Comments on ABJM free energy on $S^3$ at large $N$ and perturbative expansions in M-theory and string theory

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${}_4\times S^7\mathrm{/}{\mathbb{Z}}_k$ and to the weak string coupling expansion of type IIA effective action on AdS${}_4\times {\mathrm{C}\mathrm{P}}^3$. 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${}_4\times S^7\mathrm{/}{\mathbb{Z}}_k$ 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.