Behaviour at the non-positive integers of Dirichlet series associated to polynomials of several variables

Abstract

In this paper, we relate the special values at a non-positive integer ${\underline{\mathbf{s}}=(s_{1},\ldots, s_{r})= -\underline{\mathbf{N}}= (-N_{1},\ldots, -N_{r})}$ obtained by meromorphic continuation of the multiple Dirichlet series ${{Z(\underline{\mathbf{P}}, \underline{\mathbf{s}})=\sum_{\underline{m} \in {\mathbb{N}}^{*n}}{\frac{1}{\prod_{i=1}^{r}{P_{i}^{ s_{i}}(\underline{m})}}}}}$ to special values of the function ${Y(\underline{\mathbf{P}}, \underline{\mathbf{s}})=\int_{[1, +\infty[^{n}} {\prod_{i=1}^{r}{P_{i}^{- s_{i}}(\underline{\mathbf{x}})} \; d{\underline{\mathbf{x}}}}}$ where ${\underline{\mathbf{P}}=(P_{1},..., P_{r}),\; (r\geq 1)}$ are elliptic polynomials in “${n}$ ” variables. We prove a simple relation between ${Z(\underline{\mathbf{P}}_{\underline{\mathbf{a}}}, -\underline{\mathbf{N}})}$ and ${Y(\underline{\mathbf{P}}_{\underline{\mathbf{a}}}, -\underline{\mathbf{N}})}$, such that for all ${\underline{\mathbf{a}} \in {\mathbb{R}}^{n}_{+}}$, we denote ${\underline{\mathbf{P}}_{\underline{\mathbf{a}}}:=(P_{1 \underline{\mathbf{a}}},\ldots, P_{r \underline{\mathbf{a}}})}$, where ${P_{i\;\underline{\mathbf{a}}}(\underline{\mathbf{x}}):= P_i(\underline{\mathbf{x}}+ \underline{\mathbf{a}})\; (1\leq i\leq r)}$ is the shifted polynomial.