| Project/Area Number |
15K04799
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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 | Nihon University |
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Principal Investigator |
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| Research Collaborator |
DAVID Sinnou
BUGEAUD Yann
LUCA Florian
KAWASHIMA Makoto
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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 |
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2018: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2015: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
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| Keywords | 多重対数 / パデ近似 / エルミート-パデ近似 / ディオファントス近似 / 多重対数予想 / 漸近展開 / 直交多項式 / 数論的近似 / 対数関数 / リーマンゼータ関数 / Lerch関数 / 無理数 / １次独立性 / 定量的無理数度 / p進解析 / 超越数 / 楕円対数 |
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| Outline of Final Research Achievements |
Let us consider a polylogarithmic function in one variable as is defined by generalized logarithmic function. Lots of important conjectures have been considered related to the values of this function, containing so-called the polylogarithms conjecture. T. Rivoal obtained a lower bound for the dimension of the linear subspace spanned by the polylogarithms over the rational number field, however, his result does not yield any irrationality nor linear independence over the rational number field of the polylogarithms themselves. In contrast, we determine when takes the polylogarithmic function irrational values at algebraic numbers. We succeeded in giving a new criterion for the linear independence of polylogarithms over number fields as well as in making new concrete examples of linear independent polylogarithms. Our method is also useful to investigate other arithmetical properties of series with periodic coefficients.
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| Academic Significance and Societal Importance of the Research Achievements |
素数は人を魅了してやまないが，素数を解明する大切な関数にリーマンゼータ関数がある．リーマンゼータ関数と類似の性質をもち，工学にも頻繁に登場する多重対数関数という関数がある．これは対数関数の一般化であり，対数関数を表す積分表示の反復で記述される一変数関数である．多重対数関数を研究することはリーマンゼータ関数のみならず，数の性質および種々の自然現象を解明することに資する．多重対数関数の値の考察に効果的なパデ近似という手法を発展させて，我々は新しい無理数を発見したが，このように整数論の予想に近づく成果を得たことは，我が国の基礎科学の地位を高め，数学や整数論そのものに対しても新たな道を拓くものである．
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