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arxiv: 2411.16777 · v3 · submitted 2024-11-25 · ⚛️ physics.gen-ph

Equivalence between the zero distributions of the Riemann zeta function and a two-dimensional Ising model with randomly distributed competing interactions

Zhidong Zhang This is my paper

Pith reviewed 2026-05-23 17:36 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords Riemann zeta functionIsing modelFisher zerosRiemann hypothesispartition functionMöbius functionDirichlet L-functioncritical line
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The pith

The zero distributions of the Riemann zeta function match the Fisher zeros of a two-dimensional Ising model with mixed ferromagnetic and random competing interactions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a 2D Ising model in which spins interact ferromagnetically along one lattice direction and with randomly placed competing ferromagnetic or antiferromagnetic couplings along the other. It demonstrates that the energy eigenvalues of this model are all real and follow the same distribution as the Möbius function, which in turn governs the Dirichlet L-functions and the Riemann zeta function. The eigenvectors of the model are built from those of a related 1D Ising chain with phases set by the zeta function, forming a Hilbert-Pólya space. The partition function of the model has all its zeros on the unit circle in the complex-temperature plane, and these Fisher zeros map exactly onto the critical line of the zeta function.

Core claim

All energy eigenvalues of the 2D Ising model M_(FI+SGI)^2D are real and randomly distributed exactly as the Möbius function μ(n) and the Dirichlet L(s, χ_k) functions including ζ(s). The eigenvectors are constructed from 1D Ising eigenvectors with phases ω(γ_2j) tied to the energy eigenvalues γ_2j. All zeros of the model's partition function lie on the unit circle in the complex temperature plane and map directly to the nontrivial zeros of ζ(s) on the critical line, proving the closure of that zero distribution.

What carries the argument

The 2D Ising model M_(FI+SGI)^2D whose partition-function zeros (Fisher zeros) are shown to lie on the unit circle and map onto the critical line of the Riemann zeta function.

If this is right

  • All nontrivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2.
  • The same closure holds for the zeros of every Dirichlet L-function L(s, χ_k).
  • The eigenvectors of the Ising model realize the Hilbert-Pólya space whose eigenvalues are the imaginary parts of the zeta zeros.
  • The random distribution of interactions in the model reproduces the statistical properties encoded by the Möbius function.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mapping supplies a statistical-mechanics route to numerical study of zeta zeros via finite-size Ising lattices.
  • The construction may generalize to other L-functions whose zeros are conjectured to lie on a critical line.
  • Links between random competing interactions and the distribution of zeta zeros suggest possible transfer of bounds from spin-glass theory.

Load-bearing premise

The construction of the 2D Ising model with randomly distributed competing interactions is assumed to produce energy eigenvalues and partition-function zeros that match the zeta-function properties independently rather than by choices that embed those properties.

What would settle it

An explicit calculation showing that the partition-function zeros of the described Ising model fail to lie on the unit circle or fail to map onto the line Re(s) = 1/2 would disprove the claimed equivalence.

read the original abstract

In this work, we prove the equivalence between the zero distributions of the Riemann zeta function {\zeta}(s) and a two-dimensional (2D) Ising model with a mixture of ferromagnetic and randomly distributed competing interactions. At first, we review briefly the characteristics of the Riemann hypothesis and its connections to physics, in particular, to statistical physics. Second, we build a 2D Ising model, M_(FI+SGI)^2D, in which interactions between the nearest neighboring spins are ferromagnetic along one crystallographic direction while competing ferromagnetic/antiferromagnetic interactions are randomly distributed along another direction. Third, we prove that all energy eigenvalues of this 2D Ising model M_(FI+SGI)^2D are real and randomly distributed as the M\"obius function {\mu}(n), the Dirichlet L(s,\c{hi}_k ) function as well as the Riemann zeta function {\zeta}(s). Fourth, we prove that the eigenvectors of the 2D Ising model M_(FI+SGI)^2D are constructed by the eigenvectors of the 1D Ising model with phases related to the Riemann zeta function {\zeta}(s), via the relation {\omega}({\gamma}_2j) between the angle {\omega} and the energy eigenvalues {\gamma}_2j, which form the Hilbert-P\'olya space. Fifth, we prove that all the zeros of the partition function of the 2D Ising model M_(FI+SGI)^2D lie on an unit circle in a complex temperature plane (i.e. Fisher zeros), which can be mapped to the zero distribution of the Dirichlet L(s,\c{hi}_k ) function and also the Riemann zeta function {\zeta}(s) in the critical line. In a conclusion, we have proven the closure of the nontrivial zero distribution of the L(s,\c{hi}_k ) function (including the Riemann zeta function {\zeta}(s)).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to prove an equivalence between the zero distributions of the Riemann zeta function ζ(s) (and Dirichlet L-functions) and the Fisher zeros of a 2D Ising model M_(FI+SGI)^2D with ferromagnetic nearest-neighbor interactions along one axis and randomly distributed competing ferromagnetic/antiferromagnetic interactions along the other. It further asserts that the model's energy eigenvalues are real and distributed according to the Möbius function μ(n), that its eigenvectors are built from 1D Ising eigenvectors with zeta-related phases forming a Hilbert-Pólya space, and that the partition-function zeros lie on the unit circle in the complex-temperature plane, thereby establishing closure of the nontrivial zeros on the critical line.

Significance. If the claimed mappings were shown to arise independently from the Hamiltonian without presupposing the target distributions, the result would constitute a novel physical realization of the Riemann hypothesis and could stimulate further work at the interface of statistical mechanics and analytic number theory.

major comments (3)
  1. [Model-construction section] Model-construction section (following the abstract's second point): the 2D Ising model is introduced with 'randomly distributed competing interactions' that are immediately asserted to produce energy eigenvalues distributed exactly as μ(n); no explicit form of the interaction matrix, probability measure on the random couplings, or derivation of the spectrum from the Hamiltonian is supplied, so it is impossible to determine whether the Möbius distribution emerges or is imposed by construction.
  2. [Eigenvalue-distribution section] Eigenvalue-distribution section (abstract's third point): the claim that 'all energy eigenvalues ... are real and randomly distributed as the Möbius function μ(n)' is stated without an explicit characteristic polynomial, small-lattice diagonalization, or proof that the randomness measure yields the required distribution independently of the zeta-function target.
  3. [Partition-function-zeros section] Partition-function-zeros section (abstract's fifth point): the mapping of Fisher zeros from the unit circle in the complex-temperature plane onto the critical line Re(s)=1/2 of ζ(s) is asserted after the eigenvalue identification; because the latter identification is not shown to be independent, the closure argument is circular and does not constitute an independent proof.
minor comments (2)
  1. [Abstract] Abstract contains repeated LaTeX artifacts (e.g., {ζ}(s), {μ}(n)) and undefined symbols (M_(FI+SGI)^2D, ω(γ_{2j})) that should be introduced in the main text.
  2. [Introductory review] The brief review of RH-physics connections would benefit from explicit citations to prior literature on Ising models and zeta-function zeros.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments below and agree that additional explicit details will improve the clarity of our derivations.

read point-by-point responses

  1. Referee: [Model-construction section] Model-construction section (following the abstract's second point): the 2D Ising model is introduced with 'randomly distributed competing interactions' that are immediately asserted to produce energy eigenvalues distributed exactly as μ(n); no explicit form of the interaction matrix, probability measure on the random couplings, or derivation of the spectrum from the Hamiltonian is supplied, so it is impossible to determine whether the Möbius distribution emerges or is imposed by construction.

    Authors: The referee correctly identifies that the model-construction section would benefit from more explicit details. While the manuscript derives the spectrum from the Hamiltonian with the random interactions, we will revise to include the explicit form of the interaction matrix, the probability measure on the couplings, and a clear derivation showing the emergence of the Möbius distribution from the model without presupposing the target. revision: yes

  2. Referee: [Eigenvalue-distribution section] Eigenvalue-distribution section (abstract's third point): the claim that 'all energy eigenvalues ... are real and randomly distributed as the Möbius function μ(n)' is stated without an explicit characteristic polynomial, small-lattice diagonalization, or proof that the randomness measure yields the required distribution independently of the zeta-function target.

    Authors: We will add the explicit characteristic polynomial and results from small-lattice diagonalizations in the revised manuscript to demonstrate that the distribution arises independently from the randomness in the interactions. revision: yes

  3. Referee: [Partition-function-zeros section] Partition-function-zeros section (abstract's fifth point): the mapping of Fisher zeros from the unit circle in the complex-temperature plane onto the critical line Re(s)=1/2 of ζ(s) is asserted after the eigenvalue identification; because the latter identification is not shown to be independent, the closure argument is circular and does not constitute an independent proof.

    Authors: The derivations in the manuscript establish the eigenvalue properties from the Hamiltonian first, followed by the partition function analysis. To address the concern about potential circularity, we will revise the structure to more clearly separate the steps and emphasize the independence of the eigenvalue identification. revision: yes

Circularity Check

1 steps flagged

Model interactions defined as 'randomly distributed' then asserted to yield exactly μ(n) eigenvalues; equivalence is by construction.

specific steps
  1. self definitional [Abstract, paragraphs 2-3]
    "we build a 2D Ising model, M_(FI+SGI)^2D, in which interactions between the nearest neighboring spins are ferromagnetic along one crystallographic direction while competing ferromagnetic/antiferromagnetic interactions are randomly distributed along another direction. Third, we prove that all energy eigenvalues of this 2D Ising model M_(FI+SGI)^2D are real and randomly distributed as the Möbius function μ(n), the Dirichlet L(s,χ_k ) function as well as the Riemann zeta function ζ(s)."

    The model is explicitly built with 'randomly distributed' interactions; the subsequent 'proof' states that the eigenvalues are 'randomly distributed as the Möbius function μ(n)'. The randomness is therefore defined to embed the target distribution, so the claimed equivalence between model spectrum and zeta zeros holds by the initial construction rather than by independent derivation from the Hamiltonian.

full rationale

The paper's central derivation begins by constructing the 2D Ising model with 'randomly distributed competing interactions' and immediately asserts that its energy eigenvalues are distributed precisely as the Möbius function μ(n) and the zeta zeros. No independent derivation from the Hamiltonian is exhibited; the randomness appears selected to reproduce the target spectral statistics, after which the Fisher-zero mapping is claimed to prove RH closure. This matches the self-definitional pattern exactly, with the load-bearing step reducing the claimed proof to a tautology. No external benchmark or parameter-free emergence is shown.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on the definition of the specific Ising model whose interactions are asserted to reproduce Möbius and zeta statistics, together with standard analytic properties of the zeta and L-functions; no independent evidence is supplied that the random-interaction distribution is fixed by physics rather than chosen to fit the number-theoretic target.

free parameters (1)
  • random distribution of competing interactions
    The strength and sign pattern of the antiferromagnetic couplings along one axis are described as randomly distributed but no explicit probability measure or parameter values are given; these must be chosen to produce the claimed eigenvalue statistics.
axioms (2)
  • domain assumption The Möbius function μ(n) governs the distribution of energy eigenvalues of the constructed Ising model
    Invoked when stating that all energy eigenvalues are real and randomly distributed as μ(n) and the zeta function.
  • domain assumption The zeros of the partition function of the 2D Ising model map onto the nontrivial zeros of ζ(s) and L(s,χ_k)
    Central mapping asserted without derivation in the abstract.
invented entities (1)
  • 2D Ising model M_(FI+SGI)^2D no independent evidence
    purpose: To realize the zero distribution of the Riemann zeta function via its energy eigenvalues and Fisher zeros
    New model introduced whose interactions are defined to produce the claimed equivalence; no independent physical motivation or falsifiable prediction outside the zeta context is supplied.

pith-pipeline@v0.9.0 · 5893 in / 1757 out tokens · 42636 ms · 2026-05-23T17:36:10.745910+00:00 · methodology

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Exact solution of a two-dimensional (2D) Ising model with the next nearest interactions

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