1998 Fiscal Year Annual Research Report

ゼータ関数,L関数の挙動とその整数論的意味

Research Project

Project/Area Number	09440009
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, 助教授 (60192754)
Co-Investigator(Kenkyū-buntansha)	桂田昌紀鹿児島大学, 理学部, 助教授 (90224485) 秋山茂樹新潟大学, 理学部, 助教授 (60212445) 伊藤博名古屋大学, 大学院・多元数理科学研究科, 教授 (30168372) 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) 北岡良之名古屋大学, 大学院・多元数理科学研究科, 教授 (40022686)
Keywords	Riemannゼータ関数 / Dirichlet L関数 / 二乗平均 / Voronoiの公式 / 約数問題 / Rankin-Selberg L関数 / Lerchゼータ関数 / universality
Research Abstract	種々のゼータ関数,L関数について,主として平均値理論と値分布論の立場からの研究をすすめた。平均値については,まず短い区間での二乗平均を調べるため,準備として約数問題の残余項の短い区間での差の平均値について,Voronoi公式を用いるJutilaの方式を援用して考察し,さらにその方法を発展させて,Riemannゼータ関数の二乗平均の残余項Eσ(T)について同様の平均値の漸近公式を証明した。この場合もσ=3/4において挙動の変化が観察されることは著しい DirichletのL関数については,重みつき積分にAtkinsonのdissectipn methodを適用し,Weilの評価をも援用することで,O((qT)^<1/3+ε>)の形のerroe termをもつ精密な二乗平均値公式を得た。また,Voronoiの方法を正則尖点形式に付随するRankin-SelbregのL関数の係数和に用い,係数和の1次Riesz平均に関するMeurman型の二乗平均値公式を示した。またその残余項の評価が改良されれば,係数和の残余項についてのRanlinの古典的な評価が改良できるのであろう,ということを注意した。次に値分布については,Bohr-Jessenの確率測度Rの挙動について,Joyner型不等式の壁を破って,漸近等式を確立した。その証明にはKaramataのregulary carying functimの理論が用いられる。さらに,Lerch zetaのjoint cniversalityを示すため,準備としてjoint極限定理を,Bagchiのエルゴード論的手法によって証明した。保型L関数のiniversalityについても結果を得た。

Research Products

(19 results)

All Other

All Publications (19 results)

[Publications] K.Matsumoto: "Asynptatic series for double zeta,double gamma,and Hecke L-functions" Math.Proc.Cambridge Phil.Soc.123. 385-405 (1998)
[Publications] A.Laurincikas and K.Matsumoto: "Joint value-distribution theorens on Lerch zeta-functions" Liet.Mat.Rink.38. 312-326 (1998)
[Publications] T.Hattori and K.Matsumoto: "A limit theorem for Bohr-Jessen's probability measures of the Riemam zeta-function" J.Reine Angew.Math.(to appear).
[Publications] A.Ivic,K.Matsumoto and Y.Tanigawa: "On Riesz means of the coefficients of the Rankin-Selberg series" Math.Proc.Cambridge Phil.Soc.(to appear).
[Publications] M.Katsurada and K.Matsumoto: "A weighted integrad approach to the mean square of Dirichlet L-functions" in "Namber Theory and its Applications",K.Gyory and S Kanemitsu (eds.) Klower. (to appear).
[Publications] K.Matsumoto and Y.Tanigawa: "On the zeros of higher derivatives of Hardys Z-function" J.Number Theory. (to appear).
[Publications] M.Ishikawa and Y.Kitaoka: "On the clistribution of units modulo prime ideals in real quadratic fields" J.Reine.Angew.Math.494. 65-72 (1998)
[Publications] I.Kiuchi and Y.Tanigawa: "The mean value theorem of the divisor problem for short intirvals" Arch.Math.71. 445-453 (1998)
[Publications] I.Kiuchi and Y.Tanigawa: "The mean velue theorem of the Riemam zeta-function in the critical strip for short intervals" in "Number Thetry and its Applications",K.Gyory and S.Kanemitsu (eds.),Kiuwer. (to appear).
[Publications] S.Akiyama and Y.Tanigawa: "Caleulation of values of L-functions associated to elliptic curves" Math.Comput.(to appear).
[Publications] S.Akiyama: "Almost uniform distribution mod 1 and the distribution of primes" Acta Math.Hungar.78. 39-44 (1998)
[Publications] S.Akiyama: "Pisot numbers and greedy algorithm" in "Number Theory,Diophantine,Computational and Algebrat Aspects" K.Gyery et al.(eds.). 9-21 (1998)
[Publications] S.Akiyama and T.Sasahiro: "A self-similar tiling generated by the minimal-Pisot namber" Acta Math.Inform.Univ.Ostrav.(to appear).
[Publications] S.Akiyama: "Self affine tiling and Pisot numeration system" in "Number Theory and its Applications",K.Gyory and S.Kanemitsu (eds.),Kluwer. (to appear).
[Publications] M.Katsurada: "An application of Mellin-Barnes type of integrals to the mean squore of L-functions" Liet.Mati.Rink.38. 98-112 (1998)
[Publications] M.Katsurada: "Power series and asymptotic series associated with the Lerch zeta-function" Proc.Japan Acad.Ser.A. 74. 167-170 (1998)
[Publications] M.Katsurada: "Rapidly convergent series representations for S(2n+1) and their X-analogue" Acta Arith.(to appear).
[Publications] M.Amou and M.Katsurada: "Irrationality results for values of generalized Tschakaloff series" J.Number Theory. (to appear).
[Publications] S.Kanemitsu,M.Katsurada and M.Yoshimoto: "On the Hurwitz-Lerch zeta-function" Aequations Math.(to appear).

1998 Fiscal Year Annual Research Report

ゼータ関数,L関数の挙動とその整数論的意味

Principal Investigator

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

Research Products

[Publications] K.Matsumoto: "Asynptatic series for double zeta,double gamma,and Hecke L-functions" Math.Proc.Cambridge Phil.Soc.123. 385-405 (1998)

[Publications] A.Laurincikas and K.Matsumoto: "Joint value-distribution theorens on Lerch zeta-functions" Liet.Mat.Rink.38. 312-326 (1998)

[Publications] T.Hattori and K.Matsumoto: "A limit theorem for Bohr-Jessen's probability measures of the Riemam zeta-function" J.Reine Angew.Math.(to appear).

[Publications] A.Ivic,K.Matsumoto and Y.Tanigawa: "On Riesz means of the coefficients of the Rankin-Selberg series" Math.Proc.Cambridge Phil.Soc.(to appear).

[Publications] M.Katsurada and K.Matsumoto: "A weighted integrad approach to the mean square of Dirichlet L-functions" in "Namber Theory and its Applications",K.Gyory and S Kanemitsu (eds.) Klower. (to appear).

[Publications] K.Matsumoto and Y.Tanigawa: "On the zeros of higher derivatives of Hardys Z-function" J.Number Theory. (to appear).

[Publications] M.Ishikawa and Y.Kitaoka: "On the clistribution of units modulo prime ideals in real quadratic fields" J.Reine.Angew.Math.494. 65-72 (1998)

[Publications] I.Kiuchi and Y.Tanigawa: "The mean value theorem of the divisor problem for short intirvals" Arch.Math.71. 445-453 (1998)

[Publications] I.Kiuchi and Y.Tanigawa: "The mean velue theorem of the Riemam zeta-function in the critical strip for short intervals" in "Number Thetry and its Applications",K.Gyory and S.Kanemitsu (eds.),Kiuwer. (to appear).

[Publications] S.Akiyama and Y.Tanigawa: "Caleulation of values of L-functions associated to elliptic curves" Math.Comput.(to appear).

[Publications] S.Akiyama: "Almost uniform distribution mod 1 and the distribution of primes" Acta Math.Hungar.78. 39-44 (1998)

[Publications] S.Akiyama: "Pisot numbers and greedy algorithm" in "Number Theory,Diophantine,Computational and Algebrat Aspects" K.Gyery et al.(eds.). 9-21 (1998)

[Publications] S.Akiyama and T.Sasahiro: "A self-similar tiling generated by the minimal-Pisot namber" Acta Math.Inform.Univ.Ostrav.(to appear).

[Publications] S.Akiyama: "Self affine tiling and Pisot numeration system" in "Number Theory and its Applications",K.Gyory and S.Kanemitsu (eds.),Kluwer. (to appear).

[Publications] M.Katsurada: "An application of Mellin-Barnes type of integrals to the mean squore of L-functions" Liet.Mati.Rink.38. 98-112 (1998)

[Publications] M.Katsurada: "Power series and asymptotic series associated with the Lerch zeta-function" Proc.Japan Acad.Ser.A. 74. 167-170 (1998)

[Publications] M.Katsurada: "Rapidly convergent series representations for S(2n+1) and their X-analogue" Acta Arith.(to appear).

[Publications] M.Amou and M.Katsurada: "Irrationality results for values of generalized Tschakaloff series" J.Number Theory. (to appear).

[Publications] S.Kanemitsu,M.Katsurada and M.Yoshimoto: "On the Hurwitz-Lerch zeta-function" Aequations Math.(to appear).

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