  The exact expressions for integrated maximal $U(1)_Y$ violating (MUV)
$n$-point correlators in $SU(N)$ ${\mathcal N}=4$ supersymmetric Yang--Mills
theory are determined. The analysis generalises previous results on the
integrated correlator of four superconformal primaries and is based on
supersymmetric localisation. The integrated correlators are functions of $N$
and $\tau=\theta/(2\pi)+4\pi i/g_{_{YM}}^2$, and are expressed as
two-dimensional lattice sums that are modular forms with holomorphic and
anti-holomorphic weights $(w,-w)$ where $w=n-4$. The correlators satisfy
Laplace-difference equations that relate the $SU(N+1)$, $SU(N)$ and $SU(N-1)$
expressions and generalise the equations previously found in the $w=0$ case.
The correlators can be expressed as infinite sums of Eisenstein modular forms
of weight $(w,-w)$. For any fixed value of $N$ the perturbation expansion of
this correlator is found to start at order $( g_{_{YM}}^2 N)^w$. The
contributions of Yang--Mills instantons of charge $k>0$ are of the form $q^k\,
f(g_{_{YM}})$, where $q=e^{2\pi i \tau}$ and $f(g_{_{YM}})= O(g_{_{YM}}^{-2w})$
when $g_{_{YM}}^2 \ll 1$. Anti-instanton contributions have charge $k<0$ and
are of the form $\bar q^{|k|} \, \hat f(g_{_{YM}})$, where $\hat f(g_{_{YM}}) =
O(g_{_{YM}}^{2w})$ when $g_{_{YM}}^2 \ll 1$. Properties of the large-$N$
expansion are in agreement with expectations based on the low energy expansion
of flat-space type IIB superstring amplitudes. We also comment on the
identification of $n$-point free-field MUV correlators with the integrands of
$(n-4)$-loop perturbative contributions to the four-point correlator. In
particular, we emphasise the important r\^ole of $SL(2, \mathbb{Z})$-covariance
in the construction.
