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On the arithmetic nature of the values of the gamma function, Euler’s constant, and Gompertz’s constant. (English) Zbl 1288.11073
For \(\alpha\in{\mathbb C}\) and \(z\in{\mathbb C}\) with \(z\not\in {\mathbb R}_{\leq 0}\), define \[ {\mathcal G}_\alpha(z) =z^{-\alpha} \int_0^\infty (t+z)^{\alpha-1} e^{-t} \,dt. \] The first part of Theorem 1 of the paper under review is a sharp measure of simultaneous rational approximation to the two numbers \[ \xi_1=\frac{\Gamma(\alpha)}{z^\alpha} \quad\text{and}\quad \xi_2={\mathcal G}_\alpha(z), \] where \(\alpha\in{\mathbb Q}\setminus{\mathbb Z}\) and \(z\in{\mathbb Q}\), \(z>0\): for any \(\varepsilon>0\), there exists a constant \(c\) such that, for any \(p,q,r\) in \({\mathbb Z}\) with \(q\not=0\), \[ \left|\xi_1-\frac{p}{q}\right| + \left|\xi_2-\frac{r}{q}\right| >\frac{c}{H^{3+\varepsilon}}, \] where \(H=\max\{|p|, |q|, |r|\}\). The second part of Theorem 1 states that the same measure holds for the two numbers \[ \xi_1=\gamma+\log z \quad\text{and}\quad \xi_2={\mathcal G}_0(z), \] where \(z\in{\mathbb Q}\), \(z>0\). As a consequence, in both cases, one at least of the two numbers \(\xi_1\), \(\xi_2\) is irrational. Here, \(\gamma\) is Euler’s constant. Gompertz constant is \({\mathcal G}_0(1)\) [A. I. Aptekarev, “On linear forms containing the Euler constant”, arxiv:0902.1768]. The proof rests on Hermite–Padé approximants.
The second result is proved using Shidlovskiĭ’s theorem on algebraic independence of values of \(E\)–functions: for any algebraic number \(z\) with \(z\not\in {\mathbb R}_{\leq 0}\) and any algebraic number \(\alpha\) with \(\alpha\not\in{\mathbb Z}\), two at least of the three numbers \[ e^z,\quad \frac{\Gamma(\alpha)}{z^\alpha}, \quad {\mathcal G}_\alpha(z) \] are algebraically independent, and two at least of the three numbers \[ \gamma+\log z, \quad e^z,\quad {\mathcal G}_0(z) \] are algebraically independent. The author explains the connection of his second theorem with earlier results due to [K. Mahler, Proc. R. Soc. Lond., Ser. A 305, 149–173 (1968; Zbl 0164.05702)].

MSC:
11J91 Transcendence theory of other special functions
11J13 Simultaneous homogeneous approximation, linear forms
11J82 Measures of irrationality and of transcendence
33B15 Gamma, beta and polygamma functions
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References:
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