Problem 233

Erdős Problem #233: declared status 'open'. Formalized: yes. Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Prove that$$\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.$$ Current best: Cramer [Cr36] proved an upper bound of $O(N(\log N)^4)$ conditional on the Riemann hypothesis. Selberg [Se43] improved this slightly (still assuming the Riemann hypothesis) to$$\sum_{1\leq n\leq N}\frac{d_n^2}{n}\ll (\log N)^4.$$The prime number theorem immediately implies a lower bound of$$\sum_{1\leq n\leq N}d_n^2\gg N(\log N)^2.$$The values of the sum are listed at A074741 on the OEIS. Prize: no. OEIS: A074741. Tags: number theory, primes.