Rational approximations to the dilogarithm

Hata, Masayoshi

doi:10.1090/S0002-9947-1993-1147401-5

Rational approximations to the dilogarithm
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by Masayoshi Hata

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

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

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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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