Euler products associated to multivariate rational functions: maximal domain of meromorphy, zeros and poles
Abstract
We determine the maximal domain of meromorphy of any Euler product $$f(s_1,\dots,s_k) = \prod_p h(p^{-s_1}, \ldots, p^{-s_k}),$$ where $h$ is the quotient of two polynomials in $k$ variables, with integer coefficients and with constant term equal to $1$. More precisely, we define a domain $\Gamma \subseteq \mathbb C^k$, on which $f$ admits a meromorphic extension, and such that for any $\boldsymbol z$ in $\partial \Gamma$, there is no neighborhood of $\boldsymbol v z$ on which $f$ admits a meromorphic extension. This maximal domain $\Gamma$ is described as $\mathring K + \mathrm i \mathbb R^k$, where $K \in \mathbb R^k$ is a rational cone, computed from the set of exponents of the two polynomials defining $h$. We also describe the divisor of $f$ over $\Gamma$, which comes from the local factors $h(p^{-s_1}, \ldots, p^{-s_k})$, and from the zeta factors $\zeta (\alpha_1 s_1 + \cdots +\alpha_k s_k)^{-c_{\boldsymbol{\alpha}}}$, where $\zeta$ denotes the Riemann $\zeta$-function, corresponding to terms in the expansion of $h$ as a formal infinite product $\prod_{\boldsymbol{\alpha}} (1-X_1^{\alpha_1}\cdots X_k^{\alpha_k})^{c_{\boldsymbol{\alpha}}}$. We focus our study on the hyperplanes in the divisor, allowing us to use tools developed for the single variable case. We complete our study by giving a geometric and arithmetic description of the set of exponents occurring in the infinite product expansion of $h$, and by showing a new result on the geometric nature of the set of singular points of a holomorphic function defined over a tubular domain.
