The Differences Between Consecutive Primes, V

We show that
$$\begin{equation*}\sum_{\substack{p_n\le x\\ p_{n+1}-p_n\ge\sqrt{p_n}}}(p_{n+1}-p_n) \ll_{\varepsilon} x^{3/5+\varepsilon}\end{equation*}$$
or any fixed
$\varepsilon>0$
. This improves a result of Matomäki, in which the exponent was
$2/3$
.