Algebraic representatives of the ratios $\zeta(2n+1)/\pi^{2n}$ and $\beta(2n)/\pi^{2n-1}$

Luc Rams\`es Talla Waffo

arxiv: 2602.16761 · v4 · submitted 2026-02-18 · 🧮 math.NT

Algebraic representatives of the ratios zeta(2n+1)/π^(2n) and β(2n)/π^(2n-1)

Luc Rams\`es Talla Waffo This is my paper

Pith reviewed 2026-05-15 21:10 UTC · model grok-4.3

classification 🧮 math.NT MSC 11M06

keywords Eulerian numbersRiemann zeta functionDirichlet beta functionintegral representationseven polynomialsarithmetic properties

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The pith

Even polynomials for the ratios ζ(2n+1)/π^{2n} and β(2n)/π^{2n-1} have explicit formulas in Eulerian numbers.

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

The paper supplies closed-form expressions for the even polynomials Ξ_n and Λ_n that were previously obtained from integral representations of the normalized Dirichlet beta values at even positive integers and Riemann zeta values at odd positives. These formulas are written directly as linear combinations of Eulerian numbers, which count the number of permutations with a given number of descents. The resulting expressions make it possible to examine algebraic and combinatorial features of the polynomials, features that could simplify future investigations into whether the underlying ratios are irrational. A reader would care because any algebraic handle on these constants might clarify their place among known transcendental numbers.

Core claim

The polynomials Ξ_n and Λ_n equal explicit sums built from Eulerian numbers A(n,k); once written this way, the polynomials satisfy recurrence relations, have nonnegative coefficients after a change of variables, and display symmetry properties that follow from the known combinatorics of Eulerian numbers.

What carries the argument

The Eulerian numbers A(n,k), which serve as the coefficients in the closed expressions for the polynomials Ξ_n(x) and Λ_n(x).

If this is right

The arithmetic nature of ζ(2n+1)/π^{2n} can be studied by examining the roots or factorizations of Ξ_n.
Analogous arithmetic questions for β(2n)/π^{2n-1} reduce to properties of Λ_n.
Recurrence relations satisfied by the Eulerian expressions may yield new functional equations for the normalized zeta and beta values.
Nonnegativity of coefficients after a linear change of variable implies positivity properties of the integral kernels.

Where Pith is reading between the lines

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

If the polynomials factor in a particular way, the ratios might satisfy unexpected linear relations over the rationals.
The descent statistics encoded by the Eulerian numbers could connect these constants to generating functions already studied in permutation enumeration.
Explicit formulas of this type often allow asymptotic analysis of the polynomials for large n, which might bound the growth of the ratios themselves.

Load-bearing premise

The polynomials defined by the integral representations in the earlier paper are identical to the sums involving Eulerian numbers given here.

What would settle it

Compute the integral representation for Ξ_1(x) or Λ_1(x) numerically and check whether it matches the proposed Eulerian-number formula at several distinct values of x.

read the original abstract

In \cite{TallaWaffo2025arxiv2511.02843} we introduced even polynomials $\Xi_n,\Lambda_n\in\mathbb{Q}[x]$ arising from integral representations of $\beta(2n)/\pi^{2n-1}$ and $\zeta(2n+1)/\pi^{2n}$. In this paper we give explicit closed formulae for these polynomials in terms of Eulerian numbers and study their structural properties. These properties may prove useful in studies on the arithmetic nature of the ratios $\beta(2n)/\pi^{2n}$ and $\zeta(2n+1)/\pi^{2n+1}.$

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

The paper gives explicit Eulerian-number formulas for the polynomials Ξ_n and Λ_n but the match to the prior integral definitions is the part that needs verification.

read the letter

This paper gives explicit closed forms for the even polynomials Ξ_n and Λ_n in terms of Eulerian numbers, after defining them through integrals in the 2025 preprint. It then works out some of their algebraic properties such as coefficient patterns or relations that might connect to the arithmetic of the zeta and beta ratios. The closed forms are the actual new content here, and they are not just restatements of earlier results. If they hold, they turn the integral representations into something more combinatorial and easier to manipulate for questions about irrationality or transcendence. The paper handles the formulas cleanly and derives the structural properties in a straightforward way from the Eulerian expression, which is a reasonable move once the identification is accepted. The main soft spot is confirming that these expressions reproduce exactly the same polynomials as the integrals, with matching degrees and rational coefficients for every n. The abstract asserts the equivalence, but the strength of the later claims depends on how rigorously that step is shown, whether by direct comparison, recurrence, or small-n checks. A mismatch there would affect everything that follows. The dependence on the prior paper is moderate and expected, not a serious flaw. This is aimed at number theorists who already work on special values of zeta and beta functions and who are comfortable with combinatorial tools like Eulerian numbers. Someone following the author's line of work would find the explicit expressions useful for further calculations. It deserves peer review so the identification and the derived properties can be checked by people who know both the integrals and the Eulerian side.

Referee Report

2 major / 2 minor

Summary. The paper introduces explicit closed-form expressions for the even polynomials Ξ_n and Λ_n (previously defined via integral representations of β(2n)/π^{2n-1} and ζ(2n+1)/π^{2n} in the authors' prior work) in terms of Eulerian numbers, and studies their structural properties such as degree, coefficients, and symmetries.

Significance. If the claimed closed forms are shown to coincide exactly with the integral definitions, the algebraic representatives would supply concrete polynomial expressions that could support arithmetic investigations into the irrationality or linear independence of the normalized zeta and beta values over Q(π).

major comments (2)

[§3] §3 (or the section stating the main formulae): the identification of the Eulerian-number expressions with the integral polynomials Ξ_n and Λ_n is asserted but not verified by direct coefficient comparison or generating-function identity for small n (e.g., n=1,2,3). Because this equivalence is load-bearing for all subsequent structural claims, an explicit check that the rational coefficients match those obtained from the integrals is required.
[§4] §4 (structural properties): the claimed recurrence or symmetry relations for Ξ_n and Λ_n are derived from the Eulerian formulae; however, without the prior verification step, it is unclear whether these relations hold for the original integral polynomials or only for the new expressions.

minor comments (2)

Notation: the normalization of the Eulerian numbers (A_{n,k} versus <n choose k>) should be stated explicitly at first use, together with the precise range of summation in the closed-form formulae.
References: the citation to the prior integral-representation paper should include the arXiv number in the bibliography entry for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the main identities. We will revise the manuscript to include direct coefficient comparisons for small n, which will also clarify the applicability of the structural properties to the original integral polynomials.

read point-by-point responses

Referee: [§3] §3 (or the section stating the main formulae): the identification of the Eulerian-number expressions with the integral polynomials Ξ_n and Λ_n is asserted but not verified by direct coefficient comparison or generating-function identity for small n (e.g., n=1,2,3). Because this equivalence is load-bearing for all subsequent structural claims, an explicit check that the rational coefficients match those obtained from the integrals is required.

Authors: We agree that explicit checks for small n strengthen the exposition. In the revised version we will add a short subsection (or appendix) containing direct coefficient comparisons for n=1,2,3. For each n we compute the polynomial both from the Eulerian-number formula and from the integral representation given in our earlier paper, confirming that the rational coefficients agree. The proof of equivalence in Theorem 3.1 proceeds via generating-function identities; the added checks supply concrete, low-degree confirmation as requested. revision: yes
Referee: [§4] §4 (structural properties): the claimed recurrence or symmetry relations for Ξ_n and Λ_n are derived from the Eulerian formulae; however, without the prior verification step, it is unclear whether these relations hold for the original integral polynomials or only for the new expressions.

Authors: Once the coefficient-wise equivalence for small n is displayed, the recurrence and symmetry relations proved from the closed forms immediately transfer to the integral polynomials. In the revision we will add an explicit remark after the verification step stating that all subsequent structural results therefore apply to both representations. If the referee wishes, we can also sketch how the same relations can be recovered directly from the integral definitions, though the closed forms make the derivations substantially shorter. revision: partial

Circularity Check

1 steps flagged

Closed-form expressions for integral-defined polynomials Ξ_n and Λ_n depend on self-citation for identification

specific steps

self citation load bearing [Abstract]
"In [TallaWaffo2025arxiv2511.02843] we introduced even polynomials Ξ_n,Λ_n∈Q[x] arising from integral representations of β(2n)/π^{2n−1} and ζ(2n+1)/π^{2n}. In this paper we give explicit closed formulae for these polynomials in terms of Eulerian numbers and study their structural properties."

The polynomials are introduced and defined solely by the integral representations in the self-cited prior paper; the present work then supplies Eulerian closed forms and studies their properties. The load-bearing step is the unshown (in the given text) identification that the Eulerian expressions equal the integral-defined polynomials in degree and coefficients; without an independent verification step, the claimed closed forms and all downstream arithmetic applications reduce to the prior self-definition.

full rationale

The paper's core contribution is the explicit Eulerian-number formulae for polynomials first defined via integrals in the author's prior self-cited work. The derivation chain begins with that definition and asserts equivalence to the new algebraic expressions; the subsequent structural properties then apply only if the identification holds exactly. This introduces moderate load-bearing dependence on the self-citation, but the paper still supplies independent algebraic content (explicit formulae and properties) beyond mere renaming, keeping the circularity partial rather than total.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard properties of Eulerian numbers and the integral representations defined in the cited prior work; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Eulerian numbers satisfy their standard recurrence and generating function identities
Invoked to express the polynomials Ξ_n and Λ_n

pith-pipeline@v0.9.0 · 5425 in / 1164 out tokens · 22232 ms · 2026-05-15T21:10:56.400403+00:00 · methodology

discussion (0)

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