A symmetry perspective of the Riemann zeros

TL;DR

This paper explores the connection between the zeros of the Riemann zeta function and physical systems with supersymmetry, PT symmetry, and SU(2) symmetry, revealing how these symmetries relate to the distribution of zeros.

Contribution

It introduces a novel framework linking Riemann zeros to physical symmetries, demonstrating unbroken supersymmetry correlates with non-trivial zeros and analyzing symmetry breaking.

Findings

Unbroken supersymmetry correlates with non-trivial zeros of the zeta function.

PT symmetry invariance is observed within the supersymmetric system.

An SU(2) symmetry with a two-level Hilbert space structure is identified.

Abstract

We study the relationship between the zeros of the Riemann zeta function and physical systems exhibiting supersymmetry, $PT$ symmetry and $SU(2)$ group symmetry. Our findings demonstrate that unbroken supersymmetry is associated with the presence of non-trivial zeros of the zeta function. However, in other cases, supersymmetry is spontaneously broken and the ground state energy of the system is not zero. Moreover, we have established the manifestation of PT symmetry invariance within our supersymmetric system. In addition, our findings provide insights into a $SU(2)$ symmetry that arises within these systems, with the Hilbert space having a two-level structure.