Heath-Brown shows the sum of (p_{n+1}-p_n) over p_n ≤ x with gap ≥ √p_n is ≪_ε x^{3/5+ε}, improving Matomäki's 2/3 exponent.
Jutila, Zero density estimates for L-functions, Acta Arith., 32 (1977), 55–62
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The Differences Between Consecutive Primes. V
Heath-Brown shows the sum of (p_{n+1}-p_n) over p_n ≤ x with gap ≥ √p_n is ≪_ε x^{3/5+ε}, improving Matomäki's 2/3 exponent.