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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 perturbativeexpansion of M-theory partition function on AdS$_{4}\timesS^{7}/\mathbb{Z}_{k}$ and to the weak string coupling expansion of type IIAeffective action on AdS$_{4}\times {\rm CP}^3$. We show that the general formof the perturbative expansions of $F$ on the two sides of the AdS/CFT dualityis indeed the same. Moreover, the transcendentality properties of thecoefficients in the large $N$, large $k$ expansion of $F$ match those in thecorresponding M-theory or string theory expansions. To shed light on thestructure of the 1-loop M-theory partition function on AdS$_{4}\timesS^{7}/\mathbb{Z}_{k}$ we use the expression for the 1-loop 4-gravitonscattering amplitude in the 11d supergravity. We also use the known informationabout the transcendental coefficients of the leading curvature invariants inthe low-energy effective action of type II string theory. Matching of theremaining rational factors in the coefficients requires a precise informationabout currently unknown RR field strength terms in the correspondingsuperinvariants.

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