Approximations to Euler's constant

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We study a problem of finding good approximations to Euler's constant γ=ₙ→∞Sₙ, where Sₙ=∑ₖ₌₁ⁿ1/n-(n+1), by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow convergence of the sequence Sₙ can be significantly improved if Sₙ is replaced by linear combinations of Sₙ with integer coefficients. In this paper, considering more general linear transformations of the sequence Sₙ we establish new accelerating convergence formulae for γ. Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results.