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Zeros and discrete value distribution of the Riemann zeta function and its derivatives

Research Project

Project/Area Number 18K13400
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 11010:Algebra-related
Research InstitutionKyushu University (2019-2022)
Institute of Physical and Chemical Research (2018)

Principal Investigator

Suriajaya Ade Irma  九州大学, 数理学研究院, 助教 (50804241)

Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywordsゼータ関数 / 零点 / 値分布 / L関数 / 素数定理 / ゴールドバッハ表現 / 例外的零点 / 数論的関数 / リーマンゼータ関数 / 零点の対相関 / 単純零点の割合 / 代替予想 / ゴールドバッハ平均 / 実軸に近い零点 / ゴールドバッハ問題 / ディリクレL関数 / セルバーグクラス / 関数等式 / 素数 / Hardy-Littlewoodの特異級数 / 平均 / a-点 / 分布 / 臨界領域 / 導関数 / 零点個数の誤差項 / ローラン係数 / 離散的半群 / フルヴィッツゼータ関数 / スティルチェス定数 / Schemmel関数 / 平均値 / 零点の分布 / 値の分布
Outline of Final Research Achievements

The Riemann zeta-function is an important function that solves number theoretical problems, starting from counting prime numbers. During the period of research, we considered not only the distribution of zeros of the Riemann zeta-function, but also distribution of values in various settings, distribution of zeros of its derivatives, and also distribution of values of general zeta and L-functions. In terms of its connection to number theoretical problems, we also considered its connection to Goldbach's problem and distribution of prime numbers in short intervals. Not limited to those results, we also studied a generalized Euler phi-function and a sum of negative degrees of the gaps values in numerical semigroups in connection to the Riemann zeta-function.

Academic Significance and Societal Importance of the Research Achievements

研究代表者らが得た研究成果は社会への直接的な応用が困難であるが、数論分野の発展に大きく貢献した。いくつかの結果に対してのみ紹介する。ゼータ関数とL関数の各々値分布の結果によって、ゼータ関数とL関数の挙動に関する理解が深まり、数論的問題への新たな応用に繋がる可能性もある。研究代表者らの研究でゴールドバッハ表現の個数の平均と素数定理の誤差は密接に関係しているのが明らかになったため、よりたくさん知られている素数の情報を用いてゴールドバッハ表現の個数の平均に対して挙動を調べることが可能になった。研究代表者らは更に、これらの問題と重要なリーマンゼータ関数の零点の対相関の関係を明らかにした。

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

    (66 results)

All 2023 2022 2021 2020 2019 Other

All Int'l Joint Research (25 results) Journal Article (19 results) (of which Int'l Joint Research: 17 results,  Peer Reviewed: 18 results,  Open Access: 4 results) Presentation (22 results) (of which Int'l Joint Research: 14 results,  Invited: 12 results)

  • [Int'l Joint Research] San Jose State University/Rutgers University(米国)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] University of Wuerzburg(ドイツ)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] University of Toronto Scarborough(カナダ)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] IIIT Delhi/Indian Institute of Technology Kanpur(インド)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] TU Graz(オーストリア)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] サンノゼ州立大学/ラトガーズ大学/美国数学研究所(米国)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] ヴュルツブルク大学(ドイツ)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] グラーツ工科大学(オーストリア)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] トロント大学/ライアソン大学(カナダ)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] インドラプラズサ情報技術研究所、デリー/インド工科大学カンプール校(インド)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research]

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] サンノゼ州立大学/College of Williams and Mary(米国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] ヴュルツブルク大学(ドイツ)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] グラーツ工科大学/リンツ大学(オーストリア)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] 浙江理工大学(中国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] テクニオン - イスラエル工科大学(イスラエル)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] ヴュルツブルク大学(ドイツ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] College of William and Mary(米国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] JKU Linz(オーストリア)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] サンノゼ州立大学(米国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] ヴュルツブルク大学(ドイツ)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] ウォータールー大学(カナダ)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Technion(イスラエル)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] 武漢大学(中国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] リンツ大学(オーストリア)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Zeros of derivatives of L-functions in the Selberg class on Re(s)<1/22023

    • Author(s)
      Chaubey Sneha、Khurana Suraj、Suriajaya Ade
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: 151 Pages: 1855-1866

    • DOI

      10.1090/proc/16251

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] ON AN AVERAGE GOLDBACH REPRESENTATION FORMULA OF FUJII2023

    • Author(s)
      GOLDSTON DANIEL A.、SURIAJAYA ADE IRMA
    • Journal Title

      Nagoya Mathematical Journal

      Volume: FirstView Pages: 1-22

    • DOI

      10.1017/nmj.2022.44

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Exceptional zeros and the Goldbach problem2022

    • Author(s)
      Friedlander J.B.、Goldston D.A.、Iwaniec H.、Suriajaya A.I.
    • Journal Title

      Journal of Number Theory

      Volume: 233 Pages: 78-86

    • DOI

      10.1016/j.jnt.2021.06.004

    • Related Report
      2022 Annual Research Report 2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Riemann-type functional equations: Dirichlet polynomial approximations and a weak Gram law2022

    • Author(s)
      Sourmelidis Athanasios、Steuding Joern、Suriajaya Ade Irma
    • Journal Title

      Acta Arithmetica

      Volume: 204 Issue: 2 Pages: 97-113

    • DOI

      10.4064/aa210111-13-4

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Riemann-type functional equations2022

    • Author(s)
      Sourmelidis Athanasios、Steuding Joern、Suriajaya Ade Irma
    • Journal Title

      Indagationes Mathematicae

      Volume: 33 Issue: 6 Pages: 1236-1262

    • DOI

      10.1016/j.indag.2022.08.002

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Weighted one-level density of low-lying zeros of Dirichlet L-functions2022

    • Author(s)
      Shingo Sugiyama, Ade Irma Suriajaya
    • Journal Title

      Research in Number Theory

      Volume: 8 Issue: 3 Pages: 55-55

    • DOI

      10.1007/s40993-022-00359-0

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The prime number theorem and pair correlation of zeros of the Riemann zeta-function2022

    • Author(s)
      Goldston D. A.、Suriajaya Ade Irma
    • Journal Title

      Research in Number Theory

      Volume: 8 Issue: 4 Pages: 71-71

    • DOI

      10.1007/s40993-022-00371-4

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Dirichlet Series with Periodic Coefficients and Their Value-Distribution near the Critical Line2021

    • Author(s)
      Sourmelidis Athanasios、Steuding Joern、Suriajaya Ade Irma
    • Journal Title

      Proceedings of the Steklov Institute of Mathematics

      Volume: 314 Issue: 1 Pages: 238-263

    • DOI

      10.1134/s0081543821040118

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The error term in the Cesaro mean of the prime pair singular series2021

    • Author(s)
      Goldston D.A.、Suriajaya Ade Irma
    • Journal Title

      Journal of Number Theory

      Volume: 227 Pages: 144-157

    • DOI

      10.1016/j.jnt.2021.03.004

    • Related Report
      2021 Research-status Report 2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A singular series average and the zeros of the Riemann zeta-function2021

    • Author(s)
      Goldston D. A.、Suriajaya Ade Irma
    • Journal Title

      Acta Arithmetica

      Volume: 200 Issue: 1 Pages: 71-90

    • DOI

      10.4064/aa200821-24-2

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A Sum of Negative Degrees of the Gaps Values in Two-Generated Numerical Semigroups and Identities for the Hurwitz Zeta Function2021

    • Author(s)
      Fel Leonid G.、Komatsu Takao、Suriajaya Ade Irma
    • Journal Title

      Nathanson M.B. (eds) Combinatorial and Additive Number Theory IV. CANT 2020. Springer Proceedings in Mathematics & Statistics

      Volume: 347 Pages: 151-160

    • DOI

      10.1007/978-3-030-67996-5_8

    • ISBN
      9783030679958, 9783030679965
    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Stieltjes constants of L-functions in the extended Selberg class2021

    • Author(s)
      Inoue Shota、Eddin Sumaia Saad、Suriajaya Ade Irma
    • Journal Title

      The Ramanujan Journal

      Volume: 55 Issue: 2 Pages: 609-621

    • DOI

      10.1007/s11139-021-00391-1

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A singular series average and the zeros of the Riemann zeta-function2021

    • Author(s)
      D. A. Goldston and Ade Irma Suriajaya
    • Journal Title

      Acta Arithmetica

      Volume: -

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Value-Distribution of the Riemann Zeta-Function along its Julia Lines2020

    • Author(s)
      J. Steuding and A. I. Suriajaya
    • Journal Title

      CMFT Journal

      Volume: - Issue: 3-4 Pages: 389-401

    • DOI

      10.1007/s40315-020-00316-x

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] The values of the Riemann zeta-function on discrete sets2020

    • Author(s)
      Lee Junghun、Sourmelidis Athanasios、Steuding Joern、Suriajaya Ade Irma
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 84 Pages: 315-334

    • DOI

      10.2969/aspm/08410315

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A sum of negative degrees of the gaps values in 2 and 3-generated numerical semigroups2020

    • Author(s)
      Fel Leonid G.、Komatsu Takao、Suriajaya Ade Irma
    • Journal Title

      Annales Mathematicae et Informaticae

      Volume: 52 Pages: 85-95

    • DOI

      10.33039/ami.2020.08.001

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Note on the number of zeros of $\zeta^{(k)}(s)$2020

    • Author(s)
      F. Ge and A. I. Suriajaya
    • Journal Title

      Ramanujan J.

      Volume: - Issue: 2 Pages: 661-672

    • DOI

      10.1007/s11139-019-00219-z

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The Values of the Riemann Zeta-Function on Discrete Sets2020

    • Author(s)
      J. Lee, A. Sourmelidis, J. Steuding and A. I. Suriajaya
    • Journal Title

      ASPM Series

      Volume: -

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] リーマンゼータ関数の臨界領域内等差数列における値の分布 (Values of the Riemann zeta function on vertical arithmetic progressions in the critical strip)2019

    • Author(s)
      A. I. Suriajaya
    • Journal Title

      RIMS Kyokyuroku (for RIMS Workshop 2018 "Analytic Number Theory and Related Topics")

      Volume: 2131-19

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Presentation] Weighted One-level Density of Low-lying Zeros of Dirichlet L-Functions2022

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      Arithmetik an der A7
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Goldbach's Conjecture, the Riemann Hypothesis and problems on twin primes in Number Theory, and recent results relating Goldbach and prime pair problems to zeros of L-functions2022

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      Women in Mathematics
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Pair Correlation of Zeta-Zeros and Two Problems on Primes2022

    • Author(s)
      Daniel A. Goldston
    • Organizer
      Number Theory Conference in honour of Kalman Gyory - Janos Pintz - Andras Sarkoezy
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The one-level density for Dirichlet L-functions weighted by L-values2022

    • Author(s)
      Shingo Sugiyama
    • Organizer
      9th Kyoto conference on automorphic forms
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] ゴールドバッハ表現とディリクレL関数の例外的零点2021

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      第14回数論女性の集まり
    • Related Report
      2021 Research-status Report
  • [Presentation] Goldbach representations and Landau-Siegel zeros2021

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      Arithmetik an der A7
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] The number of Goldbach representations and pair correlation of zeros of the Riemann zeta function2021

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      DMV-OEMG Annual Meeting 2021
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] ゴールドバッハ表現と素数定理の誤差評価、リーマンゼータ関数の零点のペアコリレーション (対相関)2021

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      2021大分整数論研究集会
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] The Hardy-Littlewood Goldbach Conjecture and Landau-Siegel zeros2021

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      Incheon National University 13th International Symposium on Natural Sciences
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The average number of Goldbach representations, pair correlation of zeros of the Riemann zeta function and error term of the prime number theorem2021

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      RIMS Workshop 2021: Analytic Number Theory and Related Topics
    • Related Report
      2021 Research-status Report
  • [Presentation] Error term of the Riesz mean of Hardy-Littlewood singular series (Der Fehlerterm des Riesz-Mittel der singular series von Hardy und Littlewood)2021

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      Arithmetik an der A7
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] An Upper Bound for Stieltjes Constants of L-functions in the Extended Selberg Class2020

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      FCK COVID-19
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] L関数のリーマン型関数等式のJulia線の周りでの値分布2020

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      2020大分整数論研究集会
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Error term of the Riesz mean of Hardy-Littlewood singular series2020

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      RIMS Workshop: Problems and Prospects in Analytic Number Theory
    • Related Report
      2020 Research-status Report
  • [Presentation] Distribution of values of L-functions arising from a Riemann-type functional equation2020

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      International Conference on Analytic Number Theory dedicated to 75th anniversary of G.I. Arkhipov and S.M. Voronin
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] Value distribution of the Riemann zeta function around its Julia lines2020

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      The 13th Young Mathematicians Conference on Zeta Functions
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Generalized Schemmel's function and its associated mean-values2019

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      The fifth mini symposium of the Roman Number Theory Association
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Schemmel関数の平均値 ― 後編2019

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      第12回数論女性の集まり(WINJ12)
    • Related Report
      2019 Research-status Report
  • [Presentation] Distribution of zeros of the derivatives of the Riemann zeta function and its relations to zeros of the zeta function itself2019

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      Boston University/Keio University Workshop 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Improved error estimate for the number of zeros of the derivatives of the Riemann zeta function2019

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      Analytic Number Theory and Related Topics
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] ゼータ関数とL関数、そして、それらに作られた世界2019

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      理研-九大ジョイントワークショップ「数理が紡ぐ素粒子・原子核・宇宙」
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Immersed in math and numbers -- entering the world of Zeta functions2019

    • Author(s)
      Ade Irma Suriajaya
    • Organizer
      Lecture on Pure and Applied Mathematics
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited

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

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