⚛️ quantum physics

Eigenvalue-by-Eigenvalue Comparison of a Sierra--Rodríguez-Laguna-Type Spectrum with the Riemann Zeros

This paper demonstrates that by correcting the treatment of the fluctuating term in the Riemann counting function, the eigenvalues of a specific self-adjoint extension of the Sierra--Rodríguez-Laguna $H=xp$ Hamiltonian match the ordinates of the first 606 Riemann zeta zeros to within 0.03%, validating a closed-form prediction derived via the Lambert- $W$ function.

Original authors: Mi-Ra Hwang, Eylee Jung, MuSeong Kim, DaeKil Park

Published 2026-09-22

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CC0 1.0

Original authors: Mi-Ra Hwang, Eylee Jung, MuSeong Kim, DaeKil Park

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

Deep within the landscape of mathematics lies a puzzle that has resisted solution for more than a century: the Riemann Hypothesis. At its heart is a specific pattern hidden inside a complex function known as the Riemann zeta function. This function produces a list of special numbers, called zeros, which appear to follow a very strict rule. If this rule holds true, it would unlock a profound understanding of how prime numbers—the building blocks of all arithmetic—are distributed. For decades, mathematicians have wondered if these mysterious zeros are not just abstract numbers, but the actual energy levels of a physical system, much like the distinct notes a guitar string can play. This idea, known as the Hilbert–Pólya conjecture, suggests that if we can find the right physical machine, its natural vibrations would match the zeros of the zeta function perfectly.

In a recent study, researchers set out to test this idea with a specific mathematical model designed to mimic such a physical machine. They focused on a system described by a particular type of equation that had been proposed as a candidate for generating these zeros. The team, led by Mi-Ra Hwang and DaeKil Park, took this theoretical model and solved it to find its first 606 distinct energy levels. They then compared these calculated levels, one by one, against the first 606 known zeros of the Riemann zeta function. The goal was to see if the two lists matched up exactly, or if there was a persistent gap between the predicted energy levels and the actual mathematical zeros.

The researchers began by solving a complex equation that defines the allowed energy states of their model. They found that the model produces a discrete list of numbers that look remarkably similar to the list of zeta zeros. When they plotted the two lists side by side, the match was so close that the difference between them was barely visible to the naked eye. However, when they zoomed in to measure the tiny gaps between each pair, a small but consistent discrepancy appeared. The calculated energy levels were not identical to the zeros; they were slightly offset. The team needed to understand why this offset existed and whether it was a flaw in their model or a misunderstanding of how to count the zeros.

To investigate, the team used a powerful mathematical technique to approximate the behavior of their model at high energy levels. They analyzed the core function driving the system, which involves a special type of curve known as a modified Bessel function. A common concern in such studies is that the mathematical tools used to approximate these curves might introduce hidden errors or "ghost" phases that distort the results. The researchers rigorously checked their approximation against known, exact formulas for these curves at high energies. They confirmed that their method was sound and that the Bessel function itself was not the source of the mismatch. The mathematical machinery was working correctly; the error lay elsewhere.

The breakthrough came when the team re-examined how the zeros are counted. In the standard way mathematicians count these special numbers, there is a convention for handling the exact moment a zero is reached. The usual method treats the count as a midpoint between the number before and the number after the zero occurs. The researchers realized that this standard convention shifts the effective position of the quantum number by a tiny amount, specifically by one-half. When they corrected for this shift, the formula for predicting the zeros changed. Instead of the previous estimate, the corrected formula aligned the counting method with the physical model in a new way.

With this correction applied, the researchers compared their new prediction against the first 105 zeros listed in a famous database compiled by Andrew Odlyzko. The results were striking. The corrected prediction matched the actual data with an accuracy of within 0.03 percent. The previous mismatch, which had been about 24 percent off, vanished. The remaining tiny variations in the data were consistent with known statistical fluctuations that occur in the distribution of these numbers, rather than a fundamental flaw in the theory. The study also identified and removed a single false reading in their initial data, which had been a numerical artifact where two close values were mistakenly merged into one.

The paper concludes that the energy levels of this specific physical model do indeed track the Riemann zeta zeros with high precision. The small differences between the two lists are not random errors but are explained by a subtle detail in how the counting of the zeros is defined at the exact moment they occur. The researchers found that the gap between the model's predictions and the actual zeros shrinks very slowly as the numbers get larger, following a specific mathematical pattern. This work provides strong evidence that the model is a valid candidate for the physical system behind the Riemann Hypothesis, provided the counting convention is adjusted correctly. It resolves a previous numerical puzzle by showing that the discrepancy was not in the physics of the model, but in the way the mathematical target was being measured.

Technical Summary: Eigenvalue-by-Eigenvalue Comparison of a Sierra–Rodríguez-Laguna-Type Spectrum with the Riemann Zeros

Problem Statement
The paper investigates the validity of the Hilbert–Pólya conjecture, which posits that the nontrivial zeros of the Riemann zeta function ( $1/2 + i\gamma_n$ ) correspond to the eigenvalues of a self-adjoint operator. Specifically, the authors analyze the Sierra–Rodríguez-Laguna (SR) model, a modification of the Berry–Keating Hamiltonian ( $H_{BK} = xp$ ). While $H_{BK}$ reproduces the mean density of Riemann zeros, it possesses a continuous spectrum. The SR model, defined by $H_{SR} = x(p + \ell_p^2/p)$ with a hard wall at $x = \ell_x$ , yields a discrete spectrum. The central question addressed is whether the individual eigenvalues ( $E_n$ ) of the SR Hamiltonian can be matched one-to-one with the ordinates of the Riemann zeros ( $\gamma_n$ ), and if so, what the asymptotic behavior of the difference $E_n - \gamma_n$ is as $n \to \infty$ .

Methodology
The authors employ a combination of numerical computation and rigorous asymptotic analysis:

Numerical Spectrum Generation: The self-adjoint extension of the SR Hamiltonian leads to a transcendental equation for the eigenvalues: $\text{Re}[e^{-i\theta/2}K_{1/2+iE/2}(2\pi)] = 0$ , where $K_\nu$ is the modified Bessel function and $\theta$ is a free parameter. The authors solved this equation numerically for $\theta = 1.417\pi$ to obtain the first 606 eigenvalues ( $E_1, \dots, E_{606}$ ).
Asymptotic Analysis (Steepest-Descent): To derive a closed-form prediction for $E_n$ , the authors applied the steepest-descent (saddle-point) method to the integral representation of the modified Bessel function $K_{1/2+iE/2}(2\pi)$ for large $E$ . They identified the dominant saddle points and derived the leading-order phase behavior.
Verification against Rigorous Asymptotics: The authors cross-verified their saddle-point derivation against known rigorous asymptotic formulas for $K_\nu(x)$ with large imaginary order (citing Temme, Balogh, Booker, Strömbergsson, Then, and Tseng) to ensure no hidden phase terms were omitted.
Quantization and Lambert-W Solution: The quantization condition was solved using the Lambert-W function to express $E_n$ explicitly in terms of the quantum number $n$ .
Comparison with Riemann–von Mangoldt Formula: The authors compared their derived $E_n$ with the Riemann–von Mangoldt formula for the counting function $N(T)$ . They critically analyzed the treatment of the fluctuating term $S(T)$ exactly at the zeros ( $T = \gamma_n$ ), specifically addressing the convention used for the jump discontinuity in $N(T)$ .
Empirical Validation: The theoretical predictions were tested against the first 105 Riemann zeros tabulated by A. Odlyzko.

Key Contributions and Results

Asymptotic Matching: The authors demonstrated that the leading-order phase of the modified Bessel function derived via the saddle-point method matches exactly with rigorous asymptotic results. This rules out the Bessel function itself as the source of any numerical mismatch between the SR spectrum and the Riemann zeros.
Resolution of the Phase Shift: A discrepancy was identified between the standard prediction (assuming $N(\gamma_n) = n$ $N (γ_{n}) = n$ ) and the numerical data. The authors traced this to the treatment of the counting function at the zeros. They argue that the standard "midpoint convention" for the fluctuating term $S(T)$ $S (T)$ at a jump shifts the effective quantum number by $1/2$ $1/2$ .
- Standard approach: $\gamma_n \sim g(n - 7/8)$ .
- Corrected approach: $\gamma_n \sim g(n - 11/8)$ .
Closed-Form Prediction: Using the corrected counting function, the authors derived a closed-form expression for the difference $E_n - \gamma_n$ :
$E_n - \gamma_n \sim \frac{\theta + 15\pi/4}{1 + W(n/e)}$
where $W$ is the Lambert-W function. This predicts that the difference decays as $O(1/\ln n)$ .
Numerical Agreement: When comparing the first 105 zeros, the corrected prediction (black curve in Fig. 2) aligns with the measured data to within 0.03%. In contrast, the uncorrected "standard" prediction deviates by approximately 24%. The residual scatter in the data is consistent with Selberg's central limit theorem for $S(T)$ .
Artifact Identification: The authors identified and removed a spurious root near $E \approx 947.5$ (originally labeled $E_{607}$ ) which was a numerical artifact where two roots merged, confirmed by an anomalous level spacing ratio.

Significance
The paper claims to provide a satisfying resolution to the question of whether the SR Hamiltonian's eigenvalues track the Riemann zeros. The significance lies in:

Precision: Establishing that the SR model reproduces the Riemann zeros not just statistically, but eigenvalue-by-eigenvalue, with a predictable error term that vanishes as $n \to \infty$ .
Theoretical Clarity: Clarifying that the mismatch previously observed was not due to the physics of the Hamiltonian or the properties of the Bessel function, but rather a subtle convention in the definition of the Riemann counting function $N(T)$ at the zeros.
Validation of the Model: The work strengthens the case for the SR Hamiltonian as a candidate for the operator underlying the Hilbert–Pólya conjecture, showing that the leading-order asymptotics are robust and the remaining discrepancies are fully understood and correctable.

The authors maintain a modest tone, noting that while the agreement is strong and the resolution satisfying, the argument relies on specific conventions for $S(T)$ that, while standard in the literature, could be further scrutinized. They do not claim to have proven the Riemann Hypothesis, but rather to have established a precise correspondence between a specific quantum model and the known zeros.

Technical Summary: Eigenvalue-by-Eigenvalue Comparison of a Sierra–Rodríguez-Laguna-Type Spectrum with the Riemann Zeros

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