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As in all previous posts in this series, we adopt the following 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 ${O(), o(), \ll}$ 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 ${X \lessapprox Y}$ for ${X \ll x^{o(1)} Y}$, and ${X \sim Y}$ for ${X \ll Y \ll X}$.

The purpose of this (rather technical) post is both to roll over the polymath8 research thread from this previous post, and also to record the details of the latest improvement to the Type I estimates (based on exploiting additional averaging and using Deligne’s proof of the Weil conjectures) which lead to a slight improvement in the numerology.

In order to obtain this new Type I estimate, we need to strengthen the previously used properties of “dense divisibility” or “double dense divisibility” as follows.

Definition 1 (Multiple dense divisibility) Let ${y \geq 1}$. For each natural number ${k \geq 0}$, we define a notion of ${k}$-tuply ${y}$-dense divisibility recursively as follows:

• Every natural number ${n}$ is ${0}$-tuply ${y}$-densely divisible.
• If ${k \geq 1}$ and ${n}$ is a natural number, we say that ${n}$ is ${k}$-tuply ${y}$-densely divisible if, whenever ${i,j \geq 0}$ are natural numbers with ${i+j=k-1}$, and ${1 \leq R \leq n}$, one can find a factorisation ${n = qr}$ with ${y^{-1} R \leq r \leq R}$ such that ${q}$ is ${i}$-tuply ${y}$-densely divisible and ${r}$ is ${j}$-tuply ${y}$-densely divisible.

We let ${{\mathcal D}^{(k)}_y}$ denote the set of ${k}$-tuply ${y}$-densely divisible numbers. We abbreviate “${1}$-tuply densely divisible” as “densely divisible”, “${2}$-tuply densely divisible” as “doubly densely divisible”, and so forth; we also abbreviate ${{\mathcal D}^{(1)}_y}$ as ${{\mathcal D}_y}$.

Given any finitely supported sequence ${\alpha: {\bf N} \rightarrow {\bf C}}$ and any primitive residue class ${a\ (q)}$, we define the discrepancy

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

We now recall the key concept of a coefficient sequence, with some slight tweaks in the definitions that are technically convenient for this post.

Definition 2 A coefficient sequence is a finitely supported sequence ${\alpha: {\bf N} \rightarrow {\bf R}}$ that obeys the bounds

$\displaystyle |\alpha(n)| \ll \tau^{O(1)}(n) \log^{O(1)}(x) \ \ \ \ \ (1)$

for all ${n}$, where ${\tau}$ is the divisor function.

• (i) A coefficient sequence ${\alpha}$ is said to be located at scale ${N}$ for some ${N \geq 1}$ if it is supported on an interval of the form ${[cN, CN]}$ for some ${1 \ll c < C \ll 1}$.
• (ii) A coefficient sequence ${\alpha}$ located at scale ${N}$ for some ${N \geq 1}$ is said to obey the Siegel-Walfisz theorem if one has

$\displaystyle | \Delta(\alpha 1_{(\cdot,q)=1}; a\ (r)) | \ll \tau(qr)^{O(1)} N \log^{-A} x \ \ \ \ \ (2)$

for any ${q,r \geq 1}$, any fixed ${A}$, and any primitive residue class ${a\ (r)}$.

• (iii) A coefficient sequence ${\alpha}$ is said to be smooth at scale ${N}$ for some ${N > 0}$ is said to be smooth if it takes the form ${\alpha(n) = \psi(n/N)}$ for some smooth function ${\psi: {\bf R} \rightarrow {\bf C}}$ supported on an interval of size ${O(1)}$ and obeying the derivative bounds

$\displaystyle |\psi^{(j)}(t)| \lesssim \log^{O(1)} x \ \ \ \ \ (3)$

for all fixed ${j \geq 0}$ (note that the implied constant in the ${O()}$ notation may depend on ${j}$).

Note that we allow sequences to be smooth at scale ${N}$ without being located at scale ${N}$; for instance if one arbitrarily translates of a sequence that is both smooth and located at scale ${N}$, it will remain smooth at this scale but may not necessarily be located at this scale any more. Note also that we allow the smoothness scale ${N}$ of a coefficient sequence to be less than one. This is to allow for the following convenient rescaling property: if ${n \mapsto \psi(n)}$ is smooth at scale ${N}$, ${q \geq 1}$, and ${a}$ is an integer, then ${n \mapsto \psi(qn+a)}$ is smooth at scale ${N/q}$, even if ${N/q}$ is less than one.

Now we adapt the Type I estimate to the ${k}$-tuply densely divisible setting.

Definition 3 (Type I estimates) Let ${0 < \varpi < 1/4}$, ${0 < \delta < 1/4+\varpi}$, and ${0 < \sigma < 1/2}$ be fixed quantities, and let ${k \geq 1}$ be a fixed natural number. We let ${I}$ be an arbitrary bounded subset of ${{\bf R}}$, let ${P_I := \prod_{p \in I} p}$, and let ${a\ (P_I)}$ a primitive congruence class. We say that ${Type^{(k)}_I[\varpi,\delta,\sigma]}$ holds if, whenever ${M, N \gg 1}$ are quantities with

$\displaystyle M N \sim x \ \ \ \ \ (4)$

and

$\displaystyle x^{1/2-\sigma} \lessapprox N \lessapprox x^{1/2-2\varpi-c} \ \ \ \ \ (5)$

for some fixed ${c>0}$, and ${\alpha,\beta}$ are coefficient sequences located at scales ${M,N}$ respectively, with ${\beta}$ obeying a Siegel-Walfisz theorem, we have

$\displaystyle \sum_{q \in {\mathcal S}_I \cap {\mathcal D}_{x^\delta}^{(k)}: q \leq x^{1/2+2\varpi}} |\Delta(\alpha * \beta; a\ (q))| \ll x \log^{-A} x \ \ \ \ \ (6)$

for any fixed ${A>0}$. Here, as in previous posts, ${{\mathcal S}_I}$ denotes the square-free natural numbers whose prime factors lie in ${I}$.

The main theorem of this post is then

Theorem 4 (Improved Type I estimate) We have ${Type^{(4)}_I[\varpi,\delta,\sigma]}$ whenever

$\displaystyle \frac{160}{3} \varpi + 16 \delta + \frac{34}{9} \sigma < 1$

and

$\displaystyle 64\varpi + 18\delta + 2\sigma < 1.$

In practice, the first condition here is dominant. Except for weakening double dense divisibility to quadruple dense divisibility, this improves upon the previous Type I estimate that established ${Type^{(2)}_I[\varpi,\delta,\sigma]}$ under the stricter hypothesis

$\displaystyle 56 \varpi + 16 \delta + 4 \sigma < 1.$

As in previous posts, Type I estimates (when combined with existing Type II and Type III estimates) lead to distribution results of Motohashi-Pintz-Zhang type. For any fixed ${\varpi, \delta > 0}$ and ${k \geq 1}$, we let ${MPZ^{(k)}[\varpi,\delta]}$ denote the assertion that

$\displaystyle \sum_{q \in {\mathcal S}_I \cap {\mathcal D}_{x^\delta}^{(k)}: q \leq x^{1/2+2\varpi}} |\Delta(\Lambda 1_{[x,2x]}; a\ (q))| \ll x \log^{-A} x \ \ \ \ \ (7)$

for any fixed ${A > 0}$, any bounded ${I}$, and any primitive ${a\ (P_I)}$, where ${\Lambda}$ is the von Mangoldt function.

Corollary 5 We have ${MPZ^{(4)}[\varpi,\delta]}$ whenever

$\displaystyle \frac{600}{7} \varpi + \frac{180}{7} \delta < 1 \ \ \ \ \ (8)$

Proof: Setting ${\sigma}$ sufficiently close to ${1/10}$, we see from the above theorem that ${Type^{(4)}_{II}[\varpi,\delta]}$ holds whenever

$\displaystyle \frac{600}{7} \varpi + \frac{180}{7} \delta < 1$

and

$\displaystyle 80 \varpi + \frac{45}{2} \delta < 1.$

The second condition is implied by the first and can be deleted.

From this previous post we know that ${Type^{(4)}_{II}[\varpi,\delta]}$ (which we define analogously to ${Type'_{II}[\varpi,\delta], Type''_{II}[\varpi,\delta]}$ from previous sections) holds whenever

$\displaystyle 68 \varpi + 14 \delta < 1$

while ${Type^{(4)}_{III}[\varpi,\delta,\sigma]}$ holds with ${\sigma}$ sufficiently close to ${1/10}$ whenever

$\displaystyle 70 \varpi + 5 \delta < 1.$

Again, these conditions are implied by (8). The claim then follows from the Heath-Brown identity and dyadic decomposition as in this previous post. $\Box$

As before, we let ${DHL[k_0,2]}$ denote the claim that given any admissible ${k_0}$-tuple ${{\mathcal H}}$, there are infinitely many translates of ${{\mathcal H}}$ that contain at least two primes.

Corollary 6 We have ${DHL[k_0,2]}$ with ${k_0 = 632}$.

This follows from the Pintz sieve, as discussed below the fold. Combining this with the best known prime tuples, we obtain that there are infinitely many prime gaps of size at most ${4,680}$, improving slightly over the previous record of ${5,414}$.