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Source. Theorem 2, display (1.8), p. 2, of D. A. Goldston, J. Pintz and C. Y. Yıldırım, Primes in tuples I, Ann. of Math. (2) 170 (2009), no. 2, 819--862, with label and page as printed in the arXiv preprint arXiv:math/0508185v1 (10 August 2005), the edition read for the source card.

Statement

Let $p_n$ be the $n$th prime.

Theorem 2 (p. 2). Unconditionally,

$$E_1:=\liminf_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=0$$

(display (1.8)).

No hypothesis is assumed: the proof uses only the level of distribution $1/2$ that the Bombieri--Vinogradov theorem supplies. Since $E_1\le1$ follows from the prime number theorem, the theorem says that consecutive primes are infinitely often closer than any fixed positive multiple of the average spacing $\log p_n$.

Proof pointer

Section 3, p. 10. Instead of one tuple, the argument sums the weight of Theorem 1 over all $k$-tuples of distinct shifts in $[1,h]$ and compares $\sum_{1\le h_0\le h}\theta(n+h_0)$ with $r\log 3N$, for a positive integer $r$ (display (3.5)). The asymptotics from Propositions 1 and 2 and Gallagher's average of the singular series (display (3.7)) show that some interval $(n,n+h]$ with $N<n\le2N$ holds at least $r+1$ primes once $h>(r-2\vartheta+4\varepsilon+O(k^{-1/2}))\log N$ (display (3.10)), with $\ell=[\sqrt k/2]$ and $k$ large. This proves the bound $E_r\le\max(r-2\vartheta,0)$ of display (1.11) (p. 4), and Theorem 2 is its case $r=1$, $\vartheta=1/2$.

Dependencies

Propositions 1 and 2 of the paper (pp. 7--8), the Bombieri--Vinogradov theorem and Gallagher's theorem (3.7). Read depth: claims checked; the statement was read on p. 2 and Section 3 for the structure of the proof.

Bears on

• Problem 5: settles the case $C=0$. A strictly increasing sequence $n_i$ along which $(p_{n_i+1}-p_{n_i})/\log p_{n_i}\to0$ exists by the theorem, and $\log p_n\sim\log n$ turns it into $(p_{n_i+1}-p_{n_i})/\log n_i\to0$. The theorem says nothing about any $C>0$. The case is recorded on its claim page.