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

This paper provides explicit closed formulae for the even polynomials $\Xi_n$ and $\Lambda_n$, which represent the ratios $\beta(2n)/\pi^{2n-1}$ and $\zeta(2n+1)/\pi^{2n}$, by expressing them in terms of Eulerian numbers and analyzing their structural properties to aid future investigations into the arithmetic nature of these ratios.

The Gist

Imagine you are a detective trying to solve a mystery about numbers. Specifically, you are looking at two famous, stubborn families of numbers: the Riemann Zeta function (which sums up powers of all integers) and Dirichlet's Beta function (which sums up powers of odd integers with alternating signs).

For a long time, mathematicians knew the "even" members of these families (like $2, 4, 6...$) were very well-behaved. They could be written down easily using a famous number,$\pi$ (the ratio of a circle's circumference to its diameter), and some other standard constants. It was like having a perfect recipe for a cake.

However, the "odd" members (like $3, 5, 7...$) were the mystery. No one knew a simple recipe for them. We didn't even know if they were rational numbers (fractions) or irrational (like$\pi$or$\sqrt{2}$). This has been a huge puzzle in mathematics for centuries.

The Detective's New Tool: The "Shape-Shifting" Polynomials

In this paper, the author, Luc Ramsès Talla Waffo, introduces a new set of tools to help solve this mystery. Think of these tools as specialized, shape-shifting molds called polynomials (specifically named $\Xi_n$ and $\Lambda_n$).

Here is the simple breakdown of what the paper does:

1. The Problem: The "Hidden Recipe"

Imagine you have a delicious, complex soup (the value of $\zeta(2n+1)$ or $\beta(2n)$). You know it contains $\pi$, but you don't know the exact ratio.

• The Old Way: Trying to taste the soup directly is hard.
• The New Way: The author says, "Let's pour this soup into a special mold." If we can describe the shape of the mold perfectly, we might be able to figure out exactly what's inside the soup.

These "molds" are the polynomials $\Xi_n$ and $\Lambda_n$. The paper proves that if you integrate (sum up) these shapes in a specific way, you get the exact values of those mysterious odd-number ratios.

2. The Ingredients: Eulerian Numbers

To build these molds, the author uses a specific set of mathematical "bricks" called Eulerian numbers.

• Analogy: Imagine you are building a tower. You could use random rocks, but these Eulerian numbers are like Lego bricks that snap together in a very specific, predictable pattern.
• The paper gives a "blueprint" (a closed formula) showing exactly how to stack these Lego bricks to build the mold for any size $n$. Before this, we only knew how to build the small molds; now we have the instructions for the giant ones.

3. The Shape of the Molds: The "Real Root" Secret

Once the molds are built, the author studies their shape. He discovers some amazing properties:

• They are "Real-Rooted": Imagine the mold is a wave. The points where the wave touches the ground (the "zeros") are all real, physical points. They aren't imaginary or floating in a void.
• They are "Interlaced": If you take the mold for size $n$ and the mold for size $n+1$, their waves weave together like the teeth of two combs or the fingers of two hands clasping. They never cross in a messy way; they are perfectly ordered.
• They stay within bounds: All the "touching points" of these waves stay safely inside a specific box (between -1 and 1). They don't run off to infinity.

4. Why Does This Matter? (The "Irrationality" Hunt)

Why do we care about these shapes?

• The Goal: We want to prove that the odd zeta and beta values are irrational (they can't be written as simple fractions).
• The Strategy: The author suggests that if we understand the "shape" of these molds perfectly, we can use them to trap the values of the odd numbers.
• The Analogy: Imagine trying to prove a ghost exists. You can't see the ghost, but you can see the footprints it leaves in the sand. These polynomials are the footprints. By analyzing the footprints (the zeros and the shape of the polynomials), we get closer to proving the ghost (the irrationality of the numbers) is real.

5. The "Vanishing" Act

The paper also shows that as the molds get bigger and bigger (as $n$ goes to infinity), they start to flatten out and disappear, getting closer and closer to zero. This is a crucial clue. It tells us that the "footprints" are getting smaller and more precise, which helps in mathematical proofs about how these numbers behave.

Summary in a Nutshell

Think of this paper as an architect's manual for a new type of building.

1. The Building: A set of mathematical shapes (polynomials) that represent the most mysterious numbers in math.
2. The Blueprint: The author finally wrote down the exact instructions for how to build these shapes using "Lego bricks" (Eulerian numbers).
3. The Inspection: The author inspected the buildings and found they are perfectly stable, ordered, and fit within a specific area.
4. The Future: Now that we have the blueprints and know the buildings are stable, we can use them to finally solve the century-old mystery of whether these specific numbers are "fraction-friendly" or "mysteriously irrational."

This work doesn't solve the mystery yet, but it provides the most detailed map and the best tools we have ever had to try and solve it.

Technical Summary

Here is a detailed technical summary of the paper "Algebraic representatives of the ratios $\zeta(2n + 1)/\pi^{2n}$ and $\beta(2n)/\pi^{2n-1}$" by Luc Ramsès Talla Waffo.

1. Problem Statement and Motivation

The paper addresses a fundamental asymmetry in analytic number theory regarding the special values of the Riemann zeta function $\zeta(s)$ and the Dirichlet beta function $\beta(s)$:

• Even Zeta Values: Euler established a closed formula for $\zeta(2n)$ in terms of Bernoulli numbers and powers of $\pi$.
• Odd Zeta Values: No analogous closed formula is known for $\zeta(2n+1)$. The arithmetic nature (rationality/irrationality) of the normalized ratios $\zeta(2n+1)/\pi^{2n+1}$ remains a deep, unresolved problem.
• Dirichlet Beta Function: Conversely, $\beta(s)$ has explicit evaluations at odd integers (involving Euler numbers) but remains mysterious at even integers $\beta(2n)$.

The Core Problem: The author investigates the arithmetic nature of the normalized ratios:
$$\frac{\zeta(2n+1)}{\pi^{2n}} \quad \text{and} \quad \frac{\beta(2n)}{\pi^{2n-1}}$$
Building on previous work [11], the paper focuses on even polynomials $\Xi_n(x)$ and $\Lambda_n(x)$ with rational coefficients that arise from integral representations of these ratios. The goal is to derive explicit closed forms for these polynomials and analyze their structural properties to potentially aid in proving the irrationality of the aforementioned ratios.

2. Methodology

The paper employs a synthesis of analytic number theory, combinatorics, and the theory of real-rooted polynomials.

• Integral Representations: The study starts from Malmsten-type integrals involving polylogarithms of negative order ($\text{Li}_{-m}$). By performing integration by parts and variable substitutions (specifically $x = \tanh u$ and $x = \frac{1-z^2}{1+z^2}$), the author transforms these integrals into forms involving the polynomials $\Xi_n$ and $\Lambda_n$.
• Combinatorial Identities: The coefficients of these polynomials are expressed using Eulerian numbers of Type A ($\left\langle \begin{smallmatrix} n \\ k \end{smallmatrix} \right\rangle$) and Type B ($\left\langle \begin{smallmatrix} n \\ k \end{smallmatrix} \right\rangle_B$). The author utilizes generating functions and Worpitzky-type identities to link negative-index polylogarithms to these combinatorial numbers.
• Polynomial Analysis: The paper treats the sequences of polynomials $(\Xi_n)_n$ and $(\Lambda_n)_n$ as objects of study in their own right. It applies techniques from the theory of real-rooted polynomials, including:
  • Factorization and root location analysis.
  • Interlacing properties of zeros.
  • Log-concavity and unimodality of coefficient sequences (via Newton's inequalities).
  • Asymptotic analysis of zeros using endpoint ratios and Stirling's approximation.

3. Key Contributions and Results

A. Explicit Closed-Form Expressions

The author derives explicit formulas for the coefficients of $\Xi_n(x)$ and $\Lambda_n(x)$.

• Coefficients: The coefficients $C_{n,t}^{(B)}$ and $C_{n,t}^{(A)}$ are given as finite sums involving Eulerian numbers and binomial coefficients.
• Polynomial Forms: The polynomials are expressed in terms of Eulerian polynomials $A_n(z)$ and $B_n(z)$ via a Möbius transformation:
  $$\Xi_n(x) \propto \frac{(1+x)^{2n-1}}{x} B_{2n-1}\left(\frac{-1-x}{1+x}\right), \quad \Lambda_n(x) \propto \frac{(1+x)^{2n-1}}{x} A_{2n}\left(\frac{-1-x}{1+x}\right)$$
  This reveals a direct bridge between the analytic ratios and classical combinatorial polynomials.

B. Structural Properties of the Polynomials

The paper establishes several rigorous properties for $n \geq 2$:

1. Degree and Leading Coefficients: Both polynomials are even and have degree exactly $2n-2$. The leading coefficients are explicitly calculated.
2. Real-Rootedness: All zeros of $\Xi_n(x)$ and $\Lambda_n(x)$ are real, simple, and lie strictly within the interval $(-1, 1)$.
3. Interlacing: The zeros of consecutive polynomials ($\Xi_n$ and $\Xi_{n+1}$, or $\Lambda_n$ and $\Lambda_{n+1}$) interlace.
4. Sign and Log-Concavity:
   • The polynomials do not vanish for $|x| > 1$.
   • The sequences of absolute values of their coefficients are log-concave and therefore unimodal.
5. Boundary Values: Explicit formulas are provided for $\Xi_n(0)$, $\Lambda_n(0)$, $\Xi_n(1)$, and $\Lambda_n(1)$, relating them to Bernoulli and Euler numbers.

C. Asymptotic Behavior

• Uniform Convergence: As $n \to \infty$, both $\Xi_n(x)$ and $\Lambda_n(x)$ converge uniformly to 0 on the interval $(0, 1)$.
• Zero Distribution:
  • The largest zeros of the associated polynomials (in terms of $x^2$) converge to 1.
  • The smallest positive zeros converge to 0.
  • This implies the zeros become dense in the interval $(0, 1)$ as $n$ increases.

D. Integral Identities

The paper provides exact evaluations for the integrals of these polynomials against the weight $\frac{1}{\sqrt{1-x^2}}$:
$$\int_0^1 \frac{\Xi_n(x)}{\sqrt{1-x^2}} dx = \pi \cdot (\text{rational expression involving } B_{2n})$$
$$\int_0^1 \frac{\Lambda_n(x)}{\sqrt{1-x^2}} dx = \pi \cdot (\text{rational expression involving } E_{2n})$$
Crucially, these integrals are shown to be strictly positive for all $n \geq 1$.

4. Significance and Implications

• New Perspective on Irrationality: The paper suggests a novel approach to the irrationality of $\zeta(2n+1)/\pi^{2n+1}$ and $\beta(2n)/\pi^{2n}$. By establishing that the polynomials converge uniformly to zero and possess specific positivity properties, the author proposes that these polynomials could serve as test functions in a "robust approach" (similar to Apéry's method) to prove that the ratios cannot be rational. If the ratios were rational, specific integral identities would imply a contradiction with the asymptotic behavior of these polynomials.
• Combinatorial-Number Theoretic Bridge: The work uncovers a deep connection between the special values of Dirichlet series and the theory of real-rooted polynomials generated by Eulerian numbers. It demonstrates that the "mystery" of odd zeta values and even beta values is encoded in the algebraic structure of these specific polynomial families.
• Unification: It unifies the treatment of $\zeta$ and $\beta$ functions through a single analytic framework involving polylogarithms and hyperbolic kernels, providing a unified algebraic representative for both cases.

In summary, this paper moves beyond the existence of these polynomials (established in [11]) to provide a complete algebraic and analytic characterization. It transforms the problem of evaluating special values into a study of the zeros and coefficients of real-rooted polynomials, offering new tools for attacking long-standing problems in the arithmetic nature of $\pi$ and zeta values.