  We derive the exact longitudinal plasmon dispersion relations, $\omega(k)$ of
classical one and two dimensional Wigner crystals at T=0 from the real space
equations of motion, of which properly accounts for the full unscreened Coulomb
interactions. We make use of the polylogarithm function in order to evaluate
the infinite lattice sums of the electrostatic force constants. From our exact
results we recover the correct long-wavelength behavior of previous approximate
methods. In 1D, $\omega(k) \sim | k |\log ^{1/2} (1/k)$, validating the known
RPA and bosonization form. In 2D $\omega(k) \sim \sqrt k$, agreeing remarkably
with the celebrated Ewald summation result. Additionally, we extend this
analysis to calculate the band structure of tight-binding models of
non-interacting electrons with arbitrary power law hopping.
