Conference "Zeta Functions"

Analytic continuation of zeta functions and self-similarity

Driss Essouabri (Saint-Étienne)

Tuesday 22 June, 11:30 - 12:30

Abstract

Let $A$ be an arithmetical subset of a euclidean space $(E,q)$ (eg. $A$ is a subset of a Lattice defined by arithmetical conditions). Several arithmetic and geometric information of $A$ can be deduced from the analytic properties of its zeta function $\zeta(A;s)=\sum_{m\in A}' q(m)^{-s}$; more precisely from its meromorphic continuation, the distribution of its poles, etc.. If $A$ has some algebraic or analytic regularity, one can then use the analytic or algebraic machinery to extend analytically $\zeta(A;s)$. The purpose of this talk is to introduce a method to analytically continue the zeta functions $\zeta(A;s)$ associated to arithmetic sets $A$ possibly irregulars, but with additional fractal structures. The idea is to exploit self-similarity instead of algebraic or analytic regularity.