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Let $d_n=p_{n+1}-p_n$. Prove that \[\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.\]
Cramer proved an upper bound of $O(N(\log N)^4)$ conditional on the Riemann hypothesis. The prime number theorem immediately implies a lower bound of $\gg N(\log N)^2$.