Geodesic
A multidimensional analog of the Weierstrass $\zeta$-function in the problem of the number of integer points in a~domain
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 5 (2012) no. 4, pp. 480-484

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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.
Keywords: Weierstrass $\zeta$-function, integer lattice, Bochner–Martinelli kernel, Gauss circle problem. 
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Elena N. Tereshonok; Alexey V. Shchuplev. A multidimensional analog of the Weierstrass $\zeta$-function in the problem of the number of integer points in a~domain. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 5 (2012) no. 4, pp. 480-484. http://geodesic.mathdoc.fr/item/JSFU_2012_5_4_a4/