Relation between the Riemann zeta-function and the Casimir operator

黎曼 zeta 函数与卡西米尔算子之间的关系

• 批准号: 12640043
• 负责人: MOTOHASHI Yoichi
• 金额: $ 2.37万
• 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640043/
• 关键词: zeta-function; automorphic forms; representation theory; distribution of prime numbers; Casimir operator; spectral decomposition; ゼータ関数; リー群; 表現論

The principal purpose of our research was to develop a theory to grasp the Riemann zeta-function ζ(s) in the geometric perspective of Lie groups, dispensing with the theory of Kloosterman sums. We, under a joint research activity with R.W. Bruggeman, achieved this aim. We could embed the fourth power moment of ζ(s) in the space L^2(PSL_2(Z)\PSL_2R) as a special value of a certain Poincare series, and retrieved its known spectral decomposition in a new way, i.e., completely without recourse to the theory of Kloosterman sums. This work of ours revealed also a deep relation between ζ(s) and the Jacquet-Langlands local functional equation. This fact will possibly make it feasible to extend our theory to higher dimensional situation, which otherwise should have been almost impossible. Our result could be assessed as to be a basic achievement in the theory of zeta-functions.

我们研究的主要目的是发展一种理论，以从李群的几何角度理解黎曼 zeta 函数 ζ(s)，而无需使用 Kloosterman 和的理论。我们与 R.W. Bruggeman 的联合研究活动取得了成果。我们可以将 ζ(s) 的四次方矩嵌入到空间 L^2(PSL_2(Z)\PSL_2R) 中作为某个特定值。庞加莱级数，并以一种新的方式检索其已知的谱分解，即完全不求助于 Kloosterman 和的理论。我们的这项工作还揭示了 z(s) 和 Jacquet-Langlands 局部函数方程之间的深层关系。事实可能使我们的理论扩展到更高维度的情况成为可能，否则我们的结果可以被评估为理论的基本成就。 zeta 函数。

项目成果

Y. Motohashi: "New analytic problems over imaginary quadratic number fields"Proc. Number Theory Conference in Memory of K. Inkeri, de Gruyter. 255-279 (2001)

Y. Motohashi：“虚数二次数域的新分析问题”Proc。

本橋洋一(Y.Motohashi): "Riemann予想(英文:The Riemann Hyporhesis)"American Math. Soc. Expositions. (掲載決定). (2003)

Y. Motohashi：“黎曼猜想”美国数学会（2003 年出版）。

R.W. Bruggeman and Y. Motohashi: "Fourth moment of Dedekind zeta-functions of real quadratic number fields of class number one"Functiones et Approximatio. 29. 41-79 (2001)

R.W. Bruggeman 和 Y. Motohashi：“第一类实二次数域的 Dedekind zeta 函数的第四矩”函数和近似。

R.W.Bruggeman, Y.Motohashi: "A new approach to the spectral theory of the fourth moment of the Riemann zeta-function"Proc. MPIM-Bonn Special Workshop on ANT. (掲載決定). (2003)

R.W.Bruggeman、Y.Motohashi：“黎曼 zeta 函数第四矩谱理论的新方法”Proc. MPIM-Bonn 特别研讨会（2003 年出版）。

R.W.Bruggeman, Y.Motohashi: "A note on the mean value of the zeta and L-functions X"Proc.Japan Academy. 77A. 111-114 (2001)

R.W.Bruggeman、Y.Motohashi：“关于 zeta 和 L 函数 X 平均值的注释”Proc.Japan Academy。