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Following arXiv:1308.2337, we carry out one loop tests of higher spinAdS$_{d+1}$/CFT$_d$ correspondences for $d\geq 2$. The Vasiliev theories inAdS$_{d+1}$, which contain each integer spin once, are related to the $U(N)$singlet sector of the $d$-dimensional CFT of $N$ free complex scalar fields;the minimal theories containing each even spin once -- to the $O(N)$ singletsector of the CFT of $N$ free real scalar fields. Using analytic continuationof higher spin zeta functions, which naturally regulate the spin sums, wecalculate one loop vacuum energies in Euclidean AdS$_{d+1}$. In even $d$ wecompare the result with the $O(N^0)$ correction to the $a$-coefficient of theWeyl anomaly; in odd $d$ -- with the $O(N^0)$ correction to the free energy $F$on the $d$-dimensional sphere. For the theories of integer spins, thecorrection vanishes in agreement with the CFT of $N$ free complex scalars. Forthe minimal theories, the correction always equals the contribution of one realconformal scalar field in $d$ dimensions. As explained in arXiv:1308.2337, thisresult may agree with the $O(N)$ singlet sector of the theory of $N$ realscalar fields, provided the coupling constant in the higher spin theory isidentified as $G_N\sim 1/(N-1)$. Our calculations in even $d$ are closelyrelated to finding the regularized $a$-anomalies of conformal higher spintheories. In each even $d$ we identify two such theories with vanishing$a$-anomaly: a theory of all integer spins, and a theory of all even spinscoupled to a complex conformal scalar. We also discuss an interacting UV fixedpoint in $d=5$ obtained from the free scalar theory via an irrelevantdouble-trace quartic interaction. This interacting large $N$ theory is dual tothe Vasiliev theory in AdS$_6$ where the bulk scalar is quantized with thealternate boundary condition.

Higher Spin AdS$_{d+1}$/CFT$_d$ at One Loop