A note on primes and Goldbach numbers in short intervals

Let $J(N,H)$ be the Selberg integral and $E(x,T)$ the error term in Kaczorowski-Perelli's weighted form of the classical explicit formula. We prove that the estimate $J(N,H)=o(H^2N)$ is connected with an appropriate estimate of $\int_N^{2N}\vert E(x,T)\vert ^2 dx$, uniformly for $H$ and $T$ in some ranges. Moreover, assuming a suitable bound for the quantity $\int_N^{2N}\vert E(x,T)\vert ^2 dx$, we also obtain, for all sufficiently large $N$ and $H\gg(\log N)^{11/2}$, that every interval $[N,N+H]$ contains $\gg H$ Goldbach numbers.