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

TL;DR

This paper extends Estermann's 1928 theorem to multivariate polynomials, characterizing the domain of meromorphy of associated Euler products and identifying their natural boundaries, with applications to toric varieties.

Contribution

It generalizes Estermann's theorem from one variable to multivariate polynomials, providing a detailed description of the natural boundary of the associated Euler products.

Findings

Determined the maximal domain of meromorphy for multivariate Euler products.

Precisely described the natural boundary when it exists.

Applied results to compute natural boundaries for toric varieties.

Abstract

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