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Irrationality of certain numbers that contain values of the di- and trilogarithm. (English) Zbl 1105.11021
The authors prove that, for \(z \in \{1/2, 2/3, 3/4, 4/5\}\), at least one of the two numbers
\[ \text{Li}_2(z) + \log(1 - z) \log(z), \quad \text{Li}_3(z) + \frac 12 \log(1 - z) \log(z)^2, \]
is irrational, where \(\text{Li}_s(z) = \sum_{k=1}^{\infty} z^k/k^s\). In the case \(z = 1/2\), this improves the previous result of S. Fischler and the reviewer in [“Approximants de Padé et séries hypergéométriques équilibrées,” J. Math. Pures Appl. 82,No. 10, 1369–1394 (2003; Zbl 1064.11053)] that at least one of the three numbers \[ \text{Li}_2(1/2) + \log(1/2)^2, \quad \text{Li}_3(1/2) + \frac 12 \log(1/2)^3, \quad \text{Li}_4(1/2) + \frac 16 \log(1/2)^4 \] is irrational. The proof essentially uses two tools. Firstly, the method introduced by Fischler and the reviewer (in the above mentioned article): one can explicitly compute certain simultaneous Padé approximants for the families \((\text{Li}_s(z), s = 1,\dots, A)\) and \((\log^s(z), s = 1, \dots, A)\) at the point \(z = 0\) and \(z = 1\) respectively and combine them by multiplying one of the linear forms by \(\text{Li}_1(z) = -\log(1-z)\). Secondly, and this is the main new ingredient, the quality of the linear form is improved thanks to a now classical arithmetical method, which consists in finding and removing large common prime factors in the coefficients of the linear forms.
MSC:
11J72 Irrationality; linear independence over a field
11G55 Polylogarithms and relations with \(K\)-theory
33C20 Generalized hypergeometric series, \({}_pF_q\)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
Citations:
Zbl 1064.11053
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References:
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