A distribution related to Farey series - II
https://doi.org/10.22405/2226-8383-2025-26-3-136-173
Abstract
We continue to study some arithmetical properties of Farey sequences by the method introduced by F. Boca, C. Cobeli and A. Zaharescu (2001). Let Φ𝑄 be the classical Farey sequence of order 𝑄. Having the fixed integers 𝐷 ⩾ 2 and 0 ⩽ 𝑐0 ⩽ 𝐷 − 1, we colour to the red
the fractions in Φ𝑄 with denominators ≡ 𝑐0 (mod𝐷). Consider the gaps in Φ𝑄 with coloured endpoints, that do not contain the fractions 𝑎/𝑞 with 𝑞 ≡ 𝑐0 (mod𝐷) inside. The question is to find the limit proportions 𝜈(𝑟;𝐷, 𝑐0) (as 𝑄 → +∞) of such gaps with precisely 𝑟 fractions inside in the whole set of the gaps under considering (𝑟 = 0, 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 (2012). However, such formula expresses 𝜈(𝑟;𝐷, 𝑐0) in the terms of areas of some polygons related to a special geometrical transform. In the present paper, we obtain explicit formulas for 𝜈(𝑟;𝐷, 𝑐0) for the cases 𝐷 = 3 and 𝑐0 = 1, 2. Thus this and previous author’s papers cover the case 𝐷 = 3.
About the Author
Maxim Alexandrovich Korolev
Steklov Mathematical Institute of Russian Academy of Sciences
Russian Federation
Russian Federation
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Review
For citations:
Korolev M.A. A distribution related to Farey series - II. Chebyshevskii Sbornik. 2025;26(3):136-173. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-3-136-173
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