Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-99-02027-9
Published electronically: April 20, 1999
MathSciNet review: 1443872
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

  • 1. Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer-Verlag, 1989. MR 90b:01039
  • 2. P. Cassou-Nouguès, Valeurs aux entiers négatifs des functions zêta $p$-adiques, Invent. Math. 51 (1979), 25-59.
  • 3. -, Valeurs aux entiers négatifs des séries de Dirichlet associées à un polynôme I, J. Number Theory 14 (1982), 32-64.MR 83e:12012
  • 4. -, Séries de Dirichlet et intégrales associées à un polynôme a deux indéterminés, J. Number Theory 23 (1986), 1-54.MR 87j:11086
  • 5. Minking Eie, On a Dirichlet series associated with a polynomial, Proc. Amer. Math. Soc. 110 (1990), 583-590.MR 91m:11071
  • 6. -, The special values at negative integers of Dirichlet series associated with polynomials of several variables, Proc. Amer. Math. Soc. 119 (1993), 51-61.MR 93k:11082
  • 7. -, A note on Bernoulli numbers and Shintani generalized Bernoulli polynomial, Trans. Amer. Math. Soc. 348 (1996), 1117-1136.MR 96h:11011
  • 8. I. M. Gelfand and G. E. Shilov, Generalized function, vol. 1, Academic Press, 1964.MR 55:8786a
  • 9. T. Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), 393-417.MR 55:266
  • 10. -, On the values at $s=1$ of certain $L$-functions of totally real algebraic number fields, Algebraic number theory, International Symposium, Kyoto, 1976.MR 56:11962
  • 11. D. Zagier, Valeurs des fonctions zêta des corps quadratiques réels aux entiers négatifs, Astérisque (1977), 41-42. MR 52:10684

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11M06

Retrieve articles in all journals with MSC (1991): 11M06


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: https://doi.org/10.1090/S0002-9947-99-02027-9
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

American Mathematical Society