  We investigate the R\'enyi entanglement entropies for the one-dimensional
massless free boson compactified on a circle, which describes the low energy
sector of several interacting many-body 1d systems (Luttinger Liquid). We focus
on systems on a finite segment with open boundary conditions and possible
inhomogeneities in the couplings. We provide expressions for the R\'enyi
entropies of integer indices in terms of Fredholm determinant-like expressions.
Within the homogeneous case, we reduce the problem to the solution of linear
integral equations and the computation of Riemann Theta functions. We mainly
focus on a single interval in the middle of the system, but results for generic
bipartitions are given as well.
