The pseudomoments of the Riemann zeta function are the moments of the
partial sums associated with zeta on the critical line.
Using probabilistic methods of Harper, we provide bounds which give the order
of magnitude of all pseudomoments. We also
provide upper and lower bounds for the pseudomoments of the powers of zeta that
are almost-matching when combined with
previous bounds of Bondarenko, Heap
and Seip, and —somewhat surprisingly— behave in a rather different
manner.