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

这些成果可分为两类：l -函数理论和素数分布理论。主要的结果是证明了一个基本的断言，即任何自同构l函数的林德洛夫常数都小于或等于1/3。这些细节作为与M.Jutila的合作成果发表在瑞典皇家科学院数学学报（卷195(2005)）上，这是纯数学领域最受国际认可的期刊之一。对Rankin-Selberg l -函数的扩展很快就得到了，还是和Jutila一起；这一次，常数小于或等于2/3，尽管在某种程度上不均匀，但它在很大程度上取代了迄今为止由普林斯顿数论学派得出的最佳结果。该研究成果由Jutila在“多重ζ函数”国际会议（2005）上口头发表；元桥因为他的教学职责而无法参加。这些细节与Motohashi的exp（关于ζ函数均值的更多历史文章）一起发表在著名的美国数学学会论文集（2006）上。这些常数或指数的成就现在被大多数专家认为是很难超越的。近年来，在素数分布理论方面有了一些非凡的发现。其中之一是D.A.Goldston， J.Pintz和c.y.y yildirim。他们发现了一种非常有前途的方法，通过展示素数之间存在小间隙来接近著名的“双素数猜想”的解。Motohashi通过设计一个特别简短的证明来证明他们的结果的核心部分，并将其细节作为四人合著的论文发表在《日本科学院学报》（2006）上。后来，Motohashi和Pintz对相关筛法进行了理想的改进；该研究结果将在哥伦比亚大学（纽约）举行的一次国际会议上发表，前提是元桥的教学日程允许他参加会议。尽管这是在研究结束之后，最近Motohashi广为人知的关于zeta和l函数均值的系列的14^< >文章（2004）被v.b omer和g.h orcos以一种基本的方式利用。受到这一进展的启发，Motohashi为他们的结果设计了另一种证明；在此基础上，他以一种统一的方式建立了附属于酉主级数中任何不可约表示的自同构l函数的均方的完全谱分解。这是一个长期存在的关于l函数的问题的解决方案。元桥设想了一个关于这些平均值及其在素数分布中的应用的宏大理论的出现。少

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