  We study the weak-coupling limit of the $t-t^\prime-U$ Hubbard model on a
two-dimensional square lattice using a direct perturbative approach. Aided by
symbolic computational tools, we compute the longitudinal density-density
correlation functions in the $\chi_{\uparrow \uparrow}$ and $\chi_{\uparrow
\downarrow}$ basis from which we can obtain the dynamical spin and charge
susceptibilities at arbitrary doping and temperature. We find that for non-zero
$t^\prime$, the zero frequency commensurate $\mathbf{q} = (\pi, \pi)$ spin and
charge excitations are each strongest at different densities and we observe a
clear behavioral change that appears tied to the van Hove singularity of the
non-interacting dispersion upon which the perturbative expansion is built. We
find a strongly reduced compressibility in the vicinity of the van Hove
singularity as well as a behavioral change in the double occupancy. For finite
$t^\prime$, the observed van Hove singularity occurs away from half-filling
leading us to conclude that that this reduction in compressibility is distinct
from Mott insulating physics that one expects in the strong-coupling regime. We
compute the full dynamical spin and charge excitations and observe distinct
structure for electron and hole doped scenarios in agreement with experiments
on cuprate materials. Finally, we observe a peculiar splitting in spin and
charge excitations in the vicinity of the van Hove singularity, the origin of
which is traced to a splitting near the bottom of the band.
