Problem 233

Let $d_n=p_{n+1}-p_n$ , where $p_n$ is the $n$ th prime. Prove that

\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.

Status

Open

No claim settles this problem.

Cr36: Cramér, Harald, On the order of magnitude of the difference between consecutive prime numbers. Acta Arithmetica (1936), 23-46.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp. Section A8 "Gaps between primes. Twin primes.", printed pp. 33--34: Cramér's conditional bound $\sum_{n<x}d_n^2<cx(\ln x)^4$ , and "Erdős conjectures that the right-hand side should be $cx(\ln x)^2$ , but thinks that there is no hope of a proof". Library home: Guy 2004.
Se43: Selberg, Atle, On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid. (1943), 87-105.