%0 Journal Article
%T Complex dimensions of fractals and meromorphic extensions of fractal zeta functions
%A Michel L. Lapidus
%A Goran Radunovi£¿
%A Darko £¿ubrini£¿
%J Physics
%D 2015
%I arXiv
%X 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 $\{\textrm{Re}\ s=D\}$.
%U http://arxiv.org/abs/1508.04784v1