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We present the results of a calculation of the frequency- and wave-vector-dependent longitudinal dielectric function of a quasi-one-dimensional electron gas. The electrons are taken to be localized to the chains and both the tight-binding and free-electron extremes are considered along the chain axis. Local-field effects are included. Dispersion curves for plasmons and single-particle-excitation spectra are presented. We find that the plasmon modes are not Landau damped and that for long wavelengths these modes have eigenfrequencies ranging continuously from the usual three-dimensional plasma frequency for propagation along the chain axis to zero for propagation perpendicular to it. Finally, we discuss the effects these excitations should have on the optical properties. The absorption in the free-electron extreme contains both single-particle and plasmon contributions throughout the optical spectrum. In the tight-binding limit, the plasmon contributions persist to frequencies larger than the single-particle bandwidth. In no event is the absorption of the Drude form.