REVIEW 1 major objections 2 minor 1 cited by
Transcendence Results for $\Gamma^{(n)}(1)$ and Related Sequences of Generalized Constants
T0 review · 1 major / 2 minor · reviewed 2026-05-18 · grok-4.3
Pith's one-line read The η^(n) from Gumbel(0,1) moments are algebraically independent for all n ≥ 0.
desk verdict Paper defines Gumbel moment sequences and applies Shidlovskii to get algebraic independence of all η^(n), which forces the transcendence claims, but the E-function DE verification is the part that needs explicit checking. 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
Shidlovskii’s theorem applied to the E-functions constructed from the generating functions of the partial moments η^(n) of the Gumbel(0,1) distribution.
What would settle it
Explicitly exhibiting a nonzero polynomial with rational coefficients that is satisfied by any finite collection of the numerical values η^(n) would disprove the claimed algebraic independence.
Extended reading notes
Core claim
Via generating functions from Gumbel(0,1) moments, γ^(2n) belongs to the rational field extension generated by γ and the preceding odd-indexed terms for n ≥ 2. The η^(n) are shown to be algebraically independent using Shidlovskii’s theorem on E-functions, hence each is transcendental. This independence implies that for every n ≥ 1 at least one member of the pairs {γ^(n), δ^(n)/e} and {γ^(n), δ^(n)} is transcendental, and at least two members of the triple {γ^(n), δ^(n)/e, δ^(n)} are transcendental. Both δ^(n)/e and δ^(n) are transcendental for infinitely many n, each with lower asymptotic density at least 1/2. Parallel transcendence results hold for the tilde versions satisfying the non-altr
Load-bearing premise
The generating functions or E-functions arising from the Gumbel(0,1) moments must satisfy the hypotheses of Shidlovskii’s theorem.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines sequences γ^(n), δ^(n), and η^(n) from raw, conditional, and partial moments of the Gumbel(0,1) distribution, generalizing the Hardy relation γ + δ/e = Ein(1) to γ^(n) + δ^(n)/e = η^(n) for n ≥ 0. It derives polynomial relations showing γ^(2n) lies in Q[γ, γ^(2), …, γ^(2n-1)] for n ≥ 2 and that γ^(n) is transcendental for infinitely many n. The central result applies Shidlovskii's theorem to establish algebraic independence of all η^(n) (n ≥ 0), implying that at least one element of each pair {γ^(n), δ^(n)/e} and {γ^(n), δ^(n)} and at least two elements of the triple {γ^(n), δ^(n)/e, δ^(n)} are transcendental for n ≥ 1. Parallel results are given for the non-alternating analogues, along with lower asymptotic densities of at least 1/2 for the transcendence of δ^(n)/e and δ^(n).
Significance. If the application of Shidlovskii's theorem is fully justified, the algebraic independence of the η^(n) would constitute a substantial advance in transcendence theory for generalized Euler-Mascheroni and Gompertz constants, yielding the first results on algebraic independence for these sequences and strengthening the known fact that at least one of γ or δ is transcendental. The polynomial relations among the γ^(n) and the density statements for transcendence of the δ sequences are also of interest.
major comments (1)
- [Application of Shidlovskii's theorem to the η^(n)] The section (or paragraph) applying Shidlovskii's theorem to the η^(n) series does not explicitly derive the linear differential equation system satisfied by the associated generating functions or E-functions, nor does it verify that the coefficients lie in Q(z) (rather than a larger extension) and that the Siegel-Shidlovskii irreducibility condition holds. These verifications are load-bearing for the algebraic-independence claim and the consequent transcendence statements for the pairs and triples involving γ^(n) and δ^(n).
minor comments (2)
- The explicit integral or moment definitions of γ^(n), δ^(n), and η^(n) should be stated in the introduction for immediate clarity before the generating-function setup is introduced.
- Notation for the non-alternating sequences ~δ^(n) and ~η^(n) is introduced late; a brief forward reference in the abstract or introduction would help readers.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for highlighting the need for more explicit details in the application of Shidlovskii's theorem. We address this comment below and will revise the paper to include the requested verifications.
read point-by-point responses
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Referee: The section (or paragraph) applying Shidlovskii's theorem to the η^(n) series does not explicitly derive the linear differential equation system satisfied by the associated generating functions or E-functions, nor does it verify that the coefficients lie in Q(z) (rather than a larger extension) and that the Siegel-Shidlovskii irreducibility condition holds. These verifications are load-bearing for the algebraic-independence claim and the consequent transcendence statements for the pairs and triples involving γ^(n) and δ^(n).
Authors: The referee correctly identifies that the manuscript does not explicitly derive the linear differential equation system or verify the conditions for Shidlovskii's theorem in full detail. This is a valid point, as these steps are indeed crucial for the rigor of the algebraic independence result. In the revised manuscript, we will add an appendix or expanded section that: derives the system of linear differential equations satisfied by the generating functions for the η^(n) from the integral representations or recurrence relations coming from the Gumbel(0,1) distribution; confirms that the coefficients are in Q(z); and verifies the Siegel-Shidlovskii irreducibility condition. With these additions, the application of the theorem will be fully justified, supporting the algebraic independence of the η^(n) and the consequent transcendence results. We do not anticipate any changes to the main theorems themselves. revision: yes
Circularity Check
No circularity: external Shidlovskii theorem applied to internally constructed E-functions
full rationale
The paper constructs the sequences γ^(n), δ^(n), η^(n) from Gumbel(0,1) moment generating functions and derives the polynomial relations γ^(2n) ∈ Q[γ, γ^(2), …, γ^(2n-1)] directly from those generating-function identities. It then invokes the independent external theorem of Shidlovskii to obtain algebraic independence of the η^(n). Because Shidlovskii’s result is a pre-existing theorem whose hypotheses are checked against the paper’s explicit series (rather than assumed or fitted from the target transcendence statements), and no step equates a claimed output to its own input by definition or self-citation chain, the derivation chain contains no circular reduction.
Assumptions & free parameters
assumptions (2)
- domain assumption The relation γ^(n) + δ^(n)/e = η^(n) holds for the sequences derived from raw, conditional, and partial moments of the Gumbel(0,1) distribution
- domain assumption Shidlovskii's theorem applies to the generating functions or E-functions associated with the η^(n)
invented entities (1)
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Sequences γ^(n), δ^(n), η^(n) and their non-alternating analogues
Cite this review
Pith. "Pith review of Transcendence Results for $\Gamma^{(n)}(1)$ and Related Sequences of Generalized Constants." pith.science (2026). https://pith.science/paper/2511.01849
@misc{pith2026251101849,
author = {Pith},
title = {Pith review of: Transcendence Results for $\Gamma^(n)(1)$ and Related Sequences of Generalized Constants},
year = {2026},
howpublished = {\url{https://pith.science/paper/2511.01849}},
note = {Machine review of arXiv:2511.01849}
}
abstract
Neither the Euler-Mascheroni constant, $\gamma=0.577215...$, nor the Euler-Gompertz constant, $\delta=0.596347...$, is currently known to be irrational. However, it has been proved that at least one of them is transcendental. The two constants are related through a well-known equation of Hardy, equivalent to $\gamma+\delta/e=\textrm{Ein}(1)$, which recently has been generalized to $\gamma^{(n)}+\delta^{(n)}/e=\eta^{(n)},\:n\geq0$ for sequences of constants $\gamma^{(n)}$, $\delta^{(n)}$, and $\eta^{(n)}$ (derived respectively from raw, conditional, and partial moments of the $\textrm{Gumbel}(0,1)$ probability distribution). Investigating $\gamma^{(n)}=(-1)^{n}\Gamma^{(n)}(1),\:n\geq1$ through $\textrm{Gumbel}(0,1)$ generating functions, we find that $\gamma^{(2n)}\in\mathbb{Q}[\gamma,\gamma^{(2)}$, $\gamma^{(3)},...,\gamma^{(2n-1)}]$ for $n\geq2$ and $\gamma^{(n)}$ is transcendental infinitely often. We then show, via a theorem of Shidlovskii, that the $\eta^{(n)}$ are algebraically independent, and therefore transcendental, for all $n\geq0$, implying that at least one element of each pair, $\left\{\gamma^{(n)},\delta^{(n)}/e\right\}$ and $\left\{\gamma^{(n)},\delta^{(n)}\right\}$, and at least two elements of the triple $\left\{\gamma^{(n)},\delta^{(n)}/e,\delta^{(n)}\right\}$ are transcendental for all $n\geq1$. Further analysis of the $\gamma^{(n)}$ and $\eta^{(n)}$ reveals that both the $\delta^{(n)}/e$ and $\delta^{(n)}$ are transcendental infinitely often with lower asymptotic densities of at least 1/2. Finally, we provide parallel results for the sequences $\widetilde{\delta}^{(n)}$ and $\widetilde{\eta}^{(n)}$ satisfying the "non-alternating analogue" equation $\gamma^{(n)}+\widetilde{\delta}^{(n)}/e=\widetilde{\eta}^{(n)}$.
Forward citations
Cited by 1 Pith paper
-
"Infinitely Often" Transcendence of Gamma-Function Derivatives
For q in half-integers excluding non-positive integers, Gamma^(n)(q) is transcendental for a positive proportion of n up to N, at least max(0, sqrt(N)-5/2)/N many.
Reference graph
Works this paper leans on
-
[1]
On linear forms containing the Euler constant
Aptekarev, A. I., 2009, “On Linear Forms Containing the Euler Constant”, arXiv:0902.1768v2
work page Pith review arXiv 2009
-
[2]
Shidlovskii, A. B., 1962, “On Transcendency and Algebraic Independence of the Values of E-Functions Connected with an Arbitrary Number of Algebraic Equations in the Field of Rational Functions”,Proceedings of the USSR Academy of Sciences, Mathematics Series, 26, 877-910 (in Russian)
work page 1962
-
[3]
Applications of a Theorem by A. B. Shidlovski
Mahler, K., 1968, “Applications of a Theorem by A. B. Shidlovski”,Proceedings of the Royal Society A, 305, 1481, 149-173
work page 1968
-
[4]
Rivoal, T., 2012, “On the Arithmetic Nature of the Values of the Gamma Function, Euler’s Constant, and Gompertz’s Constant”,Michigan Mathematical Journal, 61, 2, 239-254
work page 2012
-
[5]
Euler’s Constant: Euler’s Work and Modern Developments
Lagarias, J. C., 2013, “Euler’s Constant: Euler’s Work and Modern Developments”,Bulletin of the American Mathematical Society, 50, 4, 527-628
work page 2013
-
[6]
H., 1949,Divergent Series, Clarendon Press, Oxford, UK
Hardy, G. H., 1949,Divergent Series, Clarendon Press, Oxford, UK
work page 1949
-
[7]
Conjunctions of Three ’Euler Constants’ in Poisson-Related Expressions
Powers, M. R., 2025, “Conjunctions of Three ’Euler Constants’ in Poisson-Related Expressions”, arXiv:2503.02054v4
-
[8]
Pinkus, A., 2010,Totally Positive Matrices, Cambridge University Press, Cambridge, UK
work page 2010
Show all 18 references
-
[9]
Liu, Q., 2002,Algebraic Geometry and Arithmetic Curves, Oxford University Press, Oxford, UK
2002
-
[10]
Rational Approximations for Values of Derivatives of the Gamma Function
Rivoal, T., 2009, “Rational Approximations for Values of Derivatives of the Gamma Function”,Transactions of the American Mathematical Society, 361, 11, 6115-6149
2009
-
[11]
Relations between Values of Arithmetic Gevrey Series, and Applications to Values of the Gamma Function
Fischler, S. and Rivoal, T., 2024, “Relations between Values of Arithmetic Gevrey Series, and Applications to Values of the Gamma Function”,Journal of Number Theory, 261, 36-54
2024
-
[12]
Many Odd Zeta Values are Irrational
Fischler, S., Sprang, J., and Zudilin, W., 2019, “Many Odd Zeta Values are Irrational”,Compositio Mathematica, 155, 938-952
2019
-
[13]
On the Zeroes of Certain Classes of Integral Taylor Series, Part II
Hardy, G. H., 1905, “On the Zeroes of Certain Classes of Integral Taylor Series, Part II”,Proceedings of the London Mathematical Society, s2-2, 1, 401-431
1905
-
[14]
B., 1989,Transcendental Numbers, Walter de Gruyter, Berlin, Germany
Shidlovskii, A. B., 1989,Transcendental Numbers, Walter de Gruyter, Berlin, Germany
1989
-
[15]
A Refined Version of the Siegel-Shidlovskii Theorem
Beukers, F., 2006, “A Refined Version of the Siegel-Shidlovskii Theorem”,Annals of Mathematics, 163, 369-379
2006
-
[16]
On the Uniqueness of Ein(1) among Linear Combinations of the Euler-Mascheroni and Euler-Gompertz Constants
Powers, M. R., 2025, “On the Uniqueness of Ein(1) among Linear Combinations of the Euler-Mascheroni and Euler-Gompertz Constants”, arXiv:2510.00315v1
2025
-
[17]
Berndt, B. C. and Rankin, R. A., 1995,Ramanujan: Letters and Commentary, American Mathematical Society, Providence, RI, USA
1995
-
[18]
Extensions by Antiderivatives, Exponentials of Integrals, and by Iterated Logarithms
Srinivasan, V. R., 2008, “Extensions by Antiderivatives, Exponentials of Integrals, and by Iterated Logarithms”, arXiv:0811.3004v1. Appendix Proof of Proposition 1. We consider the three parts of the proposition in the order: (iii), (ii), (i). (iii) From (7), it is easy to see...
2008 arXiv
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