OP-00015 Open

Convergence of the Flint Hills series

Does the series $$\sum_{n=1}^{\infty} \frac{1}{n^3 \sin^2 n}$$ converge?

Source: C. A. Pickover, The Mathematics of Oz (2002), ch. 25; Alekseyev (2011) related it to the irrationality measure of π.

Classification

Primary: CA — Classical Analysis and ODEs
Secondary: NT
Keywords: irrationality measure, series convergence

References

@book{pickover2002,
  author = {Pickover, Clifford A.},
  title = {The Mathematics of {O}z: Mental Gymnastics from Beyond the Edge},
  publisher = {Cambridge University Press},
  year = {2002}
}

@misc{alekseyev2011,
  author = {Alekseyev, Max A.},
  title = {On convergence of the {F}lint {H}ills series},
  eprint = {1104.5100},
  archivePrefix = {arXiv},
  year = {2011}
}

@article{zeilbergerzudilin2020,
  author = {Zeilberger, Doron and Zudilin, Wadim},
  title = {The irrationality measure of $\pi$ is at most 7.103205334137\ldots},
  journal = {Moscow Journal of Combinatorics and Number Theory},
  volume = {9},
  pages = {407--419},
  year = {2020}
}

@misc{mathworld-flinthills,
  title = {Flint Hills Series},
  howpublished = {MathWorld},
  url = {https://mathworld.wolfram.com/FlintHillsSeries.html}
}

@misc{formalconjectures,
  author = {{The Formal Conjectures Authors}},
  title = {Formal Conjectures},
  url = {https://github.com/google-deepmind/formal-conjectures/blob/0771383387505c96d1b2f6a3d35088ad00892c5c/FormalConjectures/Wikipedia/FlintCooksonHills.lean#L34},
  note = {Lean statement under Apache-2.0; problem text under CC BY 4.0, or CC BY-SA 4.0 where it is based on Wikipedia}
}

Canonical Lean statement

A submission resolves this problem by proving the constant Problem_OP_00015.

/-- OP-00015: Convergence of the Flint Hills series -/
def Problem_OP_00015 : Prop :=
  Summable fun (n : ℕ) ↦
      (1 : ℝ) / (((↑n : ℝ) + (1 : ℝ)) ^ (3 : ℕ) * Real.sin ((↑n : ℝ) + (1 : ℝ)) ^ (2 : ℕ))

Formalised by The Formal Conjectures Authors (Apache-2.0); reviewed by clemens-admin.

Available to match against in: Lean v4.33.1 with Mathlib v4.33.1.

Resolving this problem in Lean

Add the problems package as a dependency, import this problem's module, and prove the constant. Any published revision works.

In lakefile.toml:

[[require]]
name = "RegistryProblems"
git = "https://github.com/cthalhammer/registry-problems"
rev = "<published revision>"

In your proof:

import RegistryProblems.OP_00015

theorem resolution : RegistryProblems.Problem_OP_00015 := by
  sorry

In preprint.toml, mapping the claim to the declaration that establishes it:

[[statements]]
label = "Theorem 1"
lean = "resolution"
latex = "..."
problem = "OP-00015"
claim = "full"

To disprove it instead, prove the negation and claim "disproof", or "counterexample" if the proof exhibits one:

theorem disproof : ¬ RegistryProblems.Problem_OP_00015 := by
  sorry

You can also write the statement out in full; the verifier checks it matches up to definitional equality. Partial results and reductions are not matched against this statement.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00015, "Convergence of the Flint Hills series", clopen.solutions.