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.

Abstract

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+log⁡q (q∈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+log⁡q. It is shown that for every q∈ℚ>0 and every integer d≥42 there are infinitely many rationals am/bm for m=1,2,… such that |γ+log⁡q−am/bm|≪((1−1/d)d/(d−1)4d)m and bm∣Zm with log⁡Zm~12d2m2 for m tending to infinity.