Unified and mutually evolving study of various multiple zeta values

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14294
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Aichi Prefectural University
• Principal Investigator: Tasaka Koji 愛知県立大学, 情報科学部, 准教授 (30780762)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 多重ゼータ値 / 有限多重ゼータ値 / q類似 / モジュラー形式 / 多重Eisenstein級数 / 金子-Zagier予想 / Broadhurst-Kreimer予想 / 多重モジュラー値

Outline of Final Research Achievements

In the study of number theory, special values of zeta functions are an important subject of research involving various fields of mathematics and theoretical physics. In this study, we have obtained several results concerning the study of multiple zeta values, which are multivariable version of the Riemann zeta function, and their relationship with finite multiple zeta values and modular forms. In particular, in the development of a unified theory applicable to multiple zeta-valued variants and other variants introduced in various backgrounds, we have constructed a basic theory of multiple Eisenstein series and found applications to the generalized Kaneko-Zagier conjecture with supercongruences of q-analogue of multiple zeta values.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

今回の成果から，金子-Zagier予想は混合Tateモチーフの周期である数のクラスについても期待できる現象であることがわかり，金子-Zagier予想の真髄に一歩近づくことができた。また，四半世紀未解決であるBroadhurst-Kreimer予想を部分的に解析できたことも今後につながる材料となりうる。こういった予想の解析から次の時代の数学がどんどん芽生えているという意味では，本研究の学術的意義は高い。