Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions

This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of &quote;fractal drums&quote; (and especially of &quote;fractal strings&quote;). In this work, the authors extend previous results in this area by using the notion of generalized Minkowski content which is defined through some suitable &quote;gauge functions&quote; other than power functions. (This content is used to measure the irregularity (or &quote;fractality&quote;) of the boundary of an open set in $R^n$ by evaluating the volume of its small tubular neighborhoods.) In the situation when the power function is not the natural &quote;gauge function&quote;, this enables the authors to obtain more precise estimates, with a broader potential range of applications than in previous papers of the second author and his collaborators.