OP-5722Status unassessedNumber theory

Irrationality of odd positive integer values of the Riemann zeta function

TheoremDB P8674 · Candidate awaiting source review

Inferred from the community field; finer classification has not been reviewed.

Source candidate · not reviewedAs of 2026-09-08Source & evidence ↓

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Problem statement

Let $\zeta(s)$ denote the Riemann zeta function, defined for $\Re(s) > 1$ by $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ and extended by analytic continuation to $\mathbb{C} \setminus \{1\}$ . A real number is called irrational if it is not equal to any ratio of two integers. For each positive integer $n$ , determine whether there exists an irrational number $x$ such that $\zeta(2n+1) = x$ .

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Formal statement

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

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Progress & source assessment

Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.

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References

[1]TheoremDB P8674 link
[2]Source formal statement (not locally verified) link

Recorded progress

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Discussion

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