2000 年度実績報告書

整数論的なL関数の解析的拳動の研究

研究課題

研究課題/領域番号	12440004
研究機関	名古屋大学
研究代表者	松本耕二名古屋大学, 大学院・多元数理科学研究科, 教授 (60192754)
研究分担者	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)
キーワード	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
研究概要	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。

研究成果
(38件)

すべてその他

すべて文献書誌 (38件)

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

2000 年度 実績報告書

整数論的なL関数の解析的拳動の研究

研究代表者

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

研究成果

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

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

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

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

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

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

[文献書誌] 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)

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

[文献書誌] 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).

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

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

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

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

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

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

[文献書誌] 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)

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

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

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

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

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

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

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

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

[文献書誌] 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)

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

[文献書誌] 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)

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

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

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

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

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

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

[文献書誌] 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).

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

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

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

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

2000 年度実績報告書

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