  For a fixed integer $k\ge 3$ and fixed $1/2 < \sigma > 1$ we consider $$
\int_1^T |\zeta(\sigma + it)|^{2k}dt = \sum_{n=1}^\infty d_k^2(n)n^{-2\sigma}T
+ R(k,\sigma;T), $$ where $R(k,\sigma;T) = o(T) (T\to\infty)$ is the error term
in the above asymptotic formula. Hitherto the sharpest bounds for
$R(k,\sigma;T)$ are given for certain ranges of $\sigma$. We also obtain new
mean value results for the zeta-functions of holomorphic cusp forms and the
Rankin-Selberg series.
