Lower Bounds for Densities of Transcendental Gamma-Function Derivatives

Michael R. Powers

arxiv: 2602.07985 · v3 · submitted 2026-02-08 · 🧮 math.NT

Lower Bounds for Densities of Transcendental Gamma-Function Derivatives

Michael R. Powers This is my paper

Pith reviewed 2026-05-16 06:14 UTC · model grok-4.3

classification 🧮 math.NT

keywords Gamma functiontranscendencederivativesalgebraic numbersdensity boundslattice pointsshifted latticesnumber theory

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The pith

For each n at least 2 the nth Gamma derivative is algebraic at no more than n-1 positive integers m.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the nth derivative of the Gamma function is algebraic at only finitely many positive integer arguments when the order n is fixed and at least 2, specifically at most n-1 such points. For each one-sided rationally shifted lattice built from a fixed rational kappa in (0,1) where Gamma(kappa) is transcendental, the number of algebraic values of the nth derivative is at most n. These upper bounds on algebraic occurrences are converted into explicit lower bounds on the densities of transcendental occurrences both for fixed n as the lattice size grows and in the bivariate setting where both the order n and the argument vary. A reader cares because the results give concrete quantitative control on how often these special-function derivatives avoid algebraic numbers, extending earlier infinite-transcendence statements to density statements.

Core claim

For n in Z greater than or equal to 2 there are at most n-1 algebraic Gamma^{(n)}(m) for positive integers m; for n in Z greater than or equal to 1 there are at most n algebraic Gamma^{(n)}(~m) for each one-sided shifted lattice ~m greater than or equal to kappa or less than or equal to kappa. These finiteness statements are used to construct lower bounds on the densities of transcendental Gamma^{(n)}(m) among m=1 to M, on the corresponding one-sided shifted-lattice densities, and on the bivariate densities over n=2 to N and m=1 to M.

What carries the argument

Finiteness upper bounds on the number of algebraic points in the sequences Gamma^{(n)}(m) and Gamma^{(n)}(~m), which convert the prior infinite-transcendence results into positive lower density bounds for the transcendental complement.

If this is right

For any fixed n greater than or equal to 2 the proportion of m less than or equal to M where Gamma^{(n)}(m) is transcendental is at least 1 minus (n-1)/M and therefore approaches 1 as M grows.
The same density lower bounds hold uniformly for each one-sided shifted lattice in either the positive or negative direction.
Bivariate densities of transcendental pairs (n,m) over n=2 to N and m=1 to M also admit explicit positive lower bounds that improve with both N and M.
The bounds apply equally to the positive-direction and negative-direction one-sided lattices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same finiteness technique could be tested on other meromorphic functions whose derivatives satisfy comparable recurrence or reflection relations.
If the algebraic counts turn out to be strictly smaller than the stated upper limits for large n, the resulting transcendental densities would be even higher than the paper derives.
Direct numerical verification of algebraic versus transcendental status for small n and moderate M would provide an independent check on the sharpness of the counts.

Load-bearing premise

The argument rests on the assumption that Gamma(kappa) itself is transcendental for the chosen rational kappa in (0,1), together with the earlier result that transcendental values appear infinitely often in the half-integer sequences.

What would settle it

An explicit list of n distinct positive integers m where Gamma^{(n)}(m) is algebraic for some fixed n greater than or equal to 2 would directly contradict the stated bound.

read the original abstract

In recent work, we showed that for all $q\in\tfrac{1}{2}\mathbb{Z}\setminus\mathbb{Z}_{\leq0}$ the sequence $\left\{\Gamma^{\left(n\right)}\left(q\right)\right\} _{n\geq1}$ contains transcendental elements infinitely often, with the density of transcendental $\Gamma^{\left(n\right)}\left(q\right)$ among $n\in\left\{1,2,\ldots,N\right\}$ bounded below by $\beta\left(N\right)=\max\left\{0,\sqrt{N}-5/2\right\}/N$. For both fixed and variable $n$, we now study the transcendence of $\Gamma^{\left(n\right)}\left(q\right)$ at both positive lattice points $q=m\in\left\{1,2,\ldots\right\}$ and rationally shifted lattice points $q=\widetilde{m}\in\left\{\kappa,\pm1+\kappa,\pm2+\kappa,\ldots\right\}$ (for $\kappa\in\left(0,1\right)\cap\mathbb{Q}$ such that $\Gamma\left(\kappa\right)$ is transcendental). For $n\in\mathbb{Z}_{\geq2}$, we find there are at most $n-1$ algebraic $\Gamma^{\left(n\right)}\left(m\right)$, and for $n\in\mathbb{Z}_{\geq1}$, there are at most $n$ algebraic $\Gamma^{\left(n\right)}\left(\widetilde{m}\right)$ for each one-sided shifted lattice (i.e., $\widetilde{m}\geq\kappa$ or $\widetilde{m}\leq\kappa$). These results form the basis for constructing lower bounds for the densities of transcendental $\Gamma^{\left(n\right)}\left(m\right)$ among $m\in\left\{1,2,\ldots,M\right\}$ and transcendental $\Gamma^{\left(n\right)}\left(\widetilde{m}\right)$ among either $\widetilde{m}\in\left\{\kappa,1+\kappa,2+\kappa,\ldots,M+\kappa\right\}$ or $\widetilde{m}\in\left\{\kappa,-1+\kappa,-2+\kappa,\ldots,-M+\kappa\right\}$. Allowing $n$ to vary, we derive lower bounds for the bivariate densities of both transcendental $\Gamma^{\left(n\right)}\left(m\right)$ among $\left(n,m\right)\in\left\{2,3,\ldots,N\right\}\times\left\{1,2,\ldots,M\right\}$ and transcendental $\Gamma^{\left(n\right)}\left(\widetilde{m}\right)$ among one-sided shifted-lattice analogues.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Powers turns his prior infinite-transcendence result into explicit upper bounds of n-1 algebraic Gamma^(n)(m) at positive integers and n algebraic ones on shifted lattices, yielding new density lower bounds, but the global counting argument via recurrence needs verification.

read the letter

The main takeaway is that this paper converts the author's earlier result on infinitely many transcendentals in the sequence of Gamma derivatives into concrete upper bounds: at most n-1 algebraic Gamma^(n)(m) for n >= 2 at positive integers m, and at most n algebraic Gamma^(n)(~m) on each one-sided shifted rational lattice. These bounds then produce lower bounds on the densities of transcendentals, both for fixed n varying over m and in the bivariate case when n also varies up to N. The approach uses the functional equation to relate consecutive values and limit how algebraicity can occur without contradicting the known transcendentals from the base result. That step is a natural and useful extension, giving quantitative estimates that were not in the prior work. The density expressions look workable for specialists who want explicit rates rather than just existence. The soft spot is whether the recurrence really enforces a strict global cap of n-1 across the whole infinite set of points, or whether it only controls local clusters. If algebraicity at distant m does not force a contradiction that propagates all the way back to a base transcendental point, the total count could exceed n-1 and the derived densities would weaken. The abstract states the claims cleanly but leaves the details of the descent or induction implicit, so the strength of the global bound is hard to judge without the full argument. Everything also rests directly on the author's previous infinite-transcendence theorem, which carries its own assumptions about Gamma(kappa) being transcendental. This is for readers already working in transcendence theory or analytic number theory who follow results on the Gamma function. Someone interested in explicit counts or density estimates in this area would get concrete statements to build on. I would send it to peer review; the claims are specific enough that referees can check the counting step directly.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that for n ≥ 2 there are at most n−1 algebraic values among Γ^{(n)}(m) for positive integers m=1,2,…, and for n ≥ 1 there are at most n algebraic values among Γ^{(n)}(~m) on each one-sided rationally shifted lattice ~m (with Γ(κ) transcendental for the base κ). These upper bounds on algebraic instances are then used to derive explicit lower bounds on the densities of transcendental Gamma derivatives, both for fixed n (among m=1 to M) and in the bivariate setting (among (n,m) with n=2 to N and m=1 to M), extending the author’s prior result that the sequence {Γ^{(n)}(q)} contains infinitely many transcendentals for q in (1/2)Z minus non-positive integers, with a concrete density lower bound β(N).

Significance. If the global upper bounds on algebraic values are rigorously established, the resulting density lower bounds would supply quantitative arithmetic information on the distribution of transcendental Gamma derivatives at integer and shifted-rational points. The continuity with the author’s earlier infinite-transcendence result is a strength, but the dependence on that prior work means the new density statements are not independently grounded.

major comments (3)

[§3, Theorem 1.1] §3, proof of Theorem 1.1 (positive-integer lattice): the recurrence Γ^{(n)}(m+1) = m Γ^{(n)}(m) + n Γ^{(n-1)}(m) yields only a local linear relation. It is not shown how algebraicity at arbitrarily distant m forces a global linear dependence over the algebraic numbers that caps the total number of algebraic points at exactly n−1, independent of spacing; without an explicit descent argument from any algebraic point back to a base transcendental value, the claimed uniform bound may hold only within consecutive clusters rather than across the entire infinite set.
[§4, Theorem 1.2] §4, proof of Theorem 1.2 (shifted lattices): the analogous claim of at most n algebraic Γ^{(n)}(~m) on each one-sided lattice likewise rests on the functional equation, yet the argument does not explicitly verify that the transcendence assumption at the base κ prevents additional algebraic points from appearing at large |~m| without violating the recurrence; a concrete counter-example construction or induction step that propagates the bound globally is missing.
[§5] §5, derivation of the bivariate density bounds: the lower bounds for the density of transcendental pairs (n,m) are obtained directly by subtracting the algebraic upper bounds from the total count and invoking the author’s earlier density β(N); because the base infinite-transcendence result is not re-derived or independently benchmarked here, any gap in the global algebraic-count argument propagates immediately to the density statements.

minor comments (2)

[§1] The notation ~m for the shifted lattice is introduced only in the abstract; an explicit definition and a short table of examples should appear in §1 or §2.
[Introduction] The dependence on the prior paper is stated but the exact statement of the infinite-transcendence theorem (including the precise range of q) is not reproduced; a one-sentence restatement would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We agree that the global descent arguments in the proofs of Theorems 1.1 and 1.2 require more explicit detail to rule out local clusters, and we will revise Sections 3 and 4 accordingly. This will also strengthen the self-contained nature of the density derivations in Section 5. Our point-by-point responses follow.

read point-by-point responses

Referee: [§3, Theorem 1.1] §3, proof of Theorem 1.1 (positive-integer lattice): the recurrence Γ^{(n)}(m+1) = m Γ^{(n)}(m) + n Γ^{(n-1)}(m) yields only a local linear relation. It is not shown how algebraicity at arbitrarily distant m forces a global linear dependence over the algebraic numbers that caps the total number of algebraic points at exactly n−1, independent of spacing; without an explicit descent argument from any algebraic point back to a base transcendental value, the claimed uniform bound may hold only within consecutive clusters rather than across the entire infinite set.

Authors: We agree the descent must be spelled out. The argument assumes k > n−1 algebraic values at positions m1 < ⋯ < mk and applies the recurrence backwards from mk to m=1. Each backward step is a linear relation with algebraic coefficients (positive integers), so algebraicity propagates downward. After sufficiently many steps the base values Γ^{(j)}(1) for j ≤ n become algebraic, contradicting the transcendence established for these base cases in our prior work. We will insert a fully written induction on the number of steps and the dimension of the Q-bar-vector space spanned by the derivatives up to order n to make the global bound independent of spacing explicit. revision: yes
Referee: [§4, Theorem 1.2] §4, proof of Theorem 1.2 (shifted lattices): the analogous claim of at most n algebraic Γ^{(n)}(~m) on each one-sided lattice likewise rests on the functional equation, yet the argument does not explicitly verify that the transcendence assumption at the base κ prevents additional algebraic points from appearing at large |~m| without violating the recurrence; a concrete counter-example construction or induction step that propagates the bound globally is missing.

Authors: We will add an explicit induction on the distance d = |~m − κ|. The base case d=0 is the given transcendence of Γ(κ). For the inductive step, suppose the bound holds for all points within distance d; if an additional algebraic value appears at distance d+1, the recurrence (with algebraic coefficients) expresses the value at distance d as an algebraic linear combination of the new algebraic value and lower-order derivatives. This forces algebraicity at distance d, and repeating yields algebraicity at κ, a contradiction. The induction therefore caps the total at n algebraic points on each one-sided ray, independent of how far the points lie. revision: yes
Referee: [§5] §5, derivation of the bivariate density bounds: the lower bounds for the density of transcendental pairs (n,m) are obtained directly by subtracting the algebraic upper bounds from the total count and invoking the author’s earlier density β(N); because the base infinite-transcendence result is not re-derived or independently benchmarked here, any gap in the global algebraic-count argument propagates immediately to the density statements.

Authors: The density lower bounds are obtained by subtracting the newly proved algebraic upper bounds (at most n−1 for each fixed n on the positive integers, at most n on each shifted ray) from the total cardinality of the finite sets under consideration and then invoking the earlier density β(N) only for the variable-n direction. In the revision we will insert a concise paragraph summarizing the key steps of the prior infinite-transcendence result that produces β(N), thereby making the present paper self-contained while preserving the natural dependence on the published predecessor. With the clarified global bounds in §§3–4, no gap propagates. revision: partial

Circularity Check

1 steps flagged

Self-citation on infinite transcendence occurrences is load-bearing for the new density lower bounds

specific steps

self citation load bearing [Abstract, first paragraph]
"In recent work, we showed that for all $q∈(1/2)Z∖Z≤0 the sequence {Γ^{(n)}(q)} _{n≥1} contains transcendental elements infinitely often, with the density of transcendental Γ^{(n)}(q) among n∈{1,2,…,N} bounded below by β(N)=max{0,√N−5/2}/N. For both fixed and variable n, we now study the transcendence of Γ^{(n)}(q) at both positive lattice points q=m∈{1,2,…} and rationally shifted lattice points q=~m∈{κ,±1+κ,±2+κ,…} (for κ∈(0,1)∩Q such that Γ(κ) is transcendental). For n∈Z≥2, we find there are at most n−1 algebraic Γ^{(n)}(m), and for n∈Z≥1, there are at most n algebraic Γ^{(n)}(~m) for each"

The upper bounds 'at most n-1 algebraic' and the subsequent density lower bounds are explicitly constructed from the cited prior result on infinite transcendence. The paper states that 'these results form the basis for constructing lower bounds for the densities', so the central claims depend on the self-citation without independent grounding supplied inside this manuscript.

full rationale

The paper opens by citing its own prior result on infinite transcendence (for fixed q, varying n) and then derives upper bounds on algebraic counts via the recurrence Gamma^{(n)}(m+1) = m Gamma^{(n)}(m) + n Gamma^{(n-1)}(m) together with the assumption that Gamma(kappa) is transcendental. These algebraic-count bounds are then used to construct the claimed density lower bounds. The recurrence step supplies independent content, but the overall density claims reduce to the self-cited foundation without external verification or machine-checked support in the present manuscript. This produces moderate circularity (score 4) while leaving the local recurrence argument non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on a prior result by the same author establishing infinite transcendence for the relevant q, plus the domain assumption that Gamma(kappa) is transcendental for the base rational kappa; no free parameters or new entities are introduced.

axioms (2)

domain assumption For q in (1/2)Z excluding non-positive integers, the sequence Gamma^(n)(q) contains transcendental elements infinitely often.
Invoked as the foundation from recent work on which the new counting and density bounds are built.
domain assumption Gamma(kappa) is transcendental for kappa in (0,1) cap Q.
Required to apply the shifted-lattice results.

pith-pipeline@v0.9.0 · 5756 in / 1420 out tokens · 36995 ms · 2026-05-16T06:14:19.873642+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1: for n>=2, # {m : Gamma^(n)(m) algebraic} <= n-1, proved via invertible Tn(M) and contradiction with transcendence of gamma or pi
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_add unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Identity (3): Gamma^(n)(m) = sum Tn,ell(m) Gamma^(ell)(1) with rational coefficients from elementary symmetric polynomials

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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Forward citations

Cited by 1 Pith paper

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

"Infinitely Often" Transcendence of Gamma-Function Derivatives
math.NT 2026-01 unverdicted novelty 7.0

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.