  We study perturbative behavior of free energies on a d-dimensional sphere S^d
for theories with marginal interactions. The free energies are interpreted as
the "dilaton effective action" with the dilaton having a nontrivial background
vacuum expectation value. We compute the dependence of the free energies on the
radius of the sphere by using dimensional regularization. It is shown that the
first (second) derivative of the free energies in odd (even) dimensions with
respect to the radius of the sphere are proportional to the square of the beta
functions of coupling constants. The result is consistent with the c, F and
a-theorems in two, three, four and six dimensions. The result is also used to
rule out a large class of scale invariant theories which are not conformally
invariant.
