Research on the special values of various zeta functions

各种zeta函数特殊值的研究

基本信息

批准号：
14540021
负责人：
TANIGAWA Yoshio
金额：
$ 2.5万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

Our research started from the Ramanujan formula for the Riemann zeta-values at positive odd integers. We found that his formula is based on Bochner's modular relations and the Inverse Mellin transform with shifted arguments. As an application of this new viewpoints, we derived the analytic expressions of various zeta-values, jfor examples, the Dirichlet L-function values at integer points, the multiple Hurwitz zeta-values at rational arguments, automorphic L-function values at all positive integers. We also found that the derivation of Ramanujan's formula with respect to the parameter x gives Guinand's formula and automorphy of holomorphic Eisenstein series. We also tried to generalize modular relations using G-and H-functions.The Madelung constants play an important role in crystal chemistry. We regard them as special values of Epstein zeta function and elucidated the previous works of Glasser, Zucker, Chaba-Pathria and others. Applying various formulas for quadratic forms, e.g. Chowla-Selberg formulas, we derived various relations among Madelung constants.Furthermore, we studied the mean square formula for general Dirichlet series with a functional equation and estimated the error term in great detail. We also studied the mean square of the values |L(1,x)|, where we employed the classical formula for digamma function, and derived a very close formula for it. Finally we studied the analytic continuation of multiple zeta function of general type, and order estimate of double zeta function of Euler-Zagier type using the theory of exponential sums.

我们的研究始于Ramanujan公式，用于奇数整数的Riemann Zeta-Values。我们发现他的公式基于Bochner的模块化关系，而梅林逆变换则以转变的论点为基础。作为这种新观点的应用，我们得出了各种Zeta值的分析表达式，示例示例，dirichlet l功能值，在整数点，有理参数的多个hurwitz zeta值整数。我们还发现，Ramanujan公式相对于参数X的推导得出了吉南的公式和全体形态Eisenstein系列的公式和汽车。我们还试图使用G和H功能来概括模块化关系。Madelung常数在晶体化学中起着重要作用。我们认为它们是爱泼斯坦Zeta功能的特殊值，并阐明了格拉瑟，扎克，查巴·帕尔里亚等先前的作品。为二次形式应用各种公式，例如Chowla-selberg公式，我们在Madelung常数之间得出了各种关系。Furthermore，我们研究了具有功能方程的常规Dirichlet系列的均方根公式，并详细估计了错误项。我们还研究了值| l（1，x）|的均方根，在那里我们采用了digamma函数的经典公式，并为其得出了非常紧密的公式。最后，我们研究了通用类型的多个ZETA功能的分析延续，并使用指数总和理论来研究了Euler-Zagier类型的双重Zeta函数的顺序估计。