The problem of an interacting electron gas in a static lattice (an inhomogeneous electron gas) at zero temperature is treated by use of field theoretic perturbation theory. The unperturbed system is taken to be a non-interacting electron gas in a static lattice. The electron-electron interactions are then treated as a perturbation or;. this system. One-particle basis functions, Bloch functions, are introduced explicitly into the perturbation series for the Green functions, and diagrams associated with the various terms are defined. Certain formal results in integral equation form are noted. Using the method of DuBois, the plasmon excited state is studied. In particular, plasmon energies and lifetimes are calculated in the pair approximation for several insulators.