Ernvall-Hytönen, Anne-Maria; Matala-aho, Tapani; Seppälä, Louna Euler’s factorial series, Hardy integral, and continued fractions. (English) Zbl 1514.11043 J. Number Theory 244, 224-250 (2023). Let function \(E_p(t)\) be defined by Euler series \[ E_p(t):=\sum_{k=0}^\infty k! t^k \] which converges in the \(p\)-adic metric \(|p|_p=p^{-1}\) for a prime \(p\) in the interior of disk \(\{ t \in \mathbb{C}_p\, \mid\, |t|_p <p^{\frac{1}{p-1}}\}\).Let \[ \mathcal{H}(t)=\int_0^\infty \frac{e^{-s}}{1-ts} ds \] be the Hardy integral. The \(p\)-adic lower bounds for the linear form \(cE_p(\pm p^a)-d\) with integer coefficients \(c,d\) and a positive integer \(a\) are getting in the paper. The following result is proven using the Padé approximation. Let \(p\) be a prime number and \(H \in \mathbb{Z}_{\geq 4}\). Suppose that \(a\) is a positive integer such that \(p^a >c_1 \log (c_2 H),\) where \(c_1, c_2\) are constants. Then, for all \(c,d \in \mathbb{Z}, c\ne 0\) with \(|c|+|d| \leq H\) there holds \[ |cE_p(\pm p^a)-d|_p > \Big(2He^{\frac{11}{16}}\Big)^{-\frac{32}{11} a \log p}. \] An interconnection between \(E(t)\) and \(\mathcal{H}(t)\) via continued fractions is presented. Reviewer: Mykhaylo Pahirya (Uzhhorod) Cited in 3 Documents MSC: 11J61 Approximation in non-Archimedean valuations 11J70 Continued fractions and generalizations 11J85 Algebraic independence; Gel’fond’s method Keywords:\(p\)-adic; Diophantine approximation; Padé approximation; continued fractions × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Bertrand, D.; Chirskiĭ, G. V.; Yebbou, J., Effective estimates for global relations on Euler-type series, Ann. Fac. Sci. Toulouse Math. (6), 13, 2, 241-260 (2004) · Zbl 1176.11036 [2] Borwein, J.; van der Poorten, A.; Shallit, J.; Zudilin, W., Neverending Fractions: An Introduction to Continued Fractions (2014), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1307.11001 [3] Chirskiĭ, G. V., Non-trivial global relations, Vestn. Mosk. Univ. Ser. I Mat. Mekh.. Vestn. Mosk. Univ. Ser. I Mat. Mekh., Mosc. Univ. Math. Bull., 44, 5, 41-44 (1989), (in Russian); English translation in · Zbl 0699.10052 [4] Chirskiĭ, G. V., Product formula, global relations, and polyadic numbers, Russ. J. Math. Phys., 26, 3, 286-305 (2019) · Zbl 1425.11135 [5] Ernvall-Hytönen, A.-M.; Matala-aho, T.; Seppälä, L., Euler’s divergent series in arithmetic progressions, J. Integer Seq., 22, Article 19.2.2 pp. (2019) · Zbl 1443.11138 [6] Hardy, G. H., Divergent Series (1973), Clarendon Press: Clarendon Press Oxford · Zbl 0032.05801 [7] Lagarias, J. C., Euler’s constant: Euler’s work and modern developments, Bull. Am. Math. Soc., 50, 4, 527-628 (2013) · Zbl 1282.11002 [8] Leppälä, K.; Matala-aho, T.; Törmä, T., Rational approximations of the exponential function at rational points, J. Number Theory, 179, 220-239 (2017) · Zbl 1418.11110 [9] Mahler, K., Applications of a theorem by A. B. Shidlovskii, Proc. R. Soc. Lond. Ser. A, 305, 149-173 (1968) · Zbl 0164.05702 [10] Matala-aho, T.; Zudilin, W., Euler’s factorial series and global relations, J. Number Theory, 186, 202-210 (2018) · Zbl 1444.11037 [11] Remmal, S.-E., Problèmes de transcendance liés aux E-fonctions et aux G-fonctions p-adiques (1981), Thèse de troisième cycle, Paris VI; Publ. Math. Univ. Pierre Marie Curie 43, Groupe Etud. Prob. Diophantines 1980/81, Exp. No. 3, 44 p. · Zbl 0523.10021 [12] Remmal, S.-E., Mesure de transcendance partielle pour une famille de nombres liés aux solutions d’équations différentielles à points singuliers irréguliers, (Groupe de travail d’analyse ultramétrique, tome 7-8. Groupe de travail d’analyse ultramétrique, tome 7-8, Paris, 1979/1981 (1979-1981)), Exp. No. 9, 1-2 · Zbl 0523.10020 [13] Rivoal, T., On the arithmetic nature of values of the gamma function, Euler’s constant and the Gompertz constant, Mich. Math. J., 61, 239-254 (2012) · Zbl 1288.11073 [14] Seppälä, L., Euler’s factorial series at algebraic integer points, J. Number Theory, 206, 250-281 (2020) · Zbl 1437.11095 [15] Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23 (1975), American Mathematical Society: American Mathematical Society Providence, RI · JFM 61.0386.03 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.