Extension of Estermann's theorem to Euler products associated to a multivariate polynomial

摘要

Given a multivariate polynomial $h(X_1,...,X_n)$ with integral coefficientsverifying an hypothesis of analytic regularity (and satisfying$h(\textbf{0})=1$), we determine the maximal domain of meromorphy of the Eulerproduct $\prod_{p \ \textrm{prime}}h(p^{-s_1},...,p^{-s_n})$ and the naturalboundary is precisely described when it exists. In this way we extend a wellknown result for one variable polynomials due to Estermann from 1928. As anapplication, we calculate the natural boundary of the multivariate Eulerproducts associated to a family of toric varieties.