Distribution of primes in smooth moduli

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]\displaystyle{ MPZ[\varpi,\delta] }[/math] (and later strengthened to [math]\displaystyle{ MPZ'[\varpi,\delta] }[/math]). These estimates are obtained as a combination of three other estimates, which we will call [math]\displaystyle{ Type_I[\varpi,\delta,\sigma] }[/math], [math]\displaystyle{ Type_{II}[\varpi,\delta] }[/math], and [math]\displaystyle{ Type_{III}[\varpi,\delta,\sigma] }[/math].

Definitions

Asymptotic notation

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

Coefficient sequences

We need a fixed quantity [math]\displaystyle{ A_0\gt 0 }[/math].

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

[math]\displaystyle{ \displaystyle |\alpha(n)| \ll \tau^{O(1)}(n) \log^{O(1)}(x) }[/math]

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

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

• A coefficient sequence [math]\displaystyle{ \alpha }[/math] at scale [math]\displaystyle{ N }[/math] is said to obey the Siegel-Walfisz theorem if one has

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

for any [math]\displaystyle{ q,r \geq 1 }[/math], any fixed [math]\displaystyle{ A }[/math], and any primitive residue class [math]\displaystyle{ a\ (r) }[/math].

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

[math]\displaystyle{ \displaystyle \psi^{(j)}(t) = O( \log^{j A_0} x ) }[/math]

for all fixed [math]\displaystyle{ j \geq 0 }[/math] (note that the implied constant in the [math]\displaystyle{ O() }[/math] notation may depend on [math]\displaystyle{ j }[/math]).

Congruence class systems

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

• A singleton congruence class system on [math]\displaystyle{ I }[/math] is a collection [math]\displaystyle{ {\mathcal C} = (\{a_q\})_{q \in {\mathcal S}_I} }[/math] of primitive residue classes [math]\displaystyle{ 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{ \displaystyle a_{qr}\ (qr) = (a_q\ (q)) \cap (a_r\ (r)) }[/math]

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

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

obeys the estimate

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

for any fixed [math]\displaystyle{ C \gt 1 }[/math] and any congruence class [math]\displaystyle{ a\ (r) }[/math] with [math]\displaystyle{ r \in {\mathcal S}_I }[/math]. Here [math]\displaystyle{ \tau }[/math] is the divisor function. [Actually, in the most recent proofs of Type I, II, and III estimates, the controlled multiplicity hypothesis is no longer needed, and so this definition is no longer relevant for the project.]

Smooth and densely divisible numbers

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

We let [math]\displaystyle{ {\mathcal D}_y }[/math] denote the space of [math]\displaystyle{ y }[/math]-densely divisible numbers, and [math]\displaystyle{ {\mathcal D}_y^2 }[/math] the space of doubly densely divisible numbers, thus

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

MPZ and variants

Let [math]\displaystyle{ 0 \lt \varpi \lt 1/4 }[/math] and [math]\displaystyle{ 0 \lt \delta \lt \varpi + 1/4 }[/math] be fixed. Let [math]\displaystyle{ \Lambda }[/math] denote the von Mangoldt function.

• We say that the estimate [math]\displaystyle{ MPZ[\varpi,\delta] }[/math] holds if one has the estimate

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

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

• We say that the estimate [math]\displaystyle{ MPZ'[\varpi,\delta] }[/math] holds if one has the estimate

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

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

• We say that the estimate [math]\displaystyle{ MPZ''[\varpi,\delta] }[/math] holds if one has the estimate

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

for any fixed [math]\displaystyle{ A \gt 0 }[/math], any [math]\displaystyle{ I \subset {\mathbf R} }[/math], and any congruence class system [math]\displaystyle{ (\{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

Let [math]\displaystyle{ 0 \lt \varpi \lt 1/4 }[/math], [math]\displaystyle{ 0 \lt \delta \lt 1/4+\varpi }[/math], and [math]\displaystyle{ 0 \lt \sigma \lt 1/2 }[/math] be fixed.

• We say that [math]\displaystyle{ Type_I[\varpi,\delta,\sigma] }[/math] holds if, whenever [math]\displaystyle{ M,N }[/math] are quantities with

[math]\displaystyle{ \displaystyle MN \sim x }[/math]

and

[math]\displaystyle{ \displaystyle x^{1/2-\sigma} \ll N \ll x^{1/2-2\varpi-c} }[/math]

or equivalently

[math]\displaystyle{ \displaystyle x^{1/2+2\varpi+c} \ll M \ll x^{1/2+\sigma} }[/math]

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

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

for all fixed [math]\displaystyle{ A\gt 0 }[/math].

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

[math]\displaystyle{ \displaystyle MN \sim x }[/math]

and

[math]\displaystyle{ \displaystyle x^{1/2-2\varpi-c} \ll N \ll x^{1/2} }[/math]

or equivalently

[math]\displaystyle{ \displaystyle x^{1/2} \ll M \ll x^{1/2+2\varpi+c} }[/math]

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

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

for all fixed [math]\displaystyle{ A\gt 0 }[/math].

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

[math]\displaystyle{ \displaystyle MN \sim x }[/math]

[math]\displaystyle{ \displaystyle N_1N_2, N_2 N_3, N_1 N_3 \gg x^{1/2 + \sigma} }[/math]

[math]\displaystyle{ \displaystyle x^{2\sigma} \ll N_1,N_2,N_3 \ll x^{1/2-\sigma}, }[/math]

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

[math]\displaystyle{ \sum_{q \in {\mathcal S}_I: q \lt 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]\displaystyle{ A\gt 0 }[/math].

• We define [math]\displaystyle{ Type'_I[\varpi,\delta,\sigma] }[/math], [math]\displaystyle{ Type'_{II}[\varpi,\delta] }[/math], [math]\displaystyle{ Type_{III}[\varpi,\delta,\sigma] }[/math] analogously to [math]\displaystyle{ Type_I[\varpi,\delta,\sigma] }[/math], [math]\displaystyle{ Type_{II}[\varpi,\delta] }[/math], [math]\displaystyle{ Type_{III}[\varpi,\delta,\sigma] }[/math] but with the hypothesis [math]\displaystyle{ I \subset [1,x^\delta] }[/math] replaced with [math]\displaystyle{ I \subset \mathbf{R} }[/math], and [math]\displaystyle{ {\mathcal S}_I }[/math] replaced with [math]\displaystyle{ {\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]\displaystyle{ 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

Combinatorial lemma Let [math]\displaystyle{ 0 \lt \varpi \lt 1/4 }[/math], [math]\displaystyle{ 0 \lt \delta \lt 1/4 + \varpi }[/math], and [math]\displaystyle{ 1/10 \lt \sigma \lt 1/2 }[/math] be fixed.

• If [math]\displaystyle{ Type_I[\varpi,\delta,\sigma] }[/math], [math]\displaystyle{ Type_{II}[\varpi,\delta] }[/math], and [math]\displaystyle{ Type_{III}[\varpi,\delta,\sigma] }[/math] all hold, then [math]\displaystyle{ MPZ[\varpi,\delta] }[/math] holds.
• Similarly, if [math]\displaystyle{ Type'_I[\varpi,\delta,\sigma] }[/math], [math]\displaystyle{ Type'_{II}[\varpi,\delta] }[/math], and [math]\displaystyle{ Type'_{III}[\varpi,\delta,\sigma] }[/math] all hold, then [math]\displaystyle{ MPZ'[\varpi,\delta] }[/math] holds.
• Similarly, if [math]\displaystyle{ Type''_I[\varpi,\delta,\sigma] }[/math], [math]\displaystyle{ Type''_{II}[\varpi,\delta] }[/math], and [math]\displaystyle{ Type''_{III}[\varpi,\delta,\sigma] }[/math] all hold, then [math]\displaystyle{ MPZ''[\varpi,\delta] }[/math] holds.

This lemma is (somewhat implicitly) proven here. It reduces the verification of [math]\displaystyle{ MPZ[\varpi,\delta] }[/math] and [math]\displaystyle{ 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]\displaystyle{ \sigma \gt 1/10 }[/math].

Type I estimates

In all of the estimates below, [math]\displaystyle{ 0 \lt \varpi \lt 1/4 }[/math], [math]\displaystyle{ 0 \lt \delta \lt 1/4 + \varpi }[/math], and [math]\displaystyle{ \sigma \gt 0 }[/math] are fixed.

Level 1

Type I-1 We have [math]\displaystyle{ Type'_I[\varpi,\delta,\sigma] }[/math] (and hence [math]\displaystyle{ Type_I[\varpi,\delta,\sigma] }[/math]) whenever

[math]\displaystyle{ \displaystyle 11\varpi +3\delta + 2 \sigma \lt \frac{1}{4} }[/math].

This result is implicitly proven here. (There, only [math]\displaystyle{ Type_I[\varpi,\delta,\sigma] }[/math] is proven, but the method extends without difficulty to [math]\displaystyle{ 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

Type I-2 We have [math]\displaystyle{ Type'_I[\varpi,\delta,\sigma] }[/math] (and hence [math]\displaystyle{ Type_I[\varpi,\delta,\sigma] }[/math]) whenever

[math]\displaystyle{ \displaystyle 17\varpi +4\delta + \sigma \lt \frac{1}{4} }[/math]

and

[math]\displaystyle{ \displaystyle 20\varpi +6\delta + 3\sigma \lt \frac{1}{2} }[/math]

and

[math]\displaystyle{ \displaystyle 32\varpi +9\delta + \sigma \lt \frac{1}{2} }[/math].

This estimate is implicitly proven here. It improves upon the Level 1 estimate by using the q-van der Corput A-process in the [math]\displaystyle{ d_2 }[/math] direction. The final constraint [math]\displaystyle{ 32\varpi +9\delta + \sigma \lt \frac{1}{2} }[/math] was removed in this comment.

Level 3

Type I-3 We have [math]\displaystyle{ Type'_I[\varpi,\delta,\sigma] }[/math] (and hence [math]\displaystyle{ Type_I[\varpi,\delta,\sigma] }[/math]) whenever

[math]\displaystyle{ \displaystyle 54\varpi + 15 \delta + 5 \sigma \lt 1 }[/math].

This estimate is established here (it was previously tentatively established in this comment with an additional condition [math]\displaystyle{ 32 \varpi + 9 \delta + \sigma \lt 1/2 }[/math], which can now be dropped, thanks to an improved control on a secondary error term in the exponential sum estimates). 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, it appears that one can obtain [math]\displaystyle{ Type_I[\varpi,\delta,\sigma] }[/math] assuming a constraint of the form

[math]\displaystyle{ \displaystyle \frac{236}{3}\varpi + \frac{64}{3} \delta + 4 \sigma \lt 1 }[/math]

but this is inferior to the Level 3 estimates in practice. Details can be found 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]\displaystyle{ n,q_1 }[/math] (leaving [math]\displaystyle{ h,q_2 }[/math] to be doubled), factor [math]\displaystyle{ q_2 = r_2 s_2 }[/math] and Cauchy-Schwarz in [math]\displaystyle{ n,q_1,r_2 }[/math] and only double [math]\displaystyle{ h,s_2 }[/math]. The idea is to make the diagonal case [math]\displaystyle{ h s'_2 = h' s_2 }[/math] do more of the work and the off-diagonal case [math]\displaystyle{ hs'_2 \neq h' s_2 }[/math] do less of the work. This idea was first raised here. Preliminary computations suggest that this allows one to take [math]\displaystyle{ 56 \varpi + 16 \delta + 4 \sigma \lt 1 }[/math] for the Type I sums in [math]\displaystyle{ x^\delta }[/math]-smooth case. In this post it is shown that the same bound holds in the densely divisible case, thus [math]\displaystyle{ Type''_I[\varpi,\delta,\sigma] }[/math] holds whenever

[math]\displaystyle{ 56 \varpi + 16 \delta + 4 \sigma \lt 1 }[/math].

Level 5

(The numbering here is out of order because the Level 5 estimates proved harder to implement than the Level 6 estimates.)

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

[math]\displaystyle{ \frac{160}{3} \varpi + 16 \delta + \frac{34}{9} \sigma \lt 1 }[/math]

together with the secondary condition

[math]\displaystyle{ 64\varpi+18\delta+2\sigma \lt 1 }[/math];

see this comment.

A further averaging in the [math]\displaystyle{ l }[/math] parameter has led to a preliminary improvement of the main condition to

[math]\displaystyle{ 48 \varpi + \frac{44}{3} \delta + \frac{38}{9} \sigma \lt 1 }[/math],

see this comment.

Type II estimates

In all of the estimates below, [math]\displaystyle{ 0 \lt \varpi \lt 1/4 }[/math] and [math]\displaystyle{ 0 \lt \delta \lt 1/4 + \varpi }[/math] are fixed.

Level 1

Type II-1 We have [math]\displaystyle{ Type'_{II}[\varpi,\delta] }[/math] (and hence [math]\displaystyle{ Type_{II}[\varpi,\delta] }[/math]) whenever

[math]\displaystyle{ \displaystyle 58\varpi + 10\delta \lt \frac{1}{2} }[/math].

This estimate is implicitly proven here. (There, only [math]\displaystyle{ Type_I[\varpi,\delta,\sigma] }[/math] is proven, but the method extends without difficulty to [math]\displaystyle{ 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

Type II-1a We have [math]\displaystyle{ Type'_{II}[\varpi,\delta] }[/math] (and hence [math]\displaystyle{ Type_{II}[\varpi,\delta] }[/math]) whenever

[math]\displaystyle{ \displaystyle 48\varpi + 7\delta \lt \frac{1}{2} }[/math].

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 1b

Type II-1b We have [math]\displaystyle{ Type'_{II}[\varpi,\delta] }[/math] (and hence [math]\displaystyle{ Type_{II}[\varpi,\delta] }[/math]) whenever

[math]\displaystyle{ \displaystyle 38\varpi + 7\delta \lt \frac{1}{2} }[/math].

This refinement of the Level 1a estimate came from realising that in the Type II case, the R parameter can be selected to lie in the range [math]\displaystyle{ [x^{1/2-2\varpi-\delta-\varepsilon}, x^{1/2-2\varpi-\varepsilon}] }[/math] rather than [math]\displaystyle{ [x^{-2\varpi-\delta-\varepsilon} N, x^{-2\varpi-\varepsilon} N] }[/math]. See this comment for details.

Level 1c

Type II-1c We have [math]\displaystyle{ Type'_{II}[\varpi,\delta] }[/math] (and hence [math]\displaystyle{ Type_{II}[\varpi,\delta] }[/math] and [math]\displaystyle{ Type''_{II}[\varpi,\delta] }[/math]) whenever

[math]\displaystyle{ \displaystyle 34\varpi + 7\delta \lt \frac{1}{2} }[/math].

This further refinement of the Level 1b estimate came from realising that R can in fact range in [math]\displaystyle{ [x^{-\delta-\varepsilon} N, x^{-\varepsilon} N] }[/math] if one strengthens the controlled multiplicity hypothesis slightly; see this comment or this post for details.

Level 2

In analogy with the Type I-2 estimates, one could hope to improve the Type II estimates by using the q-van der Corput process in the [math]\displaystyle{ d_2 }[/math] direction. Interestingly, however, it appears that the Type II numerology lies outside of the range in which the van der Corput process is beneficial (at least if one only applies it once), so the Level 2 estimate looks to be inferior to the Level 1b estimate.

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. As with Level 2 estimates though, it appears that Level 3 estimates are inferior to the Level 1b estimate.

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.

Level 6

Even further improvement in the Type II sums may be possible by rebalancing the final Cauchy-Schwarz: instead of performing Cauchy-Schwarz in [math]\displaystyle{ n }[/math] (leaving [math]\displaystyle{ h,q_1, q_2 }[/math] to be doubled), factor [math]\displaystyle{ q_1 = r_1 s_1 }[/math] and Cauchy-Schwarz in [math]\displaystyle{ n,r_1 }[/math] and only double [math]\displaystyle{ h,s_1,q_2 }[/math]. The idea is to make the diagonal case [math]\displaystyle{ h s'_1 q'_2 = h' s_1 q_2 }[/math] do more of the work and the off-diagonal case [math]\displaystyle{ hs'_1 q'_2 \neq h' s_1 q_2 }[/math] do less of the work. This idea was first raised here.

Type III estimates

In all of the estimates below, [math]\displaystyle{ 0 \lt \varpi \lt 1/4 }[/math], [math]\displaystyle{ 0 \lt \delta \lt 1/4 + \varpi }[/math], and [math]\displaystyle{ \sigma \gt 0 }[/math] are fixed.

Level 1

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

[math]\displaystyle{ \displaystyle \frac{13}{2} (\frac{1}{2} + \sigma) \gt 8 (\frac{1}{2} + 2 \varpi) + \delta }[/math]

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

[math]\displaystyle{ \displaystyle \sigma \gt \frac{3}{26} + \frac{32}{13} \varpi + \frac{2}{13} \delta }[/math].

Level 2

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

[math]\displaystyle{ \displaystyle 1 + 5 (\frac{1}{2} + \sigma) \gt 8 (\frac{1}{2} + 2 \varpi) + \delta }[/math]

This estimate is implicitly proven here. It is a refinement of the Level 1 estimate that takes advantage of the [math]\displaystyle{ \alpha }[/math] averaging. The constraint may also be written as a lower bound on [math]\displaystyle{ \sigma }[/math]:

[math]\displaystyle{ \displaystyle \sigma \gt \frac{1}{10} + \frac{16}{5} \varpi + \frac{1}{5} \delta }[/math].

Level 3

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

[math]\displaystyle{ \displaystyle \frac{3}{2} (\frac{1}{2} + \sigma) \gt \frac{7}{4} (\frac{1}{2} + 2 \varpi) + \frac{3}{8} \delta }[/math].

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 [math]\displaystyle{ \sigma }[/math]:

[math]\displaystyle{ \displaystyle \sigma \gt \frac{1}{12} + \frac{7}{3} \varpi + \frac{1}{4} \delta }[/math].

Level 4

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

[math]\displaystyle{ \displaystyle \frac{1}{4} + \frac{3}{4} \frac{3}{2} (\frac{1}{2} + \sigma) \gt \frac{7}{4} (\frac{1}{2} + 2 \varpi) + \frac{1}{4} \delta }[/math].

This estimate is proven in this comment and then in this post. It modifies the Level 3 argument by exploiting averaging in the [math]\displaystyle{ \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]\displaystyle{ \sigma }[/math]:

[math]\displaystyle{ \displaystyle \sigma \gt \frac{1}{18} + \frac{28}{9} \varpi + \frac{2}{9} \delta }[/math].

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).

Level 6

An alternative way to improve upon Level 4 estimates would be to use the q-van der Corput process to bound incomplete Kloosterman correlations.

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]\displaystyle{ MPZ[\varpi,\delta] }[/math] or [math]\displaystyle{ MPZ[\varpi',\delta'] }[/math] for [math]\displaystyle{ \varpi,\delta }[/math] 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	Where optimum is obtained
Level 1	Level 1	Level 1	[math]\displaystyle{ 828\varpi + 172\delta \lt 1 }[/math]	details	Type I / Type III border
Level 1	Level 1	Level 2	[math]\displaystyle{ 348\varpi + 68\delta \lt 1 }[/math]	details	Type I / Type III border
Level 2	Level 1a	Level 1	[math]\displaystyle{ 178\varpi + 52\delta \lt 1 }[/math]	details	Type I / Type III border
Level 2	Level 1a	Level 2	[math]\displaystyle{ 148\varpi + 33\delta \lt 1 }[/math]	details	Type I / Type III border
Level 3?	Level 1a	Level 2	[math]\displaystyle{ 140 \varpi + 32\delta \lt 1 }[/math]?	details	Type I / Type III border
Level 2	Level 1a	Level 3	[math]\displaystyle{ 116\varpi + 25.5 \delta \lt 1 }[/math]	details refinement	Type I / Type III border
Level 3	Level 1a	Level 3	[math]\displaystyle{ 112 \frac{4}{7} \varpi+27 \frac{6}{7} \delta \lt 1 }[/math]	details	Type I / Type III border
Level 3	Level 1c	Level 4	[math]\displaystyle{ 108\varpi+30\delta \lt 1 }[/math]	details	Type I / combinatorial border
Level 6	Level 1c	Level 4	[math]\displaystyle{ \frac{280}{3} \varpi + \frac{80}{3} \delta \lt 1 }[/math]	details	Type I / combinatorial border
Level 5	Level 1c	Level 4	[math]\displaystyle{ \frac{600}{7} \varpi + \frac{180}{7} \delta \lt 1 }[/math]	details	Type I / combinatorial border

For simplicity, only the constraint that is relevant for near-maximal values of [math]\displaystyle{ \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 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]\displaystyle{ \varpi \gt 1/200 }[/math], in the base constraints to restrict attention to the regime of large values of [math]\displaystyle{ \varpi }[/math].

with(SolveTools[Inequality]);
base := [ sigma > 1/10, sigma < 1/2, varpi > 0, varpi < 1/4, delta > 0, delta < 1/4+varpi ];
typeI_1 := [ 11 * varpi + 3 * delta + 2 * sigma < 1/4 ];
typeI_2 := [ 17 * varpi + 4 * delta + sigma < 1/4, 20 * varpi + 6 * delta + 3 * sigma < 1/2, 32 * varpi + 9 * delta + sigma < 1/2 ];
typeI_3 := [ 54 * varpi + 15 * delta + 5 * sigma < 1 ];
typeI_4 := [ 236 * varpi/3 + 64 * delta/3 + 4*sigma < 1];
typeI_6 := [ 56 * varpi + 16*delta + 4*sigma < 1];
typeI_5 := [ 160*varpi/3 + 16*delta + 34*sigma/9 < 1, 64*\varpi+18*\delta+2*\sigma < 1];
typeII_1 := [ 58 * varpi + 10 * delta < 1/2 ];
typeII_1a := [48 * varpi + 7 * delta < 1/2 ];
typeII_1b := [38 * varpi + 7 * delta < 1/2 ];
typeII_1c := [34 * varpi + 7 * delta < 1/2 ];
typeIII_1 := [ (13/2) * (1/2 + sigma) > 8 * (1/2 + 2*varpi) + delta ];
typeIII_2 := [ 1 + 5 * (1/2 + sigma) > 8 * (1/2 + 2*varpi) + delta ];
typeIII_3 := [ 3/2 * (1/2 + sigma) > (7/4) * (1/2 + 2*varpi) + (3/8) * delta ];
typeIII_4 := [ 1/4 + (3/4) * (3/2) * (1/2 + sigma) > (7/4) * (1/2 + 2*varpi) + (1/4) * delta ];
constraints := [ op(base), op(typeI_5), op(typeII_1c), op(typeIII_4) ];
LinearMultivariateSystem(constraints, [varpi,delta,sigma]);