GENERALIZED MINKOWSKI CONTENT, SPECTRUM OF FRACTAL DRUMS, FRACTAL STRINGS AND THE RIEMANN ZETA-FUNCTION

This paper provides a detailed study of the effect of non powerlike ir regularities of (the geometry of) the fractal boundary on the spectrum of 'fractal drums? (and especially, of:fractal strings'). In earlier work [La1]-devoted to the standard case of power-type irregularities-t he second author obtained a partial resolution of the Weyl-Berry conje cture for the vibrations of 'fractal drums' (i.e., 'drums with fractal boundaries'); he thereby obtained sharp error estimates for the asymp totics of the eigenvalue distribution of the Dirichlet (or Neumann) La placian on an open subset Omega of R-n with finite volume and very irr egular ('fractal') boundary Gamma=partial derivative Omega. Futher, wh en n=1, Lapidus and Pomerance [LaPo1,2] made a detailed study of the c orresponding direct spectral problem for the vibrations of ''fractal s trings'' (i.e., one-dimensional 'fractal drums') and established in th e process some unexpected connections with the Riemann zeta-function z eta=zeta(s) in the 'critical interval' 0<s<1. Later on (still when n=1 ), using the oscillatory phenomena associated with the complex zeros o f zeta in the 'critical strip' 0 <Re s <1, Lapidus and Maier [LaMa1,2] obtained a new characterization of the Riemann hypothesis by means of an associated inverse spectral problem. In this memoir, we will extend most of these results by using, in particular, the notion of generali zed Minkowski content which is defined through some suitable 'gauge fu nctions' other than the power functions. [This content is used to meas ure the irregularity (or 'fractality') of the boundary Gamma=partial d erivative Omega by evaluating the volume of its small tubular neighbor hoods.] In the situation when the power function is not the natural 'g auge function', this will enable us to obtain more precise estimates, with a broader potential range of applications than in the above paper s.