A distribution related to Farey series
https://doi.org/10.22405/2226-8383-2023-24-4-137-190
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 ⩾ 2 is a fixed integer and denote by Φ𝑄 the classical Farey series of order 𝑄. Now let us colour to the red the fractions in Φ𝑄 with denominators divisible by 𝐷. Consider the gaps in Φ𝑄 with coloured endpoints, that do not contain the fractions 𝑎/𝑞 with 𝐷|𝑞 inside. The question is to find the limit proportions 𝜈(𝑟;𝐷) (as 𝑄 → +∞) of such gaps with precisely 𝑟 fractions inside in the whole set of the gaps under considering (𝑟 = 1, 2, 3, . . .).
In fact, the expression for this proportion can be derived from the general result obtained by C. Cobeli, M. Vˆajˆaitu and A. Zaharescu (2014). However, such formula expresses 𝜈(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 ⩽ 1. In the present paper, we obtain the precise formulas for 𝜈(𝑟;𝐷) (in terms of the parameter 𝑟, 𝑟 = 1, 2, 3, . . .) for the cases 𝐷 = 2, 3.
About the Author
Maxim Alexandrovich Korolev
Steklov Mathematical Institute of Russian Academy of Sciences
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor of Russian Academy of Sciences
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For citations:
Korolev M.A. A distribution related to Farey series. Chebyshevskii Sbornik. 2023;24(4):137-190. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-4-137-190
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