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Sum formulas for arithmetical functions and mean value theorem for the zeta functions

Research Project

Project/Area Number 11640022
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

TANIGAWA Yoshio  Nagoya Univ., Graduate School of Mathematics, Assoc.Prof., 大学院・多元数理科学研究科, 助教授 (50109261)

Co-Investigator(Kenkyū-buntansha) KIUCHI Isao  Yamaguchi Univ., Faculty of Science, Assoc.Prof., 理学部, 助教授 (30271076)
AKIYAMA Shigeki  Niigata Univ., Faculty of Science, Assoc.Prof., 理学部, 助教授 (60212445)
MATSUMOTO Kohji  Nagoya Univ., Graduate School of Mathematics, Assoc.Prof., 大学院・多元数理科学研究科, 助教授 (60192754)
北岡 良之  名古屋大学, 大学院・多元数理科学研究科, 教授 (40022686)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2000: ¥1,600,000 (Direct Cost: ¥1,600,000)
KeywordsDirichlet series / Mean value formula / Arithmetical function / Elliptic curve / Modular relation / Ramanujan formula / Multiple zeta function / Analytic continuation / Bernoulli数 / Rankin-Selberg 級数 / 近似関数等式 / 二乗平均
Research Abstract

The Riemann zeta-function, the Dirichlet L-fucntion and other Dirichlet series have a long history of researches. They are still the main oobjects of researches at present day, since many important number theoretic properties are reflected in the analytic properties of Dirichlet series, such as the analtic continuation, functional equation, the place of poles and their residues.
To study the detailed local behavior of zeta function, Kiuchi and Tanigawa derived mean value theorem for short intervals for E_σ (T)(=the error term of the mean square formula of ζ^2 (σ+it) and R (σ+it)(=the remainder term of the approximate functional equation of ζ^2 (s)). We also studied the mean value of eror term arising from the Dirichlet divisor problem with characters.
Akiyama and Tanigawa considered the L-function associated to elliptic curves. In order to compute the numerical values, we derived the approximate functional equation with incomplete gamma functions. Using our formula, we checked that the Riemann hypothesis holds in the range Im (s) 【less than or equal】400 for several elliptic curves. Furthermore, we studied the relation between Sate-Tate conjecture and the Riemann hypothesis.
The above mentioned functional equation is related to the modular relation. Kanemitsu, Yoshimoto and Tanigawa reconstruct many results related to the Ramanujan formulas from the point of modular relations. The new formula for ζ(2/3) of Ramanujan type is also obtained.
The analytic continuation of Euler-Zagier's multiple zeta-function was established by Akiyama, Egami and Tanigawa. We noted for the first time that the points of indeterminacy appear for this function. We also computes the zeta-values at non-positive integers.

Report

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

    (23 results)

All Other

All Publications (23 results)

  • [Publications] S.Akiyama and Y.Tanigawa: "Calculation of values of L-functions associated to elliptic curves"Mathematics of Computation. 68. 1201-1231 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] A.Ivic,K.Matsumoto and Y.Tanigawa: "On Riesz means of the coefficients of the Rankin-Selberg series"Math.Proc.Camb.Phil.Soc. 127. 117-131 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 谷川好男: "指標付き約数問題(Divisor problem with characters)"京都大学数理解析研究所講究録. 1091. 1-8 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] I.Kiuchi and Y.Tanigawa: "A mean value theorem of the approximate functional equation of ζ^2(s) for short intervals"J.Ramanujan Math.Soc.. 15. 53-70 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Akiyama,S.Egami and Y.Tanigawa: "Analytic continuation of multiple zeta functions and their values at non-positive in-tegers"Acta Arith.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Kanemitsu,Y.Tanigawa and M.Yoshimoto: "On zeta- and L-function values at special rational arguments via the modular relation "Proceedings of the International Conference on Special Functions and their Applications, Lucknow, India. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Akiyama and Y.Tanigawa: "Calculation of values of L-functions associated to elliptic curves"Mathematics of Computation. 68. 1201-1231 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] A.Ivic, K.Matsumoto and Y.Tanigawa: "On Riesz means of the coefficients of the Rankin-Selberg series"Math.Proc.Camb.Phil.Soc.. 127. 117-131 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Y.Tanigawa: "Divisor problem with characters"Surikaisekikenkyusho Kokyuroku. 1091. 1-8 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] I.Kiuchi and Y.Tanigawa: "A mean value theorem of the approximate functional equation of ζ^2 (s) for short intervals"J.Ramanujan Math.Soc.. 15. 53-70 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Akiyama, S.Egami and Y.Tanigawa: "Analytic continuation of multiple zeta functions and their values at non-positive integers"Acta Arith.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Kanemitsu, Y.Tanigawa and M.Yoshimoto: "On zeta-and L-function values at special rational arguments via the modular relation"Proceedings of the International Conference on Special Functions and their Applications, Lucknow, India. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Akiyama and Y.Tanigawa: "Calculation of values of L-functions associated to elliptic curves"Mathematics of Computation . 68. 1201-1231 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] A.Ivic,K.Matsumoto and Y.Tanigawa: "On Riesz means of the coefficients of the Rankin-Selberg series"Math.Proc.Camb.Phil.Soc. 127. 117-131 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] 谷川好男: "指標付き約数問題(Divisor problem with characters)"京都大学数理解析研究所講究録. 1091. 1-8 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] I.Kiuchi and Y.Tanigawa: "A mean value theorem of the approximate functional equation of ζ^2(s) for short intervals "J.Ramanujan Math.Soc.. 15. 53-70 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Akiyama,S.Egami and Y.Tanigawa: "Analytic continuation of multiple zeta functions and their values at non-positive integers"Acta Arith.. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kanemitsu,Y.Tanigawa and M.Yoshimoto: "On zeta- and L-function values at special rational arguments via the modular relation "Proceedings of the International Conference on Special Functions and their Applications, Lucknow, India . (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Akiyama and Y.Tanigawa: "Calculation of values of L-functions associated to elliptic curves"Mathematics of Computation. 68. 1201-1231 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Matsumoto and Y.Tanigawa: "On the zeros of higher derivatives of Hardy's Z-function"Journal of Number Theory. 75. 262-278 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] A.Ivic,K.Matsumoto and Y.Tanigawa: "On Riesz means of the coefficients of the Rankin-Selberg series"Math.Proc.Camb Phil Soc.. 127. 117-131 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] I..Kiuchi and Y.Tanigawa: "The mean value theorem of the Riemann Zeta-function in the critical strip for short intervals"Number Theory and its Applications (ed.by Gyery and Kanemitsu). 231-240 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] I..Kiuchi and Y.Tanigawa: "The mean value theorem of the approximate functional equation of ζ(s)^2 for short intervals"to appear in Journal of Ramanujan Math. Soc. (印刷中).

    • Related Report
      1999 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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