Uniform distribution of non-divisible vectors in an integer space

Abstract

A vector in an integer space is said to be divisible if it is the product of another vector in this space and an integer exceeding 1.The uniform distribution of a set of integer vectors means that the number of points of this set in the image of a domain in -dimensional space under -fold dilation is asymptotically proportional to the product of and the volume of the domain as .The constant of proportionality (called the density of the set) is equal to for the set of non-divisible vectors in -dimensional integer space (where ). For example, the density of the set of non-divisible vectors on the plane is equal to . It was this discovery that led Euler to the definition of the zeta-function.The proof of the uniform distribution of the set of non-divisible integer vectors is published here because there are arbitrarily large domains containing no non-divisible vectors. We shall show that such domains are situated only far from the origin and are infrequent even there. Their distribution is also uniform and has a peculiar fractal character, which has not yet been studied even at the empirical computer-guided level or even for .