Improved bounds for the Mayer-Erdős phenomenon on similarly ordered Farey fractions

2025/08/28 by Wouter van Doorn, van Doorn, Wouter · 1 citation
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2509.00121

Abstract

Let (a₁)/(b₁), (a₂)/(b₂), … be the Farey fractions of order n. We then prove that the inequality (a_l - a_k)(b_l - b_k) ≥ 0 holds for all k and l > k with l-k ≤ ((1)/(12) - o(1) )n, sharpening an old result by Erdős. On the other hand, we will show that for all n ≥ 4 there are k, l with k < l < k + (n)/(4) + 5 for which the product (a_l - a_k)(b_l - b_k) is negative.

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• Optimality of Wouter van Doorn's Upper Bound for the Mayer-Erdős Farey Problem

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