caa a22 4500 397532636 CHVBK 20180308164655.0 cr unu---uuuuu 161202e199508 xx s 000 0 eng 10.1112/jlms/52.1.15 doi (NATIONALLICENCE)oxford-10.1112/jlms/52.1.15 The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings [Elektronische Daten] [Michel L. Lapidus, Helmut Maier] Motivated in part by the first author's work [23] on the Weyl-Berry conjecture for the vibrations of ‘fractal drums' (that is, ‘drums with fractal boundary'), M. L. Lapidus and C. Pomerance [31] have studied a direct spectral problem for the vibrations of ‘fractal strings' (that is, one-dimensional ‘fractal drums') and established in the process some unexpected connections with the Riemann zeta-function ζ = ζ (s) in the ‘critical interval' 0 < s < 1. In this paper we show, in particular, that the converse of their theorem (suitably interpreted as a natural inverse spectral problem for fractal strings, with boundary of Minkowski fractal dimension D ∈ (0,1)) is not true in the ‘midfractal' case when D = 12, but that it is true for all other D in the critical interval (0,1) if and only if the Riemann hypothesis is true. We thus obtain a new characterization of the Riemann hypothesis by means of an inverse spectral problem. (Actually, we prove the following stronger result: for a given D ∈ (0,1), the above inverse spectral problem is equivalent to the ‘partial Riemann hypothesis' for D, according to which ζ = ζ (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 fractal string?” - now interpreted as a suitable converse (namely, the above inverse problem) - is intimately connected with the existence of zeros of ζ = ζ(s) in the critical strip 0 < Res < 1, and hence to the Riemann hypothesis. © The London Mathematical Society Notes and Papers nationallicence Lapidus Michel L. Department of Mathematics, Sproul Hall, The University of California Riverside, California 92521-0135, USA E-mail: lapidus@math.ucr.edu FAX: (909) 787-7314 aut Maier Helmut Department of Mathematics, Boyd Graduate Studies Research Center, The University of Georgia Athens, Georgia 30602, USA aut Journal of the London Mathematical Society Oxford University Press 52/1(1995-08), 15-34 0024-6107 52:1<15 1995 52 jlms https://doi.org/10.1112/jlms/52.1.15 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1112/jlms/52.1.15 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Lapidus Michel L. Department of Mathematics, Sproul Hall, The University of California Riverside, California 92521-0135, USA E-mail: lapidus@math.ucr.edu FAX: (909) 787-7314 aut NATIONALLICENCE 700 1- Maier Helmut Department of Mathematics, Boyd Graduate Studies Research Center, The University of Georgia Athens, Georgia 30602, USA aut NATIONALLICENCE 773 0- Journal of the London Mathematical Society Oxford University Press 52/1(1995-08), 15-34 0024-6107 52:1<15 1995 52 jlms Metadata rights reserved CC BY-NC-4.0 http://creativecommons.org/licenses/by-nc/4.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-oxford