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Izvestiya: Mathematics, 2009, Volume 73, Issue 1, Pages 21–29
DOI: https://doi.org/10.1070/IM2009v073n01ABEH002436
(Mi im2773)
 

This article is cited in 2 scientific papers (total in 2 papers)

Uniform distribution of non-divisible vectors in an integer space

V. I. Arnol'd

Steklov Mathematical Institute, Russian Academy of Sciences
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 $n$-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 $n=2$.
Keywords: crystal lattice, zeta-function, prime numbers, trigonometric sums, inclusion/exclusion, density of distribution, Legendre–Chebyshev theorem.
Received: 21.02.2008
Bibliographic databases:
Document Type: Article
UDC: 511+517.938
Language: English
Original paper language: Russian
Citation: V. I. Arnol'd, “Uniform distribution of non-divisible vectors in an integer space”, Izv. Math., 73:1 (2009), 21–29
Citation in format AMSBIB
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\by V.~I.~Arnol'd
\paper Uniform distribution of non-divisible vectors in an integer space
\jour Izv. Math.
\yr 2009
\vol 73
\issue 1
\pages 21--29
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  • https://www.mathnet.ru/eng/im2773
  • https://doi.org/10.1070/IM2009v073n01ABEH002436
  • https://www.mathnet.ru/eng/im/v73/i1/p21
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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