| Project/Area Number |
09440009
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Nagoya University |
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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)
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| Co-Investigator(Kenkyū-buntansha) |
ITO Hiroshi Nagoya Univ., Grad. Sch. Math., Prof., 大学院・多元数理科学研究所, 教授 (30168372)
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)
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)
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| Project Period (FY) |
1997 – 1999
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| Keywords | Riemann zeta-function / Dirichlet L-function / Voronoi's formula / Divisor problem / Mean square / Asymptotic expansion / Rankin-Selberg L-function / Universality |
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| 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 zeta-function 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 Voronoi-type formula for the Riesz sum of the coefficient of Rankin-Selberg L-functions, and proved their mean square formulas.
4. We developed the method of using Mellin-Barnes 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 zeta-functions, and the universality of L-functions attached to modular forms.
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