Approximations to Euler's constant

TL;DR

This paper develops new methods to improve the approximation of Euler's constant using linear transformations of a specific sequence, leading to faster convergence and more accurate estimates.

Contribution

It introduces generalized linear transformation techniques that accelerate convergence to Euler's constant, extending previous results by Elsner, Rivoal, and the author.

Findings

New convergence acceleration formulas for Euler's constant.

Sharpened and generalized previous approximation results.

Enhanced accuracy in approximating 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.