engleski

Lapidus zeta functions of arbitrary fractals and compact sets in Euclidean spaces

A new class of zeta functions, for which I propose the name indicated in the title of this talk, has been discovered by Professor Michel L. Lapidus in 2009, during my lecture delivered at a conference at the University of Catania, Italy. The discovery has been disclosed to me immediately after the lecture, and it was a starting point of our joint work. The new zeta functions represent a bridge between the geometry of fractal sets and complex analysis. In this talk I have the privilege of reporting the main results from this paper, some of them obtained very recently. Assume that $A$ is a nonempty, bounded set in $R^N$, and let $d(x, A)$ denote the Euclidean distance from $x\in R^N$ to $A$. Fixing any $\delta>0$, let $A_\delta$ be an open $\d$-neighbourhood of $A$. Then the {;\em Lapidus zeta function}; (in \cite{;LaRaZu}; we call it the {;\em distance zeta function};) is defined as follows: $$ \zeta_A(s)=\int_{;A_\delta};d(x, A)^{;s-N};dx. $$ Here $s$ is a complex number, and the integral is understood in the sense of Lebesgue. This zeta function has several remarkable properties. (50 min. lecture delivered by D. Zubrinic)

Lapidus zeta functions; distance zeta function; tube zeta function; fractal sets; minkowski content; box dimension; meromorphic function

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