• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Research on the mean values of zeta-functions and arithmetical error terms

Research Project

Project/Area Number 18K03237
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

Tanigawa Yoshio  名古屋大学, 多元数理科学研究科, 招へい教員 (50109261)

Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2020: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2019: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Keywordsゼータ関数 / ハーディ関数 / 二重ゼータ関数 / 数論的誤差項 / 2乗平均 / 混合型平均値 / 保型ハーディ関数 / ハイブリッド平均値 / 近似関数等式 / ヴォロノイ公式 / 約数問題 / ハイブリッド型平均値 / トングタイプのヴォロノイ公式 / 高次べき平均 / ハイブリッド型平均 / ハーディ関数およびその導関数 / リーマン-ジーゲル公式 / グラムポイント / 2重ゼータ関数 / Hardy関数とその導関数 / Riemann-Siegel 公式 / 二乗平均 / ペロンの公式 / ハーディー関数 / リーマンゼータ関数
Outline of Final Research Achievements

In his research of consecutive zeros of the Riemann zeta-function, Hall investigated the mean square of k-th derivative of Hardy's function and proposed to improve his error term. By using the Riemann-Siegel formula, we showed the accurate approximation of the derivative of Harsy's function and succeeded to improve Hall's result. We also derived the asymptotic formulas of several hybrid mean values of Hardy's function and the Riemann zeta-function. On the double zeta-function, we showed the detailed order of double zeta-funtion and improved the preceding results of its mean square estimate. We also studied the hybrid mean values of different kind of arithmetical functions related to the Dirichlet and Piltz divisor problems.

Academic Significance and Societal Importance of the Research Achievements

ホールの問題を解決するにあたっては, リーマン-ジーゲル公式を応用してハーディ関数の導関数の精密な近似式を導き, それによって2乗平均を改良することができた.これらは今後のハーディ関数の研究に寄与できたと思う.多重ゼータ関数は日本を中心に精力的に研究されている.二重ゼータ関数の上からの詳細な評価や2乗平均の改良は学術的に大きな意義がある. 数論的誤差項の混合型平均値では,全く異なっていると思われるものの積の平均が漸近式を持つなど, 今後の研究を示唆しているように思われる.

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (23 results)

All 2022 2021 2020 2019 2018 Other

All Int'l Joint Research (7 results) Journal Article (10 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 10 results) Presentation (6 results)

  • [Int'l Joint Research] 中国鉱業大学/北京石油化工学院(中国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] IIIT(Delhi)(インド)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] 中国鉱業大学/北京石油化工学院(中国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] IIIT (Delhi)(インド)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] IIIT(インド)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] 中国鉱業大学(中国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] 北京石油化工学院(中国)

    • Related Report
      2019 Research-status Report
  • [Journal Article] Average of Hardy's function at Gram points2021

    • Author(s)
      Xiaodong Cao, Yoshio Tanigawa, Wenguang Zhai
    • Journal Title

      Acta Arithmetica

      Volume: 199 Pages: 237-251

    • DOI

      10.4064/aa191103-13-2

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On hybrid moments of \Delta_2(x) and \Delta_3(x)2021

    • Author(s)
      Xiaodong Cao, Yoshio Tanigawa, Wenguang Zhai
    • Journal Title

      Ramanujan Journal

      Volume: 未定 Issue: 2 Pages: 597-631

    • DOI

      10.1007/s11139-021-00431-w

    • Related Report
      2021 Research-status Report 2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Mean square of double zeta-function2021

    • Author(s)
      D. Banerjee, M. Minamide, Y.Tanigawa
    • Journal Title

      to appear in Tokyo Journal of Mathematics

      Volume: 未定

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Some mean value results related to Hardy's function2021

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

      Research in Number Theory

      Volume: 7 Issue: 2

    • DOI

      10.1007/s40993-021-00255-z

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Mean square of double zeta-function2020

    • Author(s)
      D. Banerjee, T.M. Minamide, Y. Tanigawa
    • Journal Title

      Tokyo J. Math.

      Volume: 未定 Issue: -1 Pages: 83-101

    • DOI

      10.3836/tjm/1502179322

    • Related Report
      2021 Research-status Report 2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Mean square of the derivatives of Hardy's Z-function2020

    • Author(s)
      T.Makoto Minamide, Yoshio Tanigawa
    • Journal Title

      J.Math.Anal.Appl.

      Volume: 485 Issue: 1 Pages: 123772-123772

    • DOI

      10.1016/j.jmaa.2019.123772

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Bounds of double zeta-function and their applications2020

    • Author(s)
      D. Banerjee, T.M. Minamide, Y. Tanigawa
    • Journal Title

      Pacific J. Math.

      Volume: 304 Pages: 15-41

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Mean square of the derivatives of Hardy's function2020

    • Author(s)
      T.M. Minamide, Y. Tanigawa
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 未定

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Titchmarsh's method for the approximate functional equation for \zeta'(s)^2, \zeta(s)\zeta''(s) and \zeta'(s)\zeta''(s)2019

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

      Canadian J. Math.

      Volume: 71 Pages: 1465-1493

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the Wintner-Ingham-Segal summability method2018

    • Author(s)
      S. Kanemitsu, T. Kuzumaki, Y. Tanigawa
    • Journal Title

      Hardy-Ramanujan Journal

      Volume: 41 Pages: 40-47

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Presentation] モデュラー群の合同部分群の指数の平均について2022

    • Author(s)
      井川祥彰, Debika Banerjee, 南出真, 谷川好男
    • Organizer
      日本数学会秋季総合分科会(北海道大学)
    • Related Report
      2022 Annual Research Report
  • [Presentation] 合同約数和の平均に対する誤差項の平均について2021

    • Author(s)
      南出真, 矢代好克, 谷川好男
    • Organizer
      日本数学会秋季総合分科会(オンライン)
    • Related Report
      2021 Research-status Report
  • [Presentation] On partial sum of Apostol's Mobius function2021

    • Author(s)
      D. Banerjee, 藤澤雄介,南出真,谷川好男
    • Organizer
      日本数学会秋季総合分科会 熊本大学(オンライン)
    • Related Report
      2020 Research-status Report
  • [Presentation] 法 m の約数問題と整数の表現への応用2020

    • Author(s)
      出崎千晶, 南出真, 谷川好, 矢代好克
    • Organizer
      令和元年度日本数学会中国・四国支部例会(岡山理科大学, 国際学術センター・加計研修センター)
    • Related Report
      2019 Research-status Report
  • [Presentation] ハーディー関数の導関数の2乗平均について2019

    • Author(s)
      南出真, 谷川好男
    • Organizer
      令和1年日本数学会秋季総合分科会(金沢大学)
    • Related Report
      2019 Research-status Report
  • [Presentation] Mean square of double zeta-function2018

    • Author(s)
      D. Banerjee, 南出真, 谷川好男
    • Organizer
      平成30年度日本数学会秋季総合分科会
    • Related Report
      2018 Research-status Report

URL: 

Published: 2018-04-23   Modified: 2024-01-30  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi