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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A theorem on zeta functions
associated with polynomials


Authors: Minking Eie and Kwang-Wu Chen
Journal: Trans. Amer. Math. Soc. 351 (1999), 3217-3228
MSC (1991): Primary 11M06
Published electronically: April 20, 1999
MathSciNet review: 1443872
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Abstract | References | Similar Articles | Additional Information

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.


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Additional Information

Minking Eie
Affiliation: Institute of Applied Mathematics, National Chung Cheng University, Ming-Hsiung, Chia-Yi 621, Taiwan, Republic of China
Email: mkeie@math.ccu.edu.tw

Kwang-Wu Chen
Affiliation: Institute of Applied Mathematics, National Chung Cheng University, Ming-Hsiung, Chia-Yi 621, Taiwan, Republic of China

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02027-9
PII: S 0002-9947(99)02027-9
Received by editor(s): August 11, 1995
Received by editor(s) in revised form: February 4, 1997
Published electronically: April 20, 1999
Additional Notes: This work was supported by Department of Mathematics, National Chung Cheng University and National Science Foundation of Taiwan, Republic of China
Article copyright: © Copyright 1999 American Mathematical Society