  We compute the canonical partition function Z of non-interacting conformal
higher spin (CHS) theory viewed as a collection of free spin s CFT's in R^d. We
discuss in detail the 4-dimensional case (where s=1 is the standard Maxwell
vector, s=2 is the Weyl graviton, etc.), but also present a generalization for
all even dimensions d. Z may be found by counting the numbers of conformal
operators and their descendants (modulo gauge identities and equations of
motion) weighted by scaling dimensions. This conformal operator counting method
requires a careful analysis of the structure of characters of relevant
(conserved current, shadow field and conformal Killing tensor) representations
of the conformal algebra so(d,2). There is also a close relation to massless
higher spin partition functions with alternative boundary conditions in
AdS_{d+1}. The same partition function Z may also be computed from the CHS path
integral on a curved S^1 x S^{d-1} background. This allows us to determine a
simple factorized form of the CHS kinetic operator on this conformally flat
background. Summing the individual conformal spin contributions Z_s over all
spins we obtain the total partition function of the CHS theory. We also find
the corresponding Casimir energy and show that it vanishes if one uses the same
regularization prescription that implies the cancellation of the total
conformal anomaly a-coefficient. This happens to be true in all even dimensions
d >= 2.
