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

TL;DR

This paper establishes a deep mathematical connection between the zero distributions of the Riemann zeta function and a specially constructed 2D Ising model with mixed interactions, linking physics and number theory.

Contribution

It proves the equivalence of zero distributions of the Riemann zeta function and a 2D Ising model with random competing interactions, advancing understanding in both physics and mathematics.

Findings

Eigenvalues of the 2D Ising model are real and distributed like the Möbius function.

Eigenvectors relate to the Riemann zeta function and form the Hilbert-Pólya space.

Partition function zeros lie on the unit circle, corresponding to the zeros of the zeta and L-functions.

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)).