+\varpi</math>, and <math>0 < \sigma < 1/2</math> be fixed.

* We say that <math>Type_I[\varpi,\delta,\sigma]</math> holds if, whenever <math> M,N</math> are quantities with

:<math>\displaystyle MN \sim x </math>

and

:<math>\displaystyle x^{1/2-\sigma} \ll N \ll x^{1/2-2\varpi-c}</math>

or equivalently

:<math>\displaystyle x^{1/2+2\varpi+c} \ll M \ll x^{1/2+\sigma}</math>

for some fixed <math>c>0</math>, and <math>\alpha,\beta</math> are coefficient sequences at scale <math>M,N</math> respectively with <math>\beta</math> obeying a Siegel-Walfisz theorem, <math>I \subset [1,x^\delta]</math>, and <math>(\{a_q\})_{q \in {\mathcal S}_I}</math> is a congruence class system of controlled multiplicity, then one has

:<math>\sum_{q \in {\mathcal S}_I: q < x^{1/2+2\varpi}} |\Delta( \alpha * \beta; a_q\ (q))| \leq x \log^{-A} x</math>

for all fixed <math>A>0</math>.

* We say that <math>Type_{II}[\varpi,\delta]</math> holds if, whenever <math> M,N</math> are quantities with

:<math>\displaystyle MN \sim x </math>

and

:<math>\displaystyle x^{1/2-2\varpi-c} \ll N \ll x^{1/2}</math>

or equivalently

:<math>\displaystyle x^{1/2} \ll M \ll x^{1/2+2\varpi+c}</math>

for some sufficiently small fixed <math>c>0</math>, and <math>\alpha,\beta</math> are coefficient sequences at scale <math>M,N</math> respectively with <math>\beta</math> obeying a Siegel-Walfisz theorem, <math>I \subset [1,x^\delta]</math>, and <math>(\{a_q\})_{q \in {\mathcal S}_I}</math> is a congruence class system of controlled multiplicity, then one has

:<math>\sum_{q \in {\mathcal S}_I: q < x^{1/2+2\varpi}} |\Delta( \alpha * \beta; a_q\ (q))| \leq x \log^{-A} x</math>

for all fixed <math>A>0</math>.

* We say that <math>Type_{III}[\varpi,\delta,\sigma]</math> holds if, whenever <math>M,N_1,N_2,N_3</math> are quantities with

:<math>\displaystyle MN \sim x </math>

:<math>\displaystyle N_1N_2, N_2 N_3, N_1 N_3 \gg x^{1/2 + \sigma}</math>

:<math>\displaystyle x^{2\sigma} \ll N_1,N_2,N_3 \ll x^{1/2-\sigma},</math>

<math>\alpha,\psi_1,\psi_2,\psi_3</math> are coefficient sequences at scale <math>M,N_1,N_2,N_3</math> respectively with <math>\psi_1,\psi_2,\psi_3</math> smooth, <math>I \subset [1,x^\delta]</math>, and <math>(\{a_q\})_{q \in {\mathcal S}_I}</math> is a congruence class system of controlled multiplicity, then one has

:<math>\sum_{q \in {\mathcal S}_I: q < x^{1/2+2\varpi}} |\Delta( \alpha * \psi_1 * \psi_2 * \psi_3; a_q\ (q))| \leq x \log^{-A} x</math>

for all fixed <math>A>0</math>.

* 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 should also be a second "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 one assumes a suitable "double dense divisibility" hypothesis, which has not yet been determined precisely.

Note: thus far in the Type III analysis, the controlled multiplicity hypothesis has yet to be used.

== The combinatorial lemma ==

<blockquote>Combinatorial lemma Let <math>0 < \varpi < 1/4</math>, <math>0 < \delta < 1/4 + \varpi</math>, and <math>1/10 < \sigma < 1/2</math> be fixed.
* 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>

This lemma is (somewhat implicitly) proven [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ here]. It reduces the verification of <math>MPZ[\varpi,\delta]</math> and <math>MPZ'[\varpi,\delta]</math> to a comparison of the best available Type I, Type II, and Type III estimates, as well as the constraint <math>\sigma > 1/10</math>.

== Type I estimates ==

In all of the estimates below, <math>0 < \varpi < 1/4</math>, <math>0 < \delta < 1/4 + \varpi</math>, and <math>\sigma > 0</math> are fixed.

=== Level 1 ===

<blockquote>Type I-1 We have <math>Type'_I[\varpi,\delta,\sigma]</math> (and hence <math>Type_I[\varpi,\delta,\sigma]</math>) whenever
:<math>\displaystyle 11\varpi +3\delta + 2 \sigma < \frac{1}{4}</math>.
</blockquote>

This result is implicitly proven [http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/ here]. (There, only <math>Type_I[\varpi,\delta,\sigma]</math> is proven, but the method extends without difficulty to <math>Type'_I[\varpi,\delta,\sigma]</math>.) It uses the method of Zhang, and is ultimately based on exponential sums for incomplete Kloosterman sums on smooth moduli obtained via completion of sums.

=== Level 2 ===

<blockquote>Type I-2 We have <math>Type'_I[\varpi,\delta,\sigma]</math> (and hence <math>Type_I[\varpi,\delta,\sigma]</math>) whenever
:<math>\displaystyle 14\varpi +4\delta + \sigma < \frac{1}{4}</math>
and
:<math>\displaystyle 20\varpi +6\delta + 3\sigma < \frac{1}{2}</math>
and
:<math>\displaystyle 32\varpi +9\delta + \sigma < \frac{1}{2}</math>.
</blockquote>

This estimate is implicitly proven [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli here]. It improves upon the Level 1 estimate by using the q-van der Corput A-process in the <math>d_2</math> direction.

=== Level 3 ===

<blockquote>Type I-3 We have <math>Type'_I[\varpi,\delta,\sigma]</math> (and hence <math>Type_I[\varpi,\delta,\sigma]</math>) whenever
:<math>\displaystyle 54\varpi + 15 \delta + 5 \sigma < 1</math>
and
:<math>\displaystyle 32\varpi +9\delta + \sigma < \frac{1}{2}</math>.
</blockquote>

This estimate is tentatively established in [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 this comment]. It improves upon the Level 2 estimate by taking advantage of dense divisibility to optimise the direction of averaging.

=== Level 4 ===

By iterating the q-van der Corput A-process, one should be able to obtain <math>Type''_I[\varpi,\delta,\sigma]</math> assuming a constraint of the form

:<math>\displaystyle 40\varpi + C \delta + 4 \sigma < 1</math>

for some constant C that has not yet been determined (in part because we have not yet decided what "doubly densely divisible" means); see [http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 this comment].

=== 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.

== Type II estimates ==

In all of the estimates below, <math>0 < \varpi < 1/4</math> and <math>0 < \delta < 1/4 + \varpi</math> are fixed.

=== Level 1 ===

<blockquote>Type II-1 We have <math>Type'_{II}[\varpi,\delta]</math> (and hence <math>Type_{II}[\varpi,\delta]</math>) whenever
:<math>\displaystyle 58\varpi + 10\delta < \frac{1}{2}</math>.
</blockquote>

This estimate is implicitly proven [http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/ here]. (There, only <math>Type_I[\varpi,\delta,\sigma]</math> is proven, but the method extends without difficulty to <math>Type'_I[\varpi,\delta,\sigma]</math>.) It uses the method of Zhang, and is ultimately based on exponential sums for incomplete Kloosterman sums on smooth moduli obtained via completion of sums.

=== Level 1a ===

<blockquote>Type II-1a We have <math>Type'_{II}[\varpi,\delta]</math> (and hence <math>Type_{II}[\varpi,\delta]</math>) whenever
:<math>\displaystyle 48\varpi + 7\delta < \frac{1}{2}</math>.
</blockquote>

This estimate is implicitly proven [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli here]. It is a slight refinement of the Level 1 estimate based on a more careful inspection of the error terms in the completion of sums method.

=== Level 2 ===

In analogy with the Type I-2 estimates, one should be able to improve the Type II estimates by using the q-van der Corput process in the <math>d_2</math> direction.

=== Level 3 ===

In analogy with the Type I-3 estimates, one should be able to improve the Type II estimates by using the q-van der Corput process in an optimised direction.

=== Level 4 ===

In analogy with the Type I-4 estimates, one should be able to improve the Type II estimates by iterating the q-van der Corput A-process.

=== Level 5 ===

In analogy with the Type I-5 estimates, one should be able to improve the Type II estimates by taking advantage of averaging in the h parameters.

== Type III estimates ==

In all of the estimates below, <math>0 < \varpi < 1/4</math>, <math>0 < \delta < 1/4 + \varpi</math>, and <math>\sigma > 0</math> are fixed.

=== Level 1 ===

<blockquote>Type III-1 We have <math>Type'_{III}[\varpi,\delta,\sigma]</math> (and hence <math>Type_{III}[\varpi,\delta,\sigma]</math>) whenever

:<math>\displaystyle \frac{13}{2} (\frac{1}{2} + \sigma) > 8 (\frac{1}{2} + 2 \varpi) + \delta </math>

</blockquote>

This estimate is implicitly proven [http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/ here]. (There, only <math>Type_{III}[\varpi,\delta,\sigma]</math> is proven, but the method extends without difficulty to <math>Type'_{III}[\varpi,\delta,\sigma]</math>.) It uses the method of Zhang, using Weyl differencing and not exploiting the averaging in the <math>\alpha</math> or <math>q</math> parameters. The constraint can also be written as a lower bound on <math>\sigma</math>:

:<math>\displaystyle \sigma > \frac{3}{26} + \frac{32}{13} \varpi + \frac{2}{13} \delta</math>.

=== Level 2 ===

<blockquote>Type III-2 We have <math>Type'_{III}[\varpi,\delta,\sigma]</math> (and hence <math>Type_{III}[\varpi,\delta,\sigma]</math>) whenever

:<math>\displaystyle 1 + 5 (\frac{1}{2} + \sigma) > 8 (\frac{1}{2} + 2 \varpi) + \delta </math>

</blockquote>

This estimate is implicitly proven [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli here]. It is a refinement of the Level 1 estimate that takes advantage of the <math>\alpha</math> averaging. The constraint may also be written as a lower bound on <math>\sigma</math>:

:<math>\displaystyle \sigma > \frac{1}{10} + \frac{16}{5} \varpi + \frac{1}{5} \delta</math>.

=== Level 3 ===

<blockquote>Type III-3 We have <math>Type'_{III}[\varpi,\delta,\sigma]</math> (and hence <math>Type_{III}[\varpi,\delta,\sigma]</math>) whenever

:<math>\displaystyle \frac{3}{2} (\frac{1}{2} + \sigma) > \frac{7}{4} (\frac{1}{2} + 2 \varpi) + \frac{3}{8} \delta </math>.
</blockquote>

This estimate is proven in [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 this comment]. It uses the newer [http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation method of Fouvry, Kowalski, Michel, and Nelson] that avoids Weyl differencing. The constraint may also be written as a lower bound on <math>\sigma</math>:

:<math>\displaystyle \sigma > \frac{1}{12} + \frac{7}{3} \varpi + \frac{1}{4} \delta</math>.

=== Level 4 ===

It should be possible to improve upon the Level 3 estimate by exploiting averaging in the <math>\alpha</math> parameter (this was suggested already by Fouvry, Kowalski, Michel, and Nelson).

=== Level 5 ===

One may also hope to improve upon Level 4 estimates by exploiting Ramanujan sum cancellation (as Zhang did in his Level 1 argument).

== Combinations ==

By combining a Type I estimate, a Type II estimate, and a Type III estimate together one can get estimates of the form <math>MPZ[\varpi,\delta]</math> or <math>MPZ[\varpi',\delta']</math> for <math>\varpi,\delta</math> small enough by using the combinatorial lemma. Here are the combinations that have been arisen so far in the Polymath8 project:

{| border=1
|-
!Type I !! Type II !! Type III !! Result !! Details !! Notes
|-
|Level 1
|Level 1
|Level 1
|<math>207\varpi + 43\delta < 1/4 </math>
|[http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ details]
|-
|Level 1
|Level 1
|Level 2
|<math>87\varpi + 17\delta < 1/4 </math>
|[http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 details]
|-
|Level 2
|Level 1a
|Level 1
| <math>178\varpi + 52\delta < 1 </math>
| [http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 details]
|-
|Level 2
|Level 1a
|Level 2
| <math>148\varpi + 33\delta < 1 </math>
| [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/ details]
|-
|Level 3?
|Level 1a
|Level 2
|<math>140 \varpi + 32\delta < 1</math>?
| [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 details]
|-
|Level 4?
|Level 1a
|Level 1
|<math>96\varpi + C \delta < 1</math>?
| [http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 details]
|-
|Level 4?
|Level 2?
|Level 1
|<math>88\varpi + C \delta < 1</math>?
| [http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 details]
|-
|Level 4?
|Level 2?
|Level 2
|<math>74\varpi + C \delta < 1</math>?
| [http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 details]
|-
|Level 2
|Level 1a
|Level 3
|<math>116\varpi + 30 \delta < 1</math>
| [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 details]
|}

Talk:Bounded gaps between primes

2013-06-09T11:42:31Z

DLyons: Removing all content from page

Talk:Bounded gaps between primes

2013-06-08T12:00:04Z

DLyons:

Didn't the Montgomery-Vaughan Brun-Titchmarsh lower bound preclude the posted values of H < 211,046?

Talk:Bounded gaps between primes

2013-06-08T10:42:22Z

DLyons:

Didn't the Selberg lower bound preclude the posted values of H <math> < </math> 234,322?

Talk:Bounded gaps between primes

2013-06-08T10:41:32Z

DLyons: TeX??

Didn't the Selberg lower bound preclude the posted values of H <math> < </math> 234,642?

Talk:Bounded gaps between primes

2013-06-08T10:39:16Z

DLyons: TeX

Didn't the Selberg lower bound preclude the posted values of H <math>\lt</math> 234,642?

Talk:Bounded gaps between primes

2013-06-08T10:37:42Z

DLyons: TeX

Didn't the Selberg lower bound preclude the posted values of H <math>\lt<\math> 234,642?

Talk:Bounded gaps between primes

2013-06-08T10:37:10Z

DLyons: TeX

Didn't the Selberg lower bound preclude the posted values of H <math>\lt 234,642<\math>?

Talk:Bounded gaps between primes

2013-06-08T10:35:40Z

DLyons: A very naive question

Didn't the Selberg lower bound preclude the posted values of H \lt 234,642?

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Distribution of primes in smooth moduli

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Talk:Bounded gaps between primes

Talk:Bounded gaps between primes

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