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Source. Theorem 3, displays (1.12) and (1.13), p. 4, 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

For $r\ge1$ let

$$E_r=\liminf_{n\to\infty}\frac{p_{n+r}-p_n}{\log p_n}$$

(display (1.10), p. 3), where $p_n$ is the $n$th prime. Level of distribution $\vartheta$ is as defined on the page for Theorem 1.

Theorem 3 (p. 4). Assume the primes have level of distribution $\vartheta$. Then for every $r\ge2$

$$E_r\le\bigl(\sqrt r-\sqrt{2\vartheta}\bigr)^2 .$$

Unconditionally, for every $r\ge1$,

$$E_r\le\bigl(\sqrt r-1\bigr)^2 .$$

The paper notes (p. 4) that either this bound or the weaker $E_r\le\max(r-2\vartheta,0)$ of display (1.11) shows that the Elliott--Halberstam conjecture implies $E_2=0$ (display (1.14)).

Proof pointer

Section 10, pp. 31--35. A single weight, the sum of $\Lambda_R(n;\mathcal H,\ell)$ over all $k$-subsets $\mathcal H$ of $\{1,\dots,h\}$, is squared and compared with $\sum_{1\le h_0\le h}\theta(n+h_0)-\nu\log3N$ (display (10.1)). Expanding the square groups pairs of subsets by the size of their intersection; Propositions 1 and 2 and Gallagher's theorem in the form (10.5) evaluate each group in terms of $x=\log R/h$ (display (10.6)). The parameters are then chosen with $k=(\ell+1)^2$ and $\ell$ large (display (10.16)), and letting an auxiliary $\varepsilon_0$ tend to $0$ gives the bound (pp. 34--35).

Dependencies

Propositions 1 and 2 of the paper (pp. 7--8) and Gallagher's theorem. Read depth: claims checked; the statement was read on p. 4 and Section 10 for the structure of the proof only.

Bears on

No Erdős problem is linked from this page; it records the paper's third main result.