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Statement

Notation (pp. 1 and 4): $p_n$ is the $n$th prime and $\mathbb N=\{1,2,\dots\}$.

Theorem 1.1 (p. 2). Let $m\in\mathbb N$. Then

$$\liminf_{n}\,(p_{n+m}-p_n)\ll m^3e^{4m}.$$

The proof (p. 7) makes the implied constant absolute. In words: there is an absolute constant $C'$ such that for every $m\in\mathbb N$ there are infinitely many $n$ with $p_{n+m}-p_n\le C'm^3e^{4m}$, so some interval of that length holds $m+1$ primes infinitely often.

Remarks (pp. 2--3). The paper notes that Tao independently proved Theorem 1.1 with a slightly weaker bound by a similar method, that the bound is far from the size about $m\log m$ which the prime $m$-tuples conjecture predicts, and (p. 3) that under the Elliott--Halberstam conjecture the bound improves to $O(m^3e^{2m})$; no proof of that improvement is written out. For $m=1$ the theorem gives only $\liminf_n(p_{n+1}-p_n)\ll e^4$ with an unspecified constant; the explicit bound 600 is Theorem 1.3.

Source. J. Maynard, Small gaps between primes, Ann. of Math. (2) 181 (2015), no. 1, 383--413, doi:10.4007/annals.2015.181.1.7, read in the arXiv:1311.4600v3 preprint (28 October 2019) identified on the source card; the pages cited are the preprint's printed pages, not the journal's. Theorem 1.1 and the remarks on pp. 2--3, the deduction on p. 7.

Read depth. Claims checked: the statement and the deduction from Propositions 4.2 and 4.3 (pp. 5--7) were read clause by clause. The proofs of Proposition 4.1 (Sections 5 and 6, pp. 7--18) and of part (3) of Proposition 4.3 (Section 7, pp. 18--20) were read for their structure, not step by step. Nothing here is independently reviewed.

Proof pointer

P. 7. The Bombieri--Vinogradov theorem gives level of distribution $\theta=1/2-\epsilon$, and part (3) of Proposition 4.3 gives $M_k>\log k-2\log\log k-2$ for large $k$, which is the bound (4.5) for $\theta M_k/2$. Taking $\epsilon=1/k$, this exceeds $m$ once $k\ge Cm^2e^{4m}$ for an absolute constant $C$, so by Proposition 4.2 every admissible set of that size has at least $m+1$ of the $n+h_i$ prime for infinitely many $n$. The paper applies this to the $k$ consecutive primes following $p_{\pi(k)}$, an admissible set of diameter $\ll k\log k$, with $k=\lceil Cm^2e^{4m}\rceil$.

Dependencies

Proposition 4.2 and Proposition 4.3 (3) of the same paper; the Bombieri--Vinogradov theorem.

Bears on

• Problem 6: the problem asks whether $d_n<d_{n+1}<d_{n+2}$ for infinitely many $n$, with $d_n=p_{n+1}-p_n$. Theorem 1.1 bounds gaps spanning several primes and says nothing about the order of consecutive gaps, so it does not decide the question. Banks, Freiberg and Turnage-Butterbaugh answer it using as input the Maynard--Tao theorem for admissible tuples of linear forms, in Granville's formulation. Its shift case is the step of this proof recorded on Proposition 4.2; for linear forms this paper has only the unnumbered remark on p. 2.