2014/06/01 by Michel L. Lapidus, Goran Radunović, Darko Žubrinić · 18 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Arithmetic zeta function #Fractal #Fractal analysis #Fractal dimension #Fractal dimension on networks #Lacunarity #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Meromorphic function #Minkowski–Bouligand dimension #Pure mathematics #Riemann hypothesis #Riemann zeta function #advanced mathematical theories #math-ph #math.CV #math.MP #math.SP
paper · pdf · doi:10.1007/s11784-014-0207-y
published in Journal of Fixed Point Theory and Applications 15(2), 321-378 (Birkhäuser) · 60 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
openalex publication_date 2014/06/01 · arxiv created 2014/11/17 · arxiv updated 2015/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.