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On Mahler’s transcendence measure for \(e\). (English) Zbl 1435.11095
Given a transcendental number \(\xi\) one can define a transcendence measure \(w(m,H)\) by the infinum of numbers \(r>0\) such that \[ |\lambda_0+\lambda_1\xi+\dots+\lambda_m\xi^{m}|>H^{-r} \] for all integers \(\lambda_i\) with \(|\lambda_i|\leq H\) and with \(i=1,\dots,m\). In the case of \(\xi=e\) the authors provide explicit upper bounds for \(w(m,H)\) provided that \(H\) is sufficiently large. These results improve earlier results due to M. Hata [J. Number Theory 54, No. 1, 81–92 (1995; Zbl 0839.11027)].

MSC:
11J82 Measures of irrationality and of transcendence
11J72 Irrationality; linear independence over a field
41A21 Padé approximation
Software:
OEIS; SageMath
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