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As in all previous posts in this series, we adopt the following asymptotic notation: is a parameter going off to infinity, and all quantities may depend on unless explicitly declared to be “fixed”. The asymptotic notation is then defined relative to this parameter. A quantity is said to be *of polynomial size* if one has , and *bounded* if . We also write for , and for .

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 . For each natural number , we define a notion of -tuply -dense divisibility recursively as follows:

- Every natural number is -tuply -densely divisible.
- If and is a natural number, we say that is -tuply -densely divisible if, whenever are natural numbers with , and , one can find a factorisation with such that is -tuply -densely divisible and is -tuply -densely divisible.
We let denote the set of -tuply -densely divisible numbers. We abbreviate “-tuply densely divisible” as “densely divisible”, “-tuply densely divisible” as “doubly densely divisible”, and so forth; we also abbreviate as .

Given any finitely supported sequence and any primitive residue class , we define the discrepancy

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 2Acoefficient sequenceis a finitely supported sequence that obeys the boundsfor all , where is the divisor function.

- (i) A coefficient sequence is said to be
located at scalefor some if it is supported on an interval of the form for some .- (ii) A coefficient sequence located at scale for some is said to
obey the Siegel-Walfisz theoremif one has- (iii) A coefficient sequence is said to be
smooth at scalefor some is said to besmoothif it takes the form for some smooth function supported on an interval of size and obeying the derivative boundsfor all fixed (note that the implied constant in the notation may depend on ).

Note that we allow sequences to be smooth at scale without being located at scale ; for instance if one arbitrarily translates of a sequence that is both smooth and located at scale , 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 of a coefficient sequence to be less than one. This is to allow for the following convenient rescaling property: if is smooth at scale , , and is an integer, then is smooth at scale , even if is less than one.

Now we adapt the Type I estimate to the -tuply densely divisible setting.

Definition 3 (Type I estimates)Let , , and be fixed quantities, and let be a fixed natural number. We let be an arbitrary bounded subset of , let , and let a primitive congruence class. We say that holds if, whenever are quantities withfor some fixed , and are coefficient sequences located at scales respectively, with obeying a Siegel-Walfisz theorem, we have

for any fixed . Here, as in previous posts, denotes the square-free natural numbers whose prime factors lie in .

The main theorem of this post is then

Theorem 4 (Improved Type I estimate)We have wheneverand

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 under the stricter hypothesis

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 and , we let denote the assertion that

for any fixed , any bounded , and any primitive , where is the von Mangoldt function.

*Proof:* Setting sufficiently close to , we see from the above theorem that holds whenever

and

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

From this previous post we know that (which we define analogously to from previous sections) holds whenever

while holds with sufficiently close to whenever

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

As before, we let denote the claim that given any admissible -tuple , there are infinitely many translates of that contain at least two primes.

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 , improving slightly over the previous record of .

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