  We investigate the fluctuations in the number of integral lattice points on
the Heisenberg groups which lie inside a Cygan-Kor{\'a}nyi norm ball of large
radius. Let
$\mathcal{E}_{q}(x)=\big|\mathbb{Z}^{2q+1}\cap\delta_{x}\mathcal{B}\big|-\textit{vol}\big(\mathcal{B}\big)x^{2q+2}$
denote the error term which occurs for this lattice point counting problem on
the Heisenberg group $\mathbb{H}_{q}$, where $\mathcal{B}$ is the unit ball in
the Cygan-Kor{\'a}nyi norm and $\delta_{x}$ is the Heisenberg-dilation by
$x>0$. For $q\geq3$ we consider the suitably normalized error term
$\mathcal{E}_{q}(x)/x^{2q-1}$, and prove it has a limiting value distribution
which is absolutely continuous with respect to the Lebesgue measure. We show
that the defining density for this distribution, denoted by
$\mathcal{P}_{q}(\alpha)$, can be extended to the whole complex plane
$\mathbb{C}$ as an entire function of $\alpha$ and satisfies for any
non-negative integer $j\geq0$ and any $\alpha\in\mathbb{R}$,
$|\alpha|>\alpha_{q,j}$, the bound: \begin{equation*} \begin{split}
\big|\mathcal{P}^{(j)}_{q}(\alpha)\big|\leq\exp{\Big(-|\alpha|^{4-\beta/\log\log{|\alpha|}}\Big)}
{split} {equation*} where $\beta>0$ is an absolute constant. In addition, we
give an explicit formula for the $j$-th integral moment of the density
$\mathcal{P}_{q}(\alpha)$ for any integer $j\geq1$.
