On some mean square estimates for the zeta-function in short intervals

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi)$ with $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - 1/2\Delta(4x)$ and we set $\int_0^T E^*(t)\,dt = 3\pi T/4 + R(T)$, then we obtain $$\int_T^{T+H}(E^*(t))^2\,dt \gg HT^{1/3}\log^3T$$ and $$HT\log^3T \ll \int_T^{T+H}R^2(t)\,dt \ll HT\log^3T,$$ for $T^{2/3+\epsilon}\le H \le T$.