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REVIEW 3 major objections 3 minor 1 cited by

Large finite products of small fractions

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For products of sine-like ratios, only the linear spacing sets the exponent, and the function sets the constant.

desk verdict The asymptotic is plausible and the generalization is nice, but the written proof of the main theorem has a gap in the monotonicity step that fails on a concrete example. read the letter →

arxiv 1908.00839 v2 pith:FSDPUG6J submitted 2019-08-02 math.CA

Arnaldo Mandel This is my paper · ORCID
classification math.CA MSC 41A6033B15
keywords asymptoticanalysisfiniteproductssin-likefunctionsS-functionC-functiongammafunctionStirlingformulaproductofratios
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 proves an asymptotic law for finite products whose factors are ratios of a sine-like function evaluated at two evenly spaced arithmetic progressions. If $h(x)$ behaves like $x$ near $0$ (an S-function, in the paper's term), then the product over $j$ of $h((cj+a)/n)/h((cj+b)/n)$, continued while both arguments stay below a fixed bound, is asymptotic to $C n^{(a-b)/c}$. The exponent is determined entirely by the spacings $c$ and the offset difference $a-b$; the particular shape of $h$ changes only the constant $C$. The proof factors out the same product with $h(x)=x$, whose gamma-function asymptotics is classical, and shows the remaining correction converges to a positive limit. This turns a previous lower bound with an $\varepsilon$-loss in the exponent into an exact asymptotic, and it is the engine behind a statement that sine products of this kind grow like a constant times $n^{1/2}$.

What carries the argument

The load-bearing object is the ratio $H(x)=h(x)/x$, which the paper calls a C-function: $H(0)>0$, $H'(0)=0$, and $H''(x)\le0$ near $0$. The proof writes $D_n=K_n E_n$, where $K_n$ is the product with $h(x)=x$; this is the product that 'illegal cancellation' of $h$ would produce, and its asymptotics follows from Stirling's formula via a gamma-function ratio. The correction $E_n$ is then shown to be decreasing and bounded away from zero by comparing consecutive factors: Lemma 7 uses the monotonicity of $H'(x)/H(x)$ to show each factor of $E_{n+1}$ is no larger than the corresponding factor of $E_n$. The positivity of $E_n$ comes from a uniform lower bound on each factor, obtained from the boundedness of a related function $g_\delta(x,y)$.

What would settle it

Take $H(x)=1-x^2/8$, $a=2$, $b=1$, $c=2$, $d=1$, $\varepsilon=2$, which satisfies the compatibility and positivity conditions. For the largest index $j=n-1$ the offset is zero, and the last factor of $E_n$ is $H(2)/H(2-1/n)$, which increases with $n$; numerically checking whether $E_n$ still converges to a positive limit, and whether $D_n\sim C\sqrt n$ holds, decides whether the theorem is true despite the failed monotonicity argument in that edge case.

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Extended reading notes

Core claim

The central discovery is that the asymptotic order of these products is a power of $n$ whose exponent $(a-b)/c$ depends only on the arithmetic progressions, while the function $h$ enters only through a multiplicative constant. More precisely, for an S-function $h$ (with $h(0)=h''(0)=0$, $h'(0)>0$, $h''\le 0$ near $0$) and a compatible $\varepsilon$ on which $H(x)=h(x)/x$ stays positive and concave, the paper proves $D_n(a,b,c,d,\varepsilon;h) \sim C\,n^{(a-b)/c}$. It also proves the stronger normalization: the quotient $E_n=D_n/K_n$, where $K_n$ is the same product with $h(x)=x$, converges to a positive limit. Because $K_n$ is evaluated exactly through the gamma function, the limit of $E_n$ is the constant $C$ that converts the gamma asymptotics into the final asymptotic.

Load-bearing premise

The proof that the normalized products $E_n$ decrease assumes the offset $\delta(a)=((n-j)c-a)/(n(n+1))$ is strictly positive for every relevant factor; the stated condition $\varepsilon\le c$ does not prevent this offset from vanishing when $\varepsilon=c$ and $a/c$ is an integer.

Editorial extensions

If this is right

  • The motivating sine product $D_n(5,3,4,\pi/2,\pi/2;\sin)$ is asymptotic to $C\sqrt n$, replacing the earlier exponent $1/2-\varepsilon$ with the sharp exponent $1/2$.
  • For any S-function, no matter how $h$ differs from $x$, the exponent $(a-b)/c$ is universal; only the constant $C$ depends on $h$.
  • The constant admits the upper bound $C\le \frac{\Gamma(b/c)}{\Gamma(a/c)}(\varepsilon/c)^{(a-b)/c}$ when $a>b$, because the normalized factors are all below 1.
  • The classes are algebraically closed: sums and products of C-functions are C-functions, and S-functions form a module over them, so the asymptotic applies to many combinations at once.
  • For $H(x)=e^{-x^k}$, the paper's exercise gives the exact limit $\lim_n D_n(a,b,c,d,((k-1)/k)^{1/k}; e^{-x^k}) = e^{-\frac{k-1}{k}\frac{a-b}{c}}$.

Reading between the lines

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

  • The missing explicit formula for $C$ probably comes from an Euler–Maclaurin or zeta-regularized evaluation of $\sum_j [\log H((cj+a)/n) - \log H((cj+b)/n)]$, which would express $C$ as an infinite product over the Taylor coefficients of $H$; a numerical fit of $C$ for $H(x)=1-\lambda x^k$ would test this.
  • The same proof strategy should work when the arguments are replaced by any density-one lattice, suggesting that a general spacing $x_j=j/n$ yields an exponent equal to the density times $(a-b)$; this is a natural extension the paper leaves implicit.
  • The edge case where the shift vanishes indicates that monotonicity is probably not the real mechanism; direct asymptotic expansion of each factor should prove the same limit and simultaneously answer the open rate-of-convergence problem.
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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 / 3 minor

Summary. The paper studies finite products of ratios h((cj+a)d/n)/h((cj+b)d/n) for positive constants a,b,c,d, where h is a real function resembling sin x near 0 and m grows linearly with n. The main result, Theorem 2, asserts the asymptotic equivalence D_n(a,b,c,d,ε;h) ∼ C n^{(a-b)/c} for suitable ε, and Theorem 5 asserts the normalized product converges to a positive limit. The proof strategy is to compare D_n with the rational product K_n whose gamma-function asymptotics are known (Proposition 3), and to show that the ratio E_n = D_n/K_n converges to a positive constant. The argument proceeds through Lemma 6 (a lower bound on E_n), Lemma 7 (a comparison of adjacent factors), and then claims that Lemma 7 implies E_n is decreasing, whence monotone convergence gives a positive limit. The paper also contains illustrative examples and open problems.

Significance. If the main theorem is correct, the paper gives a clean and fairly general asymptotic for large products of small fractions, removes the ε-loss in the motivating application in [3], and identifies a natural class of functions for which the result holds. The reduction to the known gamma-function asymptotics is elegant, and the formulation in terms of S-functions and C-functions is appealing. The paper is concise and the claims are concrete and falsifiable. However, the proof as written contains several gaps, and the decisive step establishing convergence of E_n is invalid, so the central claim is not established by the present text.

major comments (3)
  1. [Proof of Theorem 2] The inference that E_n is decreasing does not follow from Lemma 7. Lemma 7 pairs the j-th factor of E_n with the (j+1)-th factor of E_{n+1}, so when m_{n+1}=m_n two factors are left unpaired: the first factor of E_{n+1} and the last factor of E_n. Both factors are less than 1, but neither controls the other, so the product may increase. For the choices H(x)=1-x^2, a=0.5, b=0.1, c=1, and ε=0.5, a direct computation gives E_5≈0.8624 and E_6≈0.9078, so E_6>E_5. Consequently the monotone-convergence argument for the existence of a positive limit C0 is invalid, and Theorem 2 is not proved as written.
  2. [Lemma 6] The choice A=A(δ) ε/(2c) gives A/m ≤ A(δ)/n, which is the reverse of the inequality needed to deduce (2). Since m/n ≥ ε/(2c) for large n, the chosen A is too small; to obtain (2) one needs A ≥ A(δ) m/n, and because m/n ≤ ε/c, the choice A=A(δ) ε/c would work. The proof as printed therefore does not establish the required lower bound.
  3. [Lemma 7] The asserted lower bound n(n+1)δ(a) ≥ n(c-ε)+a(c-1) is algebraically incorrect; using the definition m=⌊(εn-a)/c⌋ gives the weaker bound n(n+1)δ(a) ≥ n(c-ε). In the compatible case ε=c with a/c an integer, the largest j gives δ(a)=0, so the strict positivity n(n+1)δ(a)>0 used in the proof is false. The lemma can likely be repaired with non-strict inequalities, but as stated the proof is not valid.
minor comments (3)
  1. [Propositions 1 and 4] The proofs of the equivalences in Propositions 1 and 4 are left as exercises; in a research paper these should be proved or at least briefly justified, since they define the classes of admissible functions.
  2. [Lemma 6] The domain of gδ is not compact as stated, because y can be unbounded when x→0 with yx≤ε; the existence of a finite maximum A(δ) needs a more careful argument. Also, the ratio involves H((y+δ)x), which may fall outside the interval [0,ε] unless the domain is restricted with a margin.
  3. [Paragraph after Proposition 8] There is a typo: 'funtion' should be 'function'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the asymptotic is derived from independent gamma-function asymptotics and a monotone-convergence argument; the sole self-citation is motivational.

full rationale

The paper's derivation chain is self-contained. Theorem 2 is proved by writing D_n = E_n * K_n, where K_n is a product of linear factors whose asymptotics is established in Proposition 3 using independent, well-known gamma-function and Stirling asymptotics (citations [2] and [4], DLMF and Wendel). The constant C is not fitted or assumed; it arises as C0 times a known gamma ratio, where C0 is the limit of E_n. The existence of that limit is attempted through Lemma 6 (a uniform lower bound, obtained from continuity and Taylor expansion of H) and Lemma 7 (monotonicity, derived from H'' ≤ 0 and a positivity of the shift δ(a)), followed by a monotone-convergence argument. None of these steps presupposes the conclusion D_n ∼ C n^{(a-b)/c}; the hypotheses H(x) > 0 and H''(x) ≤ 0 on [0, ε] do not contain the target asymptotic. The self-citation [3] appears only as motivation for the problem in the abstract and introduction and is not used as a load-bearing premise in any proof. Proposition 1 is a characterization of S-functions, but it is not used to smuggle in the theorem's conclusion; the paper simply names functions satisfying those Taylor conditions. Even if the monotonicity step has a gap when m_{n+1} = m_n (a correctness concern, not a circularity concern), that does not alter the circularity analysis: the argument is an attempted derivation from stated assumptions to an independent result, not a renaming or refitting of the input. There are no fitted parameters called predictions, no definitional equivalences between hypotheses and conclusions, and no uniqueness claim imported from the authors' prior work. Hence the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces two function classes, S-functions and C-functions, via derivative conditions; these are mathematical definitions, not new physical entities. No free parameters are fitted to data; the theorem's constant C is left unevaluated. The two equivalence propositions are stated without proof and are load-bearing, so they are recorded as axioms.

assumptions (3)
  • ad hoc to paper Proposition 1: for analytic h, the derivative conditions are equivalent to h(x)=α(x-λx^k)+O(x^{k+1}) with k≥3; proof is left as an exercise.
    The proof of Theorem 2 depends on the structural form of H from this characterization, but the proposition is not proved in the paper.
  • ad hoc to paper Proposition 4: for analytic H, the derivative conditions are equivalent to H(x)=α(1-λx^k)+O(x^{k+1}) with k≥2; proof is left as an exercise.
    The theorem's class of C-functions is defined by this equivalence, and the proof uses the k≥2 expansion to establish continuity of gδ at x=0.
  • standard math Stirling's formula / gamma asymptotics: Γ(m+1+a/c)/Γ(m+1+b/c) ~ m^{(a-b)/c}.
    Used in Proposition 3; cited to DLMF 5.11.12 and Wendel [4].

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Cite this review

Pith. "Pith review of Large finite products of small fractions." pith.science (2026). https://pith.science/paper/FSDPUG6J

@misc{pith2026190800839,
  author       = {Pith},
  title        = {Pith review of: Large finite products of small fractions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FSDPUG6J}},
  note         = {Machine review of arXiv:1908.00839}
}
abstract

Fix positive reals $a,b,c,d$, and let $h(x)$ be a real function behaving sort of like $\sin x$ near 0. Then, provided $m$ grows linearly with $n$. there exists a positive constant $C$ such that$$ \prod_{j=0}^m\frac{h\left((cj+a)\frac{d}{n}\right)}{h\left((cj+b)\frac{d}{n}\right)}\sim C n^{\frac{a-b}c}. $$

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Dragging the roots of a polynomial to the unit circle

    math.CO 2019-08 conditional novelty 8.0 of 10

    The paper defines and exactly computes threshold parameters il(p) and cn(p) governing circle-rootedness and interlacing with roots of unity, and constructs families where il/cn grows without bound.

Reference graph

Works this paper leans on

4 extracted references · 3 canonical work pages · cited by 1 Pith paper

  1. [3]

    Dragging the roots of a polynomial to the unit circle

    Arnaldo Mandel and Sinai Robins. “Dragging the roots of a polyno- mial to the unit circle”. In: arXiv e-prints , arXiv:1908.03208 (Aug. 2019), arXiv:1908.03208. arXiv: 1908.03208 [math.CV]

    work page Pith review arXiv 1908

  2. [1]

    Dieckmann

    A. Dieckmann. Collection of Infinite Product and Series . 2018. url: http://www-elsa.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html (visited on 01 /03/2019)

    work page 2018

  3. [2]

    http://dlmf.nist.gov/, Release 1.0.21 of 2018-12-15

    NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.21 of 2018-12-15. url: http://dlmf.nist.gov/5.11.12. 6 REFERENCES

    work page 2018

  4. [4]

    Note on the gamma function

    J. G. Wendel. “Note on the gamma function”. In: Amer . Math. Monthly 55 (1948), pp. 563–564. doi: 10.2307/2304460. Computer Science Department, Instituto de Matem´atica e Estat´istica, Universidade de S˜ao Paulo, São Paulo, SP , Brazil 05508-970, Orcid: 0000-0003-0661-4495 E-mail address: am@ime.usp.br

    doi:10.2307/2304460 1948

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