Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums

TL;DR

This paper reviews the development of fractal zeta functions, their connection to complex dimensions, and extends the theory to relative fractal drums, highlighting spectral properties and open problems in fractal geometry.

Contribution

It introduces and surveys the theory of fractal zeta functions, extending to relative fractal drums and exploring their spectral and geometric properties.

Findings

Zeta functions of fractal sets relate to Minkowski content and box dimension.

Spectral zeta functions of fractal drums have meromorphic extensions.

Upper bounds for the abscissa of convergence are proven to be optimal.

Abstract

The theory of 'zeta functions of fractal strings' has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called `distance zeta functions', which since then, has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of 'tube zeta functions', defined using the tube function of a fractal set. These three classes of zeta functions, under the name of 'fractal zeta functions', exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal. Finally, we conclude this survey article by proposing several open problems and directions for future research in this area.