Project/Area Number  09440009 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
Algebra

Research Institution  Nagoya University 
Principal Investigator 
MATSUMOTO Kohji Nagoya Univ., Grad. Sch. Math., Assoc. Prof., 大学院・多元数理科学研究所, 助教授 (60192754)

CoInvestigator(Kenkyūbuntansha) 
KATSURADA Masanori Keio Univ., Fac. of Eco., Assoc. Prof., 経済学部, 助教授 (90224485)
AKIYAMA Shigeki Niigata Univ., Fac. of Sci., Assoc. Prof., 理学部, 助教授 (60212445)
木内 功 山口大学, 理学部, 助教授 (30271076)
TANIGAWA Yoshio Nagoya Univ., Grad. Sch. Math., Assoc. Prof., 大学院・多元数理科学研究所, 助教授 (50109261)
KITAOKA Yoshiyuki Nagoya Univ., Grad. Sch. Math., Prof., 大学院・多元数理科学研究所, 教授 (40022686)
ITO Hiroshi Nagoya Univ., Grad. Sch. Math., Prof., 大学院・多元数理科学研究所, 教授 (30168372)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥11,600,000 (Direct Cost : ¥11,600,000)
Fiscal Year 1999 : ¥4,500,000 (Direct Cost : ¥4,500,000)
Fiscal Year 1998 : ¥3,300,000 (Direct Cost : ¥3,300,000)
Fiscal Year 1997 : ¥3,800,000 (Direct Cost : ¥3,800,000)

Keywords  Riemann zetafunction / Dirichlet Lfunction / Voronoi's formula / Divisor problem / Mean square / Asymptotic expansion / RankinSelberg Lfunction / Universality / Riemannゼータ関数 / Dirichlet L関数 / Voronoi公式 / 約数問題 / 二乗平均 / 漸近展開 / RankinSelberg L関数 / universality / Voronoiの公式 / Lerchゼータ関数 / ゼータ関数 / L関数 / 平均値定理 / Pisot数 / 二重ガンマ関数 
Research Abstract 
There is a strong analogy between the divisor problem (the evaluation of ΔィイD2aィエD2 (X)) and the evaluation of the remainder term EィイD2σィエD2 (T) in the mean square formula for the Riemann zetafunction ζ (S). The basic tools for the study of them are Voronoi's formula and Atkinson's formula, respectively. The results we have obtained on this topic are : 1. By using Voronoi's and Atkinson's formulas, we obtained the mean square formulas of the differences of ΔィイD2aィエD2 (X), or EィイD2σィエD2 (T), in short intervals. Also we proved similar results for the remainder term in the approximate functional equation for ζ ィイD12ィエD1(S). 2. We have studied the generalization to the cases with characters. 3. We showed the Voronoitype formula for the Riesz sum of the coefficient of RankinSelberg Lfunctions, and proved their mean square formulas. 4. We developed the method of using MellinBarnes type of integrals, and established the usefulness of this method for the study of analytic continuation and asymptotic expansions. Also, we could prove the joint universality for Lerch zetafunctions, and the universality of Lfunctions attached to modular forms.
