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Statement

Setting (p. 5). A $k$-tuple $\mathcal H=\{h_i\}_{i=1}^k$ of distinct non-negative integers is admissible when, for every prime $p$, the number $\nu_p(\mathcal H)$ of residue classes mod $p$ it covers is less than $p$; equivalently its singular series $\mathfrak S(\mathcal H)$ (the paper's (3.3)) is positive. $P^-(m)$ is the least prime factor of $m$ (p. 6).

Conjecture DHL*$(k,2)$ (p. 6). Let $k\ge2$, let $\mathcal H$ be any admissible $k$-tuple, $N\in\mathbb Z^+$ and $\varepsilon>0$ sufficiently small ($\varepsilon<\varepsilon_0$), with $\mathcal H\subset[0,H]$, $H\le\varepsilon\log N$ and $P_{\mathcal H}(n)=\prod_{i=1}^k(n+h_i)$. Then there are positive constants $c_1(k)$ and $c_2(k)$ such that, for $N>N_0(\mathcal H)$, at least

$$c_2(k)\,\mathfrak S(\mathcal H)\,\frac{N}{\log^k N}$$

integers $n\in[N,2N)$ have both properties: $n+\mathcal H$ contains at least two consecutive primes, and every component is an almost prime, that is $P^-(P_{\mathcal H}(n))>n^{c_1(k)}$.

Main Theorem (p. 6, quoted). "Conjecture DHL*$(k,2)$ is true for $k\ge3.5\times10^6$."

Compared with DHL$(k,2)$ (p. 5: an admissible $\mathcal H$ has at least two primes in $n+\mathcal H$ for infinitely many $n$), the paper lists what is gained (p. 6): the tuple may grow with $N$ up to $\varepsilon\log N$, the two primes can be taken consecutive, every component is an almost prime, and the number of such $n$ is bounded below by (3.6). The paper writes $P_{\mathcal H}(n)=\prod(n-h_i)$ in (3.1) and $\prod(n+h_i)$ in (3.5); the statement above uses (3.5).

Section 8 (p. 12) adds, as a sketch, that Zhang's theorem and all the paper's results become effective once the one possible exceptional Landau--Page modulus $q$ is excluded from the sieve weights, since Lemma 3 lets one discard the $n$ with $P^-(P_{\mathcal H}(n))<n^{c_1(k)}$.

Proof pointer

Pages 6--9. The paper describes only the changes to earlier work, not a self-contained proof. Zhang's method (his Theorem A, p. 1) is run with these modifications: the Goldston--Pintz--Yıldırım argument behind Theorem B allows $H\ll\log N$; the Motohashi--Pintz step that discards non-smooth moduli stays uniform under $H\ll\log N$, at the cost of the extra error (3.8); and Lemmas 1 and 2 (p. 7), from the author's 2010 paper, show that the $n$ with $P^-(P_{\mathcal H}(n))<R^\eta$ carry only an $O_k(\eta)$ share of the sieve weight. This gives (3.17): at least $c_2(k)\mathfrak S(\mathcal H)N/\log^kN$ integers $n\in[N,2N)$ with two primes in $n+\mathcal H$ and almost primes in every component. To make two of the primes consecutive, the paper fixes a pattern $V_0$ of prime positions that occurs for many $n$ (3.20), takes consecutive positions $i<j$ in it, and bounds, by Selberg's upper-bound sieve (Lemma 3, p. 7) and the author's averaged singular-series estimate (Lemma 4, p. 8), the $n$ for which some $n+h$ with $h_i<h<h_j$, $h\notin\mathcal H$, is prime: there are at most $2C_4(k)\varepsilon,\mathfrak S(\mathcal H)N/\log^kN$ of them (3.24), which is small once $\varepsilon<\varepsilon_0(k)$.

Read depth

Claims checked: the definitions, the conjecture and the Main Theorem were read clause by clause on the printed pages of arXiv:1305.6289v1. The proof is a description of modifications to the cited works of Zhang, Goldston--Pintz--Yıldırım, Motohashi--Pintz and Pintz (2010), which were not read. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: Zhang's bounded-gaps theorem and method; Goldston, Pintz and Yıldırım, Primes in tuples I; Motohashi and Pintz, A smoothed GPY sieve; Lemmas 1 to 4 (pp. 7--8), each taken from an earlier work cited there.

Source. János Pintz, Polignac numbers, conjectures of Erdős on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture, arXiv:1305.6289v1 (2013); published in From Arithmetic to Zeta-Functions, Springer (2016), 367--384, doi:10.1007/978-3-319-28203-9_22. Labels and pages here are those of arXiv v1. The edition read is named on the source card.

Bears on

No Erdős problem directly. It is the input to Theorem 3, which bears on Problem 5.