| Project/Area Number |
15K04801
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Okayama University of Science |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | Dedekind和 / 変換公式 / Dirichlet級数 / エータ関数 / Lambert級数 / Dirichlet L-関数 / Goss L-関数 / 周期関数 / Ｌａｍｂｅｒｔ級数 / 保型形式 / 同変関数 / 有限体 / 相互法則 / 合成則 |
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| Outline of Final Research Achievements |
We introduced an analog of the logarithm of the Dedekind eta function, and established the transformation formula. To describe this, we used the Dedekind sum which was defined before. This formula is similar to the classical formula for the logarithm of the Dedekind eta function. As an application, we obtain another proof of the reciprocity law for our Dedekind sums. Moreover, we introduced an analog of the Lambert series, and established the transformation formula. To describe this, we introduced the generalized Dedekind sum. This formula is similar to the classical formula for the Lambert series. As an application, we established the reciprocity law for the generalized Dedekind sums.
In the above results, we obtained an analog of the cotangent function. Using this function, we obtained an analog of A. Baker, B. Birch, and E. Wirsing's theorem, which is a result for the non-vanishing the Dirichlet series at 1.
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| Academic Significance and Societal Importance of the Research Achievements |
エータ関数の対数の変換公式やLambert級数の変換公式は、保型形式と関係がある。我々の研究成果を関数体上の保型形式に応用すれば、保型形式やDedekind和の結果が得られると推測できるので、意義があると思われる。また、関数体における我々の成果から得たアイデアを有限体に応用して、有限体上でもエータ関数の対数の変換公式やLambert級数の変換公式の類似が得られた。有限体上の数論、特に、保型形式の理論を構築する上で我々の成果は、大きな影響力をもつと思われる。
Diricchlet級数の1での値がいつ0にならないか、という問題はChowlaの問題と呼ばれ、我々の結果はその成果の一つである。
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