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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: Lamzouri@dms.umontreal.ca
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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

References

[1] Baker, R. C. and Harman, G., Shifted primes without large prime factors. Acta Arith. 83(1998), no. 4, 331–361.Google Scholar
[2] Davenport, H., Multiplicative Number Theory. Second edition. Graduate Texts in Mathematics 74, Springer-Verlag, New York, 1980.Google Scholar
[3] Erdőos, P., Some remarks on the iterates of the 𝜑 and σ functions. Colloq. Math. 17(1967), 195–202.Google Scholar
[4] Erdőos, P., On the normal number of prime factors of p–1 and some other related problems concerning Euler's ϕ–function. Quart. J. Math. 6(1935) 205–213.Google Scholar
[5] Erdőos, P., Granville, A., Pomerance, C., and Spiro, C., On the normal behavior of the iterates of some arithmetic functions. In: Analytic Number Theory. Progr. Math. 85, Birkhäuser, Boston, 1990, pp. 165–204.Google Scholar
[6] Erdőos, P. and Pomerance, C., On the normal number of prime factors of ϕ(n). Rocky Mountain Math. J. 15(1985), 343–352.Google Scholar
[7] Granville, A., Smooth numbers: computational number theory and beyond. In: Proceedings MSRI Conference on Algorithmic Number Theory, Cambridge University Press, Cambridge, to appear.Google Scholar
[8] Granville, A. and Soundararajan, K., The spectrum of multiplicative functions. Annals of Math. 153(2001), 407–470.Google Scholar
[9] Halberstam, H. and Richert, H.-E., Sieve Methods. London Mathematical Society Monographs 4, Academic Press, London, 1974.Google Scholar
[10] Hildebrand, A. and Tenenbaum, G., Integers without large prime factors. J. Théor. Nombres Bordeaux 5(1993), 411–484.Google Scholar
[11] Pomerance, C., Popular values of Euler's function. Mathematika 27(1980), 84–89.Google Scholar
[12] Shapiro, H., An arithmetic function arising from the ϕ-function. Amer. Math. Monthly 50(1943), 18–30.Google Scholar
[13] Tenenbaum, G., Introduction à la théorie analytique et probabilistique des nombres. Cours spélialisés 1, Société Mathématique de France, Paris, 1995.Google Scholar