On Rational Approximations to Euler's Constant 𝛾 and to 𝛾+log(𝑎/𝑏) - Carsten Elsner (International Journal of Mathematics and Mathematical Sciences)
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∈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∈ℚ>0 and every integer 𝑑≥42 there are infinitely many rationals 𝑎𝑚/𝑏𝑚 for 𝑚=1,2,… such that |𝛾+log𝑞−𝑎𝑚/𝑏𝑚|≪((1−1/𝑑)𝑑/(𝑑−1)4𝑑)𝑚 and 𝑏𝑚∣𝑍𝑚 with log𝑍𝑚∼12𝑑2𝑚2 for 𝑚 tending to infinity.