A conjecture by Heath-Brown: The sum of squares of the first $N$ gaps between consecutive primes behaves like $N * (log N)^2$.

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1(fun N => ∑ n ∈ Finset.range N, ↑(primeGap n) ^ 2) =O[Filter.atTop] fun N => ↑N * Real.log ↑N ^ 2

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