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Source. Theorem 1 and display (1.7), 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 labels and page as printed in the arXiv preprint arXiv:math/0508185v1 (10 August 2005), the edition read for the source card.

Statement

Let $\theta(n)=\log n$ when $n$ is prime and $\theta(n)=0$ otherwise, and let $\theta(N;q,a)$ be the sum of $\theta(n)$ over $n\le N$ with $n\equiv a\pmod q$ (p. 1). The primes have level of distribution $\vartheta$ when, for every $A>0$ and every $\varepsilon>0$,

$$\sum_{q\le Q}\ \max_{(a,q)=1}\Bigl|\theta(N;q,a)-\frac{N}{\varphi(q)}\Bigr| \ll\frac{N}{(\log N)^A}\qquad\text{with } Q=N^{\vartheta-\varepsilon}$$

(displays (1.3) and (1.4), p. 2). The Bombieri--Vinogradov theorem gives level $1/2$; the Elliott--Halberstam conjecture is level $1$.

For a set $\mathcal H=\{h_1,\dots,h_k\}$ of distinct non-negative integers, let $\nu_p(\mathcal H)$ be the number of residue classes modulo $p$ that the $h_i$ occupy. The set, and the tuple $(n+h_1,\dots,n+h_k)$, are admissible when $\nu_p(\mathcal H)<p$ for every prime $p$ (display (1.6), p. 2).

Theorem 1 (p. 2). Assume the primes have level of distribution $\vartheta>1/2$. Then there is a constant $C(\vartheta)$, depending only on $\vartheta$ and explicitly calculable, such that every admissible $k$-tuple with $k\ge C(\vartheta)$ has at least two prime components for infinitely many $n$. If $\vartheta\ge 0.971$, this holds for every $k\ge 6$.

Display (1.7) (p. 2). The $6$-tuple $(n,n+4,n+6,n+10,n+12,n+16)$ is admissible, so the Elliott--Halberstam conjecture implies

$$\liminf_{n\to\infty}\,(p_{n+1}-p_n)\le 16,$$

where $p_n$ is the $n$th prime; that is, $p_{n+1}-p_n\le16$ for infinitely many $n$.

Proof pointer

Section 3, pp. 8--12, from Propositions 1 and 2 (pp. 7--8), which are proved in Sections 6--9 (pp. 16--31); the paper credits the argument of Section 3 to Granville and Soundararajan. For a weight $\Lambda_R(n;\mathcal H_k,\ell)$, a truncated divisor sum of the polynomial $\prod_i(n+h_i)$, the two propositions give the asymptotics (3.1) and (3.2) (p. 8) of its square summed alone and against $\theta(n+h_i)$, with $R=N^{\vartheta/2-\varepsilon}$. Comparing $\sum_i\theta(n+h_i)$ with $\log 3N$ against the square of the weight on $(N,2N]$ yields the condition (3.4) (p. 9), which holds for some $k$ and $\ell$ whenever $\vartheta>1/2$ by letting $k,\ell\to\infty$ with $\ell=o(k)$; this proves the first part. Taking $\ell=1$, $k=7$ needs only $\vartheta>20/21$ (p. 9). For $k=6$ the weight is replaced by a linear combination of the $\Lambda_R(n;\mathcal H_k,\ell)$ for $\ell\le L$, which turns the problem into one about a positive eigenvalue of a quadratic form (pp. 11--12); with $L=1$ the condition becomes $\vartheta>4(8-\sqrt{19})/15=0.97096\ldots$ (display (3.16), p. 12). Tables on pp. 9 and 12 list the resulting values of $C(\vartheta)$.

Dependencies

Propositions 1 and 2 of the paper (pp. 7--8) and the lemmas of Sections 5 and 8. Read depth: claims checked; the statement and display (1.7) were read clause by clause on p. 2, and Section 3 for the structure of the proof.

Bears on

No Erdős problem is linked from this page. Theorem 2 is the paper's unconditional result on small gaps.