  We study the essential singularities of geometric zeta functions
$\zeta_{\mathcal L}$, associated with bounded fractal strings $\mathcal L$. For
any three prescribed real numbers $D_{\infty}$, $D_1$ and $D$ in $[0,1]$, such
that $D_{\infty}<D_1\le D$, we construct a bounded fractal string $\mathcal L$
such that $D_{\rm par}(\zeta_{\mathcal L})=D_{\infty}$, $D_{\rm
mer}(\zeta_{\mathcal L})=D_1$ and $D(\zeta_{\mathcal L})=D$. Here,
$D(\zeta_{\mathcal L})$ is the abscissa of absolute convergence of
$\zeta_{\mathcal L}$, $D_{\rm mer}(\zeta_{\mathcal L})$ is the abscissa of
meromorphic continuation of $\zeta_{\mathcal L}$, while $D_{\rm
par}(\zeta_{\mathcal L})$ is the infimum of all positive real numbers $\alpha$
such that $\zeta_{\mathcal L}$ is holomorphic in the open right half-plane
$\{{\rm Re}\, s>\alpha\}$, except for possible isolated singularities in this
half-plane. Defining $\mathcal L$ as the disjoint union of a sequence of
suitable generalized Cantor strings, we show that the set of accumulation
points of the set $S_{\infty}$ of essential singularities of $\zeta_{\mathcal
L}$, contained in the open right half-plane $\{{\rm Re}\, s>D_{\infty}\}$,
coincides with the vertical line $\{{\rm Re}\, s=D_{\infty}\}$. We extend this
construction to the case of distance zeta functions $\zeta_A$ of compact sets
$A$ in $\mathbb{R}^N$, for any positive integer $N$.
