Approximations to Euler's constant

Abstract

We study a problem of finding good approximations to Euler's constant $\gamma ={\lim }_{n\to \infty }{S}_n,$ 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 $\gamma .$ Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results.

arXiv comment11 pages