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arXiv · 2404.08283

Farey fractions with equal numerators and the rank of unit fractions

Rogelio Tomas Garcia
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Abstract

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 $2/k$.

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Rogelio Tomas Garcia. 2024-04-12. Farey fractions with equal numerators and the rank of unit fractions. https://doi.org/10.5281/zenodo.12685697

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