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We study meromorphic extensions of distance and tube zeta functions, as wellas 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 $\delta>0$ isfixed and $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ extends thedefinition of the zeta function associated with bounded fractal strings toarbitrary bounded subsets $A$ of $\mathbb{R}^N$. The abscissa of Lebesgueconvergence $D(\zeta_A)$ coincides with $D:=\overline\dim_BA$, the upper boxdimension of $A$. The complex dimensions of $A$ are the poles of themeromorphic continuation of the fractal zeta function of $A$ to a suitableconnected neighborhood of the "critical line" $\{\Re(s)=D\}$. We establishseveral meromorphic extension results, assuming some suitable information aboutthe second term of the asymptotic expansion of the tube function $|A_t|$ as$t\to0^+$, where $A_t$ is the Euclidean $t$-neighborhood of $A$. We payparticular attention to a class of Minkowski measurable sets, such that$|A_t|=t^{N-D}(\mathcal M+O(t^\gamma))$ as $t\to0^+$, with $\gamma>0$, and to aclass of Minkowski nonmeasurable sets, such that $|A_t|=t^{N-D}(G(\logt^{-1})+O(t^\gamma))$ as $t\to0^+$, where $G$ is a nonconstant periodicfunction and $\gamma>0$. In both cases, we show that $\zeta_A$ can bemeromorphically extended (at least) to the open right half-plane$\{\Re(s)>D-\gamma\}$. Furthermore, up to a multiplicative constant, theresidue 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 averageMinkowski content of $A$), respectively. Moreover, we construct a class offractal strings with principal complex dimensions of any prescribed order, aswell as with an infinite number of essential singularities on the critical line$\{\Re(s)=D\}$.

Complex dimensions of fractals and meromorphic extensions of fractal zeta functions