The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings

Abstract

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 = 1 2 , 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.