Numerator of ζ(4n)/ζ(2n)^2 (with a(0)=2 instead of -2)

Conjecture: if an integer n > 1 is odd, then ζ(2n)/ζ(n)^2 is irrational. Cf. W. Kohnen (link) and my conjecture in A348829. - Thomas Ordowski, Jan 05 2022

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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-114362,
  title        = {Numerator of ζ(4n)/ζ(2n)^2 (with a(0)=2 instead of -2)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-114362}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Conjecture: if an integer $n > 1$ is odd, then $ζ (2 n) / ζ (n)^{2}$ is irrational. Cf. W. Kohnen (link) and my conjecture in A348829. - Thomas Ordowski, Jan 05 2022

The ratio $ζ (4 n) / ζ (2 n)^{2}$ for $n \geq 1$ is the rational number $Q_{n} = - 2 \frac{B _{4 n}}{B _{2 n}^{2} ( 2 n 4 n )}$ where $B_{k}$ is the $k$ -th Bernoulli number. The sequence $a (n)$ is the numerator of $Q_{n}$ , with $a (0)$ defined as $2$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«114362».

theorem conjecture1 (n : ℕ) (hn_gt_one : 1 < n) (hn_odd : Odd n) :
    Irrational ((riemannZeta (2 * n : ℂ) / (riemannZeta (n : ℂ)) ^ 2).re)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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

A114362

Source and licence

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