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On rational approximations to Euler’s constant \(\gamma \) and to \(\gamma +\log (a/b)\). (English) Zbl 1168.41308

From the summary: The author continues to study series transformations for the Euler-Mascheroni constant \(\gamma \). Here, he discusses in detail recent results published in A. I. Aptekarev [Rational approximations of the Euler constant and recurrence relations. Collection of articles. Sovremennye Problemy Matematiki 9. Moskva: MatematicheskiĭInstitut im. V. A. Steklova, RAN. 82 p. (2007; Zbl 1134.41001)] and by T. Rivoal [www.-fourier.uif.grenoble.fr/ rioval] who found rational approximations to \(\gamma \) and \(\gamma +\log q(q\in \mathbb Q_{>0})\) defined by linear recurrence formulae. The main purpose of this paper is to adapt the concept of linear series transformations with integral coefficients such that rationals are given by explicit formulae which approximate \(\gamma \) and \(\gamma +\log q\). It is shown that for every \(q\in \mathbb Q_{>0}\) and every integer \(d\geq 42\) there are infinitely many rationals \(a_{m}/b_{m}\) for \(m=1,2,\dots \) such that \(|\gamma +\log q - a_{m}/b_{m}|\ll ((1 - 1/d)^d/(d - 1)4^{d})^m\) and \(b_{m}\mid Z_{m}\) with \(\log Z_{m}\sim 12d^{2}m^{2}\) for \(m\) tending to infinity.

MSC:

11Y60 Evaluation of number-theoretic constants
11J04 Homogeneous approximation to one number

Citations:

Zbl 1134.41001
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References:

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