International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 626489, 22 pages
Research Article

On Rational Approximations to Euler's Constant γ and to γ+log(a/b)

Fachhochschule für die Wirtschaft Hannover, Freundallee 15, 30173 Hannover, Germany

Received 4 December 2008; Accepted 13 April 2009

Academic Editor: Stéphane Louboutin

Copyright © 2009 Carsten Elsner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The author continues to study series transformations for the Euler-Mascheroni constant γ. Here, we discuss in detail recently published results of A. I. Aptekarev and T. Rivoal who found rational approximations to γ and γ+logq (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 γ and γ+logq. It is shown that for every q>0 and every integer d42 there are infinitely many rationals am/bm for m=1,2, such that |γ+logqam/bm|((11/d)d/(d1)4d)m and bmZm with logZm~12d2m2 for m tending to infinity.