On ideals free of large prime factors

Eira J. Scourfield[1]

  • [1] Mathematics Department, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK.

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 3, page 733-772
  • ISSN: 1246-7405

Abstract

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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 ( log log x ) ( 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.

How to cite

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Scourfield, Eira J.. "On ideals free of large prime factors." Journal de Théorie des Nombres de Bordeaux 16.3 (2004): 733-772. <http://eudml.org/doc/249280>.

@article{Scourfield2004,
abstract = {In 1989, E. Saias established an asymptotic formula for $\Psi (x,y)=\left| \left\lbrace n\le x:p\mid n\Rightarrow p\le y\right\rbrace \right| $ with a very good error term, valid for $\exp \left( (\log \log x)^\{(5/3)+\epsilon \}\right) \le y\le x$, $x\ge x_\{0\}(\epsilon )$, $\epsilon &gt;0.$ We extend this result to an algebraic number field $K$ by obtaining an asymptotic formula for the analogous function $\Psi _\{K\}(x,y)$ with the same error term and valid in the same region. Our main objective is to compare the formulae for $\Psi (x,y)$ and $\Psi _\{K\}(x,y),$ and in particular to compare the second term in the two expansions.},
affiliation = {Mathematics Department, Royal Holloway University of London, Egham, Surrey TW20 0EX, UK.},
author = {Scourfield, Eira J.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {smooth ideals; Dickman function; Dedekind zeta function},
language = {eng},
number = {3},
pages = {733-772},
publisher = {Université Bordeaux 1},
title = {On ideals free of large prime factors},
url = {http://eudml.org/doc/249280},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Scourfield, Eira J.
TI - On ideals free of large prime factors
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 3
SP - 733
EP - 772
AB - In 1989, E. Saias established an asymptotic formula for $\Psi (x,y)=\left| \left\lbrace n\le x:p\mid n\Rightarrow p\le y\right\rbrace \right| $ with a very good error term, valid for $\exp \left( (\log \log x)^{(5/3)+\epsilon }\right) \le y\le x$, $x\ge x_{0}(\epsilon )$, $\epsilon &gt;0.$ We extend this result to an algebraic number field $K$ by obtaining an asymptotic formula for the analogous function $\Psi _{K}(x,y)$ with the same error term and valid in the same region. Our main objective is to compare the formulae for $\Psi (x,y)$ and $\Psi _{K}(x,y),$ and in particular to compare the second term in the two expansions.
LA - eng
KW - smooth ideals; Dickman function; Dedekind zeta function
UR - http://eudml.org/doc/249280
ER -

References

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