Mathematics & Physics 2015, 8(2), 134–139 УДК 517.55 An Integral Formula for the Number of Lattice Points in a Domain Lev Aizenberg∗ Department of Mathematics Bar-Ilan University 52900 Ramat-Gan Israel Nikolai Tarkhanov† Institute of Mathematics University of Potsdam Am Neuen Palais, 10, Potsdam, 14469 Germany Received 06.02.2015, received in revised form 06.03.2015, accepted 14.04.2015 Using the multidimensional logarithmic residue we show a simple formula for the difference between the number of integer points in a bounded domain of Rn and the volume of this domain. <...> The difference proves to be the integral of an explicit differential form over the boundary of the domain. <...> Introduction Classical function theory is of great importance in number theory, let alone the analytical extension of the Riemann zeta function and prime number theorem, see [6,8,9], etc. <...> This work was intended as an attempt at applying the theory of functions of several complex variables to classical problems of number theory. <...> To wit, we apply the multidimensional logarithmic residue which is an efficient numerical tool of algebraic geometry, see [1]. <...> Let Z be a bounded domain with piecewise smooth boundary in the space Cn of n complex Z variables z = (z1, . . . , zn). <...> A number of classical problems of number theory, e.g. the ∗aizenbrg@gmail.com †tarkhanov@math.uni-potsdam.de Siberian Federal University. <...> All rights reserved c – 134 – Lev Aizenberg, Nikolai Tarkhanov An Integral Formula for the Number of Lattice Points in a Domain problem on the number of lattice points in a ball [10], the problems on Dirichlet divisors [4], etc. reduce to evaluating asymptotics of the difference. <...> It is worth pointing out that this asymptotics can not be found by standard methods, such as the Laplace method, stationary phase method, or saddle point method. <...> The theory of lattice points in large regions has attracted the interest ofmanymathematicians for more than eleven decades. <...> The monograph [5] presents a broad survey of the main problems and results in lattice point theory. 1. <...> The integral formula As usual, we write Rn, n 1, for the n-dimensional real Euclidean space of variables x = (x1, . . . ,xn) with xj ∈ R. Suppose X is a bounded domain in Rn whose <...>