Zeta-functions and hypergeometric functions

Zeta 函数和超几何函数

基本信息

批准号：
14540051
负责人：
KANEMITSU Shigeru
金额：
$ 2.24万
依托单位：
Kinki 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

The expected objective of studying the functional equation through the modular relation was almost completed thanks to the generous support of the research grant from the JSPS. Now we are ready to launch on the next stage of covering the theory of zeta-functions which has been studied in the last 160 years from the time of Eisenstein and Riemann.In this year we applied our results that we obtained in the previous two years to related fields of physics and chemistry. Namely we studied the fundamental subject of these disciplines-crystal (structures) and the associated constant, the Madelung constant-numerically through the associated Epstein zeta-function In three of the papers that were published in 2004, we have succeeded in incorporating all the existing results into the framework of modular relations, or the Bessel series expansion or the incomplete gamma series as their manifestations.Also in the mean square theory of zeta-and L-functions, we have extracted the core of the stuff, the Euler digamma function, and succeeded in deriving the complete asymptotic expansion for the means square of the value of the Dirichlet L-function at 1, by making full use of the special function-theoretic aspects, thus making it possible to push the situation forward further to cover the case of the Lerch zeta-function.As regards the Hurwitz zeta-function, we completed our research on the derivation of all information on it from its partial sum and now we are able to derive all formulas needed e.g. in zeta-regularization.Finally, we organized an international symposium on "Zeta functions, Topology and Quantum Physics" at Kinki University, Osaka in 2003 and as its continuation we organized another one in 2004 about "Quantum Computation". We are going to publish the proceedings of these two symposia.

在JSPS研究经费的大力支持下，通过模关系研究函数方程的预期目标已基本完成。现在我们准备开始下一阶段的 zeta 函数理论研究，该理论从爱森斯坦和黎曼时代起已经研究了 160 年。今年我们应用了前两年获得的成果物理、化学等相关领域。也就是说，我们通过相关的爱泼斯坦 zeta 函数以数值方式研究了这些学科的基础主题——晶体（结构）和相关常数马德隆常数。在 2004 年发表的三篇论文中，我们成功地将所有现有的结果进入模关系的框架，或者以贝塞尔级数展开或者不完全伽玛级数为表现形式。同样在zeta函数和L函数的均方理论中，我们提取出了核心的东西，欧拉双伽玛函数，并充分利用函数论的特殊方面，成功地推导了狄利克雷L函数在1处的均方的完全渐近展开式，从而使情况向前发展成为可能进一步讨论 Lerch zeta 函数的情况。关于 Hurwitz zeta 函数，我们完成了从其部分和推导所有信息的研究，现在我们能够推导所有信息需要的公式例如最后，我们于 2003 年在大阪近畿大学组织了一场关于“Zeta 函数、拓扑和量子物理”的国际研讨会，作为其延续，我们于 2004 年组织了另一场关于“量子计算”的国际研讨会。我们将出版这两个研讨会的会议记录。