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Number-Theoretic Indentities・Asymptotic formulas,and Special Functions

Research Project

Project/Area Number 11640051
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKinki University

Principal Investigator

KANEMATSU Shigeru  Kinki University Kyushu school of Engineering,Professor, 九州工学部, 教授 (60117091)

Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2001: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2000: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1999: ¥1,000,000 (Direct Cost: ¥1,000,000)
KeywordsModular Relations / Zeta-Functions / Functional Equations / The Riemann Hypothesis / Farey Fractions / Maillet Determinant / Divisor Problem / リーマン予想 / オイラー積 / 力学系 / ギャップフーリエ級数 / 高木函数 / ワイエルシュトラース函数 / リーマンの函数
Research Abstract

THE GREATEST RESULT THAT IS OBTAINED DURING THESE THREE YEARS IS THE DISCOVERY OF THE ROUTE PENETRATING THE MODULAR RELATIONS-MODULAR RELATION PRINCIPLE-FOR THE THEORY OF ZETA-FUNCTIONS, WHICH WE FOUND IN DEC., 2000.IN 2001, WE APPLIED THIS PRINCIPLE TO THE SPECIAL CASE OF THE FUNCTIONAL EQUATIONS OF HECKE'S TYPE-I.E. THE CASE WHERE HERE IS A SIMPLE GAMMA FACTOR, WHICH INCLUDES MANY OF THE MOST IMPORTANT SPECIAL CASES OF ZETAFUNCTIONS INCLUDING MULTIPLE HURWITZ ZETA-FUNCTION, EPSTEIN ZETA-FUNCTION, AUTOMORPHIC L-FUNCTION, ETC.ESPECIALLY, WE WERE SUCCUSSFUL IN OBTAINING THE RAMANUJAN TYPE RAPIDLY CONVERGENT FORMULAS FOR SPECIAL VALUES OF THOSE ZETA-FUNCTIONS AT RATIONAL AS WELL AS INTEGRAL ARGUMENTS. WE CAME TO THIS DISCOVERY THROUGH OUR FORMER INVESTIGATIONS ON THE SUM FORMULA EXPRESSING INFINITE SERIES WITH HURWITZ ZETA-FUNCTION COEFFICENTS IN TERMS OF THE DERIVATIVES OF THE HURWITZ ZETA-FUNCTION, WHICH WE COMPLETED IN THE FIRST LISTED PAPER, IN PARTICULAR, OUT RESULTS SUPERSEDE ALL THE CORRESPONDING RESULTS IN THE RECENTLY PUBLISHED BOOK OF SRIVASTAVA AND CHOI "SERIES ASSOCIATED WITH THE ZETA AND RELATIED FUNCTIONS".
ALONG WITH THIS WE CONTINUED OUR NEVER-STOPPING RESEARCH ON MAILLET DETERMINATS, AND SUCCEEDED IN PROVING THE DETERMINATAL EXPRESSIONS FOR THE SPECIAL VALUES OF THE DEDEKING ZETA-FUNCTION AT INTEGRAL ARGUMENTS, BY COMBINING THOSE WITH CLAUSEN FUNCTIONS, A POINT WHICH IS OVERLOOKED IN OTHER LITERATURE.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (22 results)

All Other

All Publications (22 results)

  • [Publications] S.Kanemitsu(他2名): "On rapidly convergent series expressions for zeta-, and L-fanction values, and Log Sine Integrals"The Ramanujan J.. 5. 91-104 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Kanemitsu(他1名): "On a generalization of the Maillet determinant.II"Acta Arith.. 99. 343-361 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Kanemitsu(他2名): "On rapidly convergent series for zeta-, and L-fanction values via The modular relaton"The Ramanujan J.. 1. 21-42 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Kanemitsu(他2名): "On the values of the Riemann zeta-function at rational arguments"The Hardy-Riemanujan J.. 24. 10-18 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Kanemitsu(他2名): "On rapidly convergent series for Dirichlet L-function values via the modular relation"Proc.of the Intern.Conf.on Number Theory and Discrete Mathematics in honor of Srinivasa Ramanujan. 113-133 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Kanemitsu(他1名): "On a divisor problem in Landau's framework"Proc.of the Conf.Analytic on Number Theory. 205-221 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Kanemitsu(他1名): "Number-Theoretic Methods-Future Trends"Kluwer Academic Publishers.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S.Kanemitsu(他2名): "On rapidly convergent series expressions for zata-, and L-values, and log sine integrals"The Ramanujan J.. 5. 91-104 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Kanemitsu(他1名): "On a generalization of the Maillet determinant II"Acta Arith.. 99. 343-361 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Kanemitsu(他2名): "On radply convergent series for zeta, -and L-function values via the moelnlar relation"Proc. of the Intern Conf. on Special Functions and their Applications. 1. 21-42 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Kanemitsu(他2名): "On the values of the piemarn zeta-function at rational argumants"The Hordy-Ramanujan J.. 24. 10-18 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Kanemitsu(他2名): "On rafidly convergent series for Dirichlet L-function values via the modular relation"Proc. of the Intern. Conf. On Number Theory and Discrets Math. in honor of S.Ramanujan. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Kanemitsu(他1名): "On a general divisor problem in Landan's Frameinork"Proc. of the Conf., Analytic Number Theory. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Kanemitsu(他1名): "Number Theoretic Methods -Future Trends"Kluwer Academic Pablishers.

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Kanemitsu (他2名): "On the Hurwitz-Lerch zeta-functions"Aeq.Math.. 59. 1-18 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kanemitsu (他2名): "On some sums involving Favey fractions II"J.Math.Soc.Japan. 53. 915-947 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kanemitsu (他2名): "Sums involving the Hurwitz zeta-function"The Ramanujan J.. 5. 5-19 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kanemitsu (他1名): "On a generalization of the Maillet determinant II"Acta Arith..

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kanemitsu (他1名): "On a general divisor problem in Landau's framework"Proc.Conf.Analytic Number Theory.

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kanemitsu (他2名): "On rapidly convergent series for Dirichlet, L-function value via the modular relation"Proc.Intern.Conf.Special Functions.

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kanemitsu: "Euler products,Farey series and the Rimann hypothesis"Publ.Math.(Debrecen). 56(3-4).

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Kanemitsu: "On rapidly convergent series expressions for 5-,and L-ralues and log sine integrals"The Ramanujan J..

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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