  We compare large $N$ expansion of the localization result for the free energy
$F$ in the 3d $\mathcal{N}=6$ superconformal $U(N)_k \times U(N)_{-k}$
Chern-Simons-matter theory to its AdS/CFT counterpart, i.e. to the perturbative
expansion of M-theory partition function on AdS$_{4}\times
S^{7}/\mathbb{Z}_{k}$ and to the weak string coupling expansion of type IIA
effective action on AdS$_{4}\times {\rm CP}^3$. We show that the general form
of the perturbative expansions of $F$ on the two sides of the AdS/CFT duality
is indeed the same. Moreover, the transcendentality properties of the
coefficients in the large $N$, large $k$ expansion of $F$ match those in the
corresponding M-theory or string theory expansions. To shed light on the
structure of the 1-loop M-theory partition function on AdS$_{4}\times
S^{7}/\mathbb{Z}_{k}$ we use the expression for the 1-loop 4-graviton
scattering amplitude in the 11d supergravity. We also use the known information
about the transcendental coefficients of the leading curvature invariants in
the low-energy effective action of type II string theory. Matching of the
remaining rational factors in the coefficients requires a precise information
about currently unknown RR field strength terms in the corresponding
superinvariants.
