On ideals free of large prime factors

Scourfield, Eira J.

doi:10.5802/jtnb.468

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.

Résumé
Abstract

En 1989, E. Saias a établi une formule asymptotique pour $Ψ (x, y) = |\{n \leq x : p ∣ n \Rightarrow p \leq y\}|$ avec un très bon terme d’erreur, valable si $exp ({(log log x)}^{(5 / 3) + ϵ}) \leq y \leq x$ , $x \geq x_{0} (ϵ)$ , $ϵ > 0 .$ 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 $Ψ (x, y) = |\{n \leq x : p ∣ n \Rightarrow p \leq y\}|$ with a very good error term, valid for $exp ({(log log x)}^{(5 / 3) + ϵ}) \leq y \leq x$ , $x \geq x_{0} (ϵ)$ , $ϵ > 0 .$ 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.

MR Zbl

DOI : 10.5802/jtnb.468

Affiliations des auteurs :

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. https://www.numdam.org/articles/10.5802/jtnb.468/

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