  Power moments of $$ J_k(t,G) = {1\over\sqrt{\pi}G} \int_{-\infty}^\infty
|\zeta(1/2 + it + iu)|^{2k}{\rm e}^{-(u/G)^2} du \qquad(t \asymp T, T^\epsilon
\le G \ll T),$$ where $k$ is a natural number, are investigated. The results
that are obtained are used to show how bounds for $\int_0^T|\zeta(1/2+it)|^{2k}
dt$ may be obtained.
