OP-00014 Open

Irrationality of the Euler–Mascheroni constant

Is the Euler–Mascheroni constant $\gamma = \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{1}{k} - \log n \right)$ irrational?

Source: Folklore; mentioned by Hilbert, Mathematische Probleme (ICM Paris, 1900). See Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc. 50 (2013).

Classification

Primary: NT — Number Theory
Keywords: Euler–Mascheroni constant, irrationality

References

@article{lagarias2013,
  author = {Lagarias, Jeffrey C.},
  title = {Euler's constant: {E}uler's work and modern developments},
  journal = {Bulletin of the American Mathematical Society},
  volume = {50},
  pages = {527--628},
  year = {2013}
}

@misc{wikipedia,
  title = {Irrational number: open questions},
  howpublished = {Wikipedia},
  url = {https://en.wikipedia.org/wiki/Irrational_number#Open_questions}
}

@misc{formalconjectures,
  author = {{The Formal Conjectures Authors}},
  title = {Formal Conjectures},
  url = {https://github.com/google-deepmind/formal-conjectures/blob/0771383387505c96d1b2f6a3d35088ad00892c5c/FormalConjectures/Wikipedia/Irrational.lean#L91},
  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_00014.

/-- OP-00014: Irrationality of the Euler–Mascheroni constant -/
def Problem_OP_00014 : Prop :=
  Irrational Real.eulerMascheroniConstant

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_00014

theorem resolution : RegistryProblems.Problem_OP_00014 := by
  sorry

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

[[statements]]
label = "Theorem 1"
lean = "resolution"
latex = "..."
problem = "OP-00014"
claim = "full"

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

theorem disproof : ¬ RegistryProblems.Problem_OP_00014 := 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-00014, "Irrationality of the Euler–Mascheroni constant", clopen.solutions.