整数論的なL関数の解析的挙動の研究

2001 Fiscal Year Annual Research Report

• 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): 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)
• Keywords: 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

Research Abstract

(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。

Research Products

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