Effective Estimates for a Class of Farey Fraction Sums and Bounds for Mundici-Type Constants

Abstract

Let $D_2(Q)$ denote the sum of squared distances between consecutive Farey fractions in the full interval $(0,1]$. Daniele Mundici conjectured that $C(Q):=D_2(Q)\cdot Q^2/\log Q$ is less than 3 for all $Q\ge 2$, which is confirmed true in \cite{DLN2026}. In this paper, we generalize this result to subintervals of $(0,1]$ and to $h$-spacings. As applications, we obtain Mundici-type bounds in these two settings, extending the full-interval consecutive-spacing case of Mundici's conjecture.

27 pages