On the distribution of index of Farey sequences

In this article, we study the distribution of index of Farey fractions by providing asymptotic formulas for moments of index of Farey fractions twisted by Dirichlet characters for Farey fractions with \(\mathcal {B}\) $\mathcal{B}$ -free denominators. Additionally, we reconsider the square-free case earlier done in Alkan et al. (Ramanujan J 16(2):131–161, 2008), and obtain new results for moments of indices with square-free denominators. We also obtain higher level correlation measures of the index function, generalizing earlier known results on two level correlations.