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arXiv · 2606.12907

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

Anji Dong·Huy Xuan Nguyen·Vi Anh Nguyen·Alexandru Zaharescu
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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 $Q\geq 2$, 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.

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Anji Dong, Huy Xuan Nguyen, Vi Anh Nguyen, Alexandru Zaharescu. 2026-06-11. Effective Estimates for a Class of Farey Fraction Sums and Bounds for Mundici-Type Constants. https://arxiv.org/abs/2606.12907

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