  We present a novel expression for an integrated correlation function of four
superconformal primaries in $SU(N)$ $\mathcal{N}=4$ SYM. This integrated
correlator, which is based on supersymmetric localisation, has been the subject
of several recent developments. The correlator is re-expressed as a sum over a
two dimensional lattice that is valid for all $N$ and all values of the complex
Yang-Mills coupling $\tau$. In this form it is manifestly invariant under
$SL(2,\mathbb{Z})$ Montonen-Olive duality. Furthermore, it satisfies a
remarkable Laplace-difference equation that relates the $SU(N)$ to the
$SU(N+1)$ and $SU(N-1)$ correlators. For any fixed value of $N$ the correlator
is an infinite series of non-holomorphic Eisenstein series,
$E(s;\tau,\bar\tau)$ with $s\in \mathbb{Z}$, and rational coefficients. The
perturbative expansion of the integrated correlator is asymptotic and the
$n$-loop coefficient is a rational multiple of $\zeta(2n+1)$. The $n=1$ and
$n=2$ terms agree precisely with results determined directly by integrating the
expressions in one- and two-loop perturbative SYM. Likewise, the charge-$k$
instanton contributions have an asymptotic, but Borel summable, series of
perturbative corrections. The large-$N$ expansion of the correlator with fixed
$\tau$ is a series in powers of $N^{1/2-\ell}$ ($\ell\in \mathbb{Z}$) with
coefficients that are rational sums of $E_s$ with $s\in \mathbb{Z}+1/2$. This
gives an all orders derivation of the form of the recently conjectured
expansion. We further consider 't Hooft large-$N$ Yang-Mills theory. The
coefficient of each order can be expanded as a convergent series in $\lambda$.
For large $\lambda$ this becomes an asymptotic series with coefficients that
are again rational multiples of odd zeta values. The large-$\lambda$ series is
not Borel summable, and its resurgent non-perturbative completion is
$O(\exp(-2\sqrt{\lambda}))$.
