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Smooth Values of the Iterates of the Euler Phi-Function

Published:2007-02-01
Printed: Feb 2007
• Youness Lamzouri
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Abstract

Let $\phi(n)$ be the Euler phi-function, define $\phi_0(n) = n$ and $\phi_{k+1}(n)=\phi(\phi_{k}(n))$ for all $k\geq 0$. We will determine an asymptotic formula for the set of integers $n$ less than $x$ for which $\phi_k(n)$ is $y$-smooth, conditionally on a weak form of the Elliott--Halberstam conjecture.
 MSC Classifications: 11N37 - Asymptotic results on arithmetic functions 11B37 - Recurrences {For applications to special functions, see 33-XX} 34K05 - General theory 45J05 - Integro-ordinary differential equations [See also 34K05, 34K30, 47G20]