  Recent conjectures of the c-theorem in four and higher dimensions have
suggested that the coefficient of the Euler characteristic in the trace anomaly
could measure the degrees of freedom in field theory and decrease along the
renormalization-group flow. We compute this quantity for free massless scalar,
fermion and antisymmetric tensor fields in any dimension, and analyse its
dependence on spin and space-time dimension. In the limit of large number of
dimensions, where the theories become semiclassical, we find that this quantity
does not approach the classical number of field components, but is enhanced for
spinful particles. This seemingly strange behaviour is found to be consistent
with known renormalization-group patterns and a specific c-theorem conjecture.
