  Recently, the first author has extended the definition of the zeta function
associated with fractal strings to arbitrary bounded subsets $A$ of the
$N$-dimensional Euclidean space ${\mathbb R}^N$, for any integer $N\ge1$. It is
defined by $\zeta_A(s)=\int_{A_{\delta}}d(x,A)^{s-N}\,\mathrm{d} x$ for all
$s\in\mathbb{C}$ with $\operatorname{Re}\,s$ sufficiently large, and we call it
the distance zeta function of $A$. Here, $d(x,A)$ denotes the Euclidean
distance from $x$ to $A$ and $A_{\delta}$ is the $\delta$-neighborhood of $A$,
where $\delta$ is a fixed positive real number. We prove that the abscissa of
absolute convergence of $\zeta_A$ is equal to $\overline\dim_BA$, the upper box
(or Minkowski) dimension of $A$. Particular attention is payed to the principal
complex dimensions of $A$, defined as the set of poles of $\zeta_A$ located on
the critical line $\{\mathop{\mathrm{Re}} s=\overline\dim_BA\}$, provided
$\zeta_A$ possesses a meromorphic extension to a neighborhood of the critical
line. We also introduce a new, closely related zeta function,
$\tilde\zeta_A(s)=\int_0^{\delta} t^{s-N-1}|A_t|\,\mathrm{d} t$, called the
tube zeta function of $A$. Assuming that $A$ is Minkowski measurable, we show
that, under some mild conditions, the residue of $\tilde\zeta_A$ computed at
$D=\dim_BA$ (the box dimension of $A$), is equal to the Minkowski content of
$A$. More generally, without assuming that $A$ is Minkowski measurable, we show
that the residue is squeezed between the lower and upper Minkowski contents of
$A$. We also introduce transcendentally quasiperiodic sets, and construct a
class of such sets, using generalized Cantor sets, along with Baker's theorem
from the theory of transcendental numbers.
