  The strong coupling limit of lattice QCD with staggered fermions has been
studied for decades, both via Monte Carlo and via mean field theory. In this
model, the finite density sign problem can be made mild and the full phase
diagram can be obtained, even in the chiral limit. It is however desirable to
understand the effect of a finite lattice gauge coupling $\beta$ on the phase
diagram in the $\mu-T$ plane in order to understand how it evolves into the
phase diagram of continuum QCD. Here we discuss how to construct a partition
function for non-zero lattice coupling, exact to $\mathcal{O}(\beta)$, and
present corresponding Monte Carlo results, in particular for corrections to the
chiral susceptibility and to the phase diagram.
