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Recurrence Relations for $\beta(2k)$ and $\zeta(2k + 1)$

This paper establishes that specific integrals involving hyperbolic functions can be expressed as sums of Hurwitz zeta derivatives, leading to new recurrence relations for the Dirichlet beta function $\beta(2k)$ and the Riemann zeta function $\zeta(2k+1)$ , as well as simplified digamma-based representations for their derivatives.

Original authors: Tobias Kyrion

Published 2026-07-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Tobias Kyrion

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Hidden Rhythm of Numbers

Imagine the universe of mathematics as a giant, cosmic orchestra. In this orchestra, some instruments play simple, predictable notes that we can write down easily, like the counting numbers or the famous ratio of a circle's circumference to its diameter. But then there are the "ghost notes"—complex, mysterious values that appear in the deepest calculations of physics and number theory. Two of the most famous ghost notes are the Riemann zeta function and the Dirichlet beta function. Think of these as secret codes that describe how numbers are distributed, much like how a weather map predicts storms. For centuries, mathematicians have known how to calculate these codes for even numbers (like 2, 4, 6), but the odd numbers (like 3, 5, 7) have remained stubbornly silent, refusing to reveal their simple patterns.

This paper dives into the "even" and "odd" sections of this musical score to see if we can find a new way to translate these ghost notes into something we can actually hear. The author, Tobias Kyrion, uses a special kind of mathematical tool called an "integral." You can think of an integral as a machine that takes a wiggly, complicated curve and smoothes it out into a single, meaningful number. The specific curves in this story involve hyperbolic functions (mathematical shapes that look like hanging chains or cooling towers) mixed with exponential decay (like a sound fading away). The goal is to see if running these curves through the machine reveals a hidden rhythm that connects the mysterious odd-number codes to the known even-number ones.

The Paper's Discovery: A New Recipe for Ghost Notes

In this work, Tobias Kyrion sets out to solve a specific puzzle: how to calculate certain integrals that involve the hyperbolic tangent and hyperbolic secant functions. These aren't just random shapes; they are the building blocks for understanding the behavior of the zeta and beta functions. The paper proves that these integrals can be broken down into a sum of simpler parts involving "derivatives of the Hurwitz zeta function." If that sounds like gibberish, imagine it as a recipe. Instead of trying to bake a complex, multi-layered cake from scratch, the author shows you that the cake is actually just a stack of simpler, pre-made layers (the derivatives) held together by a specific mathematical glue (polynomial coefficients).

The main finding is a set of "recurrence relations." In the world of math, a recurrence relation is like a domino effect or a family tree. If you know the value of one number in the sequence, you can use a specific rule to calculate the next one. Kyrion discovers that by evaluating these integrals for specific integer values, we can generate rules that link the values of $\beta(2k)$ (the Dirichlet beta function for even numbers) and $\zeta(2k + 1)$ (the Riemann zeta function for odd numbers) to each other. Essentially, the paper provides a new, rigorous method to calculate these elusive odd-number values by treating them as the result of a complex integration process.

The author doesn't just stop at the general formula. They show that when you plug in specific numbers (like $T = 2l$ where $l$ is an integer), the messy integrals simplify into expressions involving the "digamma function" (a cousin of the factorial function) and polynomials. This is a significant simplification, turning a difficult calculus problem into a more manageable algebraic one. The paper explicitly demonstrates this for both even and odd powers of the hyperbolic secant function, providing two distinct formulas (Theorem 1.1 and Corollary 1.2) that act as a bridge between the abstract world of infinite series and concrete calculations.

What the Paper Does and Does Not Claim

It is important to understand what this paper achieves and where it draws the line. The author has proved that these integrals can be represented as sums of Hurwitz zeta derivatives. This is a solid mathematical fact, not a guess or a simulation. The paper provides a step-by-step logical derivation, using tools like partial integration and matrix inversion (a method of solving systems of equations) to show exactly how the pieces fit together.

However, the paper also explicitly rules out the idea that these new recurrence relations immediately solve the "mystery" of the algebraic nature of these constants. While the paper gives us a new way to calculate the values, it admits that the fundamental question of whether these numbers (like $\zeta(3)$ or $\beta(2)$ ) can be written as simple combinations of $\pi$ or other basic constants remains unsolved. The author notes that previous attempts to find such simple evaluations using different identities (like those involving double zeta functions or specific integrals of cotangent) have also failed to produce simple answers. The new relations derived here are just as "mysterious" in their final form as the old ones; they are a new way to compute the numbers, but they do not magically reveal a simple, closed-form answer for them.

The paper also clarifies that while there are simpler representations for the negative T-derivative of these integrals (which only involve the digamma function), the original integrals with the $1/x$ term are more complex and require the full machinery of Hurwitz zeta derivatives. The author suggests that while their method using matrix inversions is much more efficient than previous strategies for small values, the calculations still get "increasingly tedious" for larger values.

The Bigger Picture: Why It Matters

Why should a curious teenager care about these integrals? Because mathematics is often about finding the shortest path between two distant islands. For a long time, the islands of $\zeta(2k+1)$ and $\beta(2k)$ were separated by a foggy sea of complex analysis. This paper builds a sturdy bridge. It shows that these seemingly unrelated numbers are actually connected through the geometry of hyperbolic curves.

The paper also touches on a fascinating connection to the work of the legendary mathematician Srinivasa Ramanujan. The author mentions that the starting point for some of these proofs was inspired by Ramanujan's famous formulas involving hyperbolic cotangents. By revisiting these old ideas with modern tools, the paper confirms that the "ghost notes" of the universe have a structure we can map, even if we can't yet fully decode their simplest form.

In the end, this work is a testament to the power of persistence. It doesn't claim to have found the "Holy Grail" of number theory (a simple formula for all odd zeta values), but it has handed us a better map. It shows us that if we look at the problem through the lens of these specific integrals, the path forward becomes clearer, even if the destination is still shrouded in a bit of mathematical fog. The paper concludes by suggesting that these methods could be applied to even more general integrals, hinting that the journey of discovery is far from over.

Technical Summary: Recurrence Relations for $\beta(2k)$ and $\zeta(2k + 1)$

Problem Statement
This work investigates a specific class of definite integrals involving hyperbolic functions and exponential decay:
$I_{N, L, T} = \int_0^\infty \frac{\tanh(x)}{x} \text{sech}(x)^L e^{-Tx} \, dx$
where $L \in \mathbb{N}$ and $T \in \mathbb{R}_{\ge 0}$ . The primary objective is to evaluate these integrals for specific integer parameters ( $T=2l$ ) to derive recurrence relations for the Dirichlet beta function $\beta(2k)$ and the Riemann zeta function $\zeta(2k+1)$ . A secondary goal is to analyze the $T$ -derivative of these integrals, which removes the $1/x$ factor, to find simpler representations involving only the digamma function.

Methodology
The paper employs a multi-step analytical approach combining integral representations, partial integration, and linear algebra:

Hurwitz Zeta Integral Representation: The derivation begins with the standard integral representation of the difference of Hurwitz zeta functions:
$\zeta\left(s, \frac{T+1}{4}\right) - \zeta\left(s, \frac{T+3}{4}\right) = \frac{2^{2s-1}}{\Gamma(s)} \int_0^\infty x^{s-1} \text{sech}(x) e^{-Tx} \, dx$
By applying partial integration $N$ times, the authors transform the integral into a form involving derivatives of $\text{sech}(x)$ and $\tanh(x)$ .
Linear Systems and Matrix Inversion:
- The derivatives $\frac{\partial^N}{\partial x^N} \text{sech}(x)$ and $\frac{\partial^N}{\partial x^N} \tanh(x)$ are expressed as finite sums of terms involving $\text{sech}(x)^k$ and $\tanh(x)$ .
- This creates a linear system relating the target integrals (with $\frac{\tanh(x)}{x}$ ) to the Hurwitz zeta differences.
- The authors define specific lower-triangular matrices ( $u_{N,k}$ and $x_{N,k}$ ) based on the coefficients of these derivative expansions. They prove that these matrices are invertible and explicitly construct their inverses ( $v_{N,k}$ and $y_{N,k}$ ).
- The inverse matrices involve recursively defined coefficients $g_{N,k}$ and $h_{N,k}$ , which are central to the final recurrence formulas.
Differentiation and Evaluation:
- The linear systems are differentiated with respect to the complex parameter $s$ and evaluated at non-positive integers (e.g., $s = -2N$ ).
- This process utilizes the property $\lim_{z \to 0} \frac{\psi(z)}{\Gamma(z)} = -1$ to handle singularities, converting Hurwitz zeta derivatives $\zeta'(-n, z)$ into values of $\beta$ and $\zeta$ functions.
- For the case $T=2l$ , the authors utilize elementary relations between Hurwitz zeta differences and the Dirichlet beta/Riemann zeta functions to isolate the specific values $\beta(2k)$ and $\zeta(2k+1)$ .
Alternative Approach for Derivatives:
- For the integrals without the $1/x$ factor (the $T$ -derivative of the original form), the authors derive a two-step recurrence relation for $\int_0^\infty \text{sech}(x)^L e^{-Tx} dx$ . This allows the evaluation of the target integrals directly in terms of the digamma function $\psi(z)$ , bypassing the need for Hurwitz zeta derivatives.

Key Contributions and Results

General Integral Formulas (Theorem 1.1): The paper provides explicit closed-form expressions for $\int_0^\infty \frac{\tanh(x)}{x} \text{sech}(x)^{2N+1} e^{-Tx} dx$ and $\int_0^\infty \frac{\tanh(x)}{x} \text{sech}(x)^{2N} e^{-Tx} dx$ . These are expressed as finite sums involving Hurwitz zeta derivatives $\zeta'(-n, z)$ with polynomial coefficients in $T$ and the recursively defined constants $g_{N,k}$ and $h_{N,k}$ .
Recurrence Relations for $\beta(2k)$ and $\zeta(2k+1)$ (Corollary 1.2 & 1.3): By setting $T=2l$ $T = 2 l$ (specifically $l=0$ $l = 0$ ), the authors derive recurrence relations that express these integrals as linear combinations of $\beta(2k)$ $β (2 k)$ and $\zeta(2k+1)$ $ζ (2 k + 1)$ .
- For $T=0$ , the results simplify to:
  $\int_0^\infty \frac{\tanh(x)}{x} \text{sech}(x)^{2N+1} dx = \binom{2N}{N} \frac{1}{(2N+1)2^{2N}} \sum_{k=0}^N g_{N+1, k+1} 2^{2k+2} (2k+1)! \frac{\beta(2k+2)}{\pi^{2k+1}}$
  $\int_0^\infty \frac{\tanh(x)}{x} \text{sech}(x)^{2N} dx = \frac{2^{2N}}{N^2 \binom{2N}{N}} \sum_{k=1}^N h_{N,k} (2-2^{-2k}) (2k)! \frac{\zeta(2k+1)}{\pi^{2k}}$
Simplified Representations (Theorem 2.1): The paper establishes that the integrals $\int_0^\infty \tanh(x) \text{sech}(x)^L e^{-Tx} dx$ (the $T$ -derivative of the main subject) can be represented solely using the digamma function $\psi(z)$ and rational functions of $T$ , without involving Hurwitz zeta derivatives.
Integrals of Powers of $\tanh(x)/x$ (Section 3): The authors express $\int_0^\infty (\frac{\tanh(x)}{x})^N dx$ as a linear combination of $\zeta(3), \dots, \zeta(2N-1)$ using a recursive coefficient matrix $d_N(p, 2k)$ .
Integral Representations: The work generalizes known integral representations for $\beta(2)$ and $\zeta(3)$ (found by Blagouchine) to higher orders $\beta(2N)$ and $\zeta(2N+1)$ involving $\log(\log(x))$ .

Significance and Claims
The paper claims to provide a systematic method for generating recurrence relations for odd zeta values and even beta values using hyperbolic integrals. The author notes that while previous literature (e.g., Euler, Ramanujan) has established relations involving "error terms" like double zeta functions or integrals of $z^{2m}\cot(z)$ , the relations derived here are algebraic in nature, though the constants themselves remain mysterious.

The author explicitly states that the method based on matrix inversions presented in the proof of Theorem 1.1 is "much more efficient" than alternative direct proof strategies (such as repeated differentiation of known integral identities) for larger values of $L$ . The work does not claim to solve the irrationality or transcendence of these constants but rather offers a new structural framework for relating them through definite integrals.

Limitations and Outlook
The paper acknowledges that the algebraic nature of the constants $\zeta(2k+1)$ and $\beta(2k)$ remains a mystery despite these recurrence relations. The author suggests that the matrix inversion method could potentially be extended to more general integrals of the form $\int_0^\infty \frac{\tanh(\alpha x)}{x} \text{sech}(\beta x)^L e^{-Tx} dx$ , though this is presented as an outlook rather than a completed result.

The Hidden Rhythm of Numbers

The Paper's Discovery: A New Recipe for Ghost Notes

What the Paper Does and Does Not Claim

The Bigger Picture: Why It Matters

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