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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2502.19881 |
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| _version_ | 1866910920037367808 |
|---|---|
| author | Korolev, Maxim A. |
| author_facet | Korolev, Maxim A. |
| contents | Minor corrections to previous version.
We study some arithmetical properties of Farey sequences by the method introduced by F.Boca, C.Cobeli and A.Zaharescu (2001). Let $Φ_{Q}$ be the classical Farey sequence of order $Q$. Having the fixed integers $D\geqslant 2$ and $0\leqslant c\leqslant D-1$, we colour to the red the fractions in $Φ_{Q}$ with denominators $\equiv c \pmod D$. Consider the gaps in $Φ_{Q}$ with coloured endpoints, that do not contain the fractions $a/q$ with $q\equiv c \pmod D$ inside. The question is to find the limit proportions $ν(r;D,c)$ (as $Q\to +\infty$) of such gaps with precisely $r$ fractions inside in the whole set of the gaps under considering ($r = 0,1,2,3,\ldots$). 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 $ν(r;D,c)$ in the terms of areas of some polygons related to a special geometrical transform. In the present paper, we obtain an explicit formulas for $ν(r;D,c)$ for the cases $D = 2, 3$ and $c=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19881 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A distribution related to Farey sequences -- I Korolev, Maxim A. Number Theory 11B57 Minor corrections to previous version. We study some arithmetical properties of Farey sequences by the method introduced by F.Boca, C.Cobeli and A.Zaharescu (2001). Let $Φ_{Q}$ be the classical Farey sequence of order $Q$. Having the fixed integers $D\geqslant 2$ and $0\leqslant c\leqslant D-1$, we colour to the red the fractions in $Φ_{Q}$ with denominators $\equiv c \pmod D$. Consider the gaps in $Φ_{Q}$ with coloured endpoints, that do not contain the fractions $a/q$ with $q\equiv c \pmod D$ inside. The question is to find the limit proportions $ν(r;D,c)$ (as $Q\to +\infty$) of such gaps with precisely $r$ fractions inside in the whole set of the gaps under considering ($r = 0,1,2,3,\ldots$). 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 $ν(r;D,c)$ in the terms of areas of some polygons related to a special geometrical transform. In the present paper, we obtain an explicit formulas for $ν(r;D,c)$ for the cases $D = 2, 3$ and $c=0$. |
| title | A distribution related to Farey sequences -- I |
| topic | Number Theory 11B57 |
| url | https://arxiv.org/abs/2502.19881 |