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/

[1] R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, 1994 | MR

[2] S. E. Cappell, J. L. Shaneson, Some problems in number theory. I: the circle problem, 2007, arXiv: math/0702613v3.pdf

[3] L. A. Aizenberg, “Application of multidimensional logarithmic residue to represent the difference between the number of integer points in a domain and its volume in the form of an integral”, Dokl. Nauk SSSR, 270:3 (1983), 521–523 (Russian) | MR

[4] E. Krätzel, Lattice points, Mathematics and its Applications, East European Series, 33, 1988 | MR

[5] A. N. Varchenko, “Number of lattice points in families of homothetic domains in $\mathbb R^n$”, Funkts. Anal. Prilozh., 17:2 (1983), 1–6 (Russian) | MR | Zbl

[6] R. Diaz, S. Robins, “Pick's Formula via the Weierstrass $\wp$-function”, The American Mathematical Monthly, 102:5 (1995), 431–437 | DOI | MR | Zbl

[7] P. Zappa, “Sulle classi di Dolbeault di tipo $(0,n-1)$ con singolarita in un insieme discreto”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 8:70 (1981), 87–95 (Italian)