  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 we obtain
$$ \int_0^T(E^*(t))^4 dt \ll_\epsilon T^{16/19+varepsilon}$$, which is the
first non-trivial bound for higher moments of $E^*(t)$. The method of proof
also provides an upper bound for sums of fourth powers of mean square integrals
of $|\zeta(1/2 + it)|$ over well-spaced points. This, in turn, yields a new
proof of the twelfth moment estimate for $|\zeta(1/2 + it)|$. Among the chief
ingredients in the proof is a recent result of Robert--Sargos on the
distribution of four square roots of integers, plus an approach of M. Jutila
that involves the use of Airy integrals to deal with the ensuing exponential
sums.
