1998 Fiscal Year Annual Research Report

数論的関数の和の評価とそのL関数への応用

Research Project

Project/Area Number	09640026
Research Institution	Nagoya University
Principal Investigator	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)
Co-Investigator(Kenkyū-buntansha)	鈴木浩志名古屋大学, 大学院・多元数理科学研究科, 講師 (70235993) 秋山茂樹新潟大学, 理学部, 助教授 (60212445) 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) 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	約数問題 / Voronoi公式 / Rankin-Selberg series / 2乗平均 / 楕円曲線 / Riemann予想
Research Abstract	本研究では種々の数論的関数の総和について、剰余項の評価や、二乗平均、Ω-resultをあつかった。まず保型形式のフーリエ係数について、その絶対値の2乗の和の挙動を調べた。これはRankinが1939年に始めた問題である。彼は、現在Rankin-Selberg級数と呼ばれているディリクレ級数の解析的性質を用いて、主要項の決定と剰余項の評価を出したが、彼の評価は以来改良されておらず大きな問題となっている。本研究において、我々はVoronoi公式の立場からこの問題を考えた。すなわち剰余項と、その平均ともいえる第一Riesz和に対してVoronoi公式を導き、それらの間の関係を詳しく調べた。たとえば、第一Rieaz和の二乗平均の挙動がもっと精密にわかれば、剰余項の評価も改良されることなどがわかった。Rankinの問題の困難さは剰余項の無限級数型Voronoi級数が発散しているところにあるが、第一Riesz和の無限級数型Voronoi級数は収束しているので、今後のさらなる解析が期待されるところである。局所的な挙動を調べるには短区間における平均値定理が有用である。我々は一般約数関数Σ_<d\|n>d^aやζ(s)^2の平均値の剰余項に関して短区間での平均値定理を得た。これは約数関数やcritical line上におけるJutilaの結果の類似になっている。また楕円曲線に対応するL-関数については数値実験が少なかったが、その効率的な計算方法を見いだし、いくつかの曲線で、虚部<400の範囲でRiemann予想の成立等を確かめた。また、係数の偏角の分布に関するSato-Tate予想とRiemann予想の関係についても調べた。

Research Products
(18 results)

All Other

All Publications (18 results)

[Publications] J.Furuya and Y.Tanigawa: "Estimation of a certain function related to the Dirichlet divisor problem" "Analytic and Probabilistic Methods in Number Theory", New Trends in Probability and Statistics(ed.by A.Laurincikas, et al.). 4. 171-189 (1997)
[Publications] I.Kiuchi and Y.Tanigawa: "The mean value theorem of the divisor problem for short intervals" Archiv der Math.71. 445-453 (1998)
[Publications] I.Kiuchi and Y.Tanigawa: "The mean value theorem of the Riemann zeta-function in the critical strip for short intervals" "Number Theory and its Applications", (ed.by K.Gyory and S.Kanemitsu). to appear.
[Publications] S.Akiyama and Y.Tanigawa: "Calculation of values of L-functions associated to elliptic curves" Mathematics of Computation. to appear.
[Publications] A.Ivic, K.Matsumoto and Y.Tanigawa: "On Riesz means of the coefficients of the Rankin-Selberg series" Math.Proc.Camb.Phil.Soc. to appear.
[Publications] K.Matsumoto and Y.Tanigawa: "On the zeros of higher derivatives of Hardy's Z-function" J.Number Theory. to appear.
[Publications] Y.Kitaoka: "Finite arithmetic subgroups of GL_n, V" Nagoya Math.J.146. 131-148 (1997)
[Publications] M.Ishikawa and Y.Kitaoka: "On the distribution of units modulo prime ideals in real quadratic fields" J.reine angew.Math.494. 65-72 (1998)
[Publications] I.Kiuchi and K.Matsumoto: "The resemblance of the behaviour of the remainder terms E_σ(t), △_<1-2σ>(x)and R(σ+it)" "Sieve Methods, Exponential Sums, and their Applications in Number Theory", London Math.Soc.237. 255-273 (1997)
[Publications] K.Katsurada and K.Matsumoto: "Explicit formulas and expansions for certain mean square of Hurwitz zeta-functions II" "Analytic and Probabilistic Methods in Number Theory", New Trends in Probability and Statistics, (ed.by A.Laurincikaset al.). 4. 119-134 (1997)
[Publications] K.Matsumoto: "Asymptotic 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 theorems 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 Riemann zeta-function" J.Reine Angew.Math.,. to appear.
[Publications] K.Katsurada and K.Matsumoto: "A weighted integral approach to the mean square of Dirichlet L-functions" "Number Theory and its Applications", (ed.by K.Gyory and S.Kanemitsu). to appear.
[Publications] S.Akiyama: "Almost uniform distribution modulo I and the distribution of primes" Acta Math.Hungar.78. 39-44 (1998)
[Publications] S.Akiyama: "Pisot numbers and greedy algorithm" "Number Theory, Diophantine, Computational and Algebraic Aspects" (ed.by K.Gyory, A.Petto, V.T.Sos). 9-21 (1998)
[Publications] S.Akiyama and T.Sadahiro: "A self-similar tiling generated by minimal Pisot number" Proceedings of 13th Czech and Slovak Conference on Number Theory. to appear.
[Publications] S.Akiyama: "Self affine tiling and Pisot numeration system" "Number Theory and its Applications", (ed by K.Gyory and S.Kanemitsu). to appear.

1998 Fiscal Year Annual Research Report

数論的関数の和の評価とそのL関数への応用

Principal Investigator

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)

Research Products

[Publications] J.Furuya and Y.Tanigawa: "Estimation of a certain function related to the Dirichlet divisor problem" "Analytic and Probabilistic Methods in Number Theory", New Trends in Probability and Statistics(ed.by A.Laurincikas, et al.). 4. 171-189 (1997)

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

[Publications] I.Kiuchi and Y.Tanigawa: "The mean value theorem of the Riemann zeta-function in the critical strip for short intervals" "Number Theory and its Applications", (ed.by K.Gyory and S.Kanemitsu). to appear.

[Publications] S.Akiyama and Y.Tanigawa: "Calculation of values of L-functions associated to elliptic curves" Mathematics of Computation. to appear.

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

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

[Publications] Y.Kitaoka: "Finite arithmetic subgroups of GL_n, V" Nagoya Math.J.146. 131-148 (1997)

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

[Publications] I.Kiuchi and K.Matsumoto: "The resemblance of the behaviour of the remainder terms E_σ(t), △_<1-2σ>(x)and R(σ+it)" "Sieve Methods, Exponential Sums, and their Applications in Number Theory", London Math.Soc.237. 255-273 (1997)

[Publications] K.Katsurada and K.Matsumoto: "Explicit formulas and expansions for certain mean square of Hurwitz zeta-functions II" "Analytic and Probabilistic Methods in Number Theory", New Trends in Probability and Statistics, (ed.by A.Laurincikaset al.). 4. 119-134 (1997)

[Publications] K.Matsumoto: "Asymptotic 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 theorems 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 Riemann zeta-function" J.Reine Angew.Math.,. to appear.

[Publications] K.Katsurada and K.Matsumoto: "A weighted integral approach to the mean square of Dirichlet L-functions" "Number Theory and its Applications", (ed.by K.Gyory and S.Kanemitsu). to appear.

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

[Publications] S.Akiyama: "Pisot numbers and greedy algorithm" "Number Theory, Diophantine, Computational and Algebraic Aspects" (ed.by K.Gyory, A.Petto, V.T.Sos). 9-21 (1998)

[Publications] S.Akiyama and T.Sadahiro: "A self-similar tiling generated by minimal Pisot number" Proceedings of 13th Czech and Slovak Conference on Number Theory. to appear.

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

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)