Phenomenological formula for Quantum Hall resistivity based on the Riemann zeta function

TL;DR

This paper introduces a phenomenological formula for the quantum Hall resistivity involving the Riemann zeta function, highlighting potential links between number theory and quantum physics, without deriving from a specific physical model.

Contribution

It presents a novel, elementary-function-based formula for quantum Hall resistivity that incorporates the Riemann zeta function, suggesting intriguing connections to the Riemann Hypothesis.

Findings

The formula captures detailed features of transverse resistivity in the integer quantum Hall effect.

It raises questions about the physical realization of the proposed resistivity model.

The involvement of the Riemann zeta function hints at deep mathematical-physical connections.

Abstract

We propose a formula constructed out of elementary functions that captures many of the detailed features of the transverse resistivity $\rho_{xy}$ for the integer quantum Hall effect. It is merely a phenomenological formula in the sense that it is not based on any transport calculation for a specific class of physical models involving electrons in a disordered landscape, thus, whether a physical model exists which realizes this resistivity remains an open question. Nevertheless, since the formula involves the Riemann zeta function and its non-trivial zeros play a central role, it is amusing to consider the implications of the Riemann Hypothesis in light of it.