Beneath on analytic properties ofvarious zeta-functions

下面是各种 zeta 函数的解析性质

基本信息

批准号：
17540022
负责人：
TANIGAWA Yoshio
金额：
$ 2.39万
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：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

项目摘要

In this research, we succeeded to give a new and the most general formulation of the modular relation by using Meijer's G-function and the Fox H-function as an integral kernel. In this way, we can understand more clearly the role of the functional equations of zeta functions which appeared in many previous works. Our formulation also enables us to generalize many arithmetical formulas in wider context. For example, we generalized Davenport-Segal's arithmetical Fourier series and got interesting examples. We also gave a new proof of the functional equation of the Hurwitz zeta-function and, inspired by the work of Espinosa-Moll and Mikolas, we derived many arithmetically interesting integral formulas. We are now writing a book on our results on the general modular relations.We are also interested in multiple zeta functions. Especially for double zeta function of Euler-Zagier type, we gave, by employing Krazel's theory of double exponential sum, a non-trivial upper bound in the so-called "critical strip". This is a meaningful improvement of previously known results. We can expect that our result has many applications in the theory of arithmetical functions. We also improved the upper bounds for the triple zeta functions in the critical strip. As another type of zeta functions, we investigated zeta-functions associated with polynomials. We gave the criterion of the possibility of analytic continuation and constructed an example of zeta-function with a natural boundary.

在这项研究中，我们通过使用Meijer的G功能和FOX H功能作为整体内核来成功提供模块化关系的新的，最通用的模块关系。这样，我们可以更清楚地了解Zeta函数功能方程的作用，Zeta函数出现在许多以前的作品中。我们的配方还使我们能够在更广泛的背景下概括许多算术公式。例如，我们将Davenport-Segal的算术傅立叶系列概括，并获得了有趣的例子。我们还提供了Hurwitz Zeta功能功能方程式的新证明，并受到Espinosa-Moll和Mikolas的作品的启发，我们得出了许多算术上有趣的积分公式。我们现在正在写一本关于一般模块化关系结果的书。我们也对多个Zeta功能感兴趣。特别是对于Euler-Zagier类型的双重Zeta函数，我们通过采用Krazel的双指数总和理论，即所谓的“临界条”中的非平凡的上限。这是对先前已知结果的有意义的改进。我们可以期望我们的结果在算术函数理论中有许多应用。我们还改善了临界条中三重zeta函数的上限。作为另一种类型的Zeta函数，我们研究了与多项式相关的Zeta功能。我们给出了分析延续的可能性的标准，并构建了具有自然边界的Zeta功能的例子。