Character Integral Representation of Zeta function in AdS$_{d+1}$: II. Application to partially-massless higher-spin gravities

We compute the one-loop free energies of the type-A$_\ell$ and type-B$_\ell$ higher-spin gravities in $(d+1)$-dimensional anti-de Sitter (AdS$_{d+1}$) spacetime. For large $d$ and $\ell$, these theories have a complicated field content, and hence it is difficult to compute their zeta functions using the usual methods. Applying the character integral representation of zeta function developed in the companion paper arXiv:1805.05646 to these theories, we show how the computation of their zeta function can be shortened considerably. We find that the results previously obtained for the massless theories ($\ell=1$) generalize to their partially-massless counterparts (arbitrary $\ell$) in arbitrary dimensions.