McMullen's formula and a multidimensional analog of the Weierstrass $\zeta$-function

In 1899 G. Pick found a simple formula relating the area $\mathrm{Area}\left(P\right)$ of a plane polygon $P$ with vertices in integer points with the number $I$ of its interior points and on its boundary $B$:
$$ \mathrm{Area}\left(P\right)=I+\frac{B}{2}-1. $$
However, this formula can not be simply extended even to three-dimensional case as Reeve's example demonstrates.Instead of one simple formula there exist several formulas, obtained by combinatorial or number-theoretical methods, or by methods of algebraic geometry. One such formula due to P.McMullen says that for an integer polyhedron with centrally-symmetric facets its volume is equal to the sum of all solid angles at each its integer point of the polyhedron.
A multidimensional analog of the Weierstrass $\zeta$-function
$$ \zeta(z)=\eta(z)+\sum_{\gamma\in{\mathbb Z^{2n}\smallsetminus \{0\}}}\left(\eta(z-\gamma)+\eta(\gamma)+\sum_{i=1}^{n}\left(\frac{\partial\eta}{\partial z_{i}}(\gamma)z_{i}+\frac{\partial\eta}{\partial\bar{z}_{i}}(\gamma)\bar{z}_{i}\right)\right), $$
where $\eta(z)$ is the Bochner-Martinelli differential form allows to prove this statement analytically.