REVIEW 1 major objections 2 minor 1 cited by
On the Smallest Counterexample to the Log-Concavity of the D'Arcais Polynomials
T0 review · 1 major / 2 minor · reviewed 2026-06-27 · grok-4.3
Pith's one-line read The D'Arcais polynomials first violate log-concavity at λ = 65,214,507,758,400.
desk verdict The paper turns Starr's asymptotic existence result into an explicit smallest counterexample at λ=65214507758400 by sharpening convolution estimates, but minimality depends on those bounds being fully effective. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Refined asymptotic estimates on convolutions of σ_{-1} that control the sign of the log-concavity difference for the D'Arcais polynomials.
What would settle it
Direct verification that the log-concavity difference changes sign for some λ smaller than 65214507758400, or remains non-negative at that specific λ.
Extended reading notes
Core claim
Refining the asymptotic estimates on convolutions of σ_{-1} allows the authors to prove that the D'Arcais polynomial P_λ(x) first fails to satisfy the log-concavity inequality a_n^2 ≥ a_{n-1}a_{n+1} at λ = 65214507758400, and to describe the density of later failures.
Load-bearing premise
The refined asymptotic estimates on convolutions of σ_{-1} are accurate enough to guarantee that 65214507758400 is the smallest counterexample.
Editorial extensions
If this is right
- The conjecture of Heim–Neuhauser and Abdesselam on log-concavity is false.
- The smallest explicit counterexample is λ = 65214507758400.
- Counterexamples to log-concavity occur with positive asymptotic density.
- The same refined estimates can be applied to locate further counterexamples.
Reading between the lines
- Similar asymptotic refinements could locate minimal counterexamples in other families of arithmetic polynomials.
- The size of the counterexample indicates that exhaustive search alone cannot settle the original conjecture.
- The density result suggests that violations become relatively frequent for large λ.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript refines asymptotic estimates on convolutions of the divisor sum function σ_{-1} to locate the smallest counterexample to the log-concavity conjecture for the D'Arcais polynomials at λ = 65,214,507,758,400. It also examines the asymptotic density of such counterexamples, extending prior non-constructive disproofs by Starr.
Significance. If the refined estimates supply effective error bounds sufficient to certify minimality, the work provides the first explicit counterexample to the Heim–Neuhauser–Abdesselam conjecture and supplies concrete analytic tools for studying log-concavity of related arithmetic polynomials. The explicit identification and density discussion constitute a measurable advance in the area.
major comments (1)
- [Section 3 (asymptotic estimates) and the computational verification paragraph following the statement of the main theore] The central claim that λ = 65214507758400 is the smallest counterexample rests on the refined convolution estimates excluding all smaller values. The manuscript must make explicit the constants appearing in the O-terms (or the effective range of the error bounds) and verify that these constants suffice to certify the inequality for every integer λ below the reported value; without such explicit constants the minimality statement is not fully rigorous.
minor comments (2)
- [Introduction] Notation for the D'Arcais polynomials P_λ(x) should be introduced once in the introduction with a reference to the original definition, rather than assumed from prior literature.
- [Section 5] The density statement in the final section would benefit from a brief comparison table of the predicted density versus the count of counterexamples found in a moderate range (e.g., up to 10^12).
Simulated Author's Rebuttal
We thank the referee for the careful reading and for identifying a point where the rigor of the minimality claim can be strengthened. We address the major comment below.
read point-by-point responses
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Referee: [Section 3 (asymptotic estimates) and the computational verification paragraph following the statement of the main theore] The central claim that λ = 65214507758400 is the smallest counterexample rests on the refined convolution estimates excluding all smaller values. The manuscript must make explicit the constants appearing in the O-terms (or the effective range of the error bounds) and verify that these constants suffice to certify the inequality for every integer λ below the reported value; without such explicit constants the minimality statement is not fully rigorous.
Authors: We agree that the minimality statement requires explicit constants in the error terms to be fully rigorous. In the revised manuscript we will state the explicit numerical values of all constants appearing in the O-terms of Section 3, together with the effective range on which the bounds hold. We will then add a short verification (either analytic or computational) confirming that these explicit bounds suffice to certify the required inequality for every integer λ below 65214507758400. The revised computational verification paragraph will reference these constants directly. revision: yes
Circularity Check
No circularity: refined asymptotics and explicit search yield independent identification of counterexample
full rationale
The paper refines asymptotic estimates on convolutions of σ_{-1} (building on Starr's prior non-self work) to locate and certify the minimal counterexample λ = 65214507758400 via explicit computation and error bounds. No step equates a derived quantity to its own fitted input, renames a known result, or loads the central claim on a self-citation chain; the minimality argument rests on external asymptotic machinery and direct verification rather than self-referential definition or construction. The derivation chain is therefore self-contained against the stated inputs.
Assumptions & free parameters
assumptions (1)
- standard math Standard analytic properties of the divisor function σ_{-1} and its convolutions
Cite this review
Pith. "Pith review of On the Smallest Counterexample to the Log-Concavity of the D'Arcais Polynomials." pith.science (2026). https://pith.science/paper/6C2AVMUT
@misc{pith2026260609545,
author = {Pith},
title = {Pith review of: On the Smallest Counterexample to the Log-Concavity of the D'Arcais Polynomials},
year = {2026},
howpublished = {\url{https://pith.science/paper/6C2AVMUT}},
note = {Machine review of arXiv:2606.09545}
}
abstract
Recently, Starr used asymptotic methods to disprove a conjecture by Heim--Neuhauser and Abdesselam about the log-concavity of the D'Arcais polynomials, without giving an explicit counterexample. We refine the asymptotics, to give the necessary estimates on convolutions of $\sigma_{-1}$, and identify the first counterexample at $\lambda = 65\,214\,507\,758\,400$. We also consider the asymptotic density of such counterexamples.
Forward citations
Cited by 1 Pith paper
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On the Log-Concavity of the D'Arcais Polynomials for Normalised Functions
For normalised arithmetic functions, the near-diagonal coefficients of the D'Arcais polynomials are eventually log-concave in horizontal, vertical, and a new skew sense.
Reference graph
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Reviewed June 27, 2026 · model on record in the stance chip above.
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