  Sums of the form $\sum_{n\le x}E^k(n) (k\in{\bf N}$ fixed) are investigated,
where $$ E(T) = \int_0^T|\zeta(1/2+it)|^2 dt - T\Bigl(\log {T\over2\pi} +
2\gamma -1\Bigr)$$ is the error term in the mean square formula for
$|\zeta(1/2+it)|$. The emphasis is on the case k=1, which is more difficult
than the corresponding sum for the divisor problem. The analysis requires
bounds for the irrationality measure of ${\rm e}^{2\pi m}$ and for the partial
quotients in its continued fraction expansion.
