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arXiv · 1908.07845

Essential singularities of fractal zeta functions

Michel L. Lapidus·Goran Radunović·Darko Žubrinić
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Abstract

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 $D_{\infty} \alpha\}$, 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 $N$.

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Michel L. Lapidus, Goran Radunović, Darko Žubrinić. 2019-08-21. Essential singularities of fractal zeta functions. https://arxiv.org/abs/1908.07845

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