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Lapidus zeta functions of fractal sets and their residues

Lapidus zeta functions (or distance zeta functions) have been introduced by M. L. Lapidus in 2009. They enable us to extend the notion of complex dimensions from the case of bounded fractal strings to arbitrary bounded subsets of Euclidean spaces. We will show that the residue of the Lapidus zeta function, computed at the box dimension of the set, is intimately related to the upper and lower Minkowski contents of the set. Furthermore, we shall describe some results related to the problem of existence of meromorphic extension of these zeta functions. In addition to this, we shall describe a construction of quasiperiodic sets possessing arbitrarily many quasiperiods, using suitable generalized Cantor sets along with a number theoretic result due to Alan Baker. This is a joint work with M. L. Lapidus, University of California, Riverside, and Goran Radunovic, University of Zagreb. References M. L. Lapidus, G. Radunovic and D. Zubrinic, Fractal zeta functions and complex dimensions: A general higher-dimensional theory, in: Geometry and Stochastics V (C. Bandt, K. Falconer and M. Zahle, eds.), Proc. Fifth Internat. Conf. (Tabarz, Germany, March 2014), Progress in Probability, Birkhauser, Basel, Boston and Berlin, 2015, in press. (Based on a plenary lecture given by the first author at that conference.) arXiv:1502.00878v2 [math.CV] M. L. Lapidus, G. Radunovic and D. Zubrinic, Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, research monograph, Springer, New York, 2016, to appear (approx. 660 pages)

Lapidus zeta functions; distance zeta function; tube zeta function Minkowski content; box dimension; meromorphic extension; quasiperiodicity

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