整数論的なL関数の解析的挙動の研究

2002 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): 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) ; 桂田 昌紀 慶応義塾大学, 経済学部, 教授 (90224485) ; 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 / Euler-Zagier和 / 解析接続 / 漸近展開 / Hurwitzゼータ関数 / Ramanujan formula / モジュラー関係式 / 近似関数等式

Research Abstract

(1)多重ゼータ関数の研究。Euler-Zagierの多重和とBarnesの多重ゼータ関数双方の一般化となっているような一般多重ゼータ関数を導入してその解析接続と,パラメータに関する漸近展開式を示した。さらにMordell-Tornheim型やApostol-Vu型の多重ゼータ関数,またDirichlet指標やより一般の係数を付けた多重Dirichlet級数についても,Mellin-Barnesの積分公式が有効に働き,解析的挙動が解明できることが判明した。Hurwitzゼータ関数の任意高次巾平均のq-漸次展開公式への応用もあり,種々の多重ゼータ関数をまとめあげる統一的画像が得られつつある。
(2)Ramanujan型公式,モジュラー関係式,近似関数等式(とRiemann-Siegel公式)とを結ぶ基本原理がとらえられつつある。この方向に沿って種々のL関数の特殊値の急減少級数表示,その数論的性質と多重ゼータ関数との関係,近似関数等式と平均値公式といった課題に対して新知見を得た。

Research Products

• [Publications] K.Matsumoto: "Asymptotic expansions of double gamma-functions and related remarks""Analytic Number Theory" Proc. Beijing Conf. and Kyoto Conf. (Kluwer). 243-268 (2002)
• [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)
• [Publications] S.Egami, K.Matsumoto: "Asymptotic expansions of multiple zeta-functions and power mean values of Hurwitz zeta functions"J. London Math. Soc.. (2)66. 41-60 (2002)
• [Publications] K.Matsumoto: "On the analytic continuation of various multiple zeta-functions""Number theory for the Millennium II", Proc. Mill. Conf. (A K Peters). 417-440 (2002)
• [Publications] K.Matsumoto: "The analytic continuation and the asymptutic behaviour of certain multiple zeta-functions II""Analytic and Probab. Methods in Number Theory" (TEV). 188-194 (2002)
• [Publications] K.Matsumoto: "Some problems on mean values and the universality of zeta and multiple zeta-functions""Analytic and Probab. Methods in Number Theory" (TEV). 195-199 (2002)
• [Publications] K.Matsumoto, L.Weng: "Zeta-functions defined by two polynomials""Number Theo. Methods-Future Trends", Proc. 2nd China-Japan Sem.(Kluwer). 233-262 (2002)
• [Publications] K.Matsumoto: "Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series"Nagoya Math J.. (to appear).
• [Publications] K.Matsumoto: "Functional equations for double zeta-functions"Math. Proc. Cambridge Phil. Soc.. (to appear).
• [Publications] K.Matsumoto, Y.Tanigawa: "The analytic continuation and the order estimate of multiple Dirichlet series"J. Theorie des Nombres de Bordeaux. (to appear).
• [Publications] A.Laurincikas, K.Matsumoto, J.Steuding: "The universality of L-functions associated to newforms"Izv. Ross. Akad. Nauk Ser. Mat.. (to appear).
• [Publications] S.Kondo: "The moduli space of Enriques surfaces and Borcherds product"J. Alg. Geom.. 11. 601-627 (2002)
• [Publications] S.Kondo: "The moduli of curves of genus 4 and Deligne-Mostow's complex reflection groups"Adv. Stud. Pure Math.. 36. 383-400 (2002)
• [Publications] I.Dolgachev, S.Kondo: "A supersingular K3 surface in characteristic 2 and the Leech lattice"Intern. Math. Res. Notices. 1-23 (2003)
• [Publications] I.Kiuchi, Y.Tanigawa: "On arithmetical functions related to the Ramanujan sum"Period. Math. Hung.. 45. 73-85 (2002)
• [Publications] S.Kanemitsu, Y.Tanigawa, M.Yoshimoto: "On rapidly convergent series for Dirichlet L-function values via the modular relation""Number Theory and Discrete Math." Hindustan. 113-133 (2002)
• [Publications] S.Kanemitsu, A.Sankaranarayanan, Y.Tanigawa: "A mean value theorens for Dirichlet series and a general divisor problem"Monatsh. Math.. 136. 17-34 (2002)
• [Publications] S.Kanemitsu, Y.Tanigawa, M.Yoshimoto: "On rapidly convergent series for the Riemann-zeta values via the modular relation"Abh. Math. Sem. Univ. Hamburg. 72. 187-206 (2002)
• [Publications] S.Kanemitsu, Y.Tanigawa, M.Yoshimoto: "Ramanujan's formula and modular forms""Number Theor. Methods-Future Trends" (Kluwer). 159-212 (2002)
• [Publications] S.Kanemitsu, Y.Tanigawa, M.Yoshimoto: "On multi Hurwitz-zeta function values at rational arguments"Acta Arith.. (to appear).
• [Publications] B.Hambly, K.Hattori, T.Hattori: "Self-repelling walk on the Sierpinski gasket"Probab. Th. Rel. Fields. 124. 1-25 (2002)
• [Publications] T.Hattori, T.Tsuda: "Renormalization group analysis of the self-avoiding paths on the d-dimensional Sierpinski gaskets"J. Stat. Phys.. 109. 39-66 (2002)
• [Publications] S.Kanemitsu, A.Sankaranarayanan: "On a general divisor problem in Landau's framework""Analytic Number Theory" (Kluwer). 205-221 (2002)
• [Publications] M.Amou, M.Katsurada, K.Vaananen: "On the values of certain q-hypergeometric series II""Analytic Number Theory" (Kluwer). 17-25 (2002)
• [Publications] M.Katsurada: "Asymptotic expansions of certain q-series and a formula of Ramanujan for specific values of the Riemann zeta-function"Acta Arith.. (to appear).
• [Publications] S.Egami: "On a twisted power mean of L(1,x)""Analytic Number Theory" (Kluwer). 121-125 (2002)
• [Publications] I.Kiuchi, N.Yanagisawa: "On mean value formula for the approximate functional equation of the Riemann zeta-function"Arch. Math.. 78. 378-385 (2002)
• [Publications] S.Akiyama, A.Petho: "On canonical number systems"Theo. 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" (Kluwer). 1-16 (2002)
• [Publications] S.Akiyama: "On the boundary of self affine tilings generated by Pisot numbers"J. Math. Soc. Japan. 54. 283-308 (2002)
• [Publications] S.Akiyama, H.Brunotte, A.Petho: "Cubic CNS polynomials, notes on a conjecture of W. J. Gilbet"J. Math. Anal. Appl.. (to appear).
• [Publications] C.Jia, K.Matsumoto: "Analytic Number Theory"Kluwer. 408 (2002)
• [Publications] S.Kanemitsu, C.Jia: "Number Theoretic Methods-Future Trends"Kluwer. (2002)