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Comment on "Landau Levels and Riemann Zeros"

This paper critiques the Landau-level model proposed by Sierra and Townsend for the Riemann zeros by demonstrating that higher Landau levels do not generate the necessary $\ln E$ term due to energy-independent boundary phase shifts, and by identifying the origin of the $\ln E$ term in their semiclassical equation as a first-order shift of the smooth counting function.

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

Published 2026-10-14

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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 famous puzzle concerning the distribution of prime numbers, those indivisible building blocks of arithmetic like 2, 3, and 5. For over a century, mathematicians have sought a pattern in how these numbers are spaced, a quest centered on a specific equation known as the Riemann zeta function. The most profound mystery here is the location of the function's "zeros," the points where its value drops to nothing. If these zeros follow a specific, orderly path along a vertical line in the complex plane, it would confirm a deep, hidden order in the universe of numbers. Intriguingly, physicists have long suspected that this mathematical pattern might be mirrored in the physical world, specifically in the behavior of quantum particles. The idea is that the chaotic fluctuations in the spacing of prime numbers could be explained by the way energy levels behave in a physical system, much like how the notes on a guitar string vibrate at specific frequencies. This connection has led researchers to explore whether a simple model of a charged particle moving through electric and magnetic fields could reproduce the exact statistical behavior of these mysterious zeros.

A team of researchers from Kyungnam University and Pharos iBio Co., Ltd. has taken a fresh look at this proposed connection, testing a specific model that was previously thought to hold the key. In this model, a charged particle is confined within a box and subjected to two perpendicular fields: a strong magnetic field and a weaker electric field. Under these conditions, the particle's energy does not vary smoothly but instead snaps into distinct, ladder-like steps known as Landau levels. Previous work suggested that the lowest step of this ladder, the lowest Landau level, could explain the smooth, average trend of the prime number distribution, while the higher steps might account for the jagged, fluctuating deviations. The researchers set out to verify this by calculating the precise mathematical shapes of the waves that describe the particle's position at every single step of the ladder, not just the bottom one. They examined how these waves behave at the edges of the box, a critical detail because the rules governing the waves at the boundaries determine the allowed energy levels of the system.

The team constructed the exact mathematical descriptions for the particle's wave at any given step on the energy ladder. They then analyzed how these waves behaved as the box became very large, focusing on the points where the walls of the box meet. Their calculations revealed a surprising consistency. While the waves for higher energy steps looked different from the bottom step, they all shared a fundamental property at the boundaries: the rules that determined their allowed energies were identical to the bottom step, differing only by a constant, unchanging factor that did not depend on the energy itself. This means that the higher steps of the ladder do not introduce any new, complex fluctuations into the system. Instead, they simply shift the entire set of energy levels up or down by a fixed amount. Consequently, the jagged, fluctuating part of the prime number pattern, which the previous model hoped to explain using these higher steps, does not appear in this quantum calculation. The higher levels are too orderly to produce the necessary chaos.

The researchers also revisited a specific term in the previous model that was believed to represent a fluctuation. They found that this term was actually a smooth, predictable adjustment to the energy levels, a result of the magnetic field shifting the particle's position slightly. It behaves like a steady, deterministic drift rather than a random, jittery fluctuation. Because the true fluctuations in the prime number pattern are known to change direction and vary in a way that this smooth term cannot, the authors conclude that this specific quantum model, within the approximations they used, cannot generate the required irregularities. The model works perfectly for describing the smooth, average trend of the numbers, but it fails to capture the wild variations. The study leaves open the possibility that if the particle's energy becomes so high that it grows alongside the size of the box, the rules might change, but within the standard limits of the model, the higher energy steps remain too quiet to explain the noise in the prime numbers.

Technical Summary: "Landau Levels and Riemann Zeros"

Problem Statement
This paper addresses a specific claim made by Sierra and Townsend (ST) regarding the connection between the "absorption spectrum" model of Connes and the Riemann zeros. ST proposed that the absorption spectrum arises as the lowest-Landau-level (LLL) limit of a charged particle in crossed electric and magnetic fields. Furthermore, ST suggested that higher Landau levels (n > 0) supply the fluctuation part of the Riemann-zero counting function $N(E)$ , potentially generating a logarithmic term ( $\ln E$ ) distinct from the smooth LLL contribution. The authors, Hwang, Jung, Kim, and Park, aim to test the validity of this suggestion by explicitly constructing the eigenfunctions for arbitrary Landau levels $n$ and analyzing their asymptotic behavior at the boundaries defined by ST.

Methodology
The authors work within the same approximation used by ST, setting $\hbar = \ell = 1$ and assuming a strong magnetic field limit where the cyclotron frequency $\omega_c$ is much larger than the harmonic frequency $|\omega_h|$ ( $\omega_c \gg |\omega_h|$ ). In this regime, the Hamiltonian $H$ separates into commuting components $H_c$ and $H_h$ , allowing eigenstates to be written as products of Landau-level states $|n\rangle$ and eigenstates of $H_h$ .

Construction of Eigenfunctions: The authors generalize the ST wavefunction for the LLL ( $n=0$ ) to arbitrary levels $n$ . They express the $n$ -th Landau level eigenfunction $\psi_{E,n}$ as an integral involving Hermite polynomials $H_n$ and the same spectral function $\Phi_E(Q)$ used in the LLL case. Crucially, they derive an operator form (Eq. 4) showing that $\psi_{E,n}$ can be generated by applying the operator $H_n(x - i\partial_y)$ to the LLL wavefunction $\psi_{E,0}$ .
Asymptotic Analysis: The authors analyze the asymptotic behavior of these wavefunctions at the two boundaries of the confining box, $(x, L)$ and $(L, x)$ , in the limit $L \to \infty$ for fixed $n$ . They utilize the large-argument asymptotics of the confluent hypergeometric function $M(a, b, u)$ .
Boundary Condition Derivation: By evaluating the ratio $\psi_{E,n}(x, L) / \psi_{E,n}(L, x)$ , the authors derive the quantization condition for each Landau level tower. They carefully handle the operator derivatives acting on the asymptotic phases, utilizing the Baker–Campbell–Hausdorff formula to simplify the action of the Hermite polynomial operators.

Key Contributions and Results

Exact Eigenfunctions for Arbitrary $n$ : The paper provides the explicit form of the eigenfunctions for any fixed Landau level $n$ within the ST approximation, demonstrating that they are related to the LLL solution by the operator $H_n(x - i\partial_y)$ .
Absence of $\ln E$ Fluctuations: The central result is that the boundary condition for any fixed $n$ $n$ differs from the $n=0$ $n = 0$ case only by an energy-independent phase factor, $(-i)^n$ $(- i)^{n}$ . Consequently, the quantization condition (Eq. 15) yields a counting function that is identical to the $n=0$ $n = 0$ case up to an $E$ $E$ -independent constant.
- The authors explicitly state that no $\ln E$ term arises for any fixed $n$ .
- Therefore, the higher Landau levels do not generate the fluctuation part of the Riemann-zero counting function as suggested by ST.
Reinterpretation of ST Eq. (29): The authors analyze the semiclassical equation (Eq. 29) cited by ST, which contains a $\ln E$ term. They demonstrate that this term corresponds to the first-order shift of the smooth LLL counting function due to the energy shift $E \to E + \gamma z^2$ (where $\gamma z^2$ is the cyclotron energy). They argue this is a smooth, deterministic term with a fixed sign, distinct from the oscillatory fluctuation $S(E)$ associated with the Riemann zeros.

Significance and Scope
The paper concludes that within the specific approximation ( $\omega_c \gg |\omega_h|$ ) and for fixed Landau levels $n$ as the box size $L \to \infty$ , the ST proposal that higher Landau levels supply the Riemann fluctuation is not supported. The exact quantum calculation shows that the spectral structure remains a shifted version of the LLL spectrum without generating new fluctuation terms.

The authors modestly limit the scope of their findings:

Their derivation holds strictly for fixed $n$ in the limit of large $L$ .
The regime where $n$ grows with $L$ (corresponding to large classical amplitude $z$ ) is not covered and remains an open question.
They note that beyond the $H \simeq H_c + H_h$ approximation, the exact model remains an integrable quadratic Hamiltonian, which lacks the exponentially proliferating periodic orbits required to generate a Berry-type fluctuation term.

In summary, the paper provides a rigorous quantum mechanical refutation of the specific mechanism proposed by ST for generating Riemann fluctuations via higher Landau levels, attributing the apparent $\ln E$ term in ST's semiclassical analysis to a smooth energy shift rather than a spectral fluctuation.

Technical Summary: "Landau Levels and Riemann Zeros"

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