  We develop new techniques to efficiently evaluate heat kernel coefficients
for the Laplacian in the short-time expansion on spheres and hyperboloids with
conical singularities. We then apply these techniques to explicitly compute the
logarithmic contribution to black hole entropy from an N=4 vector multiplet
about a Z(N) orbifold of the near-horizon geometry of quarter--BPS black holes
in N=4 supergravity. We find that this vanishes, matching perfectly with the
prediction from the microstate counting. We also discuss possible
generalisations of our heat kernel results to higher-spin fields over Z(N)
orbifolds of higher-dimensional spheres and hyperboloids.
