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

[2411.16777] Equivalence between the zero distributions of the Riemann zeta function and a two-dimensional Ising model with randomly distributed competing interactions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Physics > General Physics arXiv:2411.16777 (physics) [Submitted on 25 Nov 2024 ( v1 ), last revised 16 Aug 2026 (this version, v4)] Title: Equivalence between the zero distributions of the Riemann zeta function and a two-dimensional Ising model with randomly distributed competing interactions Authors: Zhidong Zhang View a PDF of the paper titled Equivalence between the zero distributions of the Riemann zeta function and a two-dimensional Ising model with randomly distributed competing interactions, by Zhidong Zhang View PDF 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 Möbius 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-Pólya 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)). Comments: 46 pages, 0 figure, mathematical rigor is improved Subjects: General Physics (physics.gen-ph) MSC classes: 82B20, 82B44, 11M26, 11A41 Cite as: arXiv:2411.16777 [physics.gen-ph] (or arXiv:2411.16777v4 [physics.gen-ph] for this version) https://doi.org/10.48550/arXiv.2411.16777 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Phys. Lett. A 591 (2026) 131910 Related DOI : https://doi.org/10.1016/j.physleta.2026.131910 Focus to learn more DOI(s) linking to related resources Submission history From: Zhidong Zhang [ view email ] [v1] Mon, 25 Nov 2024 07:29:22 UTC (979 KB) [v2] Mon, 15 Sep 2025 07:41:00 UTC (1,015 KB) [v3] Thu, 9 Apr 2026 01:46:20 UTC (1,038 KB) [v4] Sun, 16 Aug 2026 02:01:03 UTC (1,055 KB) Full-text links: Access Paper: View a PDF of the paper titled Equivalence between the zero distributions of the Riemann zeta function and a two-dimensional Ising model with randomly distributed competing interactions, by Zhidong Zhang View PDF view license Current browse context: physics.gen-ph < prev | next > new | recent | 2024-11 Change to browse by: physics References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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