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From a Windowed Square Wave to the Hyperbolic Tangent, Even Zeta Values, and Dirichlet L-Values

This paper presents a unified elementary derivation of the Mittag-Leffler expansion for the hyperbolic tangent, Fourier series for Euler polynomials, and exact values for even zeta functions and specific Dirichlet L-functions by evaluating a windowed delayed periodic square wave in both time and frequency domains.

Original authors: Peter Bevelacqua

Published 2026-07-20

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Peter Bevelacqua

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

Imagine you are listening to a radio that can only hear two things: the music playing right now (the "time" domain) and the specific mix of frequencies that make up that music (the "frequency" domain). In the world of signal processing, these are just two different ways of looking at the same sound wave. If you know the total "amount" of sound in the time domain, you can calculate the exact same amount in the frequency domain, and vice versa. This is a bit like weighing a bag of apples: you can either count the apples one by one, or you can weigh the whole bag at once. If your scale is accurate, both methods should give you the same number.

Now, imagine you have a very specific, choppy sound wave—a "square wave" that jumps instantly between high and low, like a light switch flipping on and off. Mathematicians have long been fascinated by what happens when you break these waves down into their component frequencies. They've discovered that these waves hide some of the most beautiful and mysterious numbers in mathematics, like the sum of all fractions squared (which equals a specific multiple of $\pi$ ). The big question is: how can we find these hidden numbers using simple, everyday math without getting lost in complex, scary formulas?

This paper takes a clever, almost playful approach to solve that puzzle. The author, Peter J. Bevelacqua, decides to "window" a square wave. Think of a window not as a piece of glass, but as a spotlight that gets dimmer the further you get from the center. He takes his choppy square wave and multiplies it by a signal that fades away exponentially, like a sound dying out in a large empty room. This "window" does two magical things: it makes the total energy of the signal easy to calculate in the time domain, and it smooths out the messy math in the frequency domain so that the infinite sums of numbers actually settle down to a real answer.

By calculating the total "energy" of this windowed wave in two different ways—once by adding up the pieces in time, and once by adding up the frequencies—the author forces the two answers to be equal. When he sets them equal, a powerful new identity pops out. This single identity is a master key. By simply tweaking the settings of his "spotlight" (changing how fast it fades or where the wave is delayed), he unlocks a treasure chest of famous mathematical results.

First, he shows how this method naturally leads to the formula for the hyperbolic tangent function, a shape that looks like a stretched-out S-curve. Then, by turning off the fading effect, he derives the Fourier series for a "triangle wave," which helps explain the famous "Basel problem" (the sum of $1/n^2$ ). But the fun doesn't stop there. The author climbs an "integral ladder." This is a fancy way of saying he takes his result, integrates it (which is like finding the area under the curve), and repeats the process. Each time he climbs a rung, he discovers new values for famous mathematical constants.

By choosing specific "delays" for his wave, he uncovers the value of the Dirichlet beta function at 3, which equals $\pi^3/32$ . He also finds the values for other complex number series related to moduli 6 and 8. Finally, by continuing this ladder-climbing process, he proves that this single, unified method can generate the values for $\zeta(2k)$ (the sum of $1/n^{2k}$ ) for any even power $k$ . The paper doesn't just find these numbers; it shows that they all come from the same source, derived through simple integrals and limits, without needing the heavy machinery of complex analysis. It's a unified, elegant path from a simple windowed signal to some of the deepest constants in mathematics.

Technical Summary: From a Windowed Square Wave to the Hyperbolic Tangent, Even Zeta Values, and Dirichlet L-Values

Problem Statement
The paper addresses the derivation of classical mathematical constants and series expansions, specifically the Mittag-Leffler partial-fraction expansion for the hyperbolic tangent, the Fourier expansion of the first Euler polynomial, the Basel sum $\zeta(2)$ , and higher even zeta values $\zeta(2k)$ . While these results are well-known and possess numerous existing proofs (often relying on complex analysis, Parseval's theorem, or $L^2$ theory), the author seeks a unified, elementary derivation. The core challenge is to generate these diverse results from a single, coherent calculation without invoking advanced machinery.

Methodology
The central methodology involves analyzing a "windowed" signal in both the time and frequency domains and equating the results.

Signal Construction: The author defines a zero-mean periodic square wave, $s(t)$ , delayed by a parameter $d$ . This wave is multiplied by a decaying signum function, $\text{sgn}(a, t) = \text{sgn}(t)e^{-a|t|}$ , where $a > 0$ is a decay rate. This product, $p(a, d, t)$ , serves as the windowed signal.
Dual Evaluation: The total integral of this product (equivalent to the Fourier transform evaluated at zero frequency, $f=0$ $f = 0$ ) is computed in two ways:
- Frequency Domain: The Fourier transform of the square wave is combined with the transform of the decaying signum function. The windowing function is crucial here: it improves the decay of spectral terms from $1/n$ to $1/n^2$ , ensuring that the resulting series converges absolutely. This allows for rigorous term-by-term integration and the application of the Dominated Convergence Theorem without requiring $L^2$ theory.
- Time Domain: The integral is computed directly using elementary calculus over the periodic intervals of the square wave, utilizing the odd symmetry of the wave and geometric series summation.
Equivalence and Specialization: Equating the time-domain and frequency-domain results yields a two-parameter identity (Theorem 1) involving a cosine series over odd integers. By specializing the parameters $a$ (decay rate) and $d$ (delay), and by performing termwise integration, the author derives subsequent results.

Key Contributions and Results
The paper establishes a "ladder" of results derived from a single identity:

Theorem 1: A two-parameter cosine identity over odd integers, $\sum \frac{\cos(2\pi nd)}{n^2 + 4z^2/\pi^2} = \frac{\sinh(z(1-4d))}{\cosh z}$ , where $z = a/4$ .
Corollary 2 (Hyperbolic Tangent): By setting the delay $d=0$ , the author recovers the Mittag-Leffler partial-fraction expansion for $\tanh(z)$ without complex analysis.
Corollary 3 (Euler Polynomial & Basel Sum): By sending the decay rate $a \to 0$ (removing the window), the identity yields the Fourier expansion of the first Euler polynomial $E_1(2d)$ . Setting $d=0$ in this limit recovers the Basel sum $\sum_{n \text{ odd}} 1/n^2 = \pi^2/8$ , which directly implies $\zeta(2) = \pi^2/6$ .
Climbing the Integral Ladder: The author demonstrates that termwise integration of the cosine series produces a sine series with an additional $1/n$ $1/ n$ factor.
- Corollary 4 & 5: Integrating Theorem 1 with respect to $d$ yields a sine series. Taking the limit $z \to 0$ produces a closed-form expression for $\sum \frac{\sin(2\pi nd)}{n^3}$ .
- Corollary 6 (Dirichlet L-values): Evaluating the sine series at specific delays ( $d=1/4, 1/6, 1/8$ ) yields the Dirichlet beta value $\beta(3) = \pi^3/32$ and values for Dirichlet L-functions associated with odd characters of modulus 6 and 8.
- Corollary 7 & Theorem 8 (Higher Zeta Values): Further integration returns to a cosine series. Setting $d=0$ in the fourth-order cosine series yields $\sum_{n \text{ odd}} 1/n^4 = \pi^4/96$ , leading to $\zeta(4) = \pi^4/90$ . The paper concludes with an inductive proof (Theorem 8) showing this process extends to $\zeta(2k)$ for all $k \ge 1$ , defined via a recursive polynomial sequence $Q_m(d)$ .

Significance and Claims
The paper claims that while every individual result obtained is well-known, the route taken is novel. The significance lies in the unification of these disparate results under a single "windowed signal" framework.

Elementary Nature: The derivation relies solely on elementary integrals, classical pointwise convergence, uniform boundedness of partial sums, and Maclaurin expansions. It explicitly avoids Parseval's theorem and the $L^2$ theory of Fourier series.
Convergence: The decaying signum window is highlighted as a critical technical device that ensures absolute convergence of all series involved, thereby justifying the interchange of limits and integrals rigorously.
Limitations: The author notes modestly that while this method generates $\beta(3)$ and other L-values at odd integers, it cannot generate $\zeta(3)$ (the Apéry constant). This is because the method produces sine series with coefficients that alternate or depend on $d$ , whereas $\zeta(3)$ requires a sum of all positive coefficients ( $+1$ ), which cannot be achieved with the $\sin(2\pi nd)$ term for any real $d$ .

In summary, the paper presents a unified, elementary, and rigorous pathway from a windowed square wave to the fundamental values of the Riemann zeta function at even integers and specific Dirichlet L-values, demonstrating that these results are different facets of a single underlying identity.

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