2001 年度実績報告書

整数論的なL関数の解析的挙動の研究

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	桂田昌紀慶応大学, 経済学部, 教授 (90224485) 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) 北岡良之名城大学, 理工学部, 教授 (40022686) 秋山茂樹新潟大学, 理学部, 助教授 (60212445) 木内功山口大学, 理学部, 助教授 (30271076)
キーワード	多重ゼータ関数 / 漸近展開 / Mellin-Barnes formula / Ramanujan formula / 保型L関数 / universality / Rankin-Selberg L関数 / Hurwitzゼータ関数
研究概要	(1)多重ゼータ関数の研究。Euler-Zagier型の多変数多重ゼータ関数について、Euler-Maclaurinの和公式を用いる方法あるいはMellin-Barnesの積分公式を用いる方法による解析接続を見出し、更に非正整数点での表示や漸近展開公式を得た。応用として、Hurwitzゼータ関数の任意高次巾平均のq-漸近展開公式、高次元球面上のラプラシアンの行列式の計算、実2次体のある種のHeckeのゼータ関数の特殊値の漸近公式などが得られた。 (2)Ramanujan型公式と急収束級数。ゼータ関数、L関数の種々の特殊値を表示する新しいタイプの級数の導出、とくにモジュラー関係式を背景とした統一的理論の展開に成功した。またこの種の級数の数論的性質についての新知見を得た。 (3)普遍性定理、n≧2に対するGL(n)の保型L関数の普遍性を研究する道を開拓し、とくにGL(2)の正則保型形式、及びRankin-Selberg型L関数の普遍性を証明した。Rankin-Selberg型L関数については二乗平均値に対しても新しい評価を得た。

研究成果

(36件)

すべてその他

すべて文献書誌 (36件)

[文献書誌] K. Matsumoto: "The mean values and the universality of Rankin-Selberg L-functions""Number Theory" (Proc. Turku Symp.). 201-221 (2001)
[文献書誌] A. Laurincikas, K. Matsumoto: "The universality of zeta-functions attached to certain cusp forms"Acta Arith. 98. 345-359 (2001)
[文献書誌] 松本耕二: "ゼータ関数の確率論的値分布論"数学. 53. 279-296 (2001)
[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: "Asymptotic expansions of double gamma-functions and related remarks""Analytic Number Theory" Proc. Beijing Conf. & Kyoto Conf.. (to appear).
[文献書誌] K. Matsumoto: "On the analytic continuation of various multiple zeta-functions"Number Theory for the Millenniam. (to appear).
[文献書誌] S. Egami, K. Matsumoto: "Asymptotic expansions of multiple zeta-functions and power mean values of Hurwitz zeta-functions"J. London Math. Soc. (to appear).
[文献書誌] Y. Kitaoka: "Distribution of units of a cubic field with negative discriminant"J. Number Theory. 91. 318-355 (2001)
[文献書誌] B. Hambly, K. Hattori, T. Hattori: "Self-repelling walk on the Sierpinski gasket"Probab. Th. Rel. Fields. (to appear). (2002)
[文献書誌] T. Hara, T. Hattori, H. Watanabe: "Triviality of hierarchical Ising model in four dimensions"Commun. Math. Phys.. 220. 13-40 (2001)
[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 curves of genus 4 and Deligne-Mostow's complex reflection groups"Adv. Stud. Pure Math.. (to appear).
[文献書誌] S. Kondo: "The moduli space of Enriques surfaces and Borcherds products"J. Alg. Geom.. (to appear).
[文献書誌] I. Kiuchi, Y. Tanigawa: "A mean value theorem for the approximate functional equation of 5^2(s) for short intervals"J. Ramanujan Math. Soc.. 15. 53-70 (2000)
[文献書誌] I. Kiuchi: "On the mean value formula for the non-symmetric form of the approximate functiond eq. of 5^2(s) in the critical strip"Publ. Math. Debrecen. 58. 109-119 (2001)
[文献書誌] S. Kanemitsu, H. Kumagai, M. Yoshimoto: "Sums involving the Hurwitz zeta-function"Ramanujan J.. 5. 5-19 (2001)
[文献書誌] S. Kanemitsu, H. Kumagai, M. Yoshimoto: "On rapidly convergent series expressions for zeta and L-values"Ramanujan J.. 5. 91-104 (2001)
[文献書誌] S. Kanemitsu, T. Kuzumaki: "On a generalization of the Maillet determinant II"Acta Arith.. 99. 343-361 (2001)
[文献書誌] S. Kanemitsu, Y. Tanigawa, M. Yoshimoto: "On rapidly convergent series for zeta and L-function values via the modular relation"Proc. Intern. Conf. on Special Functions and their Applications. 21-42
[文献書誌] S. Kanemitsu, Y. Tanigawa, M. Yoshimoto: "On the values of the Riemann zeta-function at rational arguments"Hardy-Ramanujan J.. 24. 10-18 (2001)
[文献書誌] S. Kanemitsu, Y. Tanigawa, M. Yoshimoto: "On rapidly convergent series for Dirichlet L-function values via the modular relation"Math. in honor of S. Ramanujan. (to appear).
[文献書誌] S. Kanemitsu, A. Sankaranarayanan: "On a general divisor problem in Landau's framework""Analytic Number Theory" Proc. Beijing Conf. & Kyoto Conf.. (to appear).
[文献書誌] S. Kanemitsu, A. Sankaranarayanan, Y. Tanigawa: "A mean value thecrem for Dirichlet series and a general divisor problem"Monatsh. Math.. (to appear).
[文献書誌] S. Kanemitsu, Y. Tanigawa, M. Yoshimoto: "Ramanujan's formula and modular forms""Number Theoretic Methods-Future Trends" Proc. China-Japan Sem.. (to appear).
[文献書誌] S. Akiyama, S. Egami, Y. Tanigawa: "Analytic continuation of multiple zeta-functions and their values at non-positive integers"Acta Arith. 98. 107-116 (2001)
[文献書誌] S. Akiyama, A. Petho: "On canonical number systems"Theor. Comp. Sci.. 270. 921-933 (2002)
[文献書誌] S. Akiyama, H. Ishikawa: "On analytic continuation of multiple L-functions and related zeta-functions""Analytic Number Theory" Proc. Beijing Conf. & Kyoto Conf.. (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] S. Akiyama, Y. Tanigawa: "Multiple zeta values at non-positive integers"Ramanujan J.. (to appear).
[文献書誌] M. Amou, M. Katsurada, K. Vaananen: "On the values of certain q-hypergeometric series""Number Theory" Proc. Turku Conf.. 5-17 (2001)
[文献書誌] M. Katsurada: "On an asymptotic formula of Ramanujan for a certain theta-type series"Acta Arith. 97. 157-172 (2001)
[文献書誌] 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)
[文献書誌] M. Amou, M. Katsurada: "Differential transcendence of a class of generalized Dirichlet series"Illinois J. Math.. 45. 939-948 (2001)
[文献書誌] M. Amou, M. Katsurada, K. Vaananen: "On the values of certain q-hypergeometric series II""Analytic Number Theory" Proc. Beijing Conf. & Kyoto Conf.. (to appear).
[文献書誌] S. Egami: "On a twisted power mean of L(1, X)""Analytic Number Theory" Proc. Beijing Conf. & Kyoto Conf.. (to appear).
[文献書誌] C. Jia, K. Matsumoto: "Analytic Number Theory"Kluwer Academic Publishers (to appear). (2002)

2001 年度 実績報告書

整数論的なL関数の解析的挙動の研究

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] K. Matsumoto: "The mean values and the universality of Rankin-Selberg L-functions""Number Theory" (Proc. Turku Symp.). 201-221 (2001)

[文献書誌] A. Laurincikas, K. Matsumoto: "The universality of zeta-functions attached to certain cusp forms"Acta Arith. 98. 345-359 (2001)

[文献書誌] 松本 耕二: "ゼータ関数の確率論的値分布論"数学. 53. 279-296 (2001)

[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: "Asymptotic expansions of double gamma-functions and related remarks""Analytic Number Theory" Proc. Beijing Conf. & Kyoto Conf.. (to appear).

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

[文献書誌] S. Egami, K. Matsumoto: "Asymptotic expansions of multiple zeta-functions and power mean values of Hurwitz zeta-functions"J. London Math. Soc. (to appear).

[文献書誌] Y. Kitaoka: "Distribution of units of a cubic field with negative discriminant"J. Number Theory. 91. 318-355 (2001)

[文献書誌] B. Hambly, K. Hattori, T. Hattori: "Self-repelling walk on the Sierpinski gasket"Probab. Th. Rel. Fields. (to appear). (2002)

[文献書誌] T. Hara, T. Hattori, H. Watanabe: "Triviality of hierarchical Ising model in four dimensions"Commun. Math. Phys.. 220. 13-40 (2001)

[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 curves of genus 4 and Deligne-Mostow's complex reflection groups"Adv. Stud. Pure Math.. (to appear).

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

[文献書誌] I. Kiuchi, Y. Tanigawa: "A mean value theorem for the approximate functional equation of 5^2(s) for short intervals"J. Ramanujan Math. Soc.. 15. 53-70 (2000)

[文献書誌] I. Kiuchi: "On the mean value formula for the non-symmetric form of the approximate functiond eq. of 5^2(s) in the critical strip"Publ. Math. Debrecen. 58. 109-119 (2001)

[文献書誌] S. Kanemitsu, H. Kumagai, M. Yoshimoto: "Sums involving the Hurwitz zeta-function"Ramanujan J.. 5. 5-19 (2001)

[文献書誌] S. Kanemitsu, H. Kumagai, M. Yoshimoto: "On rapidly convergent series expressions for zeta and L-values"Ramanujan J.. 5. 91-104 (2001)

[文献書誌] S. Kanemitsu, T. Kuzumaki: "On a generalization of the Maillet determinant II"Acta Arith.. 99. 343-361 (2001)

[文献書誌] S. Kanemitsu, Y. Tanigawa, M. Yoshimoto: "On rapidly convergent series for zeta and L-function values via the modular relation"Proc. Intern. Conf. on Special Functions and their Applications. 21-42

[文献書誌] S. Kanemitsu, Y. Tanigawa, M. Yoshimoto: "On the values of the Riemann zeta-function at rational arguments"Hardy-Ramanujan J.. 24. 10-18 (2001)

[文献書誌] S. Kanemitsu, Y. Tanigawa, M. Yoshimoto: "On rapidly convergent series for Dirichlet L-function values via the modular relation"Math. in honor of S. Ramanujan. (to appear).

[文献書誌] S. Kanemitsu, A. Sankaranarayanan: "On a general divisor problem in Landau's framework""Analytic Number Theory" Proc. Beijing Conf. & Kyoto Conf.. (to appear).

[文献書誌] S. Kanemitsu, A. Sankaranarayanan, Y. Tanigawa: "A mean value thecrem for Dirichlet series and a general divisor problem"Monatsh. Math.. (to appear).

[文献書誌] S. Kanemitsu, Y. Tanigawa, M. Yoshimoto: "Ramanujan's formula and modular forms""Number Theoretic Methods-Future Trends" Proc. China-Japan Sem.. (to appear).

[文献書誌] S. Akiyama, S. Egami, Y. Tanigawa: "Analytic continuation of multiple zeta-functions and their values at non-positive integers"Acta Arith. 98. 107-116 (2001)

[文献書誌] S. Akiyama, A. Petho: "On canonical number systems"Theor. Comp. Sci.. 270. 921-933 (2002)

[文献書誌] S. Akiyama, H. Ishikawa: "On analytic continuation of multiple L-functions and related zeta-functions""Analytic Number Theory" Proc. Beijing Conf. & Kyoto Conf.. (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] S. Akiyama, Y. Tanigawa: "Multiple zeta values at non-positive integers"Ramanujan J.. (to appear).

[文献書誌] M. Amou, M. Katsurada, K. Vaananen: "On the values of certain q-hypergeometric series""Number Theory" Proc. Turku Conf.. 5-17 (2001)

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

[文献書誌] 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)

[文献書誌] M. Amou, M. Katsurada: "Differential transcendence of a class of generalized Dirichlet series"Illinois J. Math.. 45. 939-948 (2001)

[文献書誌] M. Amou, M. Katsurada, K. Vaananen: "On the values of certain q-hypergeometric series II""Analytic Number Theory" Proc. Beijing Conf. & Kyoto Conf.. (to appear).

[文献書誌] S. Egami: "On a twisted power mean of L(1, X)""Analytic Number Theory" Proc. Beijing Conf. & Kyoto Conf.. (to appear).

[文献書誌] C. Jia, K. Matsumoto: "Analytic Number Theory"Kluwer Academic Publishers (to appear). (2002)

2001 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)

[文献書誌] 松本耕二: "ゼータ関数の確率論的値分布論"数学. 53. 279-296 (2001)