  The set of particle-hole ring diagrams for a many-fermion system in two
dimensions is studied. The complex-valued polarization function is derived in
detail and shown to be expressible in terms of square-root functions. For a
contact-interaction the perturbative contributions to the energy per particle
$\bar E(k_f)$ are calculated in closed analytical form from third up to twelfth
order. The resummation of the particle-hole ring diagrams to all orders is
studied and a pronounced dependence on the dimensionless coupling parameter
$\alpha$ is found. There is a substantial difference between the complete
ring-sum with all exchange-type diagrams included and the standard resummation
of the leading $n$-ring diagrams only. The spin factor $S_n(g)$ associated to
the $n$-th order ring diagrams is derived for arbitrary spin-degeneracy $g$.
