Riemann Zeroes and Phase Transitions via the Spectral Operator on Fractal Strings

TL;DR

This paper explores the spectral operator linked to fractal strings, establishing a new operator-theoretic reformulation of the Riemann hypothesis and identifying phase transitions at critical fractal dimensions.

Contribution

It provides a rigorous functional analytic framework for the spectral operator and links its invertibility to the zeros of the Riemann zeta function, offering a novel reformulation of the Riemann hypothesis.

Findings

Spectral operator invertibility is equivalent to the Riemann hypothesis.

Phase transitions occur at critical fractal dimensions c=1/2 and c=1.

Spectral properties change at these critical points, indicating deep connections to number theory.

Abstract

The spectral operator was introduced by M. L. Lapidus and M. van Frankenhuijsen [La-vF3] in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this survey paper, we present the rigorous functional analytic framework given by the authors in [HerLa1] and within which to study the spectral operator. Furthermore, we also give a necessary and sufficient condition for the invertibility of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral operator is invertible (or equivalently, that zero does not belong to its spectrum) if and only if the Riemann zeta function zeta(s) does not have any zeroes on the vertical line Re(s)=c. Hence, it is not invertible in the mid-fractal case when c=1/2, and it is invertible everywhere else (i.e., for all c in(0,1) with c not equal to 1/2) if and only if the Riemann hypothesis is true. We also show the existence of four types of (mathematical) phase transitions occurring for the spectral operator at the critical fractal dimension c=1/2 and c=1 concerning the shape of the spectrum, its boundedness, its invertibility as well as its quasi-invertibility.