Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator

Abstract

A spectral reformulation of the Riemann hypothesis was obtained in (LaMa2) by the second author and H. Maier in terms of an inverse spectral problem for fractal strings. The inverse spectral problem which they studied is related to answering the question Can one hear the shape of a fractal drum?and was shown in (LaMa2) to have a positive answer for fractal strings whose dimension is c ∈ (0, 1) −{ 1 } if and only if the Riemann hypothesis is true. Later on, the spectral operator was introduced semi-heuristically by M. L. Lapidus and M. van Frankenhuijsen in their development of the theory of fractal strings and their complex dimensions (La-vF2, La-vF3) as a map that sends the geometry of a fractal string onto its spectrum. In this survey article, we focus on presenting the results obtained by the authors in (HerLa1) about the invertibility (in a suitable sense) of the spectral operator, which turns out to be intimately related to the critical zeroes of the Riemann zeta function. More specifically, given any c ≥ 0, we show that the spectral operator a = ac, now precisely defined as an unbounded normal operator acting in an appropriate weighted Hilbert space Hc, is 'quasi-invertible' (i.e., its truncations are invertible) if and only if the Riemann zeta function ζ = ζ(s )d oes not have any zeroes on the vertical line Re(s )= c. It follows, in particular, that the associated inverse spectral problem has a positive answer for all possible dimensions c ∈ (0, 1), other than the mid-fractal case when c = 1 , if and only if the Riemann hypothesis is true. Therefore, in this latter result from (HerLa1), a spectral reformulation of the Riemann hypothesis is obtained from a rigorous operator theoretic point of view, thereby further extending the earlier work of the second author and H. Maier in their study of the inverse spectral problem.