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Analytic study on distribution properties of the residual order and residual index, and on the relation between the distribution and estimates on character sums

Research Project

Project/Area Number 14540048
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionMeiji Gakuin University

Principal Investigator

MURATA Leo  Meijigakuin University, Dept.of Economics, Professor, 経済学部, 教授 (30157789)

Co-Investigator(Kenkyū-buntansha) KITAOKA Yoshiyuki  Meijyou University, Dept.of Mathematics, Professor, 理工学部, 教授 (40022686)
OKAZAKI Ryutarou  Doshisha University, Dept.of Technology, Lecturer, 工学部, 専任講師 (20268113)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
Keywordsresidual order / resudual index / Artin's Conjecture on primitive roots / estimate of character sums / 剰余指数 / 剰余位数 / 原始根に関するArtinの予想
Research Abstract

Let a be a natural number, p be a prime number, D_a(p) and I_a(p) denote the residual order and the residual index of the class a(mod p), respectively.
Both D_a(p) and I_a(p) are surjective map from P(the set of all primes) to N(the set of all natural numbers). We studied here the distribution properties of the map D_a(p).
Firstly, we considered the set : Q_a(x ; s, t){p≦x ; D_a(p)≡s (mod t)}, s and t be natural numbers, and we obtained the results ;
1)when we assume the generalized Riemann Hypothesis, we can prove the existence of the natural density Δ_a(s,t) of the set Q_a(x ; s, t),
2)we can calculate the density Δ_a(s, t) effectively, and through some numerical and theoretical observations, we found out some interesting properties of the number theoretical function Δ_a(s, t).
Secondly, we considered the set : M(x) = {p ≦ x ; D_2(p) is a prime}. Then, under G.R.H., we have an estimate #M(x) << x(log x)^(-2). This estimate is most likely to be "best possible".

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report

Research Products

(33 results)

All 2006 2004 2003 2002 Other

All Journal Article (24 results) Publications (9 results)

  • [Journal Article] On a distribution property of the residual order of a(mod p) III2006

    • Author(s)
      Leo Murata, K. Chinen
    • Journal Title

      Journal of Math. Soc. Japan 58

      Pages: 693-720

    • NAID

      10018381051

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of the residual order of a(mod p) IV2006

    • Author(s)
      Leo Murata, K. Chinen
    • Journal Title

      論文集 : "Number Theory : Tradition and Modernization"(Ed. by W. Zhang, Y. Tanigawa)

      Pages: 11-22

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of the residual order of a(mod p) III.2006

    • Author(s)
      Leo Murata (with Koji Chinen)
    • Journal Title

      Journal of Math.Soc.Japan Vol.58-3

      Pages: 693-720

    • NAID

      10018381051

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of the residual order of a(mod p) IV.2006

    • Author(s)
      Leo Murata (with Koji Chinen)
    • Journal Title

      Number Theory : Tradition and Modernization (Ed.by W.Zhang, Y.Tanigawa)(Springer Science + Business Media, Inc.)

      Pages: 11-22

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of the residual order of a(mod p)2004

    • Author(s)
      Leo Murata, K. Chinen
    • Journal Title

      Journal of Number Theory 105

      Pages: 60-81

    • NAID

      40005723985

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of the residual order of a(mod p) II2004

    • Author(s)
      Leo Murata, K. Chinen
    • Journal Title

      Journal of Number Theory 105

      Pages: 82-100

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the largest prime factor of a Mersenne number2004

    • Author(s)
      Leo Murata, C. Pomerance
    • Journal Title

      Centre de Recherches Mathematiques, CRM Proceedings and Lecture Notes 36

      Pages: 209-218

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a property of the multiplicative order of a(mod p)2004

    • Author(s)
      Leo Murata
    • Journal Title

      Proceedings of the 2003 Workshop on Cryptography and Related Mathematics

      Pages: 199-207

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of a residual order of a(mod p) II2004

    • Author(s)
      Leo Murata, K. Chinen
    • Journal Title

      Proc. of Japan Acad. 80

      Pages: 182-186

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of the residual order of a(mod p) IV2004

    • Author(s)
      Leo Murata, K. Chinen
    • Journal Title

      京都大学 数理解析研究所 講究録 1384

      Pages: 169-174

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of the residual order of a(mod p).2004

    • Author(s)
      Leo Murata (with Koji Chinen)
    • Journal Title

      Journal of Number Theory Vol.105

      Pages: 60-81

    • NAID

      40005723985

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of the residual order of a(mod p) II.2004

    • Author(s)
      Leo Murata (with Koji Chinen)
    • Journal Title

      Journal of Number Theory vol.105

      Pages: 82-100

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the largest prime factor of a Mersenne number.2004

    • Author(s)
      Leo Murata (with Carl Pomerance)
    • Journal Title

      Center de Recherches Mathematiques, CRM Prceedings and Lecture Notes Volume 36

      Pages: 209-218

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a property of the multiplicative order of a(mod p).2004

    • Author(s)
      Leo Murata
    • Journal Title

      Proceedings of the 2003 Workshop on Cryptography and Related Mathematics, Chuo University

      Pages: 199-207

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of a residual order of a(mod p) II.2004

    • Author(s)
      Leo Murata (with Koji Chinen)
    • Journal Title

      Proc.of Japan Acad. Vol.80

      Pages: 182-186

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the largest prime factor of a Mersenne number2004

    • Author(s)
      Leo Murata, Carol Pomerance
    • Journal Title

      CRM Proceedings & Lecture Notes 36

      Pages: 209-218

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a distribution property of the residual order of a (mod p). IV2004

    • Author(s)
      Leo Murata, K.Chinen
    • Journal Title

      京都大学数理解析研究所 講究録 1384

      Pages: 169-174

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a distribution property of a residual order of a (mod p), (2)2004

    • Author(s)
      Leo Murata, K.Chinen
    • Journal Title

      Proceedings of the Japan Academy 80

      Pages: 182-186

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a distribution property of the residual order of a(mod p) III2003

    • Author(s)
      Leo Murata, K. Chinen
    • Journal Title

      京都大学 数理解析研究所 講究録 1319

      Pages: 139-147

    • NAID

      10018381051

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of a residual order of a(mod p)2003

    • Author(s)
      Leo Murata, K. Chinen
    • Journal Title

      Proc. of Japan Acad. 79

      Pages: 28-32

    • NAID

      40005723985

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of a residual order of a(mod p).2003

    • Author(s)
      Leo Murata (with Koji Chinen)
    • Journal Title

      Proc.of Japan Acad. Vol.79

      Pages: 28-32

    • NAID

      40005723985

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] a (mod p)の剰余位数の分布について II2002

    • Author(s)
      Leo Murata, K. Chinen
    • Journal Title

      京都大学 数理解析研究所 講究録 1274

      Pages: 62-69

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a distribution property of the residual order of a (mod p)-III

    • Author(s)
      Leo Murata (with K.Chinen)
    • Journal Title

      Journal of the Mathematical Society of Japan to appear

    • NAID

      10018381051

    • Related Report
      2005 Annual Research Report
  • [Journal Article] ON A DISTRIBUTION PROPERTY OF THE RESUDUAL ORDER OF a (mod p)-IV

    • Author(s)
      Leo Murata (with K.Chinen)
    • Journal Title

      Number Theory. Tradition and Modernization(Edited by Y.Tanigawa and W.Zhang)(Springer Veerlag) to appear

    • Related Report
      2005 Annual Research Report
  • [Publications] Leo Murata, K.Chinen: "On a distribution property of the residual order of a (mod p)"Journal of Number Theory. 105. 60-81 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Leo Murata, K.Chinen: "On a distribution property of the residual order of a (mod p), II"Journal of Number Theory. 105. 82-100 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Leo Murata, K.Chinen: "On a distribution property of the residual order of a (mod p), III"京都大学数理解析研究所講究録. 1319. 139-147 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Leo Murata: "On a property of the multiplicative order of a (mod p)"Proceedings of the Workshop on Cryptography & Related Mathematics (中央大学). 199-207 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Leo Murata, K.Chinen: "On a distribution property of a residual order of a (mod p)"Proc. of the Japan Acad. Ser. A. 79. 28-32 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 村田玲音, 知念宏司: "a (mod p)の余剰位数の分布についてII"京都大学数理解析研究所講究録. 1274. 62-69 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 村田玲音, 石井 担: "剰余位数の分布の一性質についてII"明治学院産業経済研究所年報. 19. 131-141 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Yoshiyuki Kitaoka: "Distribution of units of an algebraic number field""Galois Theory and Modulas Forms", Developments in Math (Kluwer A.P.). (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Ryotaro Okazaki: "Geometry of a cubic Thue equation"Publ. Math. Debrecen. 61. 267-314 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-03-31   Modified: 2016-04-21  

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