The Sound of Fractal Strings and the Riemann Hypothesis

TL;DR

This paper explores the deep connections between the spectral properties of fractal strings and the Riemann Hypothesis, highlighting how spectral analysis can shed light on one of mathematics' most famous unsolved problems.

Contribution

It provides an overview of the relationship between fractal string spectra and the Riemann zeta function, including recent developments in complex dimensions and spectral operators.

Findings

Spectral properties of fractal strings relate to the zeros of the Riemann zeta function.

Development of the theory of fractal complex dimensions.

Introduction of the spectral operator in number theory.

Abstract

We give an overview of the intimate connections between natural direct and inverse spectral problems for fractal strings, on the one hand, and the Riemann zeta function and the Riemann hypothesis, on the other hand (in joint works of the author with Carl Pomerance and Helmut Maier, respectively). We also briefly discuss closely related developments, including the theory of (fractal) complex dimensions (by the author and many of his collaborators, including especially Machiel van Frankenhuijsen), quantized number theory and the spectral operator (jointly with Hafedh Herichi), and some other works of the author (and several of his collaborators).