On some mean values for the divisor function and the Riemann zeta-function

Let $\Delta (x)$ and $E(x)$ denote respectively the error terms in the summatory formula for the divisor function and in the mean square formula for $\zeta (s)$ on the critical line. We consider some general mean values for $\Delta (x)$ and $E(x)$ and discover interesting differences between these two functions. In particular, this yields evidence that $E(x)$ is more negative than $\Delta (x)$.