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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 $D_{infty}<D_1le D$, 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 $N$.
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 $d(x,A)$ denote
The theory of zeta functions of fractal strings has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 20
We study fractality of unbounded sets of finite Lebesgue measure at infinity by introducing the notions of Minkowski dimension and content at infinity. We also introduce the Lapidus zeta function at infinity, study its properties and demonstrate its
In 2009, the first author introduced a new class of zeta functions, called `distance zeta functions, associated with arbitrary compact fractal subsets of Euclidean spaces of arbitrary dimension. It represents a natural, but nontrivial extension of th
From the viewpoint of quantum walks, the Ihara zeta function of a finite graph can be said to be closely related to its evolution matrix. In this note we introduce another kind of zeta function of a graph, which is closely related to, as to say, the