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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{JSFU_2015_8_2_a1,
     author = {Lev Aizenberg and Nikolai Tarkhanov},
     title = {An integral formula for the number of lattice points in a domain},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {134--139},
     publisher = {mathdoc},
     volume = {8},
     number = {2},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2015_8_2_a1/}
}
TY  - JOUR
AU  - Lev Aizenberg
AU  - Nikolai Tarkhanov
TI  - An integral formula for the number of lattice points in a domain
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2015
SP  - 134
EP  - 139
VL  - 8
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2015_8_2_a1/
LA  - en
ID  - JSFU_2015_8_2_a1
ER  - 
%0 Journal Article
%A Lev Aizenberg
%A Nikolai Tarkhanov
%T An integral formula for the number of lattice points in a domain
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2015
%P 134-139
%V 8
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2015_8_2_a1/
%G en
%F JSFU_2015_8_2_a1
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/

[1] L. A. Aizenberg, A. P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, AMS, RI, 1983

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

[3] L. Aizenberg, “Application of multidimensional logarithmic residue in number theory. An integral formula for the difference between the number of lattice points in a domain and its volume”, Annales Polonici Mathematici, 46 (1985), 395–401 | MR

[4] K. S. Chandrasekharan, Introduction to Analytic Number Theory, Springer, New York, 1968 | MR | Zbl

[5] F. Fricker, Introduction to Lattice Point Theory, Textbooks and Monographs in the Exact Sciences Mathematical Series, 73, Birkhäuser Verlag, Basel–Boston, Mass., 1982 | MR

[6] Th. W. Gamelin, Complex Analysis, Springer, New York et al., 2001 | MR | Zbl

[7] R. K. Guy, Unsolved Problems in Number Theory, Springer, New York et al., 1981 | MR | Zbl

[8] D. J. Newman, Analytic Number Theory, Springer, New York et al., 1998 | MR | Zbl

[9] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford University Press, London, 1951 | MR | Zbl

[10] I. M. Vinogradov, Particular Versions of Trigonometric Sum Method, Nauka, M., 1976 (in Russian)