A Fourier analysis approach to disprove the weak Shanks conjecture

Hugo J. Woerdeman; Jeffrey S. Geronimo

REVIEW 4 major objections 2 minor 24 references

A Fourier analysis approach to disprove the weak Shanks conjecture

T0 review · 4 major / 2 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper aims to derive hypergeometric formulas for the Fourier coefficients of |f|^2 and use them to find new counterexamples to the weak Shanks conjecture.

desk verdict Abstract promises Fourier-analysis counterexamples to the weak Shanks conjecture, but the attached text is an unrelated heavy-ion experiment; the submission is unassessable. read the letter →

arxiv 2508.15938 v1 pith:VJKVZWZ5 submitted 2025-08-21 math.CV math.CAmath.FA

Jeffrey S. Geronimo , Hugo J. Woerdeman This is my paper · ORCID

classification math.CVmath.CAmath.FA

keywords weakShanksconjectureFouriercoefficientshypergeometricfunctionstwo-variablerationalcounterexamples|f|^2parameteroptimizationcomplexanalysis

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

The paper's announced goal is to establish explicit hypergeometric formulas for the Fourier coefficients of |f|^2, where f(z1,z2)=(1-(z1+z2)/r)^(-alpha), and then use those formulas to produce additional counterexamples to the weak Shanks conjecture. That conjecture concerns how Taylor coefficients of rational functions in two complex variables behave, and it was recently shown to fail. If the formulas hold, coefficient checking becomes a closed-form computational task: one can scan alpha and r numerically and locate parameter values where the conjecture's predicted coefficient inequalities break. A sympathetic reading takes the derivation to be the contribution: it turns a hard coefficient-ratio problem into explicit special-function expressions. The attached body, however, is a different text about heavy-ion collisions, so the announced derivation is not present in the submitted manuscript and cannot currently be checked.

What carries the argument

The central object is the two-variable function f(z1,z2)=(1-(z1+z2)/r)^(-alpha) and the Fourier coefficients of its squared modulus on the unit torus. The advertised machinery is a hypergeometric-function representation of those coefficients, which is supposed to turn a check of the weak Shanks conjecture into numerical evaluation and optimization over alpha and r. Without that representation, the counterexample search would require high-dimensional coefficient computations with no closed form.

What would settle it

Take a specific parameter pair that the paper says is a counterexample and compare the advertised hypergeometric value of the relevant Fourier coefficient with direct high-precision numerical integration of |f|^2 over the unit torus. If they disagree, the formula fails. Also check the parameter regime: for positive alpha and r < 2, f is not bounded on the torus, so any claimed counterexample there would not satisfy the conjecture's hypotheses.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Fourier coefficients of |f|^2 for f(z1,z2)=(1-(z1+z2)/r)^(-alpha) are not just computable in principle but expressible in closed form in terms of hypergeometric functions. That converts a coefficient-ratio conjecture about two-variable rational functions into a parameterized special-function problem in alpha and r. The paper says this yields new counterexamples to the weak Shanks conjecture, supplementing the recent disproof. In the version of the paper supplied here, none of these formulas, parameter values, or counterexamples appear; the body is an unrelated experimental physics report. The claim is therefore a stated result witho

Load-bearing premise

The load-bearing premise is that violation of the weak Shanks conjecture for this family reduces to a checkable inequality on the Fourier coefficients of |f|^2 and that the hypergeometric formulas are valid for the r and alpha values used; the submitted body contains no derivation, so this bridge is currently unverified.

Editorial extensions

If this is right

The hypergeometric formulas, if valid, make |f|^2 coefficient computation for this family exact, so counterexample searches do not rely on numerical Fourier integration.
They give a continuous two-parameter family (alpha, r) in which to look for weak Shanks violations, going beyond isolated examples.
Each new violation found this way strengthens the recent disproof of the weak Shanks conjecture.
The same formulas may allow researchers to test nearby coefficient-ratio conjectures on the same family with little extra work.
Because the formulas are differentiable in the parameters, one can optimize to find the most extreme violations.

Reading between the lines

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

My inference: if the formulas are valid for the boundedness regime r > 2 and an integrability range of alpha, they likely connect these coefficients to known orthogonal polynomials or Appell-type functions, which could give a uniform derivation for a wider class of rational functions than the single power family.
My inference: the absence of the derivation from the body means the advertised counterexamples cannot be reproduced from the submitted text; a reader would need the full derivation or a corrected manuscript before treating them as established.
My inference: the same coefficient formulas might be used to test stronger ratio conjectures, such as monotonicity or log-concavity of coefficient sequences, not just the weak Shanks failure.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The short version: this submission does not contain the paper it claims to be. The abstract announces explicit hypergeometric formulas for Fourier coefficients of |f|^2 and new counterexamples to the weak Shanks conjecture. The body is an NA61/SHINE heavy-ion experimental report on the search for the QCD critical point. Different topic, different arXiv number embedded in the header. There is nothing here to referee on the merits.

To give credit where it is earned: the abstract itself is plausible, and the claimed formulas could be a real technical contribution if a derivation were attached. The BKS work already disproved the weak Shanks conjecture, so “additional counterexamples” is an extension, not a headline result. The possible value lies in the explicit formulas and the parameter optimization they enable. But none of that is present. No definition of f beyond the abstract, no Fourier coefficient formula, no hypergeometric function, no statement of the BKS criterion, no parameter regime (e.g., r > 2 for boundedness on the torus), no convergence conditions, no numerical search. The only equation in the body is the kaon ratio R_K.

This is not a subtle derivation gap; it is total absence of the derivation. The stress-test note is accurate: every part of the manuscript confirms the mismatch. The body’s own abstract says the main goal is the critical point in strongly interacting matter. Under the rule that all manuscript text is in-scope evidence, there is zero support for the central claim. If the correct file was uploaded, that is an honest mistake, but a referee cannot fix it by reading more carefully.

The reader’s scores track my own: soundness is near zero because there are no equations to check; novelty and significance are conditional on the abstract being true. I would reject as submitted. If the authors resubmit the actual mathematics paper, then it deserves a serious look, because explicit formulas for this family would be useful and the BKS connection gives it context. But this version should be desk-rejected.

Reading group? No, there is nothing to chew on. Cite it? No.

My recommendation: reject without external review, with a clear message that the correct manuscript would be welcome.

Referee Report

4 major / 2 minor

Summary. The abstract claims that the paper derives formulas for the Fourier coefficients of |f|^2 for f(z1,z2)=(1-(z1+z2)/r)^{-alpha}, expresses them in terms of hypergeometric functions, and uses them to provide additional counterexamples to the weak Shanks conjecture recently disproven by Bénéteau, Khavinson and Seco. The submitted full text, however, is an unrelated NA61/SHINE experimental heavy-ion physics report (arXiv:2508.15939v2 [nucl-ex]) on the search for the critical point in strongly interacting matter. None of the claimed mathematics appears anywhere in the body: there is no definition of f, no Fourier coefficient formula, no hypergeometric function, no statement of the weak Shanks conjecture or the Bénéteau-Khavinson-Seco counterexample criterion, and no numerical counterexamples. The central claim is therefore entirely unsupported by the submitted text.

Significance. If the abstract's claims were backed by a complete derivation and a verified parameter search, the paper could contribute meaningfully to the recent work on the weak Shanks conjecture, particularly by supplying a computable family of functions and explicit counterexample parameters. The paper offers no such support. It contains no equations, no convergence conditions, no proofs, no code, and no reproducible counterexample tables. As submitted, the significance cannot be assessed beyond the abstract's assertion, and the asserted result is unverified and unverifiable from the supplied manuscript.

major comments (4)

[Full text] The body of the submission is an unrelated NA61/SHINE experimental physics report. It contains no definition of f(z1,z2), no Fourier coefficient formulas, no hypergeometric functions, no mention of the weak Shanks conjecture or of Bénéteau, Khavinson and Seco, and no counterexample parameters. The abstract's central claim is thus completely absent from the manuscript. This is not a derivation gap that can be filled by local revision; the submission as it stands contains none of the claimed mathematics.
[Abstract vs. full text] The abstract states that formulas are derived and that counterexamples are provided, but the full text is a different document with its own abstract and references (the header shows arXiv:2508.15939v2 [nucl-ex]). There are no equations in the body to check, no statements of theorems, and no numerical results. Every load-bearing element of the claimed paper is missing.
[Parameter regime and convergence conditions (abstract)] Even taking the abstract as the entire mathematical content, the parameter range for which the formulas and counterexamples are valid is not stated. For the function f(z1,z2)=(1-(z1+z2)/r)^{-alpha} to be defined and integrable on the unit torus, conditions such as r > 2 and suitable restrictions on alpha are needed; the formulas for Fourier coefficients or their use in a weak-Shanks counterexample criterion require such hypotheses. Without these, no counterexample can be certified.
[Numerical counterexamples (abstract)] The abstract says the formulas 'allow for (numerical) optimization over the parameters alpha and r' and that the paper provides 'additional counterexamples', but no counterexamples, parameter values, coefficient formulas, or verification procedures are actually presented. A claim of counterexamples to a conjecture requires at least the specific functions and a check of the criterion; none appears.

minor comments (2)

[Full text header] The submitted manuscript is labeled with the NA61/SHINE experimental paper ID (arXiv:2508.15939v2 [nucl-ex]) and an entirely different set of authors and title. This is a clear mismatch with the abstract and with the arXiv listing for 2508.15938.
[Title and content] The title promises a Fourier analysis proof/disproof of the weak Shanks conjecture, but the document's content is about heavy-ion collision measurements. The title does not match the body.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning identifiable: the promised Fourier/hypergeometric derivation and weak-Shanks counterexamples are absent from the submitted body, which is an unrelated NA61/SHINE heavy-ion report; absence is a support problem, not circularity.

full rationale

The claimed derivation chain is stated only in the abstract: derive formulas for Fourier coefficients of |f|^2 with f(z1,z2) = (1-(z1+z2)/r)^{-alpha}, then use them to provide additional counterexamples to the weak Shanks conjecture. The attached full text, however, is a NA61/SHINE strong-interaction conference proceeding whose own abstract is about the search for the critical point in strongly interacting matter. It contains none of the promised mathematics: there is no definition of f, no Fourier coefficient formula, no hypergeometric function, no statement of the weak Shanks conjecture or the B\'en\'eteau-Khavinson-Seco criterion, and no parameter regime ensuring boundedness or integrability. Consequently, there is no derivation chain in the submitted text whose steps could be compared to show equivalence to inputs. The absence of the derivation is an omitted-proof / manuscript-mismatch concern, not a circularity. The abstract's mention of numerical optimization over alpha and r would not by itself be circular: counterexamples may legitimately be found by search. No self-definitional step, fitted-input-called-prediction, load-bearing self-citation, imported uniqueness theorem, ansatz-smuggling citation, or renaming of a known result is present in the supplied text. Thus the circularity score is 0, with the caveat that the submission does not actually support its stated claim.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The abstract is only three sentences, so the ledger records the assumptions a full derivation would need. The two free parameters alpha and r are the explicit optimization variables. The axioms are standard but unverifiable from the available text: the hypergeometric evaluations, the convergence conditions on the torus, the imported BKS formulation of the weak Shanks conjecture, and the productivity of the chosen parameter family. No invented entities appear. The ledger shows a small-parameter, reference-heavy computation, normal for this kind of counterexample paper, but none of these items can be confirmed because the body contains no mathematics.

free parameters (2)

alpha (exponent) = not specified in abstract
Exponent in f(z1,z2) = (1 - (z1+z2)/r)^(-alpha); the abstract states the formulas allow numerical optimization over alpha and r, so alpha is an adjustable parameter of the counterexample search.
r (singularity parameter) = not specified in abstract
Controls where the singular set z1+z2 = r sits relative to the unit torus; r > 2 is needed for f to be analytic on a neighborhood of the closed polydisk. Also targeted by the numerical optimization.

assumptions (4)

standard math Hypergeometric function evaluation identities used to turn the Fourier coefficient integrals into closed form.
The abstract claims the coefficients are expressed in terms of hypergeometric functions; the evaluation identities themselves are unstated in the available text.
domain assumption Convergence and integrability of the double Fourier series and of the integral defining the coefficients of |f|^2 on the 2-torus.
Requires conditions on alpha and r (e.g., r > 2 and an alpha range where the singular set does not touch the torus and the product is integrable). Not stated in the abstract.
domain assumption The formulation of the weak Shanks conjecture and its counterexample criterion, taken from Bénéteau, Khavinson and Seco.
The abstract cites the BKS disproof; the present counterexamples inherit the conjecture's formulation and the coefficient criterion from that reference.
ad hoc to paper The chosen family f(z1,z2) = (1 - (z1+z2)/r)^(-alpha) is broad enough in (alpha, r) to contain weak-Shanks-violating instances.
The parameter family is the paper's chosen search space; whether counterexamples exist in it is exactly what the numerical optimization would establish.

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Cite this review

Pith. "Pith review of A Fourier analysis approach to disprove the weak Shanks conjecture." pith.science (2026). https://pith.science/paper/VJKVZWZ5

@misc{pith2026250815938,
  author       = {Pith},
  title        = {Pith review of: A Fourier analysis approach to disprove the weak Shanks conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VJKVZWZ5}},
  note         = {Machine review of arXiv:2508.15938}
}

abstract

We derive formulas for the Fourier coefficients of $|f|^2$, where $f(z_1,z_2)=(1-\frac{z_1+z_2}{r})^{-\alpha}$, in terms of hypergeometric functions. Using these formulas we provide additional counterexamples to the weak Shanks conjecture, which was recently disproven by B\'en\'eteau, Khavinson and Seco. The obtained formulas allow for (numerical) optimization over the parameters $\alpha$ and $r$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Ga´ zdzicki, M.I

M. Ga´ zdzicki, M.I. Gorenstein, Acta Phys. Polon. B30, 2705 (1999)

work page 1999

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C. Alt et al. (NA49), Phys. Rev. C77, 024903 (2008)

work page 2008

[3] [3]

Adhikary et al

H. Adhikary et al. (NA61/SHINE), Eur. Phys. J. C84, 416 (2024)

work page 2024

[4] [5]

Aduszkiewicz et al

A. Aduszkiewicz et al. (NA61/SHINE), Eur. Phys. J. C76, 198 (2016)

work page 2016

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Stephanov, K

M.A. Stephanov, K. Rajagopal, E.V . Shuryak, Phys. Rev. D60, 114028 (1999)

work page 1999

[6] [7]

Grebieszkow (NA61/SHINE), PoSEPS-HEP2017, 167 (2017)

K. Grebieszkow (NA61/SHINE), PoSEPS-HEP2017, 167 (2017)

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