Conference "Zeta Functions"

Maximal domain of meromorphy of uniform multivariable Euler products.

Ludovic Delabarre (Caen)

Monday 21 June, 11:30 - 12:30

Abstract

The aim of this talk is to study the maximal domain of meromorphic extension of uniform Euler products of multivariate ``ganzvertige'' polynomials. The meromorphic continuation of this class of fonctions permits for example, using analytic tools, to obtain interesting results in arithmetic or in group theory. The problem consists first to find an expression of a meromorphic continuation of the Euler product until a certain domain precising the eventual poles or zeros that appear. By giving a necessary and sufficient condition on the polynomial which ensures the existence of a natural boundary (i.e. a boundary beyond which it does not exist meromorphic extension), we extend the classical one variable result of 1928 obtained by T. Estermann. Moreover, this work constitutes a first step towards the resolution of a conjecture of Z. Rudnick and M. du Sautoy concerning the domain of meromorphy of eulerian products associated with the counting of subgroups of a given group.