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

Abstract

In this work, we prove the equivalence between the zero distributions of the Riemann zeta function ζ(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 μ(n), the Dirichlet L(s,\c{hi}_k ) function as well as the Riemann zeta function ζ(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 ζ(s), via the relation ω(γ_2j) between the angle ω and the energy eigenvalues γ_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 ζ(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 ζ(s)).

arXiv comment46 pages, 0 figure, mathematical rigor is improved

References in corpus (15)

• Conjectures on exact solution of three - dimensional (3D) simple orthorhombic Ising lattices
• Physics of the Riemann Hypothesis
• Clifford algebra approach of 3D Ising model
• Computational complexity of spin-glass three-dimensional (3D) Ising model
• Quantum mechanical potentials related to the prime numbers and Riemann zeros
• Will a physicist prove the Riemann Hypothesis?
• Quantum hamiltonians and prime numbers
• Exact solution of two-dimensional (2D) Ising model with a transverse field: a low-dimensional quantum spin system
• Mapping between Spin-Glass Three-Dimensional (3D) Ising Model and Boolean Satisfiability Problem
• A lower bound in an approximation problem involving the zeros of the Riemann zeta function
• Nonclassical Degrees of Freedom in the Riemann Hamiltonian
• Topological Quantum Statistical Mechanics and Topological Quantum Field Theories
• Lower bound of computational complexity of knapsack problems
• Exact solution of the three-dimensional (3D) Z2 lattice gauge theory
• Exact solution of a two-dimensional (2D) Ising model with the next nearest interactions