Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums

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.

arXiv comment60 pages. Added references. Corrected typos. Expanded Thm. 2.9 to the special case when D=N, see also Rem. 2.10. Added Rem. 2.6, 3.2, 3.9, 5.17 and Rem. 8.9 corresponding to Prob. 8.8. Slightly updated Defs. 5.1, 5.20 and expanded Exercise 5.22. The paper is to appear in the Festschrift volume of the Journal of Fixed Point Theory and Applications in honor of Haim Brezis' 70th birthday

References in corpus (11)

• Riemann Zeroes and Phase Transitions via the Spectral Operator on Fractal Strings
• Spectral zeta functions of fractals and the complex dynamics of polynomials
• Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets
• Distance and tube zeta functions of fractals and arbitrary compact sets
• Box-counting fractal strings, zeta functions, and equivalent forms of Minkowski dimension
• Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis
• Fractal Strings and Multifractal Zeta Functions
• Complex dimensions of fractals and meromorphic extensions of fractal zeta functions
• Hyperfunctions and Spectral Zeta Functions of Laplacians on Self-Similar Fractals
• Fractal zeta functions and complex dimensions: A general higher-dimensional theory
• Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces

Cited by in corpus (10)

• Distance and tube zeta functions of fractals and arbitrary compact sets
• Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis
• Minkowski dimension and explicit tube formulas for $p$-adic fractal strings
• Complex dimensions of fractals and meromorphic extensions of fractal zeta functions
• Fractal zeta functions and complex dimensions: A general higher-dimensional theory
• Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces
• Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces
• Fractality and Lapidus zeta functions at infinity
• Quasiperiodic sets at infinity and meromorphic extensions of their fractal zeta functions
• Fractal zeta functions at infinity and the $\varphi$-shell Minkowski content