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      Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

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          Abstract

          We study meromorphic extensions of distance and tube zeta functions, as well as of zeta functions of fractal strings, which include perturbations of the Riemann zeta function. The distance zeta function \(\zeta_A(s):=\int_{A_\delta} d(x,A)^{s-N}\mathrm{d}x\), where \(\delta>0\) is fixed and \(d(x,A)\) denotes the Euclidean distance from \(x\) to \(A\), has been introduced by the first author in 2009, extending the definition of the zeta function \(\zeta_{\mathcal L}\) associated with bounded fractal strings \(\mathcal L=(\ell_j)_{j\geq 1}\) to arbitrary bounded subsets \(A\) of the \(N\)-dimensional Euclidean space. The abscissa of Lebesgue (i.e., absolute) convergence \(D(\zeta_A)\) coincides with \(D:=\overline\dim_BA\), the upper box (or Minkowski) dimension of \(A\). The (visible) complex dimensions of \(A\) are the poles of the meromorphic continuation of the fractal zeta function (i.e., the distance or tube zeta function) of \(A\) to a suitable connected neighborhood of the "critical line" \(\{\textrm{Re}\ s=D\}\). We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function \(|A_t|\) as \(t\to0^+\), where \(A_t\) is the Euclidean \(t\)-neighborhood of \(A\). Furthermore, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line \(\{\textrm{Re}\ s=D\}\). Finally, using an appropriate quasiperiodic version of the above construction, with infinitely many suitably chosen quasiperiods associated with a two-parameter family of generalized Cantor sets, we construct "maximally-hyperfractal" compact subsets of \(\mathbb{R}^N\), for \(N\geq 1\) arbitrary. These are compact subsets of \(\mathbb{R}^N\) such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line \(\{\textrm{Re}\ s=D\}\).

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          Author and article information

          Journal
          2015-08-19
          Article
          1508.04784
          0e74c8e0-03bf-4f8c-9ac8-5c0658b67fd0

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          51 pages, 1 figure
          math-ph math.MP

          Mathematical physics,Mathematical & Computational physics
          Mathematical physics, Mathematical & Computational physics

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