On ideals free of large prime factors
Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, p. 733-772

In 1989, E. Saias established an asymptotic formula for Ψ(x,y)=nx:pnpy with a very good error term, valid for exp(loglogx) (5/3)+ϵ yx, xx 0 (ϵ), ϵ>0. We extend this result to an algebraic number field K by obtaining an asymptotic formula for the analogous function Ψ K (x,y) with the same error term and valid in the same region. Our main objective is to compare the formulae for Ψ(x,y) and Ψ K (x,y), and in particular to compare the second term in the two expansions.

En 1989, E. Saias a établi une formule asymptotique pour Ψ(x,y)=nx:pnpy avec un très bon terme d’erreur, valable si exp(loglogx) (5/3)+ϵ yx, xx 0 (ϵ), ϵ>0. Nous étendons ce résultat à un corps de nombre K en obtenant une formule asymptotique pour la fonction analogue Ψ K (x,y) avec le même terme d’erreur et la même zone de validité. Notre objectif principal est de comparer les formules pour Ψ(x,y) et Ψ K (x,y), en particulier comparer le second terme des développements.

@article{JTNB_2004__16_3_733_0,
     author = {Scourfield, Eira J.},
     title = {On ideals free of large prime factors},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {3},
     year = {2004},
     pages = {733-772},
     doi = {10.5802/jtnb.468},
     mrnumber = {2144965},
     zbl = {1073.11061},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2004__16_3_733_0}
}
Scourfield, Eira J. On ideals free of large prime factors. Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, pp. 733-772. doi : 10.5802/jtnb.468. http://www.numdam.org/item/JTNB_2004__16_3_733_0/

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