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arXiv · 0708.2771

Approximations to Euler's constant

Kh. Hessami Pilehrood·T. Hessami Pilehrood
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Abstract

We study a problem of finding good approximations to Euler's constant $γ=\lim_{n\to\infty}S_n,$ where $S_n=\sum_{k=1}^n\frac{1}{n}-\log(n+1),$ 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 $γ.$ Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results.

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Kh. Hessami Pilehrood, T. Hessami Pilehrood. 2007-08-21. Approximations to Euler's constant. https://arxiv.org/abs/0708.2771

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