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Generalized Fractal Strings, Complex Dimensions and a Spectral Reformulation of the Riemann Hypothesis

  • Author(s): Herichi, Hafedh
  • Advisor(s): Lapidus, Michel L
  • et al.
Abstract

The spectral operator was introduced for the first time by M. L. Lapidus and his collaborator M. van Frankenhuijsen in their theory of complex dimensions in fractal geometry cite,The corresponding inverse spectral problem was first considered by M. L. Lapidus and H. Maier in their work on a spectral reformulation of the Riemann hypothesis in connection with the question "Can One Hear The Shape of a Fractal String?". The spectral operator is defined on a suitable Hilbert space as the operator mapping the counting function of a generalized fractal string to the counting function of its associated spectral measure. It relates the spectrum of a fractal string with its geometry. The inverse spectral problem for vibrating fractal strings studied by M. L. Lapidus and H. Maier has a positive answer if and only if the Riemann zeta function has no zeros on Re(s)=D, where D is in (0,1) is the dimension of the fractal string. In this work, we provide a functional analytic framework allowing us to study the spectral operator. In particular, by determining the spectrum of the spectral operator, we give a necessary and sufficient condition providing its invertibility in the critical strip. We show that such a condition is related to the location of the critical zeroes of the Riemann zeta function or equivalently that the spectral operator is invertible if and only if the Riemann hypothesis is true. As a result, the spectral operator is invertible for any D in (0,1)-{1/2} if and only if the Riemann hypothesis is true.The latter results provides a spectral reformulation of the Riemann hypotesis in terms of a rigorously defined map (the spectral operator). Hence, it sheds new light to the earlier work obtained by M. L. Lapidus and H. Maier and later revisited by M. L. Lapidus and M. van Frankenhuijsen.

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