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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 \(m\).

MSC:
11J61 Approximation in non-Archimedean valuations
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References:
[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau Of Standards Applied Mathematics Series, 55, Washington, D.C., 1964. · Zbl 0171.38503
[2] T. M. Apostol, Introduction to Analytic Number Theory, Springer, New York, 1976. · Zbl 0335.10001
[3] M. A. Bennett, G.Martin, K. O’Bryant, and A. Rechnitzer, Explicit bounds for primes in arithmetic progressions. To appear in Illinois J. Math. Available athttps://arxiv. org/abs/1802.00085. · Zbl 1440.11180
[4] D. Bertrand, V. G. Chirski˘ı, and J. Yebbou, Effective estimates for global relations on Euler-type series, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), 241–260. · Zbl 1176.11036
[5] V. G. Chirski˘ı, On algebraic relations in non-Archimedean fields, Funct. Anal. Appl. 26 (1992), 108–115.
[6] V. G. Chirski˘ı, On the arithmetic properties of generalized hypergeometric series with irrational parameters, Izv. Ross. Akad. Nauk Ser. Mat. 78 (2014), 193–210; English translation in Izv. Math. 78 (2014), 1244–1260.
[7] V. G. Chirski˘ı, Arithmetic properties of Euler series,Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2015), 59–61; English translation in Moscow Univ. Math. Bull. 70 (2015), 41–43.
[8] V. G. Chirski˘ı, Arithmetic properties of polyadic series with periodic coefficients, Izv. Ross. Akad. Nauk Ser. Mat. 81 (2017), 215–232; English translation in Izv. Math. 81 (2017), 444–461.
[9] H. Davenport, Multiplicative Number Theory, Springer-Verlag, New York, 2nd Edition, 1980. · Zbl 0453.10002
[10] K. Lepp¨al¨a, T. Matala-aho, and T. T¨orm¨a, Rational approximations of the exponential function at rational points, J. Number Theory 179 (2017), 220–239.
[11] T. Matala-aho and W. Zudilin, Euler’s factorial series and global relations, J. Number Theory 186 (2018), 202–210. · Zbl 1444.11037
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