Computing Riemann zeros with light scattering

TL;DR

This paper proposes a novel method to compute the Riemann zeta function zeros using classical light scattering, linking number theory with wave physics and optical device design.

Contribution

It introduces a new optical scattering platform for exploring the Riemann hypothesis, connecting number theory with classical wave physics.

Findings

Suppressed reflections increase with more scatterers, indicating asymptotic behavior.

Multiple scattering effects emerge as the number of scatterers grows.

The scattering landscape aligns with the perfect reflectionless condition related to Riemann zeros.

Abstract

Finding hidden order within disorder is a common interest in material science, wave physics, and mathematics. The Riemann hypothesis, stating the locations of nontrivial zeros of the Riemann zeta function, tentatively characterizes statistical order in the seemingly random distribution of prime numbers. This famous conjecture has inspired various connections with different branches of physics, recently with non-Hermitian physics, quantum field theory, trapped-ion qubits, and hyperuniformity. Here we develop the computing platform for the Riemann zeta function by employing classical scattering of light. We show that the Riemann hypothesis suggests the landscape of semi-infinite optical scatterers for the perfect reflectionless condition under the Born approximation. To examine the validity of the scattering-based computation, we investigate the asymptotic behaviours of suppressed reflections with the increasing number of scatterers and the emergence of multiple scattering. The result provides another bridge between classical physics and the Riemann zeros, exhibiting the design of wave devices inspired by number theory.