Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and theRiemann Zeta-Function

摘要

In a 1991 paper, the second author obtained a partial resolution of the Weyl-Berry conjecture for the vibrations of 'fractal drums' (i.e., 'drums with fractal boundaries'). He thereby obtaind sharp error estimates for the asymptotics of the eigenvalue distribution of the Dirichlet (or Neumann) Laplacian on an open subset Omega of R(sup n) with finite volume and very irregular ('fractal') boundary Gamma = boundary of Omega. Further, when n = 1, he and Pomerance made a detailed study of the corresponding direct spectral problem for the vibrations of 'fractal strings' (i.e., one-dimensional 'fractal drums') and established in the process some unexpected connections with the Riemann zeta-function in the 'critical strip' 0 < Re s < 1. Later on, still when n = 1, Lapidus and Maier obtained a new characterization of the Riemann hypothesis by means of an associated inverse spectral problem. In this paper, the authors will extend most of these results by using, in particular, the notion of generalized Minkowski content which is defined through some suitable 'gauge functions' other that the power functions.