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

Ludovic Delabarre (2013)

Bulletin de la Société Mathématique de France

Given a multivariate polynomial h X 1 , , X n with integral coefficients verifying an hypothesis of analytic regularity (and satisfying h ( 0 ) = 1 ), we determine the maximal domain of meromorphy of the Euler product p 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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