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Chebyshevskii Sbornik, 2023, Volume 24, Issue 4, Pages 137–190
DOI: https://doi.org/10.22405/2226-8383-2023-24-4-137-190
(Mi cheb1342)
 

A distribution related to Farey series

M. A. Korolev

Steklov Mathematical Institute of Russian Academy of Sciences (Moscow)
References:
Abstract: We study some arithmetical properties of Farey fractions by the method introduced by F. Boca, C. Cobeli and A. Zaharescu (2001). Suppose that $D\geqslant 2$ is a fixed integer and denote by $\Phi_{Q}$ the classical Farey series of order $Q$. Now let us colour to the red the fractions in $\Phi_{Q}$ with denominators divisible by $D$. Consider the gaps in $\Phi_{Q}$ with coloured endpoints, that do not contain the fractions $a/q$ with $D|q$ inside. The question is to find the limit proportions $\nu(r;D)$ (as $Q\to +\infty$) of such gaps with precisely $r$ fractions inside in the whole set of the gaps under considering ($r = 1,2,3,\ldots$).
In fact, the expression for this proportion can be derived from the general result obtained by C. Cobeli, M. Vâjâitu and A. Zaharescu (2014). However, such formula expresses $\nu(r;D)$ in the terms of areas of some polygons related to some geometrical transform of «Farey triangle», that is, the subdomain of unit square defined by $x+y>1$, $0<x,y\leqslant 1$. In the present paper, we obtain the precise formulas for $\nu(r;D)$ (in terms of the parameter $r$, $r=1,2,3,\ldots$) for the cases $D = 2, 3$.
Keywords: Farey series, Farey fractions, Farey triangle, arithmetical properties, distribution, $BCZ$-transform.
Funding agency Grant number
Russian Science Foundation 19-11-00001
Received: 20.10.2023
Accepted: 11.12.2023
Bibliographic databases:
Document Type: Article
UDC: 517
Language: Russian
Citation: M. A. Korolev, “A distribution related to Farey series”, Chebyshevskii Sb., 24:4 (2023), 137–190
Citation in format AMSBIB
\Bibitem{Kor23}
\by M.~A.~Korolev
\paper A distribution related to Farey series
\jour Chebyshevskii Sb.
\yr 2023
\vol 24
\issue 4
\pages 137--190
\mathnet{http://mi.mathnet.ru/cheb1342}
\crossref{https://doi.org/10.22405/2226-8383-2023-24-4-137-190}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4705791}
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  • https://www.mathnet.ru/eng/cheb/v24/i4/p137
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