Rational approximations to ${\@root 3 \of {2}}$ and other algebraic numbers revisited

Voutier, Paul M.

doi:10.5802/jtnb.586

Rational approximations to

\sqrt[3]{2}

and other algebraic numbers revisited

Voutier, Paul M.¹

¹ London, UK

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 263-288.

Résumé
Abstract

Dans cet article, nous améliorons des mesures effectives d’irrationalité pour certains nombres de la forme $\sqrt[3]{n}$ en utilisant des approximations obtenues à partir de fonctions hypergéométriques. Ces résultats sont très proche du mieux que peut donner cette méthode. Nous obtenons ces résultats grâce à des informations arithmétiques très précises sur les dénominateurs des coefficients de ces fonctions hypergéométriques.

Des améliorations de bornes pour les fonctions de Chebyshev $θ (k, l; x) = \sum_{\overset{p \equiv l mod k;}{p, prime, p \leq x}} log p .$ et $ψ (k, l; x) = \sum_{\overset{n \equiv l mod k;}{n \leq x}} Λ (n)$ ( $k = 1, 3, 4, 6$ ) sont aussi présentés.

In this paper, we establish improved effective irrationality measures for certain numbers of the form $\sqrt[3]{n}$ , using approximations obtained from hypergeometric functions. These results are very close to the best possible using this method. We are able to obtain these results by determining very precise arithmetic information about the denominators of the coefficients of these hypergeometric functions.

Improved bounds for the Chebyshev functions in arithmetic progressions $θ (k, l; x)$ and $ψ (k, l; x)$ for $k = 1, 3, 4, 6$ are also presented.

MR Zbl

DOI : 10.5802/jtnb.586

Affiliations des auteurs :

Voutier, Paul M. ¹

¹ London, UK

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     publisher = {Universit\'e Bordeaux 1},
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Voutier, Paul M. Rational approximations to ${\@root 3 \of {2}}$ and other algebraic numbers revisited. Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 263-288. doi : 10.5802/jtnb.586. https://www.numdam.org/articles/10.5802/jtnb.586/

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