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 函数和 Fox H 函数作为积分核，成功地给出了模关系的一个新的、最通用的公式。这样，我们可以更清楚地理解之前许多著作中出现的zeta函数的函数方程的作用。我们的公式还使我们能够在更广泛的背景下推广许多算术公式。例如，我们推广了达文波特-西格尔的算术傅里叶级数并得到了有趣的例子。我们还给出了 Hurwitz zeta 函数的函数方程的新证明，并且受到 Espinosa-Moll 和 Mikolas 工作的启发，我们推导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关系结果的书。我们也对多个 zeta 函数感兴趣。特别是对于Euler-Zagier型的双zeta函数，我们利用Krazel的双指数和理论，在所谓的“临界带”中给出了一个非平凡的上界。这是对先前已知结果的有意义的改进。我们可以预期我们的结果在算术函数理论中有很多应用。我们还改进了临界带中三重 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 函数。我们给出了解析连续可能性的判据，并构造了一个具有自然边界的zeta函数的例子。