arXiv:2510.00215v4

Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients

Henri Cohen, Wadim Zudilin

math.NTmath.CA11F1111F6711G1511J7011J8233C0533C45

Paper metadata

Original submission: September 30, 2025
Latest revision: July 12, 2026 · version 4
Claimed MSC: 11F11, 11F67, 11G15, 11J70, 11J82, 33C05, 33C45
arXiv categories: math.NT, math.CA
Paper license: arXiv non-exclusive distribution license ↗
Comments: 27 pages

Abstract

We give 39 rapidly convergent continued fractions for Chowla--Selberg gamma quotients, and deduce good irrationality measures for 20 of them, including for $\operatorname{CS}(-3)=(Γ(1/3)/Γ(2/3))^3$ , for $a^{1/4}\operatorname{CS}(-4)=a^{1/4}(Γ(1/4)/Γ(3/4))^2$ with $a=12$ and $a=1/5$ , and for $\operatorname{CS}(-7)=Γ(1/7)Γ(2/7)Γ(4/7)/(Γ(3/7)Γ(5/7)Γ(6/7))$ . These appear to be the first proved and reasonable irrationality measures for gamma quotients.

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Audit summary

Audited against arXiv v4

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

01Statements3 reported findingsContains unsupported statements

No counterexample was found to the continued-fraction identities or to the twenty irrationality-measure bounds. The general continued-fraction asymptotics and denominator estimates are correct. The exact Chowla–Selberg evaluations, and therefore the full lists in Tables 1–2 and Theorems 6.1–6.2, are not able to be verified from the available evidence because a material modular identity is attributed to an unpublished source and all but one of the CM computations are supplied only as table entries.

Theorems 4.1, 4.7, 7.1, 7.3 and Proposition 7.7Correct

General analytic and arithmetic continued-fraction machinery

Pages 7–9 and 19–24 · Sections 4 and 7 · arXiv:2510.00215v4

The hypergeometric limit formula follows from the recurrence and the asymptotic roots in Proposition 3.1. The explicit polynomial formulas in Theorem 7.1 satisfy the normalized recurrence, Theorem 7.3 controls their denominators prime by prime for $D\in\{2,3,4,6\}$ , and Proposition 7.7 gives the required least-common-multiple growth. Combined with Lemma 1.2, this chain yields the stated irrationality-measure formula whenever the exact limit identification in Table 2 has been established.

Full paper, version 4 ↗

Table 2 and Theorem 6.1Not able to verify

The claimed Chowla–Selberg continued-fraction limits are not fully verified

Pages 11–17 · Theorem 5.2, Sections 5.4–6.2, Table 2, and Theorem 6.1 · arXiv:2510.00215v4

The central claim is that the thirty-nine convergent entries of Table 2 have the displayed Chowla–Selberg limits, including the exact formula for $\operatorname{CS}(-3)$ in Theorem 6.1. Their derivation uses the four functional modular identities in Theorem 5.2 and exact CM values of the associated Hauptmoduln and modular forms. Theorem 5.2 is attributed to reference [2], which the manuscript describes as unpublished, and its proposed differential-equation verification is not carried out. Section 6 works out only tag (1.4) in detail and gives the remaining exact evaluations as table entries. These are concrete nontrivial identities, not consequences recoverable by local substitution alone. No counterexample was found, but neither a complete verification nor an independent proof is supplied here.

Full paper, version 4 ↗

Theorem 6.2Not able to verify

The full list of twenty irrationality measures remains conditional on unverified limit identities

Pages 18 and 24 · Theorem 6.2 and its concluding proof · arXiv:2510.00215v4

For each entry, the arithmetic comparison of the convergence exponent with the denominator-growth constant correctly gives the displayed upper bound once the continued fraction is known to converge to the stated Chowla–Selberg quotient. That exact target identification is verified in detail only for tag (1.4); the other nineteen bounds depend on table-only CM evaluations and the unavailable input behind Theorem 5.2. Thus the numerical irrationality calculations are internally consistent, but the theorem as a statement about the named gamma quotients is not fully established by the supplied material.

Full paper, version 4 ↗

02Proofs4 reported findingsContains incorrect or incomplete proofs

The recurrence, asymptotic, denominator, and least-common-multiple arguments are correct and complete. The proof chain identifying the limits with Chowla–Selberg gamma quotients is incomplete for the table as a whole, and its principal functional modular input is not verifiable from the cited unpublished source. One mechanically determined missing factor is reported as a typo.

Theorem 5.2Not able to verify

The functional modular evaluations rely on an unavailable cited proof

Pages 11–12 · Section 5.1 and Theorem 5.2 · arXiv:2510.00215v4

The four displayed hypergeometric identities are a material input for every exact limit in Table 2. The only proof attribution is to Beukers and Cohen, reference [2], explicitly described as unpublished or in preparation. The text says the identities can be proved by checking a second-order differential equation and initial conditions, but it supplies neither those equations nor the initial-value comparison. The precise unresolved obligation is to prove those four identities on the stated fundamental domains, including the chosen branches. Repair classification: No repair supplied.

Full paper, version 4 ↗

Sections 5.4–6.2 and Tables 1–2Incomplete as written

Thirty-eight exact CM-limit computations are omitted

Pages 13–17 · Tables 1–2 and the paragraph introducing Section 6 · arXiv:2510.00215v4

The manuscript claims a complete list of rational CM values, converts them into thirty-nine convergent continued fractions, and identifies every limit with an explicit gamma quotient. It then states that only tag (1.4) will be computed in detail and supplies the remaining results only in tables. The general recipe in Section 5.5 does not establish the individual algebraic values of $R_N$ , $D(R_N)$ , the modular forms, and their Serre derivatives at the thirty-eight remaining CM points. Those exact identities are indispensable for the named limits and for nineteen entries of Theorem 6.2. Repair classification: No repair supplied; exact derivations or independently checkable symbolic certificates for the omitted entries are required.

Full paper, version 4 ↗

Theorems 7.1 and 7.3; Proposition 7.7Correct and complete

Normalized convergents and denominator growth

Pages 19–24 · Section 7 · arXiv:2510.00215v4

The explicit formulas satisfy the normalized three-term recurrence and initial data. The prime-adic divisibility estimates cover the four permitted denominators $D$ , and the residue-class density calculation gives the constants $m_D^*$ used at the end of the proof. The resulting application of Lemma 1.2 is valid for every entry once its exact limit and convergence exponent have been established.

Full paper, version 4 ↗

Theorem 4.8Typo

A factor $K$ is missing in the nested continued fraction

Page 9 · Last displayed calculation in the proof of Theorem 4.8 · arXiv:2510.00215v4

The denominator is printed with the segment $2A-5^2/(3A-\cdots)$ . Replace $5^2$ by $5^2K$ . The preceding definition $U=B-3^2K/(2B-5^2K/(3B-\cdots))$ and the coefficient rule $b(n)=-K(2n-1)^2$ uniquely determine the missing factor. The next equality $L=b_0/(a_1-K/U)$ uses the corrected expression, so the defect does not alter the theorem.

Full paper, version 4 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2510.00215v4
Authors listed: Henri Cohen, Wadim Zudilin
Audit date: August 15, 2026

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Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

General analytic and arithmetic continued-fraction machinery

The claimed Chowla–Selberg continued-fraction limits are not fully verified

The full list of twenty irrationality measures remains conditional on unverified limit identities

The functional modular evaluations rely on an unavailable cited proof

Thirty-eight exact CM-limit computations are omitted

Normalized convergents and denominator growth

A factor KKK is missing in the nested continued fraction

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A factor $K$ is missing in the nested continued fraction