Rational approximations to the dilogarithm

Author:
Masayoshi Hata

Journal:
Trans. Amer. Math. Soc. **336** (1993), 363-387

MSC:
Primary 11J82; Secondary 11J72

DOI:
https://doi.org/10.1090/S0002-9947-1993-1147401-5

MathSciNet review:
1147401

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Abstract: The irrationality proof of the values of the dilogarithmic function ${L_2}(z)$ at rational points $z = 1/k$ for every integer $k \in ( - \infty , - 5] \cup [7,\infty )$ is given. To show this we develop the method of Padé-type approximations using Legendre-type polynomials, which also derives good irrationality measures of ${L_2}(1/k)$. Moreover, the linear independence over ${\mathbf {Q}}$ of the numbers $1$, $\log (1 - 1/k)$, and ${L_2}(1/k)$ is also obtained for each integer $k \in ( - \infty , - 5] \cup [7,\infty )$ .

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Keywords:
Dilogarithm,
irrationality measure,
Padé approximation

Article copyright:
© Copyright 1993
American Mathematical Society