分野別セミナー

講演概要：Let $A$ be a discrete subset of ${\mathbb R}^n$. For example, set $A$ may be a subset of ${\mathbb Z}^n$, defined by certain arithmetic or geometric constraints. Alternatively, it may be an open set of an algebraic variety, a discrete fractal, and so on. Let $\| \|$ be a norm or pseudo-norm of ${\mathbb R}^n$. The zeta function associated with $A$ (and the norm $\| \|$) is formally defined as follows: $\zeta_A(s):=\sum_{x \in A; x \neq 0} \|x\|^{-s}$ $(s\in {\mathbb C})$. The existence and properties of the meromorphic continuation of $\zeta_A(s)$ depend on the nature of the arithmetic set $A$ and the pseudo-norm $\| \|$. In this talk, I will outline some general methods wich enable to study zeta functions associated to certain classes of aithmetical sets of very different natures. Several arithmetic and geometric properties of $A$ can be covered indirectly from the analytical properties of $\zeta_A$ via Taubelian arguments. I will also present a multivariable Tauberian theorem that can be useful when only little information is available about $\zeta_A$. Finally, I will introduce the concept of zeta-correlation between two arithmetic sets and illustrate it with examples.