Summary: 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 \(| \zeta(\tfrac 12 +it)|\). If \(E^*(t) = E(t)- 2\pi \Delta ^*(t/2\pi )\) with \(\Delta ^*(x) = -\Delta (x)+2\Delta (2x)-\frac 1 2\Delta (4x)\) and we set \(\int^T_0 E^*(t)\, dt = 3\pi T/4 + R(T)\), then we obtain \[ \int^{T+H}_T (E^*(t))^2 \,dt \gg HT^{1/3} \log^3 T \] and \[ HT \log^3 T \ll \int^{T+H}_T R^2(t)\, dt \ll_\varepsilon HT \log^3 T+T^{5/3+\varepsilon}, \] for \(T^{2/3+\varepsilon} \leq H\leq T\).