  The pseudomoments of the Riemann zeta function, denoted $\mathcal{M}_k(N)$,
are defined as the $2k$th integral moments of the $N$th partial sum of
$\zeta(s)$ on the critical line. We improve the upper and lower bounds for the
constants in the estimate $\mathcal{M}_k(N) \asymp_k (\log{N})^{k^2}$ as
$N\to\infty$ for fixed $k\geq1$, thereby determining the two first terms of the
asymptotic expansion. We also investigate uniform ranges of $k$ where this
improved estimate holds and when $\mathcal{M}_k(N)$ may be lower bounded by the
$2k$th power of the $L^\infty$ norm of the $N$th partial sum of $\zeta(s)$ on
the critical line.
