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Diophantine approximation in low discrepancy sequences and the Kontsevich-Zagier period conjecture

Research Project

Project/Area Number 18K03225
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionNihon University

Principal Investigator

HIRATA-KOHNO Noriko  日本大学, 理工学部, 教授 (90215195)

Project Period (FY) 2018-04-01 – 2022-03-31
Project Status Discontinued (Fiscal Year 2021)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2020: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords超一様分布数列 / 多重対数 / 周期予想 / 超幾何級数 / 超越数 / パデ近似 / 一次独立性 / ディオファントス近似 / 一様分布 / 超一様分布 / Lerch関数 / リーマンゼータ関数 / 無理数性 / 多重対数関数 / Lerch 関数 / G 関数 / 数論的近似 / Pade近似 / 対数一次形式 / 暗号原理 / 周期 / 多項式写像 / Kontsevich-Zagier予想
Outline of Final Research Achievements

Diophantine approximation is one of basic methods to give a proof of the irrationality, the linear independence or to study the low discrepancy sequence in uniform distribution in Diophantine problems. It is known that Ch. Hermite constructed explicit simultaneous Diophantine approximations related to the exponential function to prove the transcendence of the number of Napier.
We now adapt this approximation method to prove a new linear independence criterion for several polylogarithms defined at several algebraic numbers. Polylogarithmic function is a natural generalization of logarithmic function, however, it has no homomorphism property like logarithmic function, then the usual tool to show the transcendence of logarithms does not work. Nevertheless, we succeeded in showing a precise criterion for the linear independence of several polylogarithms of distinct algebraic numbers, over an algebraic number field of arbitrary degree, relying on Pade approximations.

Academic Significance and Societal Importance of the Research Achievements

多重対数は数論のみならず数学や物理学のあらゆる場に現れる周期である.また超一様・一様分布数列の研究も,乱数や擬似乱数などの研究にとって重要である.多重対数関数の代数的数における値は,自然対数で表される代数体のRegulatorの一般化として現れる数としても位置付けられ,素数分布の考察に力を発揮するRiemann zeta関数の値の性質を調べる際にも登場する.本研究課題では周期予想に現れる代表的な周期の例である多重対数について考察し,異なる点での多重対数が任意次数の代数体上で一次独立になるための判定規準を与えた.また超幾何級数を用いて,アーベル多様体の周期に対する超越近似についても考察した.

Report

(4 results)
  • 2021 Final Research Report ( PDF )
  • 2020 Annual Research Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (34 results)

All 2021 2020 2019 2018 Other

All Int'l Joint Research (9 results) Journal Article (10 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 7 results,  Open Access: 4 results) Presentation (9 results) (of which Int'l Joint Research: 6 results,  Invited: 8 results) Remarks (4 results) Funded Workshop (2 results)

  • [Int'l Joint Research] Paris Sorbonne University/University of Strasbourg(フランス)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] University of Ottawa(カナダ)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] Sorbonne University (Paris 6)(フランス)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] University of Strasbourg(フランス)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Radboud University, Nijmegen(オランダ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Wits University(南アフリカ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Tata Institute of Fundamental Research(インド)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] University of Sorbonne (Paris 6)/University of Strasbourg/CIRM Institute of Mathematics(フランス)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Moscow State University(ロシア連邦)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Linear forms in Polylogarithms2021

    • Author(s)
      S. David, N. Hirata-Kohno and M. Kawashima
    • Journal Title

      Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

      Volume: -

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Ramanujan's contribution to approximate pi and visualization via Mathematica2021

    • Author(s)
      K. Kurishima, Y. Suzuki, H. Nishibayashi, K. Suzuki, S. Tonegawa, Yukiko Washio, N. Hirata-Kohno and Yusuke Washio
    • Journal Title

      RIMS Kokyuroku (Kyoto University)

      Volume: 2178 Pages: 100-108

    • Related Report
      2020 Annual Research Report
    • Open Access
  • [Journal Article] Can polylogarithms at algebraic points be linearly independent?2020

    • Author(s)
      Sinnou David, Noriko Hirata-Kohno and Makoto Kawashima
    • Journal Title

      Moscow Journal of Combinatorics and Number Theory

      Volume: 9 Issue: 4 Pages: 389-406

    • DOI

      10.2140/moscow.2020.9.389

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Diophantine Approximation2020

    • Author(s)
      Noriko Hirata-Kohno
    • Journal Title

      Exposition American Mathematical Society

      Volume: -

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] A key exchange protocol relying on polynomial maps2019

    • Author(s)
      K. Akiyama, S. Nakamura, M. Ito and Noriko Hirata-Kohno
    • Journal Title

      International Journal of Mathematics for Industry

      Volume: 11 Issue: 01 Pages: 1-11

    • DOI

      10.1142/s2661335219500035

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Minkowski Convex Body Theorem towards Polya's Visibility Problem via Mathematica2019

    • Author(s)
      N. Hirata-Kohno, Y. Ishii, Y. Kurimoto, K. Kurishima, K. Suzuki, Y. Suzuki, Y. Washio
    • Journal Title

      RIMS Kokyuroku, Kyoto University

      Volume: 2142 Pages: 42-52

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] A key exchange protocol relying on polynomial maps2019

    • Author(s)
      K. Akiyama, S. Nakamura, M. Ito and N. Hirata-Kohno
    • Journal Title

      International Journal of Mathematics for Industry

      Volume: 印刷中

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Linear dependence criterion of almost integer valued functions:a generalization of the Polya-Pisot theorem2019

    • Author(s)
      H. Furutsu and N. Hirata-Kohno
    • Journal Title

      J. Research Institute of Science and Technology, Nihon University

      Volume: 144 Pages: 12-16

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Kronecker Approximation Theorem employing Mathematica2019

    • Author(s)
      N. Hirata-Kohno, Y. Ishii, Y. Kurimoto, S. Shimawaki, K. Suzuki, Y. Washio
    • Journal Title

      RIMS Kokyuroku, Kyoto University

      Volume: 2105 Pages: 33-38

    • Related Report
      2018 Research-status Report
    • Open Access
  • [Journal Article] Schinzel’s theorem on a circle via GeoGebra2018

    • Author(s)
      Y. Kurimoto,Y. Washio, S. Nakamura, K. Suzuki and N. Hirata-Kohno
    • Journal Title

      66th JSEE ja

      Volume: 66th Jsee Pages: 58-59

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Presentation] How do we know the irrationality?2021

    • Author(s)
      N. Hirata-Kohno
    • Organizer
      日本数学会研究集会 Women in Mathematics
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Identities for pi discovered by S. Ramanujan and visualization via Mathematica2020

    • Author(s)
      K. Kurishima, Y. Suzuki, H. Nishibayashi, K. Suzuki, S. Tonegawa, Yukiko Washio, N. Hirata-Kohno and Yusuke Washio
    • Organizer
      RIMS Workshop, Kyoto University
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Linear forms in logarithms: contribution by Fel’dman and the linear independence of polylogarithms2019

    • Author(s)
      Noriko Hirata-Kohno
    • Organizer
      Transcendence and Diophantine Problems, Moscow
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Recent Progress toward Polylogarithm Conjecture2019

    • Author(s)
      Noriko Hirata-Kohno
    • Organizer
      Algebraic Number Theory and Related Topics, RIMS Workshop, Kyoto University
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Visibility problem via Minkowski convex body theorem and Mathematica2019

    • Author(s)
      N. Hirata-Kohno, Y. Ishii, Y. Kurimoto, K. Kurishima, K. Suzuki, Y. Suzuki, Y. Washio
    • Organizer
      Software and Its Effective Use for Mathematics Education, RIMS Workshop, Kyoto University
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Geometrical instructions employing GeoGebra on iPad and developments2019

    • Author(s)
      H. Furutsu, Y. Ishii and N. Hirata-Kohno
    • Organizer
      Workshop on Instructions by Tex
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Linear forms in logarithms via Pad e approximations and their applications2018

    • Author(s)
      Noriko Hirata-Kohno
    • Organizer
      Diophantine Approximation and Transcendence CIRM, Luminy, France
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Linear independence of special values of logarithms via Pade approximations2018

    • Author(s)
      M. Kawashima and N. Hirata-Kohno
    • Organizer
      Effective methods in Diophantine problems, Leiden, the Netherlands
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] An HFE-based variant of a key exchange protocol employing mulitivariate polynomial maps2018

    • Author(s)
      K. Akiyama, S. Nakamura, M. Ito and Noriko Hirata-Kohno
    • Organizer
      JSIAM Annual meeting
    • Related Report
      2018 Research-status Report
  • [Remarks] 日本大学研究業績データベース

    • URL

      https://kenkyu-web.cin.nihon-u.ac.jp/Profiles/40/0003941/profile.html

    • Related Report
      2020 Annual Research Report
  • [Remarks] 日本大学理工学部教員情報

    • URL

      http://kenkyu-web.cin.nihon-u.ac.jp/Profiles/40/0003941/profile.html

    • Related Report
      2019 Research-status Report
  • [Remarks] Noriko HIRATA-Kohno's Home Page

    • URL

      http://trout.math.cst.nihon-u.ac.jp/~hirata/

    • Related Report
      2019 Research-status Report 2018 Research-status Report
  • [Remarks] 日本大学研究者情報システム

    • URL

      http://kenkyu-web.cin.nihon-u.ac.jp/Profiles/40/0003941/profile.html

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] DARF (Diophantine Analysis and Related Fields) zoom Seminar2020

    • Related Report
      2020 Annual Research Report
  • [Funded Workshop] Diophantine Approximation and Transcendence, CIRM, Luminy, France2018

    • Related Report
      2018 Research-status Report

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Published: 2018-04-23   Modified: 2023-01-30  

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