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

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Pith's one-line read First proved irrationality measures for Chowla–Selberg gamma quotients arise from 39 simple continued fractions.

desk verdict New proved irrationality measures for Chowla–Selberg quotients, but the load-bearing modular evaluation is an unpublished manuscript co-authored by the first author. read the letter →

arxiv 2510.00215 v4 pith:LA375MSL submitted 2025-09-30 math.NT math.CA

Henri Cohen , Wadim Zudilin This is my paper · ORCID
classification math.NTmath.CA MSC 11J8211J7033C0511F03
keywords continuedfractionsirrationalitymeasureChowla–SelberggammaquotientshypergeometricfunctionsmodularformsCMvaluesapproximationofconstantsApéry-typemethods
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims to supply the first proved, reasonable bounds on how well specific products of gamma functions (Chowla–Selberg quotients) can be approximated by rationals. The authors exhibit 39 rapidly convergent continued fractions whose limits are these gamma quotients, and from these they deduce explicit upper bounds for the irrationality measure of 20 of the quotients, including CS(−3), CS(−4), and CS(−7). If correct, this opens a new route to irrationality and transcendence information for a family of classical constants previously accessible only through much heavier machinery.

What carries the argument

The central objects are continued fractions of the form [[0, a1, A(n−1)],[b0, −K(Dn−1)(D(n−1)+1)]] with D = 2,3,4,6, together with the associated linear recursion. The two key pieces are: (1) the modular hypergeometric evaluation (Theorem 5.2) that identifies the limit of such a continued fraction as a ratio of hypergeometric functions, which at CM points becomes an algebraic multiple of a Chowla–Selberg gamma quotient; (2) the denominator bound (Theorem 7.3) showing that the denominators of the scaled convergents grow only like the lcm of Dj+1 rather than their product, which is exactly what makes the irrationality measure nontrivial.

What would settle it

Check the continued fraction for CS(−3) numerically: if the convergents of [[0,31,1012(n−1)],[240,−(6n−1)(6n−5)]] do not converge to (Γ(1/3)/Γ(2/3))³ to high precision, then the modular evaluation is in error. More directly, compute the value of the hypergeometric function ₂F₁(1/2,5/6;1;3/128) numerically and compare to 2^{25/6} 3^{-3/4} 5^{-1} Γ(1/3)/Γ(2/3)²; a mismatch would refute the claimed identity.

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Extended reading notes

Core claim

The central claim is that for a family of continued fractions with linear coefficients of a very special shape, the limit can be evaluated exactly as a Chowla–Selberg gamma quotient, and the speed of convergence plus careful denominator bounds yields nontrivial upper bounds on the irrationality measure. The paper proves, for example, that µ(CS(−3)) < 5.548, µ(5^{1/6} CS(−3)) < 7.271, µ(3^{1/2} 7^{−1/6} CS(−3)) < 3.606, µ(12^{1/4} CS(−4)) < 25.733, µ(5^{−1/4} CS(−4)) < 3.452, µ(CS(−7)) < 5.283, and µ(CS(−163)) < 2.477, together with many others. The proof rests on (a) a classical hypergeometric continued fraction, (b) modular hypergeometric evaluations (the CM values of a Hauptmodul) that ide

Load-bearing premise

The most load-bearing premise is that the modular hypergeometric evaluations of Theorem 5.2, stated without proof and attributed to an unpublished manuscript, are correct; if any of those evaluations is wrong, the continued fraction limits (and hence the irrationality measures) would not be the claimed gamma quotients.

Editorial extensions

If this is right

  • If the results hold, these are the first proved nontrivial irrationality measures for any Chowla–Selberg gamma quotient, providing quantitative control on how irrational these numbers are.
  • The method yields measures for 20 distinct quotients, including CS(−3), CS(−4), CS(−7), and several with large discriminants such as CS(−163), with bounds that shrink as the discriminant grows.
  • Because the continued fractions are remarkably simple (linear a(n) and quadratic b(n)), the technique may be applicable to other families of constants whose limits can be evaluated via modular hypergeometric functions.
  • A direct corollary gives µ(Γ(1/3)/Γ(2/3)) < 16.644, so the method also supplies information about individual gamma values, not only the full quotients.
  • The paper notes that the list of 44 CM values is complete for this family, so further improvement within this approach would require new methods or different families of continued fractions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The reliance on unpublished modular hypergeometric evaluations (Theorem 5.2, attributed to Beukers–Cohen) is the least verifiable step; if any of those evaluations is incorrect, the corresponding continued fraction limit—and hence the associated measure—would fail, even though the Diophantine analysis itself may be sound.
  • The method hints at a p-adic analogue via the Gross–Koblitz formula, which could lead to proofs of irrationality of p-adic analogues of Chowla–Selberg quotients—an avenue not explored in the paper.
  • The connection to Ramanujan-type 1/π formulas (the same 44 CM values) suggests that related rational hypergeometric formulas for 1/π² might yield continued fractions for squares of Chowla–Selberg quotients, potentially extending the approach to a new class of constants.
  • The observation that the continued fractions are self-dual under the fastest Apéry acceleration—and that no usable series or integral representation is known—suggests that continued fractions are the natural, and perhaps only, tool for this particular problem, which may reshape how one searches for irrationality measures of other gamma quotients.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies a family of continued fractions of the form [[0, a1, A(n-1)], [b0, -K(Dn-1)(D(n-1)+1)]] with D in {2,3,4,6}. It identifies 39 such fractions whose limits are Chowla–Selberg gamma quotients (possibly multiplied by algebraic factors), and, using a denominator analysis for the convergents, derives explicit irrationality measures for 20 of these quotients, including CS(-3), CS(-7), and CS(-163). The main arithmetic part (Theorems 7.1–7.4, Proposition 7.7) is developed in detail and gives an Apéry-type denominator bound. The identification of the limits, however, ultimately rests on Theorem 5.2, a set of modular hypergeometric evaluations quoted from an unpublished manuscript [2] by the first author and Beukers, and the paper explicitly leaves the convergence/asymptotics of the general continued fraction to the reader (Proposition 3.1). Only one of the 39 table computations is shown in detail.

Significance. If the unproved identifications are correct, this is a substantial advance: it supplies the first proved, non-gargantuan irrationality measures for Chowla–Selberg gamma quotients, with strong numerical bounds such as mu(CS(-3))<5.548 and mu(CS(-163))<2.477. The construction via continued fractions is elegant, and the denominator analysis is a serious and apparently correct technical contribution. The paper is explicit and falsifiable: each continued fraction, each CM point, and each measure value is stated precisely, so an independent check is possible in principle. However, the central claim is conditional on modular hypergeometric evaluations that are not proved in the paper and are attributed to an unavailable source; this is a correctness risk that must be resolved before the stated theorems can be accepted.

major comments (3)
  1. [§5.3, Theorem 5.2] The four hypergeometric modular evaluations are the hinge of the paper: every continued-fraction limit L in Table 2 is obtained from them via Theorem 4.7, and hence every measure in Theorem 6.2 depends on their correctness. Yet the theorem is stated with no proof, and the text says the formulas are from the unpublished work [2], coauthored by the paper's first author. An error in a sign, exponent, or normalization in any entry would change one or more of the limits L and invalidate the corresponding irrationality measure. Please include a proof of Theorem 5.2, or at least of the special cases and CM values actually used, and update the reference if [2] is now available.
  2. [§3, Proposition 3.1] The proof is given as 'Classical and left to the reader,' but this proposition supplies the convergence rate E and the asymptotics log|q'(n)L-p'(n)| ~ -n log|E|/2, which are exactly the input F for Lemma 1.2. Without a proof or a precise reference covering this exact family (including the factor f(n)), the numerical measures in Theorem 6.2 are not fully justified. Please add a self-contained proof or a complete citation, including the claimed constants.
  3. [§6.2, Table 2 / Theorem 6.2] Only entry (1.4), corresponding to CS(-3), is computed in the text (§6.1). The other 39 continued-fraction limits in Table 2 are asserted without derivation, and all 20 measures in Theorem 6.2 use the identifications in that table. For reproducibility, please provide the intermediate values (periods, hypergeometric quotients, and resulting L) for at least the 20 rows used in Theorem 6.2, or include an electronic supplement with those computations. This is not merely a presentation issue: a misidentification in any of these rows would change the number whose irrationality measure is claimed.
minor comments (4)
  1. [Definition 1.1] The exponent in the definition of CS(D) is printed as (D/j)! w(D)/(2h(D)), which appears to contain a spurious factorial. It should presumably be (D/j) · w(D)/(2h(D)). Please correct the notation.
  2. [§1.2, Lemma 1.2] The lemma is stated without proof. Since it is used to convert the asymptotics into all the irrationality measures, a short proof or a standard reference would be helpful.
  3. [Table 2] The exclusion of rows (2.9), (3.1), (3.7), (4.6), and (4.7) is explained only by saying the corresponding continued fractions do not converge; a brief indication of why (e.g., A^2-4KD^2 < 0) in the caption would improve clarity.
  4. [References] References [2] and [8] are marked 'in preparation.' This is acceptable for background, but since [2] is load-bearing, it should either be made available or superseded by a proof in this paper.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: CF limits are computed from hypergeometric evaluations, and the irrationality measures follow from independent convergence and denominator bounds. The only caveat is reliance on the unpublished Beukers–Cohen manuscript for the modular evaluations.

full rationale

The derivation chain is not circular. For each continued fraction, the limit is obtained via Corollary 4.4 / Theorem 4.7 from the hypergeometric quotient T1/T0, and then evaluated at CM points using the modular hypergeometric identities of Theorem 5.2. The target gamma quotient is not inserted as an input; it is the output of these evaluations. For example, in Section 6.1 the authors compute explicit values of 2F1(1/2,5/6;1;3/128) and 2F1(3/2,11/6;3;3/128), and only then deduce Theorem 6.1's continued fraction for CS(−3). The irrationality measures are then consequences of Lemma 1.2 applied to the proved convergence rate F = log|E|/2 (from Proposition 3.1) and the independently established denominator asymptotics log d*_D(n) ~ m*_D n (Theorem 7.3, Proposition 7.7). The numbers in Table 2's μ column are arithmetic functions of these quantities, not fitted to known measures. The main caveat is that Theorem 5.2 and the completeness of Table 1 are stated without proof and assigned to the unpublished manuscript [2] by Beukers and Cohen, which overlaps with the present first author; this is a verification gap and a mild self-reliance risk, not a formal circularity, because the cited evaluations do not already contain the target irrationality measures and are explicit, externally checkable identities. Proposition 3.1 also has its proof left to the reader, but it affects convergence-rate constants rather than which number is the limit. Overall, the arithmetic part of the paper is self-contained and the central claim does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities: the CF coefficients are derived from the arithmetic of CM points. The main external assumptions are the unpublished modular evaluations and standard CM theory.

assumptions (4)
  • domain assumption Theorem 5.2: hypergeometric evaluations for the triangle groups (3,3,∞), (4,4,∞), (6,6,∞), and (∞,∞,∞), attributed to Beukers–Cohen [2] (unpublished)
    Used in Section 5.3 to identify the limits of the continued fractions; no proof is given in this paper.
  • domain assumption Chowla–Selberg formula / CM theory: values of modular functions of weight k at CM points are algebraic multiples of Ω(τ)^k, with Ω(τ) expressible via gamma quotients
    Used in Section 5.5 to express periods and hypergeometric values as algebraic multiples of gamma quotients.
  • domain assumption Proposition 3.1: exponential convergence and asymptotics of the continued fraction family (proof 'classical and left to the reader')
    Provides the asymptotics of q(n) and the error term that feed into Lemma 1.2; the proof is not supplied.
  • domain assumption Prime-number-theorem distribution results for lcm sums in arithmetic progressions (Nesterenko [15, Lemma 6])
    Used in Proposition 7.7 to compute the asymptotic constants m*_D for the denominator sequences.

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Pith. "Pith review of Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients." pith.science (2026). https://pith.science/paper/LA375MSL

@misc{pith2026251000215,
  author       = {Pith},
  title        = {Pith review of: Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LA375MSL}},
  note         = {Machine review of arXiv:2510.00215}
}
abstract

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

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