Extension of Estermann's theorem to Euler products associated to a multivariate polynomial
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2010
- DOI:
- 10.48550/arXiv.1001.3838
- arXiv:
- arXiv:1001.3838
- Bibcode:
- 2010arXiv1001.3838D
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Complex Variables;
- 11M32;
- 11M41;
- 32D15;
- 11N99;
- 14G05
- E-Print:
- 29 pages