Farey fractions with equal numerators and the rank of unit fractions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914868726071296 |
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| author | Garcia, Rogelio Tomas |
| author_facet | Garcia, Rogelio Tomas |
| contents | Analytical expressions are derived for the number of fractions with equal numerators in the Farey sequence of order $n$, $F_n$, and in the truncated Farey sequence $F_n^{1/k}$ containing all Farey fractions below $1/k$, with $1\leq k \leq n$. These developments lead to an expression for the rank of $1/k$ in $F_n$, or equivalently $\left|F_n^{1/k}\right|$, and to remarkable relations between the ranks of different unit fractions. Furthermore, the results are extended to Farey fractions of the form $2/k$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_08283 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Farey fractions with equal numerators and the rank of unit fractions Garcia, Rogelio Tomas Number Theory Analytical expressions are derived for the number of fractions with equal numerators in the Farey sequence of order $n$, $F_n$, and in the truncated Farey sequence $F_n^{1/k}$ containing all Farey fractions below $1/k$, with $1\leq k \leq n$. These developments lead to an expression for the rank of $1/k$ in $F_n$, or equivalently $\left|F_n^{1/k}\right|$, and to remarkable relations between the ranks of different unit fractions. Furthermore, the results are extended to Farey fractions of the form $2/k$. |
| title | Farey fractions with equal numerators and the rank of unit fractions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.08283 |