On ideals free of large prime factors
Scourfield, Eira J.1
1 Mathematics Department, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK.
Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 733-772.

En 1989, E. Saias a établi une formule asymptotique pour Ψ(x,y)=n≤x:p∣n⇒p≤y avec un très bon terme d’erreur, valable si exp(loglogx) (5/3)+ϵ ≤y≤x, x≥x 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.

In 1989, E. Saias established an asymptotic formula for Ψ(x,y)=n≤x:p∣n⇒p≤y with a very good error term, valid for exp(loglogx) (5/3)+ϵ ≤y≤x, x≥x 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.

DOI : 10.5802/jtnb.468
Scourfield, Eira J. 1

1 Mathematics Department, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK. 
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Scourfield, Eira J. On ideals free of large prime factors. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 733-772. doi : 10.5802/jtnb.468. http://www.numdam.org/articles/10.5802/jtnb.468/

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