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

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. This problem is related to 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 $c\in(0,1)-\{1/2}$ if and only if the Riemann hypothesis is true. Later on, the spectral operator was introduced heuristically by M. L. Lapidus and M. van Frankenhuijsen in their theory of complex fractal dimensions [La-vF2, La-vF3] as a map that sends the geometry of a fractal string onto its spectrum. We focus here on presenting the rigorous results obtained by the authors in [HerLa1] about the invertibility of the spectral operator. We show that given any $c\geq0$, the spectral operator $\mathfrak{a}=\mathfrak{a}_{c}$, now precisely defined as an unbounded normal operator acting in a Hilbert space $\mathbb{H}_{c}$, is `quasi-invertible' (i.e., its truncations are invertible) if and only if the Riemann zeta function $\zeta=\zeta(s)$ does not have any zeroes on the line $Re(s)=c$. It follows that the associated inverse spectral problem has a positive answer for all possible dimensions $c\in (0,1)$, other than the mid-fractal case when $c=1/2$, if and only if the Riemann hypothesis is true.