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

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

2010-01-21

arxiv.org/abs/1001.3838v3

Mathematics

Number Theory

29 pages

Scientific paper

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.

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