Multiplicative structure of values of the Euler function | Macquarie University ResearchOnline

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Title: Multiplicative structure of values of the Euler function
Related: Fields Institute communications, Vol. 41, (2004), p.29-47
Publisher: American Mathematical Society
Date: 2004
FoR/RFCD Code(s): 010100 Pure Mathematics
Author/Creator: Banks, William D
Author/Creator: Friedlander, John B
Author/Creator: Pomerance, Carl
Author/Creator: Shparlinski, Igor E
Description: We establish upper bounds for the number of smooth values of the Euler function. In particular, although the Euler function has a certain “smoothing” effect on its integer arguments, our results show that, in fact, most values produced by the Euler function are not smooth. We apply our results to study the distribution of “strong primes”, which are commonly encountered in cryptography. We also consider the problem of obtaining upper and lower bounds for the number of positive integers n ≤ x for which the value of the Euler function φ(n) is a perfect square and also for the number of n ≤ x such that φ(n) is squarefull. We give similar bounds for the Carmichael function λ(n).
Description: 19 page(s)
Subject Keyword: 010100 Pure Mathematics
Resource Type: journal article
Organisation: Macquarie University. Dept. of Computing

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