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Smooth Values of the Iterates of the Euler Phi-Function

Published online by Cambridge University Press:  20 November 2018

Youness Lamzouri
Youness Lamzouri*
Affiliation:
Départment de Mathématiques et Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, QC, H3C 3J7 e-mail: [email protected]
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Abstract

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Let $\phi (n)$ be the Euler phi-function, define ${{\phi }_{0}}\left( n \right)\,=\,n$ and ${{\phi }_{k+1}}\left( n \right)\,=\,\phi \left( {{\phi }_{k}}\left( n \right) \right)$ for all $k\ge 0$. We will determine an asymptotic formula for the set of integers $n$ less than $x$ for which ${{\phi }_{k}}\left( n \right)$ is $y$-smooth, conditionally on a weak form of the Elliott–Halberstam conjecture.

Keywords

11N3711B3734K0545J05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 1 , 01 February 2007 , pp. 127 - 147
Copyright
Copyright © Canadian Mathematical Society 2007

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