Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. 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 $A$ extends the definition of the zeta function associated with bounded fractal strings to arbitrary bounded subsets $A$ of ${\mathbb{R}}^N$. The abscissa of Lebesgue convergence $D({\zeta }_A)$ coincides with $D:={\stackrel{}{\dim }}_{B}A$, the upper box dimension of $A$. The complex dimensions of $A$ are the poles of the meromorphic continuation of the fractal zeta function of $A$ to a suitable connected neighborhood of the “critical line” $\{\mathrm{\Re }(s)=D\}$. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function $\mathrm{\mid }{A}_t\mathrm{\mid }$ as $t\to 0^{+}$, where $A_t$ is the Euclidean $t$-neighborhood of $A$. We pay particular attention to a class of Minkowski measurable sets, such that $\mathrm{\mid }{A}_t\mathrm{\mid }=t^{N-D}(\mathcal{M}+O(t^{\gamma }))$ as $t\to 0^{+}$, with $\gamma >0$, and to a class of Minkowski nonmeasurable sets, such that $|A_t|=t^{N-D}(G(\log t^{-1})+O(t^\gamma))$as$t\to0^+$,where$G$ is a nonconstant periodic function and $\gamma >0$. In both cases, we show that ${\zeta }_A$ can be meromorphically extended (at least) to the open right half-plane $\{\mathrm{\Re }(s)>D-\gamma \}$. Furthermore, up to a multiplicative constant, the residue of ${\zeta }_A$ evaluated at $s=D$ is shown to be equal to $\mathcal{M}$ (the Minkowski content of $A$) and to the mean value of $G$ (the average Minkowski content of $A$), respectively. Moreover, 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 $\{\mathrm{\Re }(s)=D\}$.