English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] bythe second author and H. Maier in terms of an inverse spectral problem forfractal strings. This problem is related to the question "Can one hear theshape of a fractal drum?" and was shown in [LaMa2] to have a positive answerfor fractal strings whose dimension is $c\in(0,1)-\{1/2}$ if and only if theRiemann hypothesis is true. Later on, the spectral operator was introducedheuristically by M. L. Lapidus and M. van Frankenhuijsen in their theory ofcomplex fractal dimensions [La-vF2, La-vF3] as a map that sends the geometry ofa fractal string onto its spectrum. We focus here on presenting the rigorousresults obtained by the authors in [HerLa1] about the invertibility of thespectral operator. We show that given any $c\geq0$, the spectral operator$\mathfrak{a}=\mathfrak{a}_{c}$, now precisely defined as an unbounded normaloperator 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 followsthat the associated inverse spectral problem has a positive answer for allpossible dimensions $c\in (0,1)$, other than the mid-fractal case when $c=1/2$,if and only if the Riemann hypothesis is true.

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