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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)].

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
Full Text: DOI Euclid
[1] G. W. Anderson, D. W. Brownawell, M. A. Papanikolas, and A. Matthew, Determination of the algebraic relations among special \(\Gamma \) -values in positive characteristic, Ann. of Math. (2) 160 (2004), 237-313. · Zbl 1064.11055 · doi:10.4007/annals.2004.160.237
[2] A. I. Aptekarev, On linear forms containing the Euler constant, preprint, 2009, http://arxiv.org/abs/0902.1768.
[3] F. Beukers, Legendre polynomials in irrationality proofs, Bull. Austral. Math. Soc. 22 (1980), 431-438. · Zbl 0436.10016 · doi:10.1017/S0004972700006742
[4] S. Bruiltet, D’une mesure d’approximation simultanée à une mesure d’irrationalité: le cas de \(\Gamma (1/4)\) et \(\Gamma(1/3),\) Acta Arith. 104 (2002), 243-281. · Zbl 1069.11029 · doi:10.4064/aa104-3-3
[5] G.V. Chudnovsky, Algebraic independence of constants connected with the exponential and the elliptic functions, Dokl. Akad. Nauk Ukrain. SSR Ser. A 8 (1976), 698-701.
[6] S. Finch, Mathematical constants, Encyclopedia Math. Appl., 94, Cambridge Univ. Press, Cambridge, 2003. · Zbl 1054.00001
[7] P. Grinspan, Approximation et indépendance algébrique de quasi-périodes de variétés abéliennes , thèse de doctorat de l’universit de Paris 6, 2000, \(\langle\)tel.archives-ouvertes.fr/docs/00/04/48/40/PDF/tel-00001328.pdf\(\rangle.\)
[8] K. Mahler, Applications of a theorem by A. B. Shidlovski, Philos. Trans. Roy. Soc. London Ser. A 305 (1968), 149-173. · Zbl 0164.05702 · doi:10.1098/rspa.1968.0111
[9] Yu. Nesterenko, Linear independence of numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 108 (1985), 46-49 (Russian); English translation in Moscow Univ. Math. Bull. 40 (1985), 69-74.
[10] T. Rivoal, Rational approximations for values of derivatives of the Gamma function, Trans. Amer. Math. Soc. 361 (2009), 6115-6149. · Zbl 1236.11061 · doi:10.1090/S0002-9947-09-04905-8
[11] —, Approximations rationnelles des valeurs de la fonction Gamma aux rationnels, J. Number Theory 130 (2010), 944-955. · Zbl 1206.11095 · doi:10.1016/j.jnt.2009.08.003
[12] —, Approximations rationnelles des valeurs de la fonction Gamma aux rationnels: le cas des puissances, Acta Arith. 142 (2010), 347-365. · Zbl 1264.11057 · doi:10.4064/aa142-4-4
[13] A. B. Shidlovskii, Transcendental numbers, de Gruyter Stud. Math., 12, de Gruyter, Berlin, 1989. · Zbl 0689.10043
[14] O. N. Vasilenko, On a construction of Diophantine approximations, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 87 (1993), 14-17 (Russian); English translation in Moscow Univ. Math. Bull. 48 (1993), 11-14. · Zbl 0848.11032
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