# Distribution of primes in smooth moduli

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

- | <blockquote><b>Combinatorial lemma</b> Let <math>0 < \varpi < 1/4</math>, <math>0 < \delta < 1/4 + \varpi</math>, and <math>1/10 < \sigma < 1/ | + | <blockquote><b>Combinatorial lemma</b> 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. | * 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. |

## Revision as of 10:25, 26 June 2013

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 (and later strengthened to . These estimates are obtained as a combination of three other estimates, which we will call , , and .

## Contents |

## Definitions

### Asymptotic notation

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

### Coefficient sequences

We need a fixed quantity *A*_{0} > 0.

A **coefficient sequence** is a finitely supported sequence that obeys the bounds

- If α is a coefficient sequence and is a primitive residue class, the (signed)
*discrepancy*of α in the sequence is defined to be the quantity

- A coefficient sequence α is said to be
*at scale*for some if it is supported on an interval of the form .*N*

- A coefficient sequence α at scale
*N*is said to*obey the Siegel-Walfisz theorem*if one has

for any , any fixed *A*, and any primitive residue class .

- A coefficient sequence α at scale
*N*is said to be*smooth*if it takes the form α(*n*) = ψ(*n*/*N*) for some smooth function supported on obeying the derivative bounds

for all fixed (note that the implied constant in the *O*() notation may depend on *j*).

### Congruence class systems

Let , and let denote the square-free numbers whose prime factors lie in *I*.

- A
*singleton congruence class system*on*I*is a collection of primitive residue classes for each </math>q \in {\mathcal S}_I</math>, obeying the Chinese remainder theorem property

whenever are coprime. We say that such a system has *controlled multiplicity* if the quantity

obeys the estimate

for any fixed *C* > 1 and any congruence class with . Here τ is the divisor function.

### Smooth and densely divisible numbers

A natural number *n* is said to be * y-smooth* if all of its prime factors are less than or equal to

*y*. We say that

*n*is

*if, for every , one can find a factor of*

*y*-densely divisible*n*in the interval [

*y*

^{ − 1}

*R*,

*R*]. Note that

*y*-smooth numbers are automatically

*y*-densely divisible, but the converse is not true in general.

### MPZ

Let and be fixed. Let Λ denote the von Mangoldt function.

- We say that the estimate holds if one has the estimate

for any fixed *A* > 0, any , and any congruence class system of controlled multiplicity.

- We say that the estimate holds if one has the estimate

for any fixed *A* > 0, any , and any congruence class system of controlled multiplicity.

### Type I, Type II, and Type III

Let , , and 0 < σ < 1 / 2 be fixed.

- We say that holds if, whenever
*M*,*N*are quantities with

and

or equivalently

for some fixed *c* > 0, and α,β are coefficient sequences at scale *M*,*N* respectively with β obeying a Siegel-Walfisz theorem, , and is a congruence class system of controlled multiplicity, then one has

for all fixed *A* > 0.

- We say that holds if, whenever
*M*,*N*are quantities with

and

or equivalently

for some sufficiently small fixed *c* > 0, and α,β are coefficient sequences at scale *M*,*N* respectively with β obeying a Siegel-Walfisz theorem, , and is a congruence class system of controlled multiplicity, then one has

for all fixed *A* > 0.

- We say that holds if, whenever
*M*,*N*_{1},*N*_{2},*N*_{3}are quantities with

α,ψ_{1},ψ_{2},ψ_{3} are coefficient sequences at scale *M*,*N*_{1},*N*_{2},*N*_{3} respectively with ψ_{1},ψ_{2},ψ_{3} smooth, , and is a congruence class system of controlled multiplicity, then one has

for all fixed *A* > 0.

- We define , , analogously to , , but with the hypothesis replaced with , and replaced with . These estimates are slightly stronger than their unprimed counterparts.

There should also be a second "double-primed" variant 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

Combinatorial lemmaLet , , and 1 / 10 < σ < 1 / 2 be fixed.

- If , , and all hold, then holds.
- Similarly, if , , and all hold, then holds.

This lemma is (somewhat implicitly) proven here. It reduces the verification of and to a comparison of the best available Type I, Type II, and Type III estimates, as well as the constraint σ > 1 / 10.

## Type I estimates

In all of the estimates below, , , and σ > 0 are fixed.

### Level 1

Type I-1We have (and hence ) whenever

- .

This result is implicitly proven here. (There, only is proven, but the method extends without difficulty to .) 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

Type I-2We have (and hence ) wheneverand

and

- .

This estimate is implicitly proven here. It improves upon the Level 1 estimate by using the q-van der Corput A-process in the *d*_{2} direction.

### Level 3

Type I-3We have (and hence ) wheneverand

- .

This estimate is tentatively established in 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 assuming a constraint of the form

for some constant C that has not yet been determined (in part because we have not yet decided what "doubly densely divisible" means); see 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, and are fixed.

### Level 1

Type II-1We have (and hence ) whenever

- .

This estimate is implicitly proven here. (There, only is proven, but the method extends without difficulty to .) 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

Type II-1aWe have (and hence ) whenever

- .

This estimate is implicitly proven 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 *d*_{2} 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, , , and σ > 0 are fixed.

### Level 1

Type III-1We have (and hence ) whenever

This estimate is implicitly proven here. (There, only is proven, but the method extends without difficulty to .) It uses the method of Zhang, using Weyl differencing and not exploiting the averaging in the α or *q* parameters. The constraint can also be written as a lower bound on σ:

- .

### Level 2

Type III-2We have (and hence ) whenever

This estimate is implicitly proven here. It is a refinement of the Level 1 estimate that takes advantage of the α averaging. The constraint may also be written as a lower bound on σ:

- .

### Level 3

Type III-3We have (and hence ) whenever

- .

This estimate is proven in this comment. It uses the newer method of Fouvry, Kowalski, Michel, and Nelson that avoids Weyl differencing. The constraint may also be written as a lower bound on σ:

- .

### Level 4

It should be possible to improve upon the Level 3 estimate by exploiting averaging in the α 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 or for small enough by using the combinatorial lemma. Here are the combinations that have been arisen so far in the Polymath8 project:

Type I | Type II | Type III | Result | Details | Notes |
---|---|---|---|---|---|

Level 1 | Level 1 | Level 1 | details | ||

Level 1 | Level 1 | Level 2 | details | ||

Level 2 | Level 1a | Level 1 | details | ||

Level 2 | Level 1a | Level 2 | details | ||

Level 3? | Level 1a | Level 2 | ? | details | |

Level 4? | Level 1a | Level 1 | ? | details | |

Level 4? | Level 2? | Level 1 | ? | details | |

Level 4? | Level 2? | Level 2 | ? | details | |

Level 2 | Level 1a | Level 3 | details |