  Several problems involving $E(T)$ and $E_2(T)$, the error terms in the mean
square and mean fourth moment formula for $|\zeta(1/2+it)}$ are discussed. In
particular, it is proved that $$ \int_0^T E(t)E_2(t)dt \ll_ T^{7/4}(\log
T)^{7/2}(\log\log T). $$
