  We study the large $N$ limit of some supersymmetric partition functions of
the $\mathrm{U}(N)_{k}\times \mathrm{U}(N)_{-k}$ ABJM theory computed by
supersymmetric localization. We conjecture an explicit expression, valid to all
orders in the large $N$ limit, for the partition function on the
$\mathrm{U}(1)\times \mathrm{U}(1)$ invariant squashed sphere in the presence
of real masses in terms of an Airy function. Several non-trivial tests of this
conjecture are presented. In addition, we derive an explicit compact expression
for the topologically twisted index of the ABJM theory valid at fixed $k$ to
all orders in the $1/N$ expansion. We use these results to derive the
topologically twisted index and the sphere partition function in the 't Hooft
limit which correspond to genus $\tt g$ type IIA string theory free energies to
all orders in the $\alpha'$ expansion. We discuss the implications of our
results for holography and the physics of AdS$_4$ black holes.
