2000 Fiscal Year Annual Research Report

整数論的なL関数の解析的拳動の研究

Research Project

Project/Area Number	12440004
Research Institution	Nagoya University
Principal Investigator	松本耕二名古屋大学, 大学院・多元数理科学研究科, 教授 (60192754)
Co-Investigator(Kenkyū-buntansha)	金光滋近畿大学, 九州工学部, 教授 (60117091) 金銅誠之名古屋大学, 大学院・多元数理科学研究科, 教授 (50186847) 谷川好男名古屋大学, 大学院・多元数理科学研究科, 助教授 (50109261) 秋山茂樹新潟大学, 理学部, 助教授 (60212445) 桂田昌紀慶応義塾大学, 経済学部, 助教授 (90224485)
Keywords	多重ゼータ関数 / 多重ガンマ関数 / 解析接続 / Riemannゼータ関数 / Hurwitzゼータ関数 / 近似関数等式 / 普遍性 / 実二次体
Research Abstract	最も重点的な研究が進められたのは多重ゼータ関数の理論である。Euler-Maclaurinの公式による方法や,Mellin-Barnesの積分表示による方法で,一般の多変数Euler-Zagier型多重ゼータ関数の解析接続の証明に成功した。さらにDirichlet指標を付けたものや,Barnes型の重みをつけたものへと理論を一般化し,その応用として多重指標和の評価,Barnesゼータ関数の重みに関する漸近展開,負の整数点での値の表示など多く新結果を得た。この理論は多重ガンマ関数や,更にそれを介して実2次体の理論に関係しており,従って多重ガンマ関数やHurwitzゼータ関数をお含む興味深い一連の等式,実2次体のHeckeのゼータ関数お特殊値の漸近表示,またRamanujan型の表示式なども得られた。一方,値分布論に関連して正密度法を導入し,SL(2,Z)の保型L関数の普遍性,またLerchのゼータ関数の同時普遍性の証明に成功した。更に一般のL関数についての普遍性の研究も進行中である。また,約数問題や近似関数等式についても一般的な,例えばLandau流の枠組での考察を進め,従来の結果を改良すると同時に,Riemannゼータ関数の近似関数等式の残余項の平均値理論についても,新しい手法を開発して詳細な漸近公式を導出することに成功した。

Research Products

(38 results)

All Other

All Publications (38 results)

[Publications] A.Laurincikas & K.Matsumoto: "The joint universality and the functional independence for Lerch zeta-functions"Nagoya Math.J.. 157. 211-227 (2000)
[Publications] K.Matsumoto: "Recent developments in the mean square theory of the Riemann zeta and other zeta-functions""Number Theory", Hindustan/Brikhauser. 241-286 (2000)
[Publications] A.Laurincikas & K.Matsumoto: "The universality of zeta-functions attached to certain cusp forms"Acta Arith.. (to appear).
[Publications] M.Katsurada & K.Matsumoto: "Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions III"Compositio Math.. (to appear).
[Publications] K.Matsumoto: "The mean values and the universality of Rankin-Selberg L-functions""Proc.Turku Symp.on Number Theory"Walter do Gruyter. (to appear).
[Publications] K.Mastumoto: "Asymptotic expansions of double gamma-functions and related remarks""Analytic Number Theory", Kluwer. (to appear).
[Publications] I.Kiuchi and Y.Tanigawa: "A mean value theorem of the approximate functional equation of ζ^2(s) for short intervals"J.Ramanujan Math.Soc.. 15. 53-70 (2000)
[Publications] S.Akiyama,S.Egami & Y.Tanigawa: "Analytic continuation of multiple zeta functionsand their values at non-positive integers"Acta Arith.. (to appear).
[Publications] S.Kanemitsu,Y.Tanigawa & M.Yoshimoto: "On zeta-and L-function values at special rational arguments via the modular relation""Proc.Intern.Conf on Special Functions and their Applications". (to appear).
[Publications] T.Hara,T.Hattori & H.Watanabe: "Triviality of hierarchical Ising model in four dimensions"Commun.Math.Phys. (to appear).
[Publications] Y.M.Chen,Y.Kitaoka & J.Yu: "Distribution of units of real quadratic fields"Nagoya Math.J.. 158. 167-184 (2000)
[Publications] S.kanemitsu,M.katsurada & M.Yoshimoto: "On the Hurwitz-Lerch zata-functions"Aeq.Math.. 59. 1-19 (2000)
[Publications] K.Matsumoto: "On analytic continuation of Various multiple zeta-functions"in "Number Theory for the Millenniam"A.K.Peters. (to appear).
[Publications] H.Ito: "On a product related to the cubic Gauss sum III"Canad.J.Math.. (to appear).
[Publications] S.Kondo: "A complex hyperbdic structure of the moduli space of curves of genus three"J.Reine Angew.Math.. 525. 219-232 (2000)
[Publications] J.Keum & S.Kondo: "The automorphism groups of Kummer surfaces associated with the product of two elliptic curves"Trans.Amer.Math.Soc. 353. 1469-1487 (2001)
[Publications] S.Kondo: "The moduli space of Enriques surfaces and Borcherds products"J.Alg.Geom.. (to appear).
[Publications] S.Kondo: "The moduli space of curves of genus 4 and Deligne-Mostow's complex reflection groups"Adv.Stud.Pure Math.. (to appear).
[Publications] S.Kanemitsu & M.Yoshimoto: "Euler products,Farey series and the Riemann hypothesis."Publ.Math.Debrecen. 59. 431-449 (2000)
[Publications] S.Kanemitsu,T.Kuzumaki & M.Yoshimoto: "On some sums involving Farey fractions II"J.Math.Soc.Japan. 53. 915-947 (2000)
[Publications] S.kanemitsu,H.Kumagai & M.Yoshimoto: "Sums involving the Hurwitz zeta-function"Ramanujan J.. (to appear).
[Publications] S.kanemitsu & A.Sankaranarayanau: "On a general divisor problem in Landan's framework"in "Analytic Number Theory".Kluwer. (to appear).
[Publications] S.Kanemitsu & T.Kuzumaki: "On a generalization of the Maillet determinant II"Acta Arith. (to appear).
[Publications] S.Kanemitsu,H.Kumagai & M.Yoshimoto: "On rapidly convergent series expressions for zeta-and L-values"Ramanujan J.. (to appear).
[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. 58. 109-119 (2001)
[Publications] I.Kiuchi & N.Yanagisawa: "On the mean value formulas for the approximate functional equation of the Riemann zeta-function"Arch.Math.. (to appear).
[Publications] 江上繁樹: "三重ゼータ関数の和の漸近展開について"数理研講究録. 1160. 1-6 (2000)
[Publications] S.Egami: "On a twisted mean of L(1,X)""Analytic Number Theory", Kluwer. (to appear).
[Publications] S.Akiyama: "Cubic Pisot units with finite beta expansions""Algebraic Number Theory and Diophantine Analysis", W.de Grayter. 11-26 (2000)
[Publications] S.Akiyama & J.thuswaldner: "Topological properties of two-dimensional number systems"J.Theor.Nombres Bordeaux. 12. 69-79 (2000)
[Publications] S.Akiyama & A.Petho": "On canonical number systems."Theor.Comp.Sci. (to appear).
[Publications] S.Akiyama & H.Ishikawa: "On analytic continuation of multiple L-functions and related zeta-functions""Analytic Number Theory", Kluwer. (to appear).
[Publications] S.Akiyama: "On the boundary of self affine tilings generated by Pisot numbers"J.Math Soc.Japan. (to appear).
[Publications] M.Amou,M.Katsurada & K Vaananeu: "On the values of certain q-hypergeometric series"in "Proc.Turku Symp.on Number Theory",W.de Gruyter,. (to appear).
[Publications] M.Katsurada: "On an asymptotic formula of Ramanujan for a certain theta-type series"Acta Arith.. (to appear).
[Publications] M.Amou,M.Katsurada & K.Vaananeu: "On the values of certain q_hypergeometric series II""Analytic Number Theory", Kluwer. (to appear).
[Publications] M.Amou & M.Katsurada: "Differential transcendence of a class of generalized Dirichlet series"Illinois J.Math. (to appear).
[Publications] M.Amou,M.Katsurada & K.Vaananeu: "Arithmetical properties of the values of functions satisfying certain functional equations of Poincare"Acta Arith.. (to appear).

2000 Fiscal Year Annual Research Report

整数論的なL関数の解析的拳動の研究

Principal Investigator

松本 耕二 名古屋大学, 大学院・多元数理科学研究科, 教授 (60192754)

Research Products

[Publications] A.Laurincikas & K.Matsumoto: "The joint universality and the functional independence for Lerch zeta-functions"Nagoya Math.J.. 157. 211-227 (2000)

[Publications] K.Matsumoto: "Recent developments in the mean square theory of the Riemann zeta and other zeta-functions""Number Theory", Hindustan/Brikhauser. 241-286 (2000)

[Publications] A.Laurincikas & K.Matsumoto: "The universality of zeta-functions attached to certain cusp forms"Acta Arith.. (to appear).

[Publications] M.Katsurada & K.Matsumoto: "Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions III"Compositio Math.. (to appear).

[Publications] K.Matsumoto: "The mean values and the universality of Rankin-Selberg L-functions""Proc.Turku Symp.on Number Theory"Walter do Gruyter. (to appear).

[Publications] K.Mastumoto: "Asymptotic expansions of double gamma-functions and related remarks""Analytic Number Theory", Kluwer. (to appear).

[Publications] I.Kiuchi and Y.Tanigawa: "A mean value theorem of the approximate functional equation of ζ^2(s) for short intervals"J.Ramanujan Math.Soc.. 15. 53-70 (2000)

[Publications] S.Akiyama,S.Egami & Y.Tanigawa: "Analytic continuation of multiple zeta functionsand their values at non-positive integers"Acta Arith.. (to appear).

[Publications] S.Kanemitsu,Y.Tanigawa & M.Yoshimoto: "On zeta-and L-function values at special rational arguments via the modular relation""Proc.Intern.Conf on Special Functions and their Applications". (to appear).

[Publications] T.Hara,T.Hattori & H.Watanabe: "Triviality of hierarchical Ising model in four dimensions"Commun.Math.Phys. (to appear).

[Publications] Y.M.Chen,Y.Kitaoka & J.Yu: "Distribution of units of real quadratic fields"Nagoya Math.J.. 158. 167-184 (2000)

[Publications] S.kanemitsu,M.katsurada & M.Yoshimoto: "On the Hurwitz-Lerch zata-functions"Aeq.Math.. 59. 1-19 (2000)

[Publications] K.Matsumoto: "On analytic continuation of Various multiple zeta-functions"in "Number Theory for the Millenniam"A.K.Peters. (to appear).

[Publications] H.Ito: "On a product related to the cubic Gauss sum III"Canad.J.Math.. (to appear).

[Publications] S.Kondo: "A complex hyperbdic structure of the moduli space of curves of genus three"J.Reine Angew.Math.. 525. 219-232 (2000)

[Publications] J.Keum & S.Kondo: "The automorphism groups of Kummer surfaces associated with the product of two elliptic curves"Trans.Amer.Math.Soc. 353. 1469-1487 (2001)

[Publications] S.Kondo: "The moduli space of Enriques surfaces and Borcherds products"J.Alg.Geom.. (to appear).

[Publications] S.Kondo: "The moduli space of curves of genus 4 and Deligne-Mostow's complex reflection groups"Adv.Stud.Pure Math.. (to appear).

[Publications] S.Kanemitsu & M.Yoshimoto: "Euler products,Farey series and the Riemann hypothesis."Publ.Math.Debrecen. 59. 431-449 (2000)

[Publications] S.Kanemitsu,T.Kuzumaki & M.Yoshimoto: "On some sums involving Farey fractions II"J.Math.Soc.Japan. 53. 915-947 (2000)

[Publications] S.kanemitsu,H.Kumagai & M.Yoshimoto: "Sums involving the Hurwitz zeta-function"Ramanujan J.. (to appear).

[Publications] S.kanemitsu & A.Sankaranarayanau: "On a general divisor problem in Landan's framework"in "Analytic Number Theory".Kluwer. (to appear).

[Publications] S.Kanemitsu & T.Kuzumaki: "On a generalization of the Maillet determinant II"Acta Arith. (to appear).

[Publications] S.Kanemitsu,H.Kumagai & M.Yoshimoto: "On rapidly convergent series expressions for zeta-and L-values"Ramanujan J.. (to appear).

[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. 58. 109-119 (2001)

[Publications] I.Kiuchi & N.Yanagisawa: "On the mean value formulas for the approximate functional equation of the Riemann zeta-function"Arch.Math.. (to appear).

[Publications] 江上繁樹: "三重ゼータ関数の和の漸近展開について"数理研講究録. 1160. 1-6 (2000)

[Publications] S.Egami: "On a twisted mean of L(1,X)""Analytic Number Theory", Kluwer. (to appear).

[Publications] S.Akiyama: "Cubic Pisot units with finite beta expansions""Algebraic Number Theory and Diophantine Analysis", W.de Grayter. 11-26 (2000)

[Publications] S.Akiyama & J.thuswaldner: "Topological properties of two-dimensional number systems"J.Theor.Nombres Bordeaux. 12. 69-79 (2000)

[Publications] S.Akiyama & A.Petho": "On canonical number systems."Theor.Comp.Sci. (to appear).

[Publications] S.Akiyama & H.Ishikawa: "On analytic continuation of multiple L-functions and related zeta-functions""Analytic Number Theory", Kluwer. (to appear).

[Publications] S.Akiyama: "On the boundary of self affine tilings generated by Pisot numbers"J.Math Soc.Japan. (to appear).

[Publications] M.Amou,M.Katsurada & K Vaananeu: "On the values of certain q-hypergeometric series"in "Proc.Turku Symp.on Number Theory",W.de Gruyter,. (to appear).

[Publications] M.Katsurada: "On an asymptotic formula of Ramanujan for a certain theta-type series"Acta Arith.. (to appear).

[Publications] M.Amou,M.Katsurada & K.Vaananeu: "On the values of certain q_hypergeometric series II""Analytic Number Theory", Kluwer. (to appear).

[Publications] M.Amou & M.Katsurada: "Differential transcendence of a class of generalized Dirichlet series"Illinois J.Math. (to appear).

[Publications] M.Amou,M.Katsurada & K.Vaananeu: "Arithmetical properties of the values of functions satisfying certain functional equations of Poincare"Acta Arith.. (to appear).

松本耕二名古屋大学, 大学院・多元数理科学研究科, 教授 (60192754)