Research related to value distribution of zeta function and infinitely divisible distribution

2022 Fiscal Year Final Research Report

• Project/Area Number: 16K05077
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo University of Science
• Principal Investigator: Nakamura Takashi 東京理科大学, 教養教育研究院野田キャンパス教養部, 准教授 (50532355)
• Project Period (FY): 2016-04-01 – 2023-03-31
• Keywords: ゼータ関数 / L関数 / 関数等式 / 実零点 / 無限分解可能性 / 臨界線上の零点

Outline of Final Research Achievements

The contents of the research can be roughly divided into five.
(1) Explicit formulas, value relational expressions, and functional relational expressions for values of multiple zeta functions. (2) Value distribution of zeta function, mainly universality. (3) Zeros of the zeta function. (4) Functional equation of the zeta function. (5) Zeta function and infinite divisibility.
In particular, (4) is new research conducted during the 16K05077 period. All of these studies are related to the zeta function and L-function, which play important roles in modern mathematics. They have a long tradition of more than 150 years, and have been actively studied all over the world since their birth.

Free Research Field

Zeta functions, L-functions

Academic Significance and Societal Importance of the Research Achievements

リーマンゼータ関数と全く同じ関数等式を持ち，かつ無限個の複素零点が臨界線上にあるゼータ関数を歴史上初めて定義した．ゼータ関数の研究はゼータ分布の研究に繋がり，保険数理への応用が期待される。さらに，ゼータ関数の普遍性はその性質から機械学習への応用が期待され、実際に福岡大の天羽氏，岡山理科大の青山氏と共同で，「制御型Loewner--Kufarev方程式の解の形－KPZ方程式の理解に向けて－」というタイトルで画像電子学会へ論文を投稿した．