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Euler’s factorial series at algebraic integer points. (English) Zbl 1437.11095
Let \(\mathbb K\) be a number field, \(n\) be a positive integer and \(\alpha_1,\dots ,\alpha_n\in\mathbb Z_{\mathbb K}\setminus \{ 0\}\). Let \(\lambda_0,\dots \lambda_n\in\mathbb Z_{\mathbb K}\) not all equal zero. Assume that \(V\) be a non-empty collection of non-Archimedian valuations of \(\mathbb K\). Then under the special conditions, the author proves that there exists valuation \(v\in V\) such that \(\lambda_0+\sum_{j=1}^n\lambda_j F_v(\alpha_j)\not= 0\) where \(F_v(t)=\sum_{i=0}^\infty i!t^i\) is a series as a function in \(v\)-adic domain \(\mathbb K\). In addition, if \(H\) is a real number such that \(H\geq \prod_{w\in V_\infty} \max_{0\leq i\leq n} \{\Vert \lambda_i \Vert _w \}\) then under other special conditions there exists prime \(p\) with \(p\in ] \log(\frac{\log H}{\log\log H}), \frac{17n\log H}{\log\log H}[\) and valuation \(v\vert p\) such that \[\Vert\lambda_0+\sum_{j=1}^n\lambda_j F_v(\alpha_j)\Vert_v> H^{-(n+1)-114n^2\frac{\log\log\log H}{\log\log H}}.\] The proofs use the method of Padé approximation.
MSC:
11J61 Approximation in non-Archimedean valuations
41A21 Padé approximation
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