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Specific values of higher derivatives of zeta functions : zeta

Research Project

Project/Area Number 13640041
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KATSURADA Masanori  Keio Univ., Econ., Professor, 経済学部, 教授 (90224485)

Co-Investigator(Kenkyū-buntansha) HIKARI Michitaka  Keio Univ., Econ., Professor, 経済学部, 教授 (30056296)
NISHIOKA Kumiko  Keio Univ., Econ., Professor, 経済学部, 教授 (80144632)
SHIOKAWA Iekata  Keio Univ., Sci. & Tech., Professor, 理工学部, 教授 (00015835)
TOSE Nobuyuki  Keio Univ., Econ., Professor, 経済学部, 教授 (00183492)
WATABE Mutsuo  Keio Univ., Bus. & Com., Professor, 商学部, 教授 (30080493)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2003: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywordszeta function / higher derivative / specific value / asymptotic expansion / メリン変換
Research Abstract

1. Specific values of higher derivatives of the Lerch zeta-function : Throughout the following, s is a complex variable, a, λ are real parameters with a > 0, and φ(s,a,λ) denotes the Lerch zeta-function defined by the Dirichlet series Σ^∞_<n=0>e^<2πin>(n+a)^<-s>, and its meromorphic continuation over the whole s-plane. At an earlier stage of the present research, the head investigator established a complete asymptotic expansion of φ(s,a+z,λ) as z → ∞ through the sector |arg z| < π, which was further applied to study the particular values of higher derivatives R_<k,m>(z,λ) = (-1)^<k+1>(δ/δs)^kφ(s,z,λ)|_<s=-m> (k,m = 0,1,...). The results obtained for R_<k,m>(z,λ) are its Taylor expansion, the formulae of the types of Gauβ, Weierstraβ and Plana ; those together with the proofs are organized in the paper "Power series and asymptotic series associated with the Lerch zeta-function : applications to higher derivatives," (preprint prepared for submission).
2. A multiple mean square of Lerch ze … More ta-functions : The Hurwitz zeta-function ζ(s,1 + x), a particular case of the Lerch zeta-function, is obtained by shifting n for n + x (n = 1,2,...) in each summand of the Riemann zeta-function ζ(s) = ζ(s,1). The head investigator recently generalized his previous result in [Collect. Math. 48 (1997)], which asserts a complete asymptotic expansion of the mean square ∫^1_0|φ(s,1+x,λ)|^2dx as t = Im s → ±∞, to show that a similar asymptotic series still exists for the multiple mean square ∫^1_0【triple bond】∫^1_0|φ(s,a+x_1+【triple bond】+x_m,λ)|^2dx_1【triple bond】dx_m (a > 0: a constant; m = 1,2,...). The results and their proofs are organized in the paper "An application of Mellin-Barnes type of integrals to the mean square of Lerch zeta-functions II," (submitted for publication).
3. Epstein zeta-functions and their integral transforms : Let z = x + iy be a parameter in the complex upper half-plane. Then the Epstein zeta-function ζ_<Z^2>(s;z), attached to the quadratic form Q(u,v) = |u + vz|^2, is defined by ζ_<Z^2>(s;z) = Σ ^</∞>_<m,n=-∞>Q(m,n)^<-s> (upon omitting the term with m = n = 0), and its meromorphic continuation over the whole s-plane ; this plays an important role in the study of (arithmetical) quadratic forms. The head investigator recently established complete asymptotic expansions, as y = Im z → +∞, of ζ_<Z^2>(s;z) and its Laplace-Mellin transform (which can be regarded as a mean value with the weight of Poisson distribution). The results obtained are organized with their proofs in the paper "Complete asymptotic expansions associated with the Epstein zeta-function," (submitted for publication). Less

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report

Research Products

(72 results)

All Other

All Publications (72 results)

  • [Publications] M.Katsurada: "On an asymptotic formula of Ramanujan for a certain theta-type series"Acta Arith.. 97. 157-172 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada, K.Vaananen: "On the values of certain q-hypergeometric series"in "Number Theory : Proc.Turku Symp.Number Theory, "M.Jutila and T.Matsankyla (Eds.) de Gruyter. 5-17 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada, K.Vaananen: "Arithmetical properties of the values of functions satisfying certain functional equations of Poincare"Acta Arith.. 99. 389-407 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada: "Differential transcendence of a class of generalized Dirichlet series"Illinois J.Math.. 45. 939-948 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada, K.Vaananen: "On the values of certain q-hypergeometric series II"in "Analytic Number Theory ; the joint Proc.China-Japan Number Theory Conf., "C.Jia and K.Matsumoto (Eds.) Kluwer. 17-25 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Katsurada, K.Matsumoto: "Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions III"Compositio Math.. 131. 239-266 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Katsurada: "Asymptotic expansions of certain q-series and a formula of Ramanujan for specific values of the Riemann zeta-function"Acta Arith.. 107. 269-298 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada: "Irrationality results for values of generalized Tschakaloff series II"J.Number Theory. 104. 132-155 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kurosawa, I.Shiokawa: "q-linear functions and algebraic independence"Tokyo J.Math.. 25. 132-155 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Okada, I.Shiokawa: "<q, r>-linear systems and algebraic independence"R.I.M.S.Kokyuroku. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Nishioka: "Algebraic independence of reciprocal sums of binary recurrences"Mh.Math.. 136. 123-141 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] D.Duverney, K.Nishioka: "An inductive method for proving the transcendence of certain series"Acta Arith.. 110. 305-330 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit difference scheme for the variable coefficients parabolic differential equation (II)"in "Proc.Para.Distr.Process.Tech.Appl.Intern.Conf." H.R.Arabnia (Ed.) IEEE. 561-569 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit difference scheme for the variable coefficients parabolic differential equation (III)"Far East J.Appl.Math.. 9. 15-29 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit differential scheme for the variable coefficients parabolic differential equation (IV)"in "Numer.Meth.Appl." Lect.Note Comp.Sci. Springer. 536-544 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit differential scheme for the variable coefficients two-dimensional parabolic differential equation"Far East J.Appl.Math.. 9. 171-188 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit differential scheme for the variable coefficients two dimensional parabolic differential equation (V)"in "Proc.9th Intern.Conf.Numer.Meth.Conti.Mech." issued in J.Mech.Eng.. 54. 327-341 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit differential schemes for the variable coefficients two dimensional parabolic differential equation (V)"J.Appl.Mech.. 54. 327-341 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Tsuboi: "The Chern number of the normalization of an algebraic threefold with ordinary singularities"in "Proc.Rencont. Singularite Franco-Japonaise" CIRM Luminy. (to appear). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Tsuboi: "The Euler numbers of the normalization of an algebraic threefold with ordinary singularities"in "Geometric Singularity Theory" Banach Center Pub. Polish Acad.Sci.. (to appear). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Tsuboi: "On certain hypersurfaces with non-isolated singularities in P^4(C)"Proc.Japan Acad.. 79 Ser.A. 1-4 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Tsuboi: "A certain degenerate ordinary singularity of dimension three"in "Finite and Infinite Dimensional Complex Analysis", issued in Shandon Sci.Tech.. 223-228 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 宮嶋公夫: "強擬凸CR多様体と正規孤立特異点の変形"数学. 53. 172-184 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Miyajima: "CR description of the formal deformations of quasi-homogeneous singularities"in "Selected Topics in Cauchy-Riemann Geometry", S.Dragomir (Ed.), issued in quademi di matematica. 9. 249-280 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Miyajima: "Analytic Approach to deformation of resolution of normal isolated singularities"J.Korean Math.Soc.. 40. 709-725 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Miyajima: "Strongly pseudoconvex CR manifolds and deformation of normal isolated singularities"Sugaku Expositions. 16. 191-206 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kurokawa: "On the closure of the Lizorkin space in spaces of Beppo Levi type"Studia.Math.. 150. 299-323 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kurokawa: "Smooth invariant classes for singular integrals"Rep.Fac.Sci.Kagoshima Univ.. 36. 29-36 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kurokawa: "Higher Riesz transforms and derivatives of the Riesz kernels"Integral Transform Spec.Funct.. 15. 51-71 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Katsurada: "On an asymptotic formula of Ramanujan for a certain theta-type series"Acta Arith.. 97. 157-172 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada, K.Vaananen: "On the values of certain q-hypergeometric series"Number Theory : Proceedings of the Turku Symposium on Number Theory in Memory of Kustaa Inkeri, May 31-June 4, 1999 (M.Jutila and T.Metsankyla (Eds.) (Gruyter, Berlin). 5-17 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada, K.Vaananen: "Arithmetical properties of the values of functions satisfying certain functional equations of Poincare"Acta Arith.. 99. 389-407 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada: "Differential transcendence of a class of generalized Dirichlet series"Illinois J.Math.. 45. 939-948 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada.K.Vaananen: "On the values of certain q-hypergeometric series II"Analytic Number Theory : the joint Proceedings of the China-Japan Number Theory Conference (C.Jia and K.Matsumoto (Eds.)) (Kluwer, Dordrecht). 17-25 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Katsurada, K.Matsumoto: "Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions III"Compositio Math.. 131. 131 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Katsurada: "Asymptotic expansions of certain q-series and a formula of Ramanujan for specific values of the Riemann zeta-function"Acta Arith.. 107. 107 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada: "Irrationality results for values of generalized Tschakaloff series II"J.Number Theory. 104. 132-155 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kurosawa, I.Shiokawa: "q-linear functions and algebraic independence"Tokyo J.Math.. 25. 459-472 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Okada, I.Shiokawa: "<q,r>-number systems and algebraic independence"Analytic Number Theory and Surrounding Areas (Kokyuroku R.I.M.S.). (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Nishioka: "Algebraic independence of reciprocal sums of binary recurrences II"Mh.Math.. 136. 123-141 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] D.Duverney, K.Nishioka: "An inductive method for proving the transcendence of certain series"Acta Arith.. 110. 305-330 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit difference schemes for the variable coefficients parabolic difference equation (II)"Proceedings of Parallel and Distributed Processing of the Techniques and Applications, International Conference, Las Vegas, Nevada, USA. June 25-28. 561-569 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit difference schemes for the variable coefficients parabolic differential equation (III)"Far East J.Appl.Math.. 9. 15-29 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit difference schemes for the variable coefficients parabolic differential equation (IV)"Numerical Methods and Applications (Lecture Notes in Computer Science) (Springer). 536-544 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit difference schemes for the variable coefficients two dimensional parabolic differential equation"Far East J.Appl.Math.. 9. 171-188 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit difference schemes for the variable coefficients two dimensional parabolic differential equation (V)"Proceedings of the 9th International Conference on Numerical Methods in Continuum Mechanics. (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nakashima: "Unconditionally stable explicit difference schemes for the variable coefficients two dimensional parabolic differential equation (V)"J.Appl.Mech.. 327-341 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Tsuboi: "The Chern number of the normalization of an algebraic threefold with ordinary singularities"Proceedings of the Rencontre "Singulariti'es Franco-Japonaise (CIRM, Luminy, Marseille, France)". (to appear). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Tsuboi: "The Euler number of the normalization of an algebraic threefold with ordinary singularities"Geometric Singularity Theory (Banach Center Publications) (Polish Academy of Sciences, Warzawa). (to appear). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Tsuboi: "On certain hypersurfaces with non-isolated singularities in P^4(C)"Proc.Japan Acad.. 79 Ser.A. 1-4 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Tsuboi: "A certain degenerate ordinary singularity of dimension three"Finite or Infinite Dimensional Complex Analysis (Shandon Science and Technology). 223-228 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 宮嶋 公夫: "強擬凸CR多様体と正規孤立特異点の変形"数学. 53巻2号. 172-184 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Miyajima: "CR description of the formal deformations of quasi-homogeneous singularities"Selected Topics in Cauchy-Riemann Geometry (Ed. S.Drago-mir) (quademi di matematica). 9. 249-280 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Miyajima: "Analytic approach to deformation of resolution of normal isolated singularities : Formal deformations"J.Korean Math.Soc.. 40. 709-725 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Miyajima: "Strongly pseudoconvex CR manifolds and deformation of normal isolated singularities"SUGAKU EXPOSITIONS. 16. 191-206 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kurokawa: "On the closure of the Lizorkin space in spaces of Beppo Levi type"Studia Math.. 150. 299-323 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kurokawa: "Smooth invariant classes for singular integrals"ReFac.Sci.Kagoshima Univ.. 36. 29-36 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kurokawa: "Higher Riesz transforms and derivatives of the Riesz kernels"Integral Transform Spec.Funct.. 15. 51-71 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Katsurada, M.: "Asymptotic expansions of certain q-series and a formula for specific values of the Riemann zeta-function"Acta Arithmetica. 107・3. 269-298 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Katsurada, M.: "Complete asymptotic expansions associated with the Epstein zeta-functions"日本数学会代数学分科会講演アブストラクト. 総合分科会. 40-41 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Katsurada, M.: "Complete asymptotic expansions associated with the Epstein zeta-functions II"日本数学会代数学分科会講演アブストラクト. 年会(掲載予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Amou, M.: "Irrationality results for values of generalized Tschakaloff series II"Journal of Number Theory. 104. 132-153 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Amou: "On the values of certain q-hypergeometric series II"in Analytic Number Theory, Kluver, Dev. Math.. 6. 17-25 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Amou: "Irrationality results for values of generalized Tschakaloff series II"Journal of Number Theory. (掲載予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Katsurada: "Explicit formulas and asymptatic expansions for certain mean square of Huraita zeta-function III"Compositio Mathematica. 131. 236-266 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Katsurada: "Asymptotic expansions of certain q-series and a formula of Ramanujan for specific values of the Riemann zeta-function"Acta Arithmetica. (掲載予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Katsurada: "On an asymptotic formula of Ramanujan for a certain theta-type 1 series"Acta Arithmetica. 97. 157-172 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Amou: "Arithmetical properties of the values of functions satisfying certain functional equations of Poincare"Acta Arithmetica. 99. 389-407 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Amou: "Differntial transcendence of a class of generalized Dirichlet series"Illinois Journal of Mathematics. 45. 939-948 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Amou: "On the values of certain q-hypergeometric series"in "Number Theory"deGruyten Berlin. proc.. 5-17 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Amou: "On the values of certain q-hypergeometric series II"in "Analytic Number Theory, "Kluwer, Doydrecht. proc.(掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Katsurada: "Explieit formulas and asymptotic expansions for certain mean square of Hurwits zeta-functions"Compositio Mathematica. (掲載予定).

    • Related Report
      2001 Annual Research Report

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Published: 2001-03-31   Modified: 2016-04-21  

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