APPROXIMATIONS TO EULER'S CONSTANT

We study a problem of finding good approximations to Euler's constant gamma = lim(n ->infinity)S(n), where S(n) = Sigma(n)(k=1) 1/k - log(n+1), by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow convergence of the sequence S(n) can be significantly improved if S(n) is replaced by linear combinations of S(n) with integer coefficients. In this paper, considering more general linear transformations of the sequence S(n) we establish new accelerating convergence formulae for gamma. Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results.