A distribution related to Farey series

M. A. Korolev

Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 137-190

Résumé

We study some arithmetical properties of Farey fractions by the method introduced by F. Boca, C. Cobeli and A. Zaharescu (2001). Suppose that $D\geqslant 2$ is a fixed integer and denote by $\Phi_{Q}$ the classical Farey series of order $Q$. Now let us colour to the red the fractions in $\Phi_{Q}$ with denominators divisible by $D$. Consider the gaps in $\Phi_{Q}$ with coloured endpoints, that do not contain the fractions $a/q$ with $D|q$ inside. The question is to find the limit proportions $\nu(r;D)$ (as $Q\to +\infty$) of such gaps with precisely $r$ fractions inside in the whole set of the gaps under considering ($r = 1,2,3,\ldots$). 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 $\nu(r;D)$ in the terms of areas of some polygons related to some geometrical transform of «Farey triangle», that is, the subdomain of unit square defined by $x+y>1$, $0$. In the present paper, we obtain the precise formulas for $\nu(r;D)$ (in terms of the parameter $r$, $r=1,2,3,\ldots$) for the cases $D = 2, 3$.

Détail

Keywords: Farey series, Farey fractions, Farey triangle, arithmetical properties, distribution, $BCZ$-transform.