Research for multiple Mahler measures via multiple L values

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K17524
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Osaka University of Health and Sport Sciences
• Principal Investigator: Sasaki Yoshitaka 大阪体育大学, 体育学部, 准教授 (20548771)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 多重ゼータ関数 / 多重L関数 / 多重Mahler測度 / 多重Bernoulli数 / 多重ポリログ / ロンサム行列

Outline of Final Research Achievements

A functional relation for the multiple polylogarithm which gives an analytic continuation of the multiple polylogarithm was obtained. We first showed the functional relation for the double polylogarithm by using a function which interpolates the harmonic numbers. After that, we give the functional relation for multiple polylogarithm by constructing a function which interpolates the multiple harmonic numbers and investigating residues of the interpolation equation. Furthermore, we try to evaluate multiple higher Mahler measures for several polynomials by using such functional relations. This research was a joint work with Kusunoki and Nakamura. Recurrence formula for poly-Bernoulli numbers was obtained. In fact, we showed similar recurrence formulas hold for the generalized poly-Bernoulli polynomials. Furthermore, we showed that the recurrence formula can be regarded as an enumeration formula for certain restricted lonesum matrices. This research was a joint work with Ohno.

Free Research Field

解析的整数論

Academic Significance and Societal Importance of the Research Achievements

多重Mahler測度は構造上多重ポリログと深く関係することから、多重ポリログの性質を深く理解するこが重要である。特に多重ポリログの明示的な関数関係式は、多重Mahler測度を具体的かつ明示的に計算するためには必要不可欠といえる。また、多重Bernoulli数は多重ゼータ関数と深く関係することから、その性質を見出すことで多重ゼータ関数の新たな性質の解明につながることが期待される。実際に、多重Bernoulli数の新たな漸化式から、ロンサム行列の個数を数え上げに関する等式が得られている。