A theorem on zeta functions associated with polynomials

Abstract

Let β = ( β 1 , … , β r ) \beta =(\beta _{1},\ldots ,\beta _{r}) be an r r -tuple of non-negative integers and P j ( X ) P_{j}(X) ( j = 1 , 2 , … , n ) (j=1,2,\ldots ,n) be polynomials in R [ X 1 , … , X r ] {\mathbb {R}}[X_{1},\ldots ,X_{r}] such that P j ( n ) > 0 P_{j}(n)>0 for all n ∈ N r n\in {\mathbb {N}}^{r} and the series ∑ n ∈ N r P j ( n ) − s \begin{equation*}\sum _{n\in {\mathbb {N}}^{r}} P_{j}(n)^{-s}\end{equation*} is absolutely convergent for Re s > σ j > 0 s>\sigma _{j}>0 . We consider the zeta functions Z ( P j , β , s ) = ∑ n ∈ N r n β P j ( n ) − s , Re s > | β | + σ j , 1 ≤ j ≤ n . \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 ( ∏ j = 1 n P j , β , s ) Z(\prod ^{n}_{j=1} P_{j},\beta ,s) and Z ( P j , β , s ) ( j = 1 , 2 , … , n ) Z(P_{j},\beta ,s)\quad (j=1,2,\ldots ,n) are analytic functions of s s when Re s \, s is sufficiently large and they have meromorphic analytic continuations in the whole complex plane. In this paper we shall prove that Z ( ∏ j = 1 n P j , β , 0 ) = 1 n ∑ j = 1 n Z ( P j , β , 0 ) . \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.