Studies on the analytic behaviour of number theoretic L-functions

数论L-函数解析行为的研究

• 批准号: 12440004
• 负责人: MATSUMOTO Kohji
• 金额: $ 8.77万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 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/en/grant/KAKENHI-PROJECT-12440004/
• 关键词: Multiple zeta-function; Euler-Zagier sum; Analytic continuation; Hurwitx zeta-function; Ramanujan's formula; Modular relation; Universality; Automorphic L-function; 漸近展開; Ramanujan formula; 近似関数等式; Mellin-Barnes formula; universality; Rankin-

(1) We introduced the notion of generalized multiple zeta-functions, which is a generalization of both the Euler-Zagier multiple sums and the Barnes multiple zeta-functions, and, by using the Mellin-Barnes integral formula, proved their analytic continuation and asymptotic expansions. As applications, we proved asymptotic expansions of higher power moments of Hurwitz zeta-functions, and also explicit formulas of determinants of the Laplacians of high-dimensional spheres.(2) We found a basic principle connecting Ramanujan's formula, modular relations, and approximate functional equations, and proved rapidly converget series expressions of various L-functions, in connection with multiple zeta-functions.(3) We introduced the positive density method in universality theory, and proved the universality of automorphic L-functions, and Rankin-Selberg L-fanctions, attached to cusp forms of SL (2, II) or its congruence sabgroups, also the joint universality of Lerch zeta-functions.

（1）我们介绍了广义多个Zeta功能的概念，这是Euler-Zagier多个总和和Barnes多个Zeta功能的概括，并且通过使用Mellin-Barnes积分公式，证明了它们的分析延续和渐进扩张。作为应用，我们证明了Hurwitz Zeta命令的更高功率力矩的渐近扩展，以及高维球体Laplacians的决定因素的明确公式。（2）我们找到了一个基本原理，将Ramanujan的基本原理连接到Ramanujan的公式，模块化关系和近似功能方程，以及各种融合的连接，并连接了各种融合的连接。 （3）我们在通用理论中介绍了正密度方法，并证明了自动形态L功能的通用性，以及与SL（2，II）或其一致性SABGROUPS的尖峰形式相连的Rankin-Selberg L-Fanctions，也是LERCH ZETA-ZETA-functions的共同性。

项目成果

S.kanemitsu,H.Kumagai & M.Yoshimoto: "Sums involving the Hurwitz zeta-function"Ramanujan J.. (to appear).

兼光·熊谷

K.Matsumoto, Y.Tanigawa: "The analytic continuation and the order estimate of multiple Dirichlet series"J.Theorie des Nombres de Bordeaux. (to appear).

K.Matsumoto，Y.Tanikawa：“多重狄利克雷级数的解析延拓和阶估计”J.Theorie des Nombres de Bordeaux。

S. Akiyama, Y. Tanigawa: "Multiple zeta values at non-positive integers"Ramanujan J.. (to appear).

S. Akiyama、Y. Tanikawa：“非正整数处的多个 zeta 值”Ramanujan J..（即将出现）。

M.Amou,M.Katsurada & K Vaananeu: "On the values of certain q-hypergeometric series"in "Proc.Turku Symp.on Number Theory",W.de Gruyter,. (to appear).

安茂先生、桂田先生

S.Kanemitsu, Y.Tanigawa, M.Yoshimoto: "Ramanujan's formula and modular forms, Number Theoretic Methods-Future Trends"Kluwer. 159-212 (2002)

S.Kanemitsu、Y.Tanikawa、M.Yoshimoto：“拉马努金的公式和模形式，数论方法 - 未来趋势”Kluwer。