⚛️ high-energy theory

Polynomials and asymptotic constants in a resurgent problem from 't Hooft

This paper resolves Gerard 't Hooft's problem of analytically continuing the series $G(z)=\sum_{n=1}^\infty\sqrt{n}\,z^n$ by providing a bilateral sum representation and characterizing the factorial growth and sinusoidal behavior of the polynomials appearing in the exponentially suppressed terms of its asymptotic expansion on the negative real axis.

Original authors: David Broadhurst, Gergő Nemes

Published 2026-08-06

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: David Broadhurst, Gergő Nemes

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 in the Chaos

Imagine you are trying to predict the weather. You have a formula that works perfectly when the sky is clear, but as soon as a storm hits, the formula breaks down, spitting out nonsense. In the world of advanced mathematics and physics, this happens all the time with "infinite series"—long lists of numbers added together to describe how things behave. Usually, when these lists stop working, scientists have to throw them away and find a new, complicated way to describe the storm. But sometimes, if you look very closely at the "broken" part, you find a secret pattern hiding in the noise.

This paper dives into a specific kind of mathematical storm involving "polylogarithms," which are fancy functions used to describe everything from the energy of particles in a quantum machine to the behavior of heat. The central character here is a problem proposed by the famous physicist Gerard 't Hooft. He asked a simple question: "If we have a formula that works for small numbers, can we stretch it to work for huge numbers, even where it seems to explode?" The answer isn't just "yes," but "yes, and there's a beautiful, rhythmic surprise waiting for you in the explosion." The paper explores how, when you push a formula to its absolute limit, the errors don't just vanish; they turn into a wave-like dance of polynomials (mathematical expressions with powers like $x^2$ or $x^3$ ) that follow a strict, predictable beat.

The Story of the Exploding Formula

The adventure begins with a formula proposed by 't Hooft, which looks like a sum of square roots multiplied by powers of a number $z$ . For small numbers, this sum is easy to handle. But 't Hooft wanted to know what happens when $z$ gets big, specifically on the negative side of the number line. When you try to calculate this for huge negative numbers, the standard formula breaks. It's like trying to drive a car at the speed of light; the engine (the math) starts to sputter.

The authors, David Broadhurst and Gergő Nemes, show that the solution isn't to abandon the formula, but to look at what's left over after you stop the calculation at the "best possible moment." This is called "optimal truncation." Imagine you are listening to a song that gets louder and louder until it distorts. If you stop the music right before the distortion becomes unbearable, you are left with a tiny, quiet whisper of sound that the main song didn't account for. In this paper, that "whisper" is an exponentially small term (something incredibly tiny, like $e^{-u}$ ) that holds the key to the mystery.

The Dancing Polynomials

Here is where it gets magical. The authors discovered that this tiny "whisper" isn't just random noise. It is made up of a sequence of mathematical shapes called polynomials, labeled $P_k(x)$ . Think of these polynomials as a series of dancers. At first, the dancers are simple and a bit clumsy. But as the sequence goes on (as $k$ gets larger), they start to move in a very specific, synchronized way.

The paper finds that for large $k$ , these polynomial dancers stop looking like jagged, messy lines and start to look almost exactly like smooth, rolling waves—sinusoids. It's as if the math, when pushed to the limit, decides to sing a song. The height of these waves (the amplitude) grows incredibly fast, multiplying by a huge number every time you take a step forward in the sequence. Meanwhile, the timing of the waves (the phase) shifts in a perfectly straight line.

The authors didn't just guess this; they proved it. They found two special numbers, which they call constants $C$ and $R$ , that act like the conductor for this orchestra.

$C$ (approximately 1.0688539158679530121571) controls the rhythm, or how fast the phase of the wave changes.
$R$ (approximately 0.5181839789815558726739) controls the size of the waves.

These numbers aren't random; they are tied to a deep mathematical relationship involving the square root of a complex number: $R \exp(i C) = \sqrt{-1/(2 + \pi i)}$ . The paper proves that this relationship is the reason the polynomials behave like waves. It's a rigid, mathematical law, not a coincidence.

The Proof and the Pattern

How do we know this is true? The authors used a mix of super-precise computer calculations and rigorous mathematical proofs. First, they used a computer to crunch numbers with thousands of digits of precision. They calculated the values of the formula for huge numbers and watched the "whisper" term. They saw the pattern emerge: the values fit the wave description perfectly. They even calculated the first 166 of these polynomial dancers and checked their coefficients, finding that the numbers involved followed strict rules about which prime numbers could appear in their denominators.

Then, to be absolutely sure, the second author provided a rigorous proof. They used a technique called "contour integration," which is like drawing a path through a complex landscape of numbers to capture the hidden values. They showed that the "whisper" term is indeed made of these wave-like polynomials and that the error (the part of the wave that doesn't fit the pattern perfectly) is so small it becomes invisible very quickly.

The paper explicitly rules out the idea that this behavior is just a fluke or a simple approximation. It proves that the wave-like nature is an intrinsic property of the function. While the authors mention that this specific case ( $p = -1/2$ ) is tricky and doesn't fit into older, general formulas, they show that this specific "storm" has a very orderly structure.

Why It Matters

So, what does a teenager need to know about this? It's a reminder that in the universe of math, chaos often hides order. When a formula seems to break, it might just be revealing a deeper, more beautiful layer of reality. The paper shows that even in the most complex quantum theories, there are hidden rhythms. The "polynomials" that describe the error aren't just messy leftovers; they are a precise, wave-like sequence governed by constants that the authors have now pinned down with mathematical certainty.

In short, 't Hooft asked, "What happens when the formula breaks?" and the answer is: "It starts to dance." And thanks to Broadhurst and Nemes, we now know the exact steps of that dance, the beat of the music, and the names of the two conductors, $C$ and $R$ , who are leading the show.

Technical Summary: Polynomials and Asymptotic Constants in a Resurgent Problem from 't Hooft

Problem Statement
This paper addresses a specific problem in the analytic continuation of fractional polylogarithms, originally proposed by Gerard 't Hooft in the context of quantum harmonic oscillators. The central object of study is the function $G(z) = \sum_{n=1}^{\infty} \sqrt{n} z^n$ , defined for $|z| < 1$ . The challenge is to determine the analytic continuation of $G(z)$ to the region $|z| \ge 1$ , excluding the branch cut along $[1, \infty)$ . While a solution exists in the form of a bilateral convergent sum involving logarithms, the authors focus on the asymptotic behavior of $G(z)$ on the negative real axis, specifically $G(-e^u)$ for large positive $u$ .

Methodology
The authors employ a hybrid approach combining high-precision numerical computation with rigorous asymptotic analysis and complex contour integration.

Numerical Discovery: The first author utilized high-precision arithmetic (6000 digits) to compute instances of $G(-e^u)$ at large even integers. By isolating the exponentially suppressed remainder term $S(u)$ after optimal truncation of the standard asymptotic series, they identified a sequence of polynomials $P_k(x)$ where $x = u/2 - \lfloor u/2 \rfloor$ . This empirical phase revealed that these polynomials approximate sinusoids with factorially growing amplitudes.
Analytic Derivation: The second author provided the rigorous foundation. The analysis begins with the representation of $G(-e^u)$ in terms of confluent hypergeometric functions and incomplete gamma functions. By utilizing terminant functions (which describe the remainder of asymptotic expansions), the authors derived an exact expansion for the remainder term $S(u)$ .
Integral Representation: A key step involves deriving a contour integral representation for the polynomials $P_k(x)$ :
$P_k(x) = \frac{1}{\sqrt{2\pi}} \frac{\Gamma(k + 1/2)}{2\pi i} \oint_{|t|=1} \frac{e^{(5/2-2x)t}}{1-e^{2t}} \frac{dt}{(e^t - t - 1)^{k+1/2}}$
This representation allows for the application of the method of steepest descent to analyze the asymptotic behavior as $k \to \infty$ .

Key Contributions and Results

Structure of the Remainder: The paper establishes that for large $u$ , $G(-e^u)$ can be expressed as an optimally truncated sign-constant asymptotic expansion in $1/u^2$ , plus an exponentially suppressed term $e^{-u} S(u)$ . The function $S(u)$ admits its own asymptotic expansion:
$S(u) \sim \sum_{k=0}^{\infty} \frac{P_k(x)}{u^k}$
where $P_k(x)$ are polynomials of degree $2k+1$ with rational coefficients.
Asymptotic Behavior of Polynomials: The authors prove that for large $k$ , the polynomials $P_k(x)$ behave as sinusoids with a specific amplitude and phase. Theorem 1 states that uniformly for bounded real $x$ :
$P_k(x) \sim \frac{1}{\sqrt{2\pi}} \sin((2k+1)C - 2\pi x) R^{2k+1} \Gamma\left(k + \frac{1}{2}\right)$
Identification of Asymptotic Constants: The paper rigorously identifies the constants $C$ and $R$ appearing in the asymptotic formula. They are solutions to the complex equation:
$R \exp(i C) = \sqrt{\frac{-1}{2 + \pi i}}$
Explicitly, $C = \frac{\pi}{2} - \frac{1}{2} \arctan(\frac{\pi}{2})$ and $R = (4 + \pi^2)^{-1/4}$ .
Arithmetic Properties: The authors demonstrate that the denominators of the coefficients of $P_k(x)$ involve no prime greater than $2k+3$ . This property was initially observed empirically and later confirmed via the integral representation.

Significance and Scope
The paper provides a significantly more precise description of the asymptotic behavior of the fractional polylogarithm $Li_{-1/2}(z)$ on the negative real axis than previously available. While general resurgence properties of polylogarithms are known, this work focuses specifically on the case $p = -1/2$ , which falls outside the validity of existing general asymptotic formulas.

The authors note that $G(-e^u)$ corresponds to a Fermi–Dirac integral of order $-3/2$ . Consequently, the analytical techniques and results presented here extend directly to the full class of Fermi–Dirac integrals. The work highlights the "resurgent" nature of the problem, where the late coefficients of an asymptotic expansion (the polynomials $P_k$ ) encode information about the singularities of the function in the complex plane, manifesting as the specific constants $C$ and $R$ . The paper concludes by suggesting that a complete asymptotic expansion for the sub-leading error terms could be developed using similar methods, though this is left for future investigation.

The Hidden Rhythm in the Chaos

The Story of the Exploding Formula

The Dancing Polynomials

The Proof and the Pattern

Why It Matters

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