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Behavior of Zeta and L-functions and their arithmetic meaning

Research Project

Project/Area Number 09440009
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

MATSUMOTO Kohji  Nagoya Univ., Grad. Sch. Math., Assoc. Prof., 大学院・多元数理科学研究所, 助教授 (60192754)

Co-Investigator(Kenkyū-buntansha) ITO Hiroshi  Nagoya Univ., Grad. Sch. Math., Prof., 大学院・多元数理科学研究所, 教授 (30168372)
TANIGAWA Yoshio  Nagoya Univ., Grad. Sch. Math., Assoc. Prof., 大学院・多元数理科学研究所, 助教授 (50109261)
KITAOKA Yoshiyuki  Nagoya Univ., Grad. Sch. Math., Prof., 大学院・多元数理科学研究所, 教授 (40022686)
KATSURADA Masanori  Keio Univ., Fac. of Eco., Assoc. Prof., 経済学部, 助教授 (90224485)
AKIYAMA Shigeki  Niigata Univ., Fac. of Sci., Assoc. Prof., 理学部, 助教授 (60212445)
木内 功  山口大学, 理学部, 助教授 (30271076)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥11,600,000 (Direct Cost: ¥11,600,000)
Fiscal Year 1999: ¥4,500,000 (Direct Cost: ¥4,500,000)
Fiscal Year 1998: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1997: ¥3,800,000 (Direct Cost: ¥3,800,000)
KeywordsRiemann zeta-function / Dirichlet L-function / Voronoi's formula / Divisor problem / Mean square / Asymptotic expansion / Rankin-Selberg L-function / Universality / Voronoiの公式 / Lerchゼータ関数 / ゼータ関数 / L関数 / 平均値定理 / Pisot数 / 二重ガンマ関数
Research Abstract

There is a strong analogy between the divisor problem (the evaluation of ΔィイD2aィエD2 (X)) and the evaluation of the remainder term EィイD2σィエD2 (T) in the mean square formula for the Riemann zeta-function ζ (S). The basic tools for the study of them are Voronoi's formula and Atkinson's formula, respectively. The results we have obtained on this topic are :
1. By using Voronoi's and Atkinson's formulas, we obtained the mean square formulas of the differences of ΔィイD2aィエD2 (X), or EィイD2σィエD2 (T), in short intervals. Also we proved similar results for the remainder term in the approximate functional equation for ζ ィイD12ィエD1(S).
2. We have studied the generalization to the cases with characters.
3. We showed the Voronoi-type formula for the Riesz sum of the coefficient of Rankin-Selberg L-functions, and proved their mean square formulas.
4. We developed the method of using Mellin-Barnes type of integrals, and established the usefulness of this method for the study of analytic continuation and asymptotic expansions.
Also, we could prove the joint universality for Lerch zeta-functions, and the universality of L-functions attached to modular forms.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (83 results)

All Other

All Publications (83 results)

  • [Publications] T. Hattori, K. Matsumoto: "A limit theorem for Bohr-Jessen's probability measures of the Riemann zeta-function"M. Reine Angew. Math.. 507. 219-232 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Matsumoto, Y. Tanigawa: "On the zeros of higher derivatives of Hardy's Z-function"J. Number Theory. 75. 262-278 (1999)

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

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Katsurada, K. Matsumoto: "A weighted integral approach to the mean square of Dirichlet L-functions""Number Theory and its Applications", Kluwer. 199-229 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Matsumoto: "Recent developments in the mean square theory of the Riemann zeta and other zeta-functions""Number Theory" Hindustan. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] A. Laurineikas, K. Matsumoto: "A joint universality and the functional independence for Lerch zeta-functions"Nagoya Math. J.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Kitaoka: "Finite arithmetic subgroups of GLn VI"Contemp. Math.. 249. 63-71 (1999)

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

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] I. Kiuchi, Y. Tanigawa: "The mean value theorem of the Riemann zeta-function in the critical strip for short intervals""Number Theory and its Applications", Kluwer. 231-240 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] I. Kiuchi, Y. Tanigawa: "A mean value theorem of the approximate functional equation of 5(5)^2 for short intervals"J. Ramanujan Math. Soc.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Egami: "On the mean square of Hecke L-functions associated to holomorphic cusp forms""Number Theory and its Applications", Kluwer. 101-110 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] I. Kiuchi: "On the mean value formula for the non-symmetric form of the approximate functional equation of 5^2(5) in the critical strip"Publ. Math. Debrecen. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Amou, M. Katsurada: "Irrationality results for values of generalized Tschakaloff series"J. Number Theory. 77. 155-169 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Katsurada: "Rapidly convergent series representations for *(2n+1) and their x-analogue"Acta Arith.. 90. 79-89 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Kanemitsu, M. Katsurada, M. Yoshimoto: "On the Hurwitz-Lerch zeta-function"Aequationes Math.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Akiyama: "Self affine tiling and Pisot numeration system""Number Theory and its Applications", Kluwer. 7-17 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Akiyama, J. Thuswaldner: "Topological properties of two-dimensional number systems"J. Th. Nomb. Bordcawx. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Hattori, K. Matsumoto: "A limit theorem for Bohr-Jessen's probability measures of the Riemann zeta-function"J. Reine Angew. Math.. 507. 219-232 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Matsumoto, Y. Tanigawa: "On the zeros of higher derivatives of Hardy's z-function"J. Number Theory. 75. 262-278 (1999)

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

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Katsurada, K. Matsumoto: "A weighted integral approach to the mean square of Dirichlet L-functions""Number Theory and its Applications", Kluwer. 199-229 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Matsumoto: "Recent developments in the mean square theory of the Riemann zeta and other zeta-functions""Number Theory" Hindustan. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] A. Laurincikas, M. Matsumoto: "A joint universality and the functional independence for Lerch zeta-functions"Nagoya Math. J.. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Kitaoka: "Finite arithmetic subgroups of GLn VI"Contemp. Math.. 249. 63-71 (1999)

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

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] I Kiuchi, Y. Tanigawa: "The mean value theorem of the Riemann zeta-function in the critical strip for short intervals""Number Theory and its Applications", Kluwer. 231-240 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] I Kiuchi, Y. Tanigawa: "A mean value theorem of the approximate functional equation of ζ(s)ィイD12ィエD1 for short intervals"J. Ramanujan Math. Soc.. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Egami: "On the mean square of Hecke L-functions associated to holomorphic cusp forms""Number Theory and its Applications", Kluwer. 101-110 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] I. Kiuchi: "On the mean value formula for the non-symmetric form of the approximate functional equation of ζ ィイD12ィエD1 (s) in the critical strip"Publ. Math. Debrecen. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Amou, M. Katsurada: "Irrationality results for values of generalized Tshakaloff series"J. Number Theory. 77. 155-169 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Katsurada: "Rapidly convergent series representations for ζ (2n+1) and their X-analogue"Acta Arith. 90. 79-89 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Kanemitsu, M. Katsurada, M. Yoshimoto: "On the Hurwitz-Lerch zeta-function"Aequationes Math.. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Akiyama: "Self affine tiling and Pisot numeration system""Number Theory and its Applications", Kluwer. 7-17 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Akiyama, J. Thuswaldner: "Topological properties of two-dimensional number systems"J. Th. Nomb. Bordeaux. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Hattori,K.Matsumoto: "A limit theorem for Bohr-Jessen's probability measures of the Riemann zeta-function"J.Reine Angew.Math. 507. 219-232 (1999)

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

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

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Katsurada,K.Matsumoto: "A weighted integral approach to the mean square of Dirichlet L-functions""Number Theory and its Applications",Kluwer. 199-229 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Matsumoto: "Recent developments in the mean square theory of the Riemann zeta and other zeta-functions""Number Theory" Hindustan. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] A.Laurineikas,K.Matsumoto: "A joint universality and the functional independence for Lerch zeta-functions"Nagoya Math. J.. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Kitaoka: "Finite arithmetic subgroups of GLn VI."Contemp.Math.. 249. 63-71 (1999)

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

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

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

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Egami: "On the mean square of Hecke L-functions associated to holomorphic cusp forms""Number Theory and its Applications" Kluwer. 101-110 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] I.Kiuchi: "On the mean value formula for the non-symmetric form of the approximate functional equation of ζ^2(s) in the critical strip"Publ.Math.Debrecen. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Amou,M.Katsurada: "Irrationality results for values of generalized Tschakaloft series"J.Number Theory. 77. 155-169 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Katsurada: "Rapidly convergent series representations for ζ(2n+1) and their x-analogue"Acta Arith.. 90. 79-89 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Kanemitsu,M.Katsurada,M.Yoshimoto: "On the Hurwitz-Lerch zeta-function"Aequationes Math.. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Akiyama: "Self Affine tiling and Pisot numeration system""Number Theory and its Applications" Kluwer. 7-17 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Akiyama,J.Thuswaldner: "Topological properties of two-dimensional number systems"J.Th.Nomb.Bordeaux. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Matsumoto: "Asynptatic series for double zeta,double gamma,and Hecke L-functions" Math.Proc.Cambridge Phil.Soc.123. 385-405 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Laurincikas and K.Matsumoto: "Joint value-distribution theorens on Lerch zeta-functions" Liet.Mat.Rink.38. 312-326 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Hattori and K.Matsumoto: "A limit theorem for Bohr-Jessen's probability measures of the Riemam zeta-function" J.Reine Angew.Math.(to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Ivic,K.Matsumoto and Y.Tanigawa: "On Riesz means of the coefficients of the Rankin-Selberg series" Math.Proc.Cambridge Phil.Soc.(to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Katsurada and K.Matsumoto: "A weighted integrad approach to the mean square of Dirichlet L-functions" in "Namber Theory and its Applications",K.Gyory and S Kanemitsu (eds.) Klower. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Matsumoto and Y.Tanigawa: "On the zeros of higher derivatives of Hardys Z-function" J.Number Theory. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Ishikawa and Y.Kitaoka: "On the clistribution of units modulo prime ideals in real quadratic fields" J.Reine.Angew.Math.494. 65-72 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] I.Kiuchi and Y.Tanigawa: "The mean value theorem of the divisor problem for short intirvals" Arch.Math.71. 445-453 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] I.Kiuchi and Y.Tanigawa: "The mean velue theorem of the Riemam zeta-function in the critical strip for short intervals" in "Number Thetry and its Applications",K.Gyory and S.Kanemitsu (eds.),Kiuwer. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Akiyama and Y.Tanigawa: "Caleulation of values of L-functions associated to elliptic curves" Math.Comput.(to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Akiyama: "Almost uniform distribution mod 1 and the distribution of primes" Acta Math.Hungar.78. 39-44 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Akiyama: "Pisot numbers and greedy algorithm" in "Number Theory,Diophantine,Computational and Algebrat Aspects" K.Gyery et al.(eds.). 9-21 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Akiyama and T.Sasahiro: "A self-similar tiling generated by the minimal-Pisot namber" Acta Math.Inform.Univ.Ostrav.(to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Akiyama: "Self affine tiling and Pisot numeration system" in "Number Theory and its Applications",K.Gyory and S.Kanemitsu (eds.),Kluwer. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Katsurada: "An application of Mellin-Barnes type of integrals to the mean squore of L-functions" Liet.Mati.Rink.38. 98-112 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Katsurada: "Power series and asymptotic series associated with the Lerch zeta-function" Proc.Japan Acad.Ser.A. 74. 167-170 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Katsurada: "Rapidly convergent series representations for S(2n+1) and their X-analogue" Acta Arith.(to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Amou and M.Katsurada: "Irrationality results for values of generalized Tschakaloff series" J.Number Theory. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Kanemitsu,M.Katsurada and M.Yoshimoto: "On the Hurwitz-Lerch zeta-function" Aequations Math.(to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] I.Kiuchi and K.Matsumoto: "The resemblance of the behaivour of the remainder terms D_δ(t),Δ_<1-2δ>(x) and R(δ+it)" “Sieve Methods,Exponential Sums,and their Applications in Number Theory",Cambridge Univ.Press. 255-273 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Katsurada and K.Matsumoto: "Explict formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions II" “Analytic and Probabilistic Methods in Number Theory"New Trends in Probab.and Statist.Vol.4.VSP/TEV. 119-134 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Matsumoto: "Asymptotic series for double zeta,double gamma,and Hecke L-functions" Math.Proc.Cambridge Phil.Soc.(to appear). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Kitaoka: "Finite arithmetic subgroups of GL_n,V" Nagoya Math.J.146. 131-148 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] F.Fujii and Y.Kitaoka: "On plain lattice points whose coordinates are reciprocals modulo a prime" Nagoya Math.J.147. 137-146 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] H.Ito: "On a prooluct related to the cubic Gauss sum II" Nagoya Math.J.148. 1-21 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] J.Furuya and Y.Tanigawa: "Estimation of a certain function related to the Dirichlet divisor problem" “Analytic and Probabilistic Methods in Number Theory"New Trends in Probab,and Statist.Vol.4,VSP/TEV. 171-189 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] S.Akiyama and Y.Tanigawa: "Calculation of values of L-functions associated to elliptic curves" Math.Comput.(to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] I.Kiuchi and Y.Tanigawa: "The mean value theorem of the divisor problem for short intervals" Areh.Math. (to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] S.Akiyama: "Almost uniform distribution modulo1 and the distribation of primes" Acta Math.Hungar.78. 39-44 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] S.Akiyama: "Pisot numbers and greedy algorithm" “Proc.Intern,conf. on Number Theory held at Eger". (to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Katsurada: "On Mellin-Barnes type of integrals and sums associated with the Riemann zeta-function" Publ.Inst.Math.(Beograd). 62. 13-25 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Katsurada: "An application of Mellin-Barnes'type integrals to the mean square of Lerch zeta-functions" Collect.Math.48. 137-153 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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