Uniform distribution of non-divisible vectors in an integer space

摘要

A vector in an integer space is said to be divisible if it is the product of another vector in this space and ail 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 n-dimensional space Under N-fold dilation is asymptotically proportional to the product of N-n and the volume of the domain as N -> infinity. The constant of proportionality (called the density of the set) is equal to 1/zeta(n) for the set of non-divisible vectors in n-dimensional integer space (where n > 1). For example, the density of the set of non-divisible vectors oil the plane is equal to 1/zeta(2) = 6/pi(2) approximate to 2/3. 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 n = 2.