All Random Solved Random Open
Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that \[\lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C?\]
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.