  We extend previous work on the perturbative expansion of the Renyi entropy,
$S_q$, around $q=1$ for a spherical entangling surface in a general CFT.
Applied to conformal scalar fields in various spacetime dimensions, the results
appear to conflict with the known conformal scalar Renyi entropies. On the
other hand, the perturbative results agree with known Renyi entropies in a
variety of other theories, including theories of free fermions and vector
fields and theories with Einstein gravity duals. We propose a resolution
stemming from a careful consideration of boundary conditions near the
entangling surface. This is equivalent to a proper treatment of
total-derivative terms in the definition of the modular Hamiltonian. As a
corollary, we are able to resolve an outstanding puzzle in the literature
regarding the Renyi entropy of ${\cal N}=4$ super-Yang-Mills near $q=1$. A
related puzzle regards the question of stationarity of the renormalized
entanglement entropy (REE) across a circle for a (2+1)-dimensional massive
scalar field. We point out that the boundary contributions to the modular
Hamiltonian shed light on the previously-observed non-stationarity. Moreover,
IR divergences appear in perturbation theory about the massless fixed point
that inhibit our ability to reliably calculate the REE at small non-zero mass.
