  We analyze the pressure and density equations of state of unpolarized
non-relativistic fermions at finite temperature in one spatial dimension. For
attractively interacting regimes, we perform a third-order lattice perturbation
theory calculation, assess its convergence properties by comparing with hybrid
Monte Carlo results (there is no sign problem in this regime), and demonstrate
agreement with real Langevin calculations. For repulsive interactions, we
present lattice perturbation theory results as well as complex Langevin
calculations, with a modified action to prevent uncontrolled excursions in the
complex plane. Although perturbation theory is a common tool, our
implementation of it is unconventional; we use a Hubbard-Stratonovich
transformation to decouple the system and automate the application of Wick's
theorem, thus generating the diagrammatic expansion, including symmetry
factors, at any desired order. We also present an efficient technique to tackle
nested Matsubara frequency sums without relying on contour integration, which
is independent of dimension and applies to both relativistic and
non-relativistic systems, as well as all energy-independent interactions. We
find exceptional agreement between perturbative and non-perturbative results at
weak couplings, and furnish predictions based on complex Langevin at strong
couplings. We additionally present perturbative calculations of up to the
fifth-order virial coefficient for repulsive and attractive couplings. Both the
lattice perturbation theory and complex Langevin formalisms can easily be
extended to a variety of situations including polarized systems, bosons, and
higher dimension.
