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Euler’s divergent series in arithmetic progressions. (English) Zbl 1443.11138
The author considers the series \(F(z)=\sum_{n=0}^\infty n!z^n\) which converges for \(|z|_p\leq p^{1/(p-1)}\) in the \(p\)-adic numbers \(\mathbb Q_p\). The corresponding function is denoted by \(F_p(z)\) for a fixed prime \(p\). From this perspective, it makes sense to ask whether \(F_p(1)\) is irrational or not. In the paper under review, the author proves that for a given rational number \(a/b\) there exist infinitely many primes \(p\) such that \(F_p(1)\neq a/b\). Moreover, let \(m\geq 3\). Then, the authors additionally show that infinitely many such primes \(p\) are contained in only \(\varphi(m)/2\) residue classes modulo \(m\).

11J61 Approximation in non-Archimedean valuations
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