Geodesic
An integral formula for the number of lattice points in a domain
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 8 (2015) no. 2, pp. 134-139

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Using the multidimensional logarithmic residue we show a simple formula for the difference between the number of integer points in a bounded domain of $\mathbb{R}^n$ and the volume of this domain. The difference proves to be the integral of an explicit differential form over the boundary of the domain.
Keywords: logarithmic residue, lattice point. 
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Lev Aizenberg; Nikolai Tarkhanov. An integral formula for the number of lattice points in a domain. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 8 (2015) no. 2, pp. 134-139. http://geodesic.mathdoc.fr/item/JSFU_2015_8_2_a1/