We observe that the partition function of the set of all free massless higher
spins
s
= 0, 1, 2, 3,... in
fl
at space is equal to one: the ghost determinants
cancel against the
‘
physical
’
ones or, equivalently, the (regularized) total
number of degrees of freedom vanishes. This re
fl
ects large underlying gauge
symmetry and suggests analogy with supersymmetric or topological theory.
The
Z
= 1 property extends also to the AdS background, i.e. the 1-loop
vacuum partition function of Vasiliev theory is equal to 1 (assuming a parti-
cular regularization of the sum over spins); this was noticed earlier as a
consistency requirement for the vectorial AdS/CFT duality. We
fi
nd that
Z
=1
is true also in the conformal higher spin theory (with higher-derivative
s
2
∂
kinetic terms) expanded near
fl
at or conformally
fl
at
S
4
background. We also
consider the partition function of free conformal theory of symmetric traceless
rank
s
tensor
fi
eld which has 2-derivative kinetic term but only scalar gauge
invariance in
fl
at 4d space. This non-unitary theory has Weyl-invariant action
in curved background and it corresponds to
‘
partially massless
’
fi
eld in
AdS
5
.
We discuss in detail the special case of
s
= 2 (or
‘
conformal graviton
’
),
compute the corresponding conformal anomaly coef
fi
cients and compare them
with previously found expressions for generic representations of conformal
group in 4 dimensions.