Phi, primorials, and Poisson
2020; Duke University Press; Volume: 64; Issue: 3 Linguagem: Inglês
10.1215/00192082-8591576
ISSN1945-6581
Autores Tópico(s)Mathematics and Applications
ResumoThe primorial p # of a prime p is the product of all primes q ≤ p . Let pr ( n ) denote the largest prime p with p # ∣ ϕ ( n ) , where ϕ is Euler's totient function. We show that the normal order of pr ( n ) is log log n / log log log n ; that is, pr ( n ) ∼ log log n / log log log n as n → ∞ on a set of integers of asymptotic density 1. In fact, we show there is an asymptotic secondary term and, on a tertiary level, there is an asymptotic Poisson distribution. We also show an analogous result for the largest integer k with k ! ∣ ϕ ( n ) .
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