Take the tour ×  I'm trying to calculate the conformal anomaly
$c$ of a free scalar on a 2-sphere. I've seen other, indirect
ways to do this, but since this is a free theory I feel like it
should be possible to see it directly as the anomalous
depenedence of the partition function on the radius of the
sphere. However, I'm having a difficult time actually doing
this.  In principle, one starts with the naive expression for
the partition function, the inverse square root of the
determinant of the Laplacian. This is obtained as a product of
its eigenvalues, $l(l+1)$, raised to their multiplicity $2l+1$.
We get:  $$Z = \prod_l (l(l+1)/\pi)^{-l-1/2}. $$  This diverges,
so needs to be regulated. For example, we might impose a cutoff
above momenta $\Lambda=l_{max}/R$. By dimensional analysis, the
regulated partition function must be a function of $\Lambda R$,
although ideally it would be cutoff independent. The point of
the conformal anomaly is then that there is some dependence
like:  $$\log Z \sim c \log (\Lambda R