  In this work we compute subleading oscillating terms in the Renyi entropy of
Fermi gases and Fermi liquids corresponding to $2k_F$-like oscillations. Our
theoretical tools are the one dimensional formulation of Fermi liquid
entanglement familiar from discussions of the logarithmic violation of the area
law and quantum Monte Carlo calculations. The main result is a formula for the
oscillating term for any region geometry and a spherical Fermi surface. We
compare this term to numerical calculations of entanglement using the
correlation function method and find excellent agreement. We also compare with
quantum Monte Carlo data on interacting Fermi liquids where we also find
excellent agreement up to moderate interaction strengths.
