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函数理论和素数分布理论。主要成果是证明了任何自守L函数的Lindeloef常数小于或等于的基本断言。 1/3 与 M.Jutila 合作发表在《瑞典皇家科学院数学学报》（第 195 卷（2005 年））上，这是国际上最受好评的纯数学期刊之一。很快又与 Jutila 联合对任何 Rankin-Selberg L 函数进行了扩展；这一次，该常数小于或等于 2/3，尽管在一定程度上不均匀，但它在很大程度上取代了迄今为止最好的成果归功于普林斯顿大学的数论学家学院，Jutila 在国际会议“多重 Zeta 函数”（2005 年）上口头发表了该成果；由于他的教学职责而无法参加，详细信息与本桥关于 zeta 函数平均值的实验文章一起发表在美国数学会的著名会议录中（2006 年）。常数或指数现在被大多数专家认为是很难超越的。最近，在素数分布理论中，D.A.Goldston 取得了非凡的发现。 J.Pintz 和 C.Y.Yildirim 通过证明素数之间存在小间隙，找到了一种非常有前途的方法来接近著名的“孪生素数猜想”的解决方案，他们通过设计一个非常简短的证明做出了贡献。他们的成果的核心部分，详细内容以四人合着论文的形式发表在《日本学士院学报》（2006）上，随后本桥和平茨对相关成果进行了理想的改进。筛法；结果将在哥伦比亚大学（纽约）举行的国际会议上发表，前提是本桥的教学安排允许他参加会议，尽管这是在研究结束后，最近的 14^< V.Blomer 和 G.Harcos 受到启发，从根本上利用了 Motohashi 广为人知的关于 zeta 和 L 函数平均值的系列文章 (2004)。通过这一进步，本桥设计了他们的结果的另一种证明；并在此基础上，他建立了自同构 L 函数的均方的完整谱分解，该函数依附于酉主级数中的任何不可约表示——以统一的形式。这是有关 L 函数的长期问题的解决方案，本桥设想出现关于此类平均值的宏大理论及其在素数分布上的应用。