Diophantine approximation in low discrepancy sequences and the Kontsevich-Zagier period conjecture
| Project/Area Number |
18K03225
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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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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Nihon University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2022-03-31
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| Project Status |
Discontinued (Fiscal Year 2021)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2020: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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| Keywords | 超一様分布数列 / 多重対数 / 周期予想 / 超幾何級数 / 超越数 / パデ近似 / 一次独立性 / ディオファントス近似 / 一様分布 / 超一様分布 / Lerch関数 / リーマンゼータ関数 / 無理数性 / 多重対数関数 / Lerch 関数 / G 関数 / 数論的近似 / Pade近似 / 対数一次形式 / 暗号原理 / 周期 / 多項式写像 / Kontsevich-Zagier予想 |
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| Outline of Final Research Achievements |
Diophantine approximation is one of basic methods to give a proof of the irrationality, the linear independence or to study the low discrepancy sequence in uniform distribution in Diophantine problems. It is known that Ch. Hermite constructed explicit simultaneous Diophantine approximations related to the exponential function to prove the transcendence of the number of Napier.
We now adapt this approximation method to prove a new linear independence criterion for several polylogarithms defined at several algebraic numbers. Polylogarithmic function is a natural generalization of logarithmic function, however, it has no homomorphism property like logarithmic function, then the usual tool to show the transcendence of logarithms does not work. Nevertheless, we succeeded in showing a precise criterion for the linear independence of several polylogarithms of distinct algebraic numbers, over an algebraic number field of arbitrary degree, relying on Pade approximations.
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| Academic Significance and Societal Importance of the Research Achievements |
多重対数は数論のみならず数学や物理学のあらゆる場に現れる周期である．また超一様・一様分布数列の研究も，乱数や擬似乱数などの研究にとって重要である．多重対数関数の代数的数における値は，自然対数で表される代数体のRegulatorの一般化として現れる数としても位置付けられ，素数分布の考察に力を発揮するRiemann zeta関数の値の性質を調べる際にも登場する．本研究課題では周期予想に現れる代表的な周期の例である多重対数について考察し，異なる点での多重対数が任意次数の代数体上で一次独立になるための判定規準を与えた．また超幾何級数を用いて，アーベル多様体の周期に対する超越近似についても考察した．
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Report
(4 results)
Research Products
(34 results)
