International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 626489, 22 pages
doi:10.1155/2009/626489

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

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.