arXiv:2607.17123v1

A Modular Form Proof of the Irrationality of $ζ\left(3\right)$

Pang Ern Thang

math.NT11J7211F1111M06

Paper metadata

Original submission: July 19, 2026
Claimed MSC: 11J72, 11F11, 11M06
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗

Abstract

We present an expository proof of the irrationality of $ζ\left(3\right)$ using modular forms of level 6. By constructing a suitable Eichler integral, we obtain a power series with controlled denominators and sufficiently large radius of convergence. Beukers' irrationality criterion then implies that $ζ\left(3\right)$ is irrational.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements1 reported findingCorrect

The central statement that $\zeta(3)$ is irrational is correct, independently established by Apéry's theorem. The manuscript's modular-form route contains an unresolved analytic-continuation step, but this does not make the theorem itself false or unsupported.

Theorem 2.5Correct

Irrationality of $\zeta(3)$

Pages 13–15 · Theorem 2.5 · arXiv:2607.17123v1

The theorem is the classical result proved by Apéry and has complete independent proofs. The present report therefore classifies the statement as correct even though the particular proof printed here does not fully justify its decisive radius-of-convergence assertion.

Beukers, A note on the irrationality of zeta(2) and zeta(3) ↗

02Proofs3 reported findingsContains incorrect or incomplete proofs

The modular-form calculations and denominator argument are largely consistent, but the proof does not establish the branch-locus and second-singularity facts needed to conclude that the key series has radius exactly $17+12\sqrt2$ and infinitely many nonzero coefficients. A harmless sign error multiplies a term subsequently shown to vanish.

Proof of Theorem 2.5Incomplete as written

The decisive radius-of-convergence assertion is not established

Pages 14–15 · Paragraph beginning with the first two positive branching values · arXiv:2607.17123v1

Proposition 2.4 proves a univalent map on the displayed region and evaluates $t$ at two distinguished points, but it does not compute the complete ramification divisor of $t$ or prove that no complex branch value of smaller modulus intervenes. The proof then calls $t_1=(\sqrt2-1)^4$ and $t_2=(\sqrt2+1)^4$ the first two branching values and concludes that invariance of $H$ at the first one makes its Taylor radius exactly $t_2$ . That conclusion needs two further facts: every obstruction inside $|t|<t_2$ must be excluded, and $H$ itself must have nontrivial monodromy or a singularity at $t_2$ . A branch point of the inverse $\tau(t)$ need not be a singularity of a composite—cancellation at $t_1$ is the argument's own example. Without a local expansion or monodromy calculation at the second point, the claims that the radius is finite and that $H$ has infinitely many nonzero coefficients do not follow. Those are required hypotheses of Proposition 1.2. Repair classification: no repair is supplied; a ramification classification plus a non-removability calculation at $t_2$ , or an independent proof of the needed radius lower bound and nonpolynomiality, would close the gap.

Full paper, version 1 ↗

Equation (12)Typo

The sign of the central-value term is reversed

Page 14 · Equation (12) · arXiv:2607.17123v1

For $k=4$ , $N=6$ , and Fricke sign $\varepsilon=-1$ , Proposition 1.3 gives $h(\tau)-D=-6\tau^2h(-1/(6\tau))$ with $D=2\pi i\tau L(F,2)$ . Hence the rearranged right-hand side contains $+2\pi i\tau L(F,2)$ , not the printed minus sign. The paper immediately proves $L(F,2)=0$ , so either sign yields the same subsequent identity and this mechanical correction has no substantive effect.

Full paper, version 1 ↗

Proposition 1.2 and the denominator calculationCorrect and complete

The arithmetic part of the criterion is correct

Pages 2–3 and 15 · Proposition 1.2 and final denominator paragraph · arXiv:2607.17123v1

The least-common-multiple estimate gives the threshold $de^r$ , and a rationality assumption turns every sufficiently small nonzero coefficient into a nonzero integer of absolute value below one. Since $t(q)\in q\mathbb Z[[q]]$ has leading coefficient one, its compositional inverse lies in $t\mathbb Z[[t]]$ . Thus the coefficient of degree $n$ in the Eichler integral has denominator dividing $\operatorname{lcm}(1,\ldots,n)^3$ , while the modular form $E$ has integral coefficients. This part would correctly apply the criterion once the missing analytic hypotheses are supplied.

Beukers, Irrationality proofs using modular forms ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.17123v1
Authors listed: Pang Ern Thang
Audit date: August 15, 2026

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A Modular Form Proof of the Irrationality of ζ(3)ζ\left(3\right)ζ(3)

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Irrationality of ζ(3)\zeta(3)ζ(3)

The decisive radius-of-convergence assertion is not established

The sign of the central-value term is reversed

The arithmetic part of the criterion is correct

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A Modular Form Proof of the Irrationality of $ζ\left(3\right)$

Irrationality of $\zeta(3)$