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The analytic theory of arithmetic L-functions and multiple zeta-functions

Research Project

Project/Area Number 18H01111
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionNagoya University

Principal Investigator

松本 耕二  名古屋大学, 多元数理科学研究科, 教授 (60192754)

Co-Investigator(Kenkyū-buntansha) 見正 秀彦  東京電機大学, システムデザイン工学部, 教授 (10435456)
鈴木 正俊  東京工業大学, 理学院, 准教授 (30534052)
小森 靖  立教大学, 理学部, 教授 (80343200)
Project Period (FY) 2018-04-01 – 2022-03-31
Project Status Granted (Fiscal Year 2021)
Budget Amount *help
¥14,560,000 (Direct Cost: ¥11,200,000、Indirect Cost: ¥3,360,000)
Fiscal Year 2021: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2020: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2019: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2018: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Keywordsゼータ関数 / L関数 / 値分布 / 多重ゼータ関数 / ルート系 / 実零点 / 漸近挙動 / 特殊値 / 普遍性 / 離散普遍性 / 混合普遍性 / ルート系のゼータ関数 / Schur 多重ゼータ関数 / L 関数 / 関数関係式 / 明示公式 / 周期積分 / Hurwitz ゼータ関数
Outline of Annual Research Achievements

今年度、特に進展したのは Euler-Zagier 型の多重ゼータ関数で、変数が全て同一になったいるような場合の、実軸上における挙動の研究である。実軸上での極の位置と、その近傍での関数の漸近的な振る舞い、また負の実軸上での零点の個数については相当に精密な結果まで到達し、さらに正の実軸上の零点の分布についても予想を提示することができた。これらの研究は松本ー東海林による二重の場合の先行研究を大幅に一般化、精密化したものになっている。他方、ルート系のゼータ関数については、共同執筆中のテキストがかなり進み、最終的な点検段階に入っている。

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

今年度はコロナウイルスの大流行により、出張等が全く不可能になった。そのため、十分な研究交流ができず、直接会って議論することによる新しい研究場の展望を得ることについてはあまり進展がなかった。だが、すでに具体化している研究テーマに関しては、その多くが、単独の努力による研究遂行の様態を取るので、そうした段階では研究の遅れはなく、順調な進展が今年度もあったと考える。特に共著テキストの執筆に関しては、普段は各自が単独で書き進めたり点検を行ったりして、特おり ZOOM 等でオンラインでの討論を行うことで、十分に執筆が捗った。

Strategy for Future Research Activity

コロナウイルスの流行の状況に大幅に依存するので予測しづらいが、できる限り直接の交流(無理なら最低限、オンラインでの意見交換)を行って、執筆中の本に関してはこの年度で完成させたい。また本科研費の最終年度であるので、総括的な集会を開き、到達した研究の状況把握と今後への展望を検討する機会としたい。

Report

(3 results)
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • 2018 Annual Research Report

Research Products

(34 results)

All 2020 2019 2018 Other

All Int'l Joint Research (6 results) Journal Article (13 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 13 results,  Open Access: 1 results) Presentation (14 results) (of which Int'l Joint Research: 9 results,  Invited: 14 results) Book (1 results)

  • [Int'l Joint Research] Novi Sad Univ(セルビア)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] Lille 大学/Lyon 大学(フランス)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Vytautas Magnus 大学(リトアニア)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Linz 工科大学(オーストリア)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Univ Lyon(フランス)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Vytautas Magnus Univ(リトアニア)

    • Related Report
      2018 Annual Research Report
  • [Journal Article] Values at non-positive integers of generalized Euler-Zagier multiple zeta-functions2020

    • Author(s)
      D. Essouabri and K. Matsumoto
    • Journal Title

      Acta Arith.

      Volume: 193 Pages: 109-131

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Zeta-functions of root systems and Poincare polynomials of Weyl groups2020

    • Author(s)
      Y. Komori, K. Matsumoto and H. Tsumura
    • Journal Title

      Tohoku Math J.

      Volume: 72 Pages: 87-126

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] An overview and supplements to the theory of functional relations for zeta-functions of root systems2020

    • Author(s)
      Y. Komori, K. Matsumoto and H. Tsumura
    • Journal Title

      Adv. Stud. Pure Math.

      Volume: 84 Pages: 263-295

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Numerical computations on the zeros of the Euler double zeta-functions II2020

    • Author(s)
      K. Matsumoto and M. Shoji
    • Journal Title

      European J. Math.

      Volume: 6 Pages: 488-507

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Laurent series expansions of multiple zeta-functions of Euler-Zagier type at integer points2020

    • Author(s)
      K. Matsumoto, T. Onozuka and I. Wakabayashi
    • Journal Title

      Math. Z.

      Volume: 295 Pages: 623-642

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On mixed joint discrete universality for a class of zeta-functions: a further generalization2020

    • Author(s)
      R. Kacinskaite and K. Matsumoto
    • Journal Title

      Math. Modell. Anal.

      Volume: 25 Pages: 569-583

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the value-distribution of symmetic power L-functions2020

    • Author(s)
      K. Matsumoto and Y. Umegaki
    • Journal Title

      Ramanujan Math. Soc. Lecture Notes

      Volume: 26 Pages: 147-167

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] GOLDBACH REPRESENTATIONS IN ARITHMETIC PROGRESSIONS AND ZEROS OF DIRICHLET L-FUNCTIONS2019

    • Author(s)
      Bhowmik Gautami、Halupczok Karin、Matsumoto Kohji、Suzuki Yuta
    • Journal Title

      Mathematika

      Volume: 65 Pages: 57-97

    • DOI

      10.1112/s0025579318000323

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the density function for the value-distribution of automorphic L-functions2019

    • Author(s)
      K. Matsumoto and Y. Umegaki
    • Journal Title

      J. Number Theory

      Volume: 198 Pages: 176-199

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On mixed joint discrete universality for a class of zeta-functions II2019

    • Author(s)
      R. Kacinskaite and K. Matsumoto
    • Journal Title

      Lithuanian Mathematical Journal

      Volume: 59 Pages: 54-66

    • DOI

      10.1007/s10986-019-09432-1

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Values at non-positive integers of partially twisted multiple zeta-functions I2019

    • Author(s)
      D. Essouabri and K. Matsumoto
    • Journal Title

      Comment. Math. Univ. St. Pauli

      Volume: 67 Pages: 83-100

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] ゼータ関数の普遍性について2019

    • Author(s)
      松本耕二
    • Journal Title

      応用数理

      Volume: 29 Pages: 135-139

    • NAID

      130007775880

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the value-distribution of the difference between logarithms of two symmetric power L-functions2018

    • Author(s)
      K. Matsumoto and Y. Umegaki
    • Journal Title

      International Journal of Number Theory

      Volume: 14 Pages: 2045-2081

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Presentation] Multiple Dirichlet series with arithmetical coefficients on the numerators2020

    • Author(s)
      K. Matsumoto
    • Organizer
      IMSc Webinar, IMS, Chennai, India
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] An M-function associated with Goldbach's problem2020

    • Author(s)
      K. Matsumoto
    • Organizer
      Intern. Conf. on Number Theory and Discrete Math., Kochi, India
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the behavior of multiple zeta-functions with identical arguments on the real line2020

    • Author(s)
      K. Matsumoto
    • Organizer
      Balu Fest, IMSc, Chennai, India
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 多重ゼータ関数の負の整数点での値の表示について2020

    • Author(s)
      松本耕二
    • Organizer
      第13回多重ゼータ研究集会&第48回関西多重ゼータ研究会
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Multiple Dirichlet series with arithmetical coefficients on the numerators2019

    • Author(s)
      Kohji Matsumoto
    • Organizer
      Mini-Symposium on Analytic Number Theory and Applications
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 多重ゼータ関数の零点集合の形状について2019

    • Author(s)
      松本耕二
    • Organizer
      東工大数論・幾何学セミナー
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] On the zero-sets of multiple zeta-functions2019

    • Author(s)
      Kohji Matsumoto
    • Organizer
      Fonctions Zeta
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] \zeta^2(s) の近似関数等式の誤差項の平均値について2019

    • Author(s)
      松本耕二
    • Organizer
      整数論・環論合同セミナー
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] Asymptotic results on mean values of logarithmic derivatives of Dirichlet L-functions2019

    • Author(s)
      K. Matsumoto
    • Organizer
      Conference in Number Theory
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] An overview of the theory of multiple zeta-functions2019

    • Author(s)
      K. Matsumoto
    • Organizer
      Value distribution of zeta and L-functions and related topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Values of general multiple zeta-functions with polynomial denominators at non-positive integer points2019

    • Author(s)
      K. Matsumoto
    • Organizer
      Value distribution of zeta and L-functions and related topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the theory of M-functions2018

    • Author(s)
      K. Katsumoto
    • Organizer
      Profinite monodromy, Galois representations, and complex functions
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] ゼータ関数の普遍性について2018

    • Author(s)
      K. Matsumoto
    • Organizer
      日本応用数理学会2018年度年会
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] On density functions in the value-distribution theory of L-functions2018

    • Author(s)
      K. Matsumoto
    • Organizer
      International Conference on Number Theory dedicated to the 70th birthdays of Professors A. Laurincikas and E. Manstavicius
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Book] ルート系のゼータ関数の理論入門2018

    • Author(s)
      松本耕二
    • Total Pages
      58
    • Publisher
      東北大学大学院理学研究科
    • Related Report
      2018 Annual Research Report

URL: 

Published: 2018-04-23   Modified: 2021-12-27  

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