Rational approximations to 2 3 and other algebraic numbers revisited
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 263-288.

In this paper, we establish improved effective irrationality measures for certain numbers of the form n 3, 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.

Dans cet article, nous améliorons des mesures effectives d’irrationalité pour certains nombres de la forme n 3 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)= p, prime ,px plmodk; logp. et ψ(k,l;x)= nx nlmodk; Λ(n) (k=1,3,4,6) sont aussi présentés.

Received: 2006-01-06
Published online: 2008-12-03
DOI: https://doi.org/10.5802/jtnb.586
@article{JTNB_2007__19_1_263_0,
     author = {Paul M. Voutier},
     title = {Rational approximations to ${\@root 3 \of {2}}$ and other algebraic numbers revisited},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     pages = {263-288},
     doi = {10.5802/jtnb.586},
     zbl = {1120.11027},
     mrnumber = {2332066},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2007__19_1_263_0/}
}
Voutier, Paul M. Rational approximations to ${\@root 3 \of {2}}$ and other algebraic numbers revisited. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 263-288. doi : 10.5802/jtnb.586. https://jtnb.centre-mersenne.org/item/JTNB_2007__19_1_263_0/

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