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A Colombeau--Beurling criterion for the Riemann hypothesis

T0 review · 2 major / 0 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read The Riemann hypothesis is equivalent to one moderate net in the Colombeau algebra G(0,1), built from damped Báez–Duarte sums by Mellin convolution, being associated with −χ_{(0,1)} and uniformly L²-bounded.

desk verdict Only the abstract of the RH/Colombeau paper is real; the full-text dump is the wrong arXiv (room EQ), so the claimed equivalence is still unauditable. read the letter →

arxiv 2606.22562 v2 pith:OPQURMCN submitted 2026-06-21 math.CV

Amaury Alvarez Cruz , Esteban A Alvarez Gutierrez This is my paper · ORCID
classification math.CV MSC 11M2646F3042A85
keywords RiemannhypothesisColombeaualgebraBáez–DuartecriterionMellinconvolutionmoderatenetsassociationuniformL²-boundednessgeneralizedfunctions
topics The Riemann Hypothesis
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 reformulates the Riemann hypothesis as a statement about generalized functions. It constructs a single moderate net in the Colombeau algebra G(0,1) from damped versions of the Báez–Duarte sums, using multiplicative (Mellin) convolution, and proves that the hypothesis holds if and only if that net is moderate, remains uniformly bounded in L², and is associated with the negative characteristic function of the unit interval. Two concrete damping schemes—one exponential with super-exponential truncation, one slowly vanishing polynomial with polynomial truncation—are shown to work in both directions. The result therefore turns the classical Beurling-type criterion into a Colombeau–Beurling criterion expressed by weak association plus uniform L² control. A reader who cares about equivalent formulations of the Riemann hypothesis gains a new analytic-function-space language in which the hypothesis becomes a statement about a single explicit net.

What carries the argument

A moderate net in the Colombeau algebra G(0,1) obtained by damping the Báez–Duarte partial sums and convolving them multiplicatively (Mellin convolution); the net’s moderation, uniform L²-boundedness, and association with −χ_{(0,1)} together encode the Riemann hypothesis.

What would settle it

Construct the damped net for either explicit scheme, verify moderation and uniform L²-boundedness by direct estimates, then test whether the net associates with −χ_{(0,1)}; if the association holds while a zero with real part larger than 1/2 is already known (or vice versa), the equivalence is false.

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Extended reading notes

Core claim

Assuming the Riemann hypothesis, the two damped nets are moderate, uniformly L²-bounded, and associated with −χ_{(0,1)}; conversely, the mere existence of any moderate net of this damped-Báez–Duarte form that is uniformly L²-bounded and associated with −χ_{(0,1)} forces the Riemann hypothesis.

Load-bearing premise

The two explicit damping-and-truncation schemes preserve exactly the information about the critical zeros that the classical undamped Báez–Duarte criterion carries; if damping destroys or creates that information, both directions of the claimed equivalence fail.

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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

2 major / 0 minor

Summary. The manuscript claims an equivalence between the Riemann hypothesis and the existence of a moderate net in the Colombeau algebra G(0,1) that is uniformly L^{2}-bounded and associated with −χ_{(0,1)}. The net is constructed from damped Báez–Duarte sums via multiplicative (Mellin) convolution. Two damping schemes are proposed: exponential damping exp(−kε^{2}) with super-exponential truncation, and polynomial damping k^{−δ(ε)} with δ(ε)=(log(1/ε))^{−α} plus polynomial truncation. Under RH the nets are asserted to be moderate, uniformly L^{2}-bounded and associated with the target; conversely, existence of any such net of this form is claimed to imply RH. The abstract presents this as a Colombeau–Beurling-type criterion reformulating RH in terms of generalized functions and weak association.

Significance. If the claimed equivalence were rigorously established, the paper would supply a genuine reformulation of RH inside the Colombeau algebra of generalized functions, linking classical L^{2} criteria of Báez–Duarte type to moderateness, association and uniform L^{2} control. Such a bridge could open new analytic tools for studying the critical zeros via nonlinear generalized-function techniques. The explicit construction of two damping families and the two-sided logical shape (RH ⇔ properties of a single net) would be a non-trivial contribution to the literature on equivalent formulations of RH. At present, however, the body of the supplied manuscript is an unrelated DAFx paper on adaptive room equalization (arXiv:2606.22563); consequently none of the estimates, lemmas or proofs that would support the claim can be examined, and the significance remains purely potential.

major comments (2)
  1. The full manuscript text supplied under the paper identifier is not the Colombeau–Beurling paper announced by the abstract and title; it is instead a complete, unrelated DAFx article on a DDSP framework for adaptive room equalization (arXiv:2606.22563). No definitions of the damped nets, no estimates establishing moderateness or association, and no proofs of either direction of the claimed equivalence appear. The central claim is therefore completely unverifiable from the document under review.
  2. Even granting the abstract’s outline, the load-bearing premise that the two explicit damping/truncation schemes (exp(−kε^{2}) with super-exponential cut-off; k^{−δ(ε)} with δ(ε)=(log(1/ε))^{−α} and polynomial cut-off) preserve exactly the information about critical zeros encoded by the classical undamped Báez–Duarte criterion cannot be audited. Both directions of the equivalence rest on this fidelity; without the missing estimates the claim cannot be accepted.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: abstract states a genuine RH equivalence criterion; wrong full text supplied, so no equation-level reduction can be exhibited.

full rationale

The claimed result is an equivalence (RH ⇔ a constructed moderate net in G(0,1) is uniformly L²-bounded and associated with −χ_{(0,1)}), built from explicitly damped Báez–Duarte sums by Mellin convolution. That logical shape is a reformulation/criterion, not a tautology forced by definition: the net is not defined as “whatever makes RH true,” and the abstract does not fit free constants to the target statement or rename a known pattern as a prediction. No self-citation chain, uniqueness import, or ansatz smuggled via overlapping authors appears in the available text. The CACHEABLE block is an unrelated DAFx room-equalization paper (arXiv:2606.22563), so the detailed estimates for moderateness, association, and the converse extraction of zeros cannot be inspected; residual risk about damping fidelity is a correctness/auditability issue, not a demonstrated circular reduction. Per the rules, only quoteable reductions raise the score; none are present. Score 0 with empty steps is therefore the honest finding.

Assumptions & free parameters 3 free parameters · 3 assumptions · 1 invented entities

Review is abstract-only for 2606.22562 (full text supplied is a different paper). Ledger entries are therefore those that the abstract itself makes load-bearing: standard Colombeau and Báez–Duarte background, the two damping constructions, and the Mellin-convolution net. No numerical fits to data appear; free parameters are the damping exponents chosen by the authors.

free parameters (3)
  • α in δ(ε)=(log(1/ε))^{-α}
    Exponent controlling the slow polynomial damping; chosen by the authors so that the net remains moderate and associated under RH. Not fixed by a uniqueness theorem in the abstract.
  • exponential damping scale ε² in exp(−kε²)
    Rate of the first damping strategy; a design choice that must be tuned so moderateness and association survive. Abstract does not derive a unique canonical rate.
  • truncation cutoffs (super-exponential / polynomial)
    How far the Báez–Duarte sums are truncated as ε→0; required for moderateness but not uniquely determined by the abstract’s statements.
assumptions (3)
  • standard math Standard theory of the Colombeau algebra G(0,1): moderateness, association, and the embedding of distributions/L² functions.
    The entire criterion is stated inside G(0,1); without the usual Colombeau calculus the claim is undefined.
  • domain assumption Báez–Duarte criterion (or its Hilbert-space formulation) as an equivalent of RH for the undamped sums.
    The nets are built from damped Báez–Duarte sums; the paper’s equivalence is a damped/Colombeau lift of that known criterion.
  • ad hoc to paper Multiplicative (Mellin) convolution preserves the analytic features needed for association with −χ_{(0,1)} under the stated damping.
    The abstract’s construction uses Mellin convolution as the glue; that this operation yields a net whose association is equivalent to RH is specific to this paper’s argument.
invented entities (1)
  • The damped Báez–Duarte moderate net in G(0,1) (two families: exponential and polynomial damping)
    purpose: Serve as the single object whose moderateness, uniform L²-boundedness, and association with −χ_{(0,1)} are claimed equivalent to RH.
    The net is constructed ad hoc for this criterion; it is not a previously named object in the literature cited by the abstract.

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Pith. "Pith review of A Colombeau--Beurling criterion for the Riemann hypothesis." pith.science (2026). https://pith.science/paper/OPQURMCN

@misc{pith2026260622562,
  author       = {Pith},
  title        = {Pith review of: A Colombeau--Beurling criterion for the Riemann hypothesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OPQURMCN}},
  note         = {Machine review of arXiv:2606.22562}
}
abstract

This paper establishes an equivalence between the Riemann hypothesis and the association, together with uniform $L^2$-boundedness, of a single moderate net in the Colombeau algebra $G(0,1)$, constructed from damped B\'aez--Duarte sums by multiplicative (Mellin) convolution. Two explicit damping strategies are introduced: an exponential damping $\exp(-k\varepsilon^2)$ combined with super-exponential truncation, and a polynomial damping $k^{-\delta(\varepsilon)}$, where $\delta(\varepsilon)=(\log(1/\varepsilon))^{-\alpha}$, combined with polynomial truncation. Assuming the Riemann hypothesis, the corresponding nets are shown to be moderate, uniformly $L^2$-bounded, and associated with the negative characteristic function of $(0,1)$. Conversely, the existence of a moderate net of this form that is uniformly $L^2$-bounded and associated with the negative characteristic function of $(0,1)$ implies the Riemann hypothesis. The result provides a Colombeau--Beurling type criterion that reformulates the Riemann hypothesis in terms of generalized functions, weak association, and uniform $L^2$ control.

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Works this paper leans on

38 extracted references · 1 linked inside Pith

  1. [1]

    INTRODUCTION Room Equalization (RE) compensates linear distortions from play- back equipment and room acoustics at the intersection of digi- tal signal processing (DSP) and active acoustics [1]. In practice, the transfer function of sound systems varies continuously with source-listener position, room geometry, crowd density, device de- viations, and envi...

    arXiv 2026

  2. [2]

    Room response equalization—a review,

    S. Cecchiet al., “Room response equalization—a review,” Applied Sciences 2018, V ol. 8, Page 16, vol. 8, p. 16, 12 2017

    2018

  3. [3]

    On the variation and invertibility of room impulse response functions,

    J. Mourjopoulos, “On the variation and invertibility of room impulse response functions,”Journal of sound and vibration, vol. 102, no. 2, pp. 217–228, 1985

    1985

  4. [4]

    Errors in real-time room acoustics dereverberation,

    P. D. Hatziantoniouet al., “Errors in real-time room acoustics dereverberation,”Journal of the Audio Engineering Society, vol. 52, no. 9, pp. 883–899, 2004

    2004

  5. [5]

    Multiple position room response equaliza- tion in frequency domain,

    A. Cariniet al., “Multiple position room response equaliza- tion in frequency domain,”IEEE/ACM Trans. Audio Speech Lang. Process., vol. 20, pp. 122–135, 2012

    2012

  6. [6]

    On adaptive inverse control,

    B. Widrowet al., “On adaptive inverse control,” inFifteenth ASILOMAR Conference On Circuits, Systems and Comput- ers. IEEE, 11 1981, pp. 185–189

    1981

  7. [7]

    Analysis of filtered-x LMS algorithm,

    E. Bjarnason, “Analysis of filtered-x LMS algorithm,”IEEE Trans. Speech Audio Process., vol. 3, pp. 504–514, 1995

    1995

  8. [8]

    Robust equalizer design for adaptive room impulse response compensation,

    R. Jariwalaet al., “Robust equalizer design for adaptive room impulse response compensation,”Applied Acoustics, vol. 125, pp. 1–6, 10 2017

    2017

Show all 38 references
  1. [9]

    Time-domain filtered-x-Newton narrowband al- gorithms for active isolation of frequency-fluctuating vibra- tion,

    Y . Liet al., “Time-domain filtered-x-Newton narrowband al- gorithms for active isolation of frequency-fluctuating vibra- tion,”Journal Sound Vibration, vol. 367, pp. 1–21, 4 2016

    2016

  2. [10]

    A biased multichannel adaptive algorithm for room equalization,

    L. Fusteret al., “A biased multichannel adaptive algorithm for room equalization,” inE. Sig. Process. Conf., 2012

    2012

  3. [11]

    Active noise control with selective perceptual equalization to shape the residual sound,

    C. Shiet al., “Active noise control with selective perceptual equalization to shape the residual sound,”Applied Acoustics, vol. 208, p. 109376, 6 2023

    2023

  4. [12]

    Combination of filtered-x adaptive filters for nonlinear listening-room compensation,

    L. Fusteret al., “Combination of filtered-x adaptive filters for nonlinear listening-room compensation,”European Sig. Pro- cess. Conf., vol. 2016-November, pp. 1773–1777, 11 2016

    2016

  5. [13]

    Adaptive filtered-x algorithms for room equalization based on block-based combination schemes,

    ——, “Adaptive filtered-x algorithms for room equalization based on block-based combination schemes,”IEEE/ACM Trans. Audio Speech Lang. Process., vol. 24, pp. 1732–1745, 10 2016

    2016

  6. [14]

    An adaptive multiple position room re- sponse equalizer,

    S. Cecchiet al., “An adaptive multiple position room re- sponse equalizer,” inEuropean Sig. Process. Conf.IEEE, 2011, pp. 1274–1278

    2011

  7. [15]

    A subband implementation of a multichannel and multiple position adaptive room response equalizer,

    ——, “A subband implementation of a multichannel and multiple position adaptive room response equalizer,”Applied Acoustics, vol. 173, p. 107702, 2 2021

    2021

  8. [16]

    A non-uniform subband implementation of an active noise control system for snoring reduction,

    S. Nobiliet al., “A non-uniform subband implementation of an active noise control system for snoring reduction,” in Conf. Dig. Audio Effects (DAFx25), Ancona, Italy, Sept. 2– 5, 2025, pp. 320–325

    2025

  9. [17]

    A multichannel and multiple position adaptive room response equalizer in warped domain: Real- time implementation and performance evaluation,

    S. Cecchiet al., “A multichannel and multiple position adaptive room response equalizer in warped domain: Real- time implementation and performance evaluation,”Applied Acoustics, vol. 82, pp. 28–37, 8 2014

    2014

  10. [18]

    Iterative adaptive frequency-domain equal- ization based on sliding window strategy over time-varying underwater acoustic channels,

    L. Jinget al., “Iterative adaptive frequency-domain equal- ization based on sliding window strategy over time-varying underwater acoustic channels,”JASA Express Letters, vol. 1, p. 76002, 7 2021

    2021

  11. [19]

    Neural parametric equalizer matching us- ing differentiable biquads,

    S. Nercessian, “Neural parametric equalizer matching us- ing differentiable biquads,” inConf. Dig. Audio Effects (DAFx20), Vienna, Austria, Sept. 9–11, 2020, pp. 265–272

    2020

  12. [20]

    Style transfer of audio effects with differentiable signal processing,

    C. J. Steinmetzet al., “Style transfer of audio effects with differentiable signal processing,”J. Audio Engineering Soci- ety, vol. 70, pp. 708–721, 7 2022

    2022

  13. [21]

    Deep Optimization of Parametric IIR Filters for Audio Equalization,

    G. Pepeet al., “Deep Optimization of Parametric IIR Filters for Audio Equalization,”IEEE/ACM Trans. Audio Speech Lang. Process., vol. 30, pp. 1136–1149, 2022

    2022

  14. [22]

    Automatic equalization for individ- ual instrument tracks using convolutional neural networks,

    F. Mockenhauptet al., “Automatic equalization for individ- ual instrument tracks using convolutional neural networks,” inIntl. Conf. Dig. Audio Effects (DAFx24), Guildford, U.K., Sept. 3–7, 2024, pp. 57–64

    2024

  15. [23]

    Biquad coefficients optimization via Kolmogorov-Arnold Networks,

    A. Maleket al., “Biquad coefficients optimization via Kolmogorov-Arnold Networks,” inIntl. Conf. Dig. Audio Ef- fects (DAFx25), Ancona, Italy, Sept. 2–5, 2025, pp. 267–274

    2025

  16. [24]

    Neural-driven multi-band processing for au- tomatic equalization and style transfer,

    P. Sarkaret al., “Neural-driven multi-band processing for au- tomatic equalization and style transfer,” inProc. Intl. Conf. Digital Audio Effects (DAFx25), Ancona, Italy, Sept. 2–5, 2025, pp. 382–389

    2025

  17. [25]

    FLAMO: An open-source library for frequency-domain differentiable audio processing,

    G. D. Santoet al., “FLAMO: An open-source library for frequency-domain differentiable audio processing,”IEEE Intl. Conf. Acous. Speech Sig. Pro., 2025

    2025

  18. [26]

    Soundcam: A dataset for finding hu- mans using room acoustics,

    M. Wanget al., “Soundcam: A dataset for finding hu- mans using room acoustics,”Adv. Neural Inf. Process. Syst., vol. 36, pp. 52 238–52 264, 2023

    2023

  19. [27]

    All about audio equalization: Solutions and frontiers,

    V . Välimäkiet al., “All about audio equalization: Solutions and frontiers,”Applied Sciences 2016, vol. 6, p. 129, 5 2016

    2016

  20. [28]

    Frequency-domain and multirate adaptive filter- ing,

    J. J. Shynk, “Frequency-domain and multirate adaptive filter- ing,”IEEE Si. Pro. Mag., vol. 9, no. 1, pp. 14–37, 2002

    2002

  21. [29]

    Fast deconvolution of multichannel sys- tems using regularization,

    O. Kirkebyet al., “Fast deconvolution of multichannel sys- tems using regularization,”IEEE Trans. Speech Audio Pro- cess., vol. 6, pp. 189–194, 1998

    1998

  22. [30]

    Nocedalet al.,Numerical optimization

    J. Nocedalet al.,Numerical optimization. Springer, 2006

    2006

  23. [31]

    Adam: A method for stochastic opti- mization,

    D. P. Kingmaet al., “Adam: A method for stochastic opti- mization,”Intl. Conf. Learn. Rep., ICLR 2015, 12 2014

    2015

  24. [32]

    The proposed homotopy analysis technique for the solution of nonlinear problems,

    S. Liao, “The proposed homotopy analysis technique for the solution of nonlinear problems,” Ph.D. dissertation, Shang- hai Jiao Tong University Shanghai, 1992

    1992

  25. [33]

    An iterative HAM approach for nonlinear boundary value problems in a semi-infinite domain,

    Y . Zhaoet al., “An iterative HAM approach for nonlinear boundary value problems in a semi-infinite domain,”Com- put. Phys. Commun., vol. 184, pp. 2136–2144, 9 2013

    2013

  26. [34]

    An attempt to apply the homotopy method to the domain of machine learning,

    Y . Liuet al., “An attempt to apply the homotopy method to the domain of machine learning,”Expert Systems with Appli- cations, vol. 234, p. 121098, 12 2023

    2023

  27. [35]

    Cartan,Differential Calculus

    H. Cartan,Differential Calculus. Hermann Paris, 1971

    1971

  28. [36]

    Medleydb: A multitrack dataset for annotation-intensive mir research

    R. M. Bittneret al., “Medleydb: A multitrack dataset for annotation-intensive mir research.” inIsmir, vol. 14, 2014, pp. 155–160

    2014

  29. [37]

    Audio Toolbox documentation,

    The MathWorks Inc., “Audio Toolbox documentation,” Mas- sachusetts, U.S., 2025

    2025

  30. [38]

    A decoupled filtered-x LMS algorithm for listening-room compensation,

    S. Goetzeet al., “A decoupled filtered-x LMS algorithm for listening-room compensation,” inWorkshop Acou. Echo Noise Ctrl., 2008. DAFx.8

    2008

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