  The $2 q$-th pseudomoment $\Psi_{2q,\alpha}(x)$ of the $\alpha$-th power of
the Riemann zeta function is defined to be the $2 q$-th moment of the partial
sum up to $x$ of $\zeta^\alpha$ on the critical line. Using probabilistic
methods of Harper, we prove upper and lower bounds for these pseudomoments when
$q \le \frac{1}{2}$ and $\alpha \ge 1$. Combined with results of Bondarenko,
Heap and Seip, these bounds determine the size of all pseudomoments with $q >
0$ and $\alpha \ge 1$ up to powers of $\log \log x$, where $x$ is the length of
the partial sum, and it turns out that there are three different ranges with
different growth behaviours. In particular, the results give the order of
magnitude of $\Psi_{2 q, 1}(x)$ for all $q > 0$.
