A theorem on zeta functions associated with polynomials

Abstract:

Let $\beta =(\beta _{1},\ldots ,\beta _{r})$ be an $r$-tuple of non-negative integers and $P_{j}(X)$ $(j=1,2,\ldots ,n)$ be polynomials in ${\mathbb {R}}[X_{1},\ldots ,X_{r}]$ such that $P_{j}(n)>0$ for all $n\in {\mathbb {N}}^{r}$ and the series \begin{equation*}\sum _{n\in {\mathbb {N}}^{r}} P_{j}(n)^{-s}\end{equation*} is absolutely convergent for Re $s>\sigma _{j}>0$. We consider the zeta functions \begin{equation*}Z(P_{j},\beta ,s)=\sum _{n\in {\mathbb {N}}^{r}}n^{\beta } P_{j}(n)^{-s},\quad \text {Re} s>|\beta |+\sigma _{j}, \quad 1\leq j\leq n.\end{equation*} All these zeta functions $Z(\prod ^{n}_{j=1} P_{j},\beta ,s)$ and $Z(P_{j},\beta ,s)\quad (j=1,2,\ldots ,n)$ are analytic functions of $s$ when Re$s$ is sufficiently large and they have meromorphic analytic continuations in the whole complex plane. In this paper we shall prove that \begin{equation*}Z(\prod _{j=1}^{n} P_{j},\beta ,0)=\frac {1}{n} \sum _{j=1}^{n} Z(P_{j},\beta ,0).\end{equation*} As an immediate application, we use it to evaluate the special values of zeta functions associated with products of linear forms as considered by Shintani and the first author.