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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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