On the divisor function and the Riemann zeta-function in short intervals

We obtain, for $T^{ϵ}\le U=U(T)\le T^{1\mathrm{/}2-ϵ}$, asymptotic formulas for $${\int }_T^{2T}(E(t+U)-E(t){)}^{2}dt,{\int }_T^{2T}(\mathrm{\Delta }(t+U)-\mathrm{\Delta }(t){)}^{2}dt,$$ where $\mathrm{\Delta }(x)$ is the error term in the classical divisor problem, and $E(T)$ is the error term in the mean square formula for $\mathrm{\mid }\zeta (1\mathrm{/}2+it)\mathrm{\mid }$. Upper bounds of the form $O_{ϵ}(T^{1+ϵ}{U}^2)$ for the above integrals with biquadrates instead of square are shown to hold for $T^{3\mathrm{/}8}\le U=U(T)\ll T^{1\mathrm{/}2}$. The connection between the moments of $E(t+U)-E(t)$ and $\mathrm{\mid }\zeta (1\mathrm{/}2+it)\mathrm{\mid }$ is also given. Generalizations to some other number-theoretic error terms are discussed.