A Geometric-Arithmetic Framework for the Flint Hills Series

TL;DR

This paper presents a geometric-arithmetic framework connecting the convergence of the Flint Hills series to the irrationality measure of pi, providing new insights into its arithmetic structure and distribution of near-multiples.

Contribution

It introduces a novel geometric-arithmetic approach that links the series' convergence to pi's irrationality measure, offering fresh perspectives on this longstanding problem.

Findings

Link between convergence and irrationality measure of pi

Insights into distribution of near-multiples of pi

New understanding of the series' arithmetic structure

Abstract

We introduce a geometric-arithmetic approach to the analysis of the Flint Hills series, linking its convergence behavior to the irrationality measure of pi. The framework highlights the interplay between the distribution of near-multiples of pi and the growth rate of denominator sequences, offering new insights into the arithmetic structure underlying this famous unsolved problem.