A multidimensional analog of the Weierstrass $\zeta$-function in the problem of the number of integer points in a domain

A multidimensional analog of the Weierstrass $\zeta$-function in $\mathbb C^n$ is a differential $(0,n-1)$-form with singularities in the points of the integer lattice $\Gamma\subset\mathbb C^n$. Using this form we construct a $\Gamma$-invariant $(n,n-1)$-form $\tau(z)\wedge dz$. The integral of this form over a domain's boundary is equal to difference between the number of integer points in the domain and its volume.