Uniform distribution of non-divisible vectors in an integer space

V. I. Arnol'd

Izvestiya. Mathematics, Tome 73 (2009) no. 1, pp. 21-29

Résumé

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 $n$-dimensional space under $N$-fold dilation is asymptotically proportional to the product of $N^n$ and the volume of the domain as $N\to\infty$. 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 on the plane is equal to $1/\zeta(2)=6/\pi^2\approx 2/3$. 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 $n=2$.

Détail

Keywords: crystal lattice, zeta-function, prime numbers, trigonometric sums, density of distribution, Legendre–Chebyshev theorem.
Mots-clés : inclusion/exclusion