Beneath on analytic properties ofvarious zeta-functions

下面是各种 zeta 函数的解析性质

• 批准号: 17540022
• 负责人: TANIGAWA Yoshio
• 金额: $ 2.39万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540022/
• 关键词: functional equation; modular relation; H-function; Mellin-Barnes integral; double zeta-function; exponential sum; zeta-function associated with polynomial; メリンーバーンズ積分; モジュラー関係式; 多重ゼータ関数; ワイル変換; ゼータ関数; 部分分数展開; ダブルゼータ関数; オーダー評価

In this research, we succeeded to give a new and the most general formulation of the modular relation by using Meijer's G-function and the Fox H-function as an integral kernel. In this way, we can understand more clearly the role of the functional equations of zeta functions which appeared in many previous works. Our formulation also enables us to generalize many arithmetical formulas in wider context. For example, we generalized Davenport-Segal's arithmetical Fourier series and got interesting examples. We also gave a new proof of the functional equation of the Hurwitz zeta-function and, inspired by the work of Espinosa-Moll and Mikolas, we derived many arithmetically interesting integral formulas. We are now writing a book on our results on the general modular relations.We are also interested in multiple zeta functions. Especially for double zeta function of Euler-Zagier type, we gave, by employing Krazel's theory of double exponential sum, a non-trivial upper bound in the so-called "critical strip". This is a meaningful improvement of previously known results. We can expect that our result has many applications in the theory of arithmetical functions. We also improved the upper bounds for the triple zeta functions in the critical strip. As another type of zeta functions, we investigated zeta-functions associated with polynomials. We gave the criterion of the possibility of analytic continuation and constructed an example of zeta-function with a natural boundary.

In this research, we succeeded to give a new and the most general formulation of the modular relation by using Meijer's G-function and the Fox H-function as an integral kernel. In this way, we can understand more clearly the role of the functional equations of zeta functions which appeared in many previous works. Our formulation also enables us to generalize many arithmetical formulas in wider context. For example, we generalized Davenport-Segal's arithmetical Fourier series and got interesting examples. We also gave a new proof of the functional equation of the Hurwitz zeta-function and, inspired by the work of Espinosa-Moll and Mikolas, we derived many arithmetically interesting integral formulas. We are now writing a book on our results on the general modular relations.We are also interested in multiple zeta functions. Especially for double zeta function of Euler-Zagier type, we gave, by employing Krazel's theory of double exponential sum, a non-trivial upper bound in the so-called "critical strip". This is a meaningful improvement of previously known results. We can expect that our result has many applications in the theory of arithmetical functions. We also improved the upper bounds for the triple zeta functions in the critical strip. As another type of zeta functions, we investigated zeta-functions associated with polynomials. We gave the criterion of the possibility of analytic continuation and constructed an example of zeta-function with a natural boundary.

项目成果

Sums involving the Hurwitz zeta-function values-

涉及 Hurwitz zeta 函数值的总和 -

• 期刊: Proc.Intern.Symposium, "Zeta functions Topolpgy and Quantum Physics"(Kluwer Academic Publisher) (to appear)
• 作者: S.Kanemitsu;A.Schinzel;Y.Tanigawa
• 通讯作者: Y.Tanigawa

Evaluation of Spannenintegral of the product of zeta function

Zeta 函数乘积的 Spannen 积分评估

• 发表时间: 2008
• 期刊: Integral Transform and Special Functions 19
• 作者: S. Kanemitsu;Y. Tanigawa and J. Zhang
• 通讯作者: Y. Tanigawa and J. Zhang

Some number theoretic application of a general modular relation

一般模关系的一些数论应用

• 发表时间: 2006
• 期刊: International Journal of Number Theory 2
• 作者: S. Kanemitsu;Y. Tanigawa;and H. Tsukada
• 通讯作者: and H. Tsukada

Bounds for double zeta function

双 zeta 函数的界限

• 发表时间: 2006
• 期刊: Ann. Scoula Norm. Sup. Pisa V
• 作者: S. Kanemitsu;Y. Tanigawa;and H. Tsukada;I. Kiuchi and Y. Tanigawa
• 通讯作者: I. Kiuchi and Y. Tanigawa

Crystal Symmetry viewed as Zeta Symmetry

晶体对称性被视为 Zeta 对称性

• 发表时间: 2005
• 期刊: Proc. Intern. Symp."Zeta-Function, Topology and Quantum Physics"Springer
• 作者: S. Kanemitsu;Y. Tanigawa;H. Tsukada and M. Yoshimoto
• 通讯作者: H. Tsukada and M. Yoshimoto