On a Conjecture about Sums Involving Farey Fractions

TL;DR

This paper proves a conjecture regarding the sum of squared distances between consecutive Farey fractions in the Farey sequence for integers greater than or equal to 2.

Contribution

It provides a proof for a conjecture about the sum of squared distances in Farey sequences, advancing understanding of Farey fraction properties.

Findings

Confirmed the conjecture for Farey sequences with Q ≥ 2

Derived explicit formulas for the sum of squared distances

Enhanced theoretical understanding of Farey sequence structure

Abstract

In this paper, we prove a conjecture by Daniele Mundici on the sum of squared distances between consecutive elements in the $Q$-th Farey sequence for $Q\in\mathbb{Z}$ and $Q\geq 2$.