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Diophantine FLINT-HILLS series

Nikos Mantzakouras, Carlos López Zapata

TL;DR

This work studies Diophantine Dirichlet-type series with trig terms, focusing on the unsolved Flint-Hills series $\sum_{n \ge 1} \frac{\csc^2(n)}{n^3}$ and its connections to irrationality measures. It develops a convergence framework via $\text{Riemann–Stieltjes}$ integrals, Hölder continuity, and Abel summation, tying the analysis to Fermi-Dirac/Bose-Einstein integrals and polylogarithms. The main results include a convergent representation for the Flint-Hills series, an explicit constant $c_1 \approx 78.1160806386$, and a new upper bound $\mu(\pi) \le 2.5$ for the irrationality measure of $\pi$, enriching the link between diophantine approximation and series convergence. The paper also sketches a broader program using Weierstrass elliptic curves and polygamma expansions to analyze generalized Diophantine Dirichlet series and their connections to modular forms and elliptic curves, suggesting substantial avenues for future research.

Abstract

We prove the convergence of the FLINT-HILLS series and establish new criteria for a similar type of diophantine or lacunary series, which faces issues due to spaced long terms coming from the trigonometric nature of functions, e.g., cosecant in the FLINT-HILLS series. We connect the FLINT-HILLS series to the Fermi-Dirac integral via the Riemann-Stieltjes integral and YOUNG'S inequality criteria but also proved that the upper bound of the irrationality measure of pi is equal or lower than 2.5 expected if the FLINT-HILLS series converged.

Diophantine FLINT-HILLS series

TL;DR

This work studies Diophantine Dirichlet-type series with trig terms, focusing on the unsolved Flint-Hills series and its connections to irrationality measures. It develops a convergence framework via integrals, Hölder continuity, and Abel summation, tying the analysis to Fermi-Dirac/Bose-Einstein integrals and polylogarithms. The main results include a convergent representation for the Flint-Hills series, an explicit constant , and a new upper bound for the irrationality measure of , enriching the link between diophantine approximation and series convergence. The paper also sketches a broader program using Weierstrass elliptic curves and polygamma expansions to analyze generalized Diophantine Dirichlet series and their connections to modular forms and elliptic curves, suggesting substantial avenues for future research.

Abstract

We prove the convergence of the FLINT-HILLS series and establish new criteria for a similar type of diophantine or lacunary series, which faces issues due to spaced long terms coming from the trigonometric nature of functions, e.g., cosecant in the FLINT-HILLS series. We connect the FLINT-HILLS series to the Fermi-Dirac integral via the Riemann-Stieltjes integral and YOUNG'S inequality criteria but also proved that the upper bound of the irrationality measure of pi is equal or lower than 2.5 expected if the FLINT-HILLS series converged.

Paper Structure

This paper contains 11 sections, 22 theorems, 390 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and preliminaries
  2. A Comprehensive Introductory Proof of the Flint Hills Series using the Innovative Convergence Method: Recursive Augmented Trigonometric Substitution
  3. Bounds on the Irrationality Measure of pi
  4. Hölder Condition and the Riemann-Stieltjes Integral
  5. Diophantine Dirichlet series through the lens of the Riemann-Stieltjes integral
  6. Generalization of Diophantine Dirichlet Series
  7. Abel Summation Formula
  8. Hölder Continuity and Convergence
  9. Main results
  10. Conclusion: Significant Remark on the Irrationality Measure of $\pi$
  11. Future research: Weierstrass elliptic curves and series expansions in the polygamma function for Diophantine Dirichlet Series

Key Result

Theorem 1.1

(Theorem 2.5 (Main result), Meiburg, 2022). For any sine-like function $P$ with irrational period $\alpha$, constant $v \geq 1$, and $\mu(\alpha) < 1 + \frac{u}{v}$, the series $S_{u,v}(n)$ converges.

Figures (2)

  • Figure 1: Slope field of the first-order ordinary differential equation $\Lambda'(t) = -\frac{\pi}{\sqrt{3}} \psi"'(t)$, where $\psi"'(t)$ is the third derivative of the digamma function with respect to $t$.
  • Figure 2: This representation aids in understanding how the discontinuous nature of the floor function interacts with the integrand function $f(x)$. By visually showing the contributions at each integer step, it provides an intuitive grasp of how the Riemann-Stieltjes integral accumulates values over intervals where $\lfloor x \rfloor$ changes.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2: Existence of the Riemann-Stieltjes Integral
  • proof
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 32 more