Farey fractions with equal numerators and the rank of unit fractions

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 $2/k$.