Particular values of the Riemann zeta function

ζ(5) is irrational.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-riemann-zeta-values,
  title        = {Particular values of the Riemann zeta function},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/riemann-zeta-values}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

irrational_five. $ζ (5)$ is irrational.

irrational_seven. $ζ (7)$ is irrational.

irrational_nine. $ζ (9)$ is irrational.

irrational_eleven. $ζ (11)$ is irrational.

irrational_odd. $ζ (2 n + 1)$ is irrational for any $n \in N^{+}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.RiemannZetaValues (5 statements).

theorem irrational_five : ∃ x, Irrational x ∧ riemannZeta 5 = x

theorem irrational_seven : ∃ x, Irrational x ∧ riemannZeta 7 = x

theorem irrational_nine : ∃ x, Irrational x ∧ riemannZeta 9 = x

theorem irrational_eleven : ∃ x, Irrational x ∧ riemannZeta 11 = x

theorem irrational_odd (n : ℕ) (hn : 0 < n) :
    ∃ x, Irrational x ∧ riemannZeta (2 * n + 1) = x

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.