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Statement

Setting (p. 4). Write $d_n=p_{n+1}-p_n$ and let $J$ be the set of limit points of $d_n/\log n$. The paper recalls Erdős's 1955 conjecture (2.8) that $J=[0,\infty]$, Westzynthius's theorem that $\infty\in J$, the Goldston--Pintz--Yıldırım theorem $\liminf d_n/\log n=0$ (Theorem C, p. 2), equivalent to $0\in J$, and the theorem of Erdős and of Ricci that $J$ has positive Lebesgue measure.

Theorem 3 (p. 4, quoted). "There is an ineffective constant $c>0$ such that $[0,c]\subset J$."

The paper calls it "a weaker form of Erdős's conjecture (2.8)" (p. 4). It is the case $f(n)=\log n$ of Theorem 4, and the paper proves it that way (p. 10).

Proof pointer

Page 10, proof of Theorems 3--4; see Theorem 4.

Read depth

Claims checked: the setting and the statement were read clause by clause on the printed pages of arXiv:1305.6289v1, and the proof on p. 10 was followed. Nothing here is independently reviewed.

Dependencies

Theorem 4 and the Main Theorem of this paper.

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

• Problem 5: the problem asks, for each $C\ge0$, for a sequence $n_i$ with $(p_{n_i+1}-p_{n_i})/\log n_i\to C$, that is whether $C\in J$. Theorem 3 gives this for every $C$ in $[0,c]$, for an ineffective $c>0$ the paper does not determine; it names no particular positive $C$ and says nothing about $C>c$. The problem's claim page for this result is 2013_05_27_pintz.