The Riemann zeta-function : its embedding into the Hilbert space over a Lie group

黎曼 zeta 函数：嵌入李群上的希尔伯特空间

基本信息

批准号：
15540047
负责人：
MOTOHASHI Yoichi
金额：
$ 2.24万
依托单位：
Nihon University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540047/
关键词：
automorphic representation zeta-function distribution of primes 保型L函数保型L-函数

项目摘要

The achievements are divided into two categories : the theory of L-functions and the theory of the distribution of prime numbers. The principal result is the proof of the fundamental assertion that the Lindeloef constant for any automorphic L-function is less than or equal to 1/3. The details was published, as a joint work with M.Jutila, in Acta Mathematica of the Royal Swedish Academy (vol.195 (2005)), one of the most internationally acclaimed periodicals in pure mathematics. An extension to any Rankin-Selberg L-function was soon made, again jointly with Jutila ; this time, the constant is less than or equal to 2/3, which, despite a non-uniformity to an extent, supersedes considerably the hitherto best result due to the school of number theorists at Princeton. The work was published orally by Jutila at the international conference 'Multiple Zeta-Functions' (2005) ; Motohashi was unable to participate because of his teaching duty. The details was published together with Motohashi's exp … More ository article on the mean values of zeta-functions, in a famed proceedings volume of the American Mathematical Society (2006). The achievement of those constants or exponents is now appreciated by most specialists as to be very hard to go beyond. Recently, in the theory of the distribution of prime numbers were made extraordinary discoveries. One of them is due to D.A.Goldston, J.Pintz, and C.Y.Yildirim. They found a highly promising way to come close to the solution of the famous 'Twin Prime Conjecture' by showing the existence of small gaps between prime numbers. Motohashi made a contribution by devising an exceptionally short proof of the core part of their result, and the details were published as a quadruple-authored paper in the Proceedings of Japan Academy (2006). Later, Motohashi and Pintz obtained an ideal improvement of the relevant sieve method ; the result is to be published at an international conference to be held at Columbia University (N.Y.), provided Motohashi's teaching schedule allows him to attend the meeting. Albeit this is after the closing of the research, very recently the 14^<th> article (2004) of Motohashi's widely known series on mean values of the zeta and L-functions was exploited in a fundamental way by V.Blomer and G.Harcos. Being inspired by this advance, Motohashi has devised an alternative proof of their result ; and based on it, he has established a complete spectral decomposition of the mean square of the automorphic L-function attached to any irreducible representation in the unitary principal series-in a unified fashion. This is a solution to a long-standing problem concerning L-functions. Motohashi envisages an emergence of a grand theory on such mean values and its applications to the distribution of prime numbers. Less

成就分为两类：L功能理论和素数分布理论。主要结果是基本断言的证据表，任何自动形态L功能的Lindelof常数小于或等于1/3。这些详细信息是与M.Jutila的联合作品发表在皇家瑞典学院的Acta Mathematica（vol.195（2005））中，是纯数学中国际知名的时期之一。很快就与Jutila共同制造了任何Rankin-Selberg L功能的扩展；这次，该常数小于或等于2/3，在某种程度上，取代的人认为这是不均匀性的，因为普林斯顿的数量理论家学校迄今为止，迄今为止的最佳结果。这项工作是由Jutila在国际会议的“多个Zeta-runtions”（2005年）上口头发表的。 Motohashi由于教学职责而无法参加。这些细节与Motohashi的Exp一起发表了……在美国数学学会的著名诉讼中，有关Zeta命令的平均值的更多文章（2006年）。现在，大多数专家都赞赏这些常数或指数的实现，因此很难超越。最近，在质数的分布理论中，已成为非凡的发现。其中之一是由于D.A. Goldston，J。Pintz和C.Y.Yildirim。他们找到了一种高度验证的方式，可以通过显示质数之间存在很小的差距，从而接近著名的“双重猜想”解决方案。 Motohashi通过设计了结果的核心部分，做出了贡献，并在日本学院会议记录（2006）的会议记录中发表了详细信息。后来，Motohashi和Pintz获得了相关筛子方法的理想改进。结果将在纽约州哥伦比亚大学（纽约州）举行的国际会议上发布，前提是Motohashi的教学时间表使他能够参加会议。尽管这是在研究结束之后，最近，Motohashi的14^<th>文章（2004）广为Zeta和L功能的平均值众所周知，以V.Blomer和G.Harcos的基本方式探索了Zeta和L功能的平均值。受到这一进步的启发，Motohashi设计了其结果的另一种证据。基于这一点，他已经建立了在统一的统一统一统一序列中附加到任何不可约代表的自动型L功能的均匀平方的完整频谱分解。这是解决有关L功能的长期问题的解决方案。 Motohashi设想了关于这种平均值及其在质数分布的应用的紧急理论。较少的