  In a remarkable paper [Phys. Rev. Lett. 96, 100503 (2006)], Dimitri Gioev and
Israel Klich conjectured an explicit formula for the leading asymptotic growth
of the spatially bi-partite von-Neumann entanglement entropy of non-interacting
fermions in multi-dimensional Euclidean space at zero temperature. Based on
recent progress by one of us (A.V.S.) in semi-classical functional calculus for
pseudo-differential operators with discontinuous symbols, we provide here a
complete proof of that formula and of its generalization to R\'enyi entropies
of all orders $\alpha>0$. The special case $\alpha=1/2$ is also known under the
name logarithmic negativity and often considered to be a particularly useful
quantification of entanglement. These formulas, exhibiting a "logarithmically
enhanced area law", have been used already in many publications.
