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Statement

Let $p_n$ denote the $n$-th prime and $G(X):=\max_{p_{n+1}\le X}(p_{n+1}-p_n)$ the maximum gap between consecutive primes less than $X$ (p. 1). Iterated logarithms are written $\log_2x=\log\log x$, $\log_3x=\log\log\log x$, and so on (footnote 1, p. 2).

Theorem 1 (Large prime gaps) (p. 2, quoted). "For any sufficiently large $X$, one has

$$G(X)\gg\frac{\log X\log_2X\log_4X}{\log_3X}.$$

The implied constant is effective."

The historical paragraph on p. 2: Westzynthius (1931) proved $G(X)/\log X\to\infty$ with $G(X)\gg\log X\log_3X/\log_4X$; Erdős (1935) sharpened this to $G(X)\gg\log X\log_2X/(\log_3X)^2$; Rankin (1938) proved $G(X)\ge(c+o(1))\log X\log_2X\log_4X/(\log_3X)^2$ with $c=1/3$, and the constant was raised by Schönhage, by Rankin, by Maier and Pomerance ($1.31256e^\gamma$) and by Pintz ($2e^\gamma$); the authors' two papers of 2014 showed that $c$ can be taken arbitrarily large, "answering in the affirmative a long-standing conjecture of Erdős", and unpublished work of Maynard gave (1.1) $G(X)\gg\log X\log_2X/\log_3X$. Theorem 1 gains the factor $\log_4X$ over (1.1).

Source. K. Ford, B. Green, S. Konyagin, J. Maynard and T. Tao, Long gaps between primes, arXiv:1412.5029v3 (14 July 2016, 40 pp.); Theorem 1 on p. 2, read on the page image and in the text layer. Published in J. Amer. Math. Soc. 31 (2018), no. 1, 65--105, DOI 10.1090/jams/876 (published online 23 February 2017; the Crossref record and the arXiv listing's journal reference were read); the journal text was not compared, so the locators are those of v3.

Read depth. Claims checked: the statement and the historical paragraph were read clause by clause on the page images of pp. 1--2. The proof was not read beyond the reduction on p. 3 described below.

Proof pointer

Section 1, p. 3: by Lemma 1.1, $G(P(x)+Y(x)+x)\ge Y(x)$; since $P(x)=e^{(1+o(1))x}$ by the prime number theorem and $Y(x)=x^{O(1)}$, this gives $G(X)\ge Y((1+o(1))\log X)$ as $X\to\infty$, so Theorem 1 is a consequence of display (1.2), $Y(x)\gg x\log x\log_3x/\log_2x$, "which we will establish later in this paper". The proof of (1.2) occupies Sections 3--8 (pp. 8--39): sieving a set of primes, a generalization of the Pippenger--Spencer hypergraph covering theorem proved by the Rödl nibble, and multidimensional sieve weights of Maynard type. Not read here.

Dependencies

Display (1.2) and Lemma 1.1 of the paper, and the prime number theorem. External premises are taken at statement level.

Bears on

• Problem 687: the prime-gap consequence of the covering bound (1.2); the paper's introduction (p. 4) is also the second-hand source for Iwaniec's upper bound $Y(x)\ll x^2$ and for the Maier--Pomerance conjecture.
• Problem 929: the site cites this theorem's paper for its upper bound $S(k)\ll k\log_3k/(\log_2k\log_4k)$. Inverting (1.2) gives the stronger $S(k)\ll k\log_2k/(\log k\log_3k)$, which implies the site's bound; the problem page records that the site's bound is not that inversion.
• Problem 4: the theorem's gap $\gg\log X\log_2X\log_4X/\log_3X$ below $X$ exceeds the problem's $C\log n\log_2n\log_4n/(\log_3n)^2$ for every $C$, with a factor $\log_3X$ to spare ($\log p_n\sim\log n$ carries the comparison from $X$ to the index); the historical paragraph records that the 2014 papers of the authors and of Maynard had already answered the problem, which this theorem improves by the factor $\log_3X$.