On the Riemann zeta-function and the divisor problem IV

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 $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - {1\over2}\Delta(4x)$, then it is proved that $$\int_0^T|E^*(t)|^3dt \ll_\epsilon T^{3/2+\epsilon},$$ which is (up to `$\epsilon$' best possible) and $\zeta(1/2+it) \ll_\epsilon t^{\rho/2+\epsilon}$ if $E^*(t) \ll_\epsilon t^{\rho+\epsilon}$.