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

Let $Δ(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|ζ(1/2+it)|$. If $E^*(t) = E(t) - 2πΔ^*(t/2π)$ with $Δ^*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x)$ and we set $\int_0^T E^*(t)\,dt = 3πT/4 + R(T)$, 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 $T^{2/3+ε}\le H \le T$.

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