The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings |
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Authors: | Lapidus Michel L; Maier Helmut |
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Institution: | Department of Mathematics, Sproul Hall, The University of California Riverside, California 92521–0135, USA E-mail: lapidus{at}math.ucr.edu FAX: (909) 787–7314
Department of Mathematics, Boyd Graduate Studies Research Center, The University of Georgia Athens, Georgia 30602, USA |
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Abstract: | Motivated in part by the first author's work 23] on the Weyl-Berryconjecture for the vibrations of fractal drums(that is, drums with fractal boundary), M. L.Lapidus and C. Pomerance 31] have studied a direct spectralproblem for the vibrations of fractal strings(that is, one-dimensional fractal drums) and establishedin the process some unexpected connections with the Riemannzeta-function = (s) in the critical interval0 < s < 1. In this paper we show, in particular, thatthe converse of their theorem (suitably interpreted as a naturalinverse spectral problem for fractal strings, with boundaryof Minkowski fractal dimension D (0,1)) is not true in themidfractal case when D = , but that it is true for all other D in the criticalinterval (0,1) if and only if the Riemann hypothesis is true.We thus obtain a new characterization of the Riemann hypothesisby means of an inverse spectral problem. (Actually, we provethe following stronger result: for a given D (0,1), the aboveinverse spectral problem is equivalent to the partialRiemann hypothesis for D, according to which = (s)does not have any zero on the vertical line Re s = D.) Therefore,in some very precise sense, our work shows that the question(à la Marc Kac) "Can one hear the shape of a fractalstring?" – now interpreted as a suitable converse (namely,the above inverse problem) – is intimately connected withthe existence of zeros of = (s) in the critical strip 0 <Res < 1, and hence to the Riemann hypothesis. |
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