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On analytic behaviour of zeta-function and its applications to the arithmetical error term

Research Project

Project/Area Number 24540015
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionNagoya University

Principal Investigator

TANIGAWA Yoshio  名古屋大学, 多元数理科学研究科, 准教授 (50109261)

Co-Investigator(Renkei-kenkyūsha) FURUYA Jun  浜松医科大学, 医学部総合人間科学講座, 教授 (10413890)
MINAMIDE Makoto  山口大学, 大学院理工学研究科, 講師 (80596552)
Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords数論的誤差項 / ディリクレ約数問題 / 多次元非対称約数問題 / Tong タイプの公式 / 2乗平均 / 微分約数問題 / 離散和と連続和 / Tongタイプの公式 / 短区間平均値定理 / 約数問題 / 非対称多次元約数問題 / ヴォロノイ公式 / チャウラ-ワールムの公式 / exponent pair / ベルヌーイ多項式 / 2乗平均値定理 / セルバーグクラス
Outline of Final Research Achievements

We derived the truncated Tong-type formula for the arithmetical error term arising from the sum of the coeffcients of a zeta function in an extended Selberg class. We applied our formula to the 3-dimensional asymetric divisor problems and got an asymptotic formula of the mean square of the error term. This gives a partial answer of the conjecture of Ivic (1987). We gave a new formula for the difference between the discrete and continuous sum of the arithmetical error term. By this formula, we gave new estimates for the difference of higher power moments case and of the Rankin-Selberg case. We considered the product of k and l-th derivatives of the Riemann zeta function. We derived a certain certain representation of the arithmetical error term arising form this product and gave an upper bound using the theory of exponent pairs. This gives an improvement of Minamide's previous result.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (16 results)

All 2015 2014 2013 2012 Other

All Journal Article (9 results) (of which Peer Reviewed: 8 results,  Acknowledgement Compliant: 1 results) Presentation (7 results) (of which Invited: 1 results)

  • [Journal Article] A generalization of Brafman-Bailey type identities2015

    • Author(s)
      H. H. Chan, Y. Tanigawa
    • Journal Title

      Proceeding of AMS

      Volume: 143 Pages: 185-195

    • Related Report
      2014 Annual Research Report
    • Acknowledgement Compliant
  • [Journal Article] Moment integrals of 1/sin t and related zeta-values2014

    • Author(s)
      Jun. Furuya, Makoto Minamide and Yoshio Tanigawa
    • Journal Title

      Ramanujan J.

      Volume: 33 Issue: 3 Pages: 423-445

    • DOI

      10.1007/s11139-013-9467-1

    • Related Report
      2014 Annual Research Report 2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the difference between two kinds of mean value formulas of number-theoretic error terms2014

    • Author(s)
      X. Cao, J. Furuya, Y. Tanigawa, W. Zhai
    • Journal Title

      International Journal of Number Theory

      Volume: 10 Issue: 05 Pages: 1143-1170

    • DOI

      10.1142/s1793042114500195

    • Related Report
      2014 Annual Research Report 2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Mean values of the error term with shifted arguments in the circile problem2014

    • Author(s)
      J. Furuya, Y. Tanigawa
    • Journal Title

      Notes on Number Theory and Discrete Mathematics

      Volume: 20 Pages: 44-51

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the integrals containing the error term in the circle problem2014

    • Author(s)
      J. Furuya, Y. Tanigawa
    • Journal Title

      Functiones et Approximatio Commentarii Mathematici

      Volume: 51.2 Issue: 2 Pages: 303-333

    • DOI

      10.7169/facm/2014.51.2.5

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A generalized divisor problem and the sum of Chowla and Walum2013

    • Author(s)
      X. Cao, J. Furuya, Y. Tanigawa, W. Zhai
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 400 Issue: 1 Pages: 15-21

    • DOI

      10.1016/j.jmaa.2012.11.027

    • Related Report
      2013 Research-status Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] A generalized divisor problem and the sum of Chowla and Walum II2013

    • Author(s)
      X. Cao, J. Furuya, Y. Tanigawa, W. Zhai
    • Journal Title

      Functiones et Approximatio Commentarii Mathematici

      Volume: 49.1 Issue: 1

    • DOI

      10.7169/facm/2013.49.1.10

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Explicit representation of the integral involving the error term of Dirichlet divisor problems II2012

    • Author(s)
      J.Furuya, Y.Tanigawa
    • Journal Title

      Glasgow Math.

      Volume: 54 Issue: 1 Pages: 133-147

    • DOI

      10.1017/s0017089511000474

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Two Dirichlet series evaluations found on page 196 of Ramanujan's Lost Notebook2012

    • Author(s)
      B.Berndt, H.H.Chan, Y.Tanigawa
    • Journal Title

      Math.Proc.Cambridge Phil.Soc.

      Volume: 153 Issue: 2 Pages: 341-360

    • DOI

      10.1017/s0305004112000151

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] On a Sum involving the Mobius function2014

    • Author(s)
      I. Kiuchi, M. Minamide, Y. Tanigawa
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      広島大学
    • Year and Date
      2014-09-25
    • Related Report
      2014 Annual Research Report
  • [Presentation] On an evaluation of Ramanujan of a certain Dirichlet series and its generalization

    • Author(s)
      谷川好男
    • Organizer
      The 7-th China-Japan seminar -- preparatory conference
    • Place of Presentation
      近畿大学産業理工学部
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Representations and evaluations of the error term in a certain divisor problem

    • Author(s)
      古屋淳,南出真, 谷川好男
    • Organizer
      日本数学会九州支部例会
    • Place of Presentation
      宮崎大学工学部
    • Related Report
      2013 Research-status Report
  • [Presentation] Mean value of the error term with shifted arguments in the circle problem

    • Author(s)
      古屋淳,谷川好男
    • Organizer
      日本数学会年会
    • Place of Presentation
      学習院大学
    • Related Report
      2013 Research-status Report
  • [Presentation] Mean square of the error term in the asymmetric many dimensional divisor problem

    • Author(s)
      X. Cao, 谷川好男, W. Zhai
    • Organizer
      日本数学会年会
    • Place of Presentation
      学習院大学
    • Related Report
      2013 Research-status Report
  • [Presentation] 4 を法とするディリクレL- および多重 L- 値の間の関係について

    • Author(s)
      谷川好男, 古屋淳, 南出真
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      九州大学伊都キャンパス
    • Related Report
      2012 Research-status Report
  • [Presentation] 整数点からずらした変数を持つ誤差項の平均値定理について

    • Author(s)
      谷川好男,古屋淳
    • Organizer
      日本数学会年会
    • Place of Presentation
      京都大学
    • Related Report
      2012 Research-status Report

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Published: 2013-05-31   Modified: 2019-07-29  

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