  We calculate the first three Gilkey-DeWitt (heat-kernel) coefficients, a0, a1
and a2, for massive particles having the spins of most physical interest in n
dimensions, including the contributions of the ghosts and the fields associated
with the appropriate generalized Higgs mechanism. By assembling these into
supermultiplets we compute the same coefficients for general supergravity
theories, and show that they vanish for many examples. One of the steps of the
calculation involves computing these coefficients for massless particles, and
our expressions in this case agree with -- and extend to more general
background spacetimes -- earlier calculations, where these exist. Our results
give that part of the low-energy effective action which depends most
sensitively on the mass of heavy fields once these are integrated out. These
results are used in hep-th/0504004 to compute the sensitivity to large masses
of the Casimir energy in Ricci-flat 4D compactifications of 6D supergravity.
