Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\log n$. This problem asks whether $S=[0,\infty]$. Although this conjecture remains unproven, a lot is known about $S$. Some highlights:

• $\infty\in S$ by Westzynthius' result [We31] on large prime gaps,


• $0\in S$ by the work of Goldston, Pintz, and Yildirim [GPY09] on small prime gaps,


• Erdős [Er55] and Ricci [Ri56] independently showed that $S$ has positive Lebesgue measure,


• Hildebrand and Maier [HiMa88] showed that $S$ contains arbitrarily large (finite) numbers,


• Pintz [Pi16] showed that there exists some small constant $c>0$ such that $[0,c]\subset S$,


• Banks, Freiberg, and Maynard [BFM16] showed that at least $12.5\%$ of $[0,\infty)$ belongs to $S$,


• Merikoski [Me20] showed that at least $1/3$ of $[0,\infty)$ belongs to $S$, and that $S$ has bounded gaps.

In [Er65b], [Er85c], and [Er97c] Erdős asks whether $S$ is everywhere dense (but Weisenberg notes that clearly $S$ is closed so this is equivalent to asking whether $S=[0,\infty]$).

See also [234].

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