Finding zeros of the Riemann zeta function by periodic driving of cold atoms

TL;DR

This paper proposes a novel method using periodically driven cold atoms to physically realize and measure the zeros of the Riemann zeta function, potentially providing new insights into the Riemann hypothesis.

Contribution

It introduces a new physical approach based on time-periodic driving to study Riemann zeros through cold atom quasienergies, inspired by the Pólya-Hilbert conjecture.

Findings

Numerical simulations show the feasibility of measuring Riemann zeros in cold atom experiments.

The quasienergies of the system can be directly governed by the zeta function.

The approach offers a new experimental avenue to explore the Riemann hypothesis.

Abstract

The Riemann hypothesis, which states that the non-trivial zeros of the Riemann zeta function all lie on a certain line in the complex plane, is one of the most important unresolved problems in mathematics. Inspired by the P\'olya-Hilbert conjecture, we propose a new approach to finding a physical system to study the Riemann zeros, which in contrast to previous examples, is based on applying a time-periodic driving field. This driving allows us to mould the quasienergies of the system (the analogue of the eigenenergies in the absence of driving), so that they are directly governed by the zeta function. We further show by numerical simulations that this allows the Riemann zeros to be measured in currently accessible cold atom experiments.