# Rational approximations to the dilogarithm

@article{Hata1993RationalAT,
title={Rational approximations to the dilogarithm},
author={Masayoshi Hata},
journal={Transactions of the American Mathematical Society},
year={1993},
volume={336},
pages={363-387}
}
• M. Hata
• Published 1993
• Mathematics
• Transactions of the American Mathematical Society
The irrationality proof of the values of the dilogarithmic function L 2 (z) at rational points z = 1/k for every integer k ∈ (−∞, −5] ∪ [7, ∞) is given. To show this we develop the method of Pade-type approximations using Legendre-type polynomials, which also derives good irrationality measures of L 2 (1/k). Moreover, the linear independence over Q of the numbers 1, log(1 − 1/k), and L 2 (1/k) is also obtained for each integer k ∈ (−∞, −5] ∪ [7, ∞)

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#### References

SHOWING 1-10 OF 12 REFERENCES
A Note on the Irrationality of ζ(2) and ζ(3)
At the “Journees Arithmetiques” held at Marseille-Luminy in June 1978, R. Apery confronted his audience with a miraculous proof for the irrationality of ζ(3) = l-3+ 2-3+ 3-3 + .... The proof wasExpand
Polylogarithms and associated functions, North-Holland
• New York,
• 1981
A proof that Euler missed—Apery's proof of the irrationality of f(3)
• Math. Intelligencer
• 1979
Padé approximations to the generalized hypergeometric functions
• I, J. Math. Pures Appl
• 1979