A Geometric-Arithmetic Framework for the Flint Hills Series
Mohammed-Adnane Garab
TL;DR
The paper reframes the unresolved Flint Hills series problem by introducing a geometric-arithmetic framework based on the distance $d(n)$ to the nearest multiple of $\pi$. By defining the auxiliary sums $L(N)=\sum_{n\le N} \frac{1}{n^3 d(n)^2}$ and $G(N)=\frac{\pi^2}{4}L(N)$, it establishes the sharp bounds $L(N)\le S_N\le G(N)$, linking convergence to the Diophantine approximation properties of $\pi$ via the irrationality exponent $\mu(\pi)$. The analysis shows spikes in $S_N$ correspond to very good rational approximations of $\pi$ (convergents), with block contributions near convergents scaling like $\text{const}/(q_k^2 q_{k+1})$, implying increasing sparsity and diminishing impact. Numerical experiments corroborate the bounds and spike behavior, while refinements (improved lower bounds, adaptive thresholds) and generalizations extend the framework to other periodic or near-periodic series. A key outcome is a convergence criterion tied to $\mu(\pi)$, with a provisional threshold $\mu_0$ in $[2.37,2.5]$ separating convergence from divergence, though the exact status of $\mu(\pi)$ remains unresolved. Overall, the work provides a precise, Diophantine-driven lens on the Flint Hills problem and actionable estimates for partial sums.
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.