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In 2009, the first author introduced a class of zeta functions, called`distance zeta functions', which has enabled us to extend the existing theoryof zeta functions of fractal strings and sprays (initiated by the first authorand his collaborators in the early 1990s) to arbitrary bounded (fractal) setsin Euclidean spaces of any dimensions. A closely related tool is the class of`tube zeta functions', defined using the tube function of a fractal set. Thesezeta functions exhibit deep connections with Minkowski contents and upper box(or Minkowski) dimensions, as well as, more generally, with the complexdimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e.,absolute) convergence of the distance zeta function coincides with the upperbox dimension of a set. We also introduce a class of transcendentallyquasiperiodic sets, and describe their construction based on a sequence ofcarefully chosen generalized Cantor sets with two auxilliary parameters. As aresult, we obtain a family of "maximally hyperfractal" compact sets andrelative fractal drums (i.e., such that the associated fractal zeta functionshave a singularity at every point of the critical line of convergence).Finally, we discuss the general fractal tube formulas and the Minkowskimeasurability criterion obtained by the authors in the context of relativefractal drums (and, in particular, of bounded subsets of the N-dimensionalEuclidean space).

Fractal zeta functions and complex dimensions: A general higher-dimensional theory