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September 2020 Phi, primorials, and Poisson
Paul Pollack, Carl Pomerance
Illinois J. Math. 64(3): 319-330 (September 2020). DOI: 10.1215/00192082-8591576

Abstract

The 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 ) .

Citation

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Paul Pollack. Carl Pomerance. "Phi, primorials, and Poisson." Illinois J. Math. 64 (3) 319 - 330, September 2020. https://doi.org/10.1215/00192082-8591576

Information

Received: 21 January 2020; Revised: 26 March 2020; Published: September 2020
First available in Project Euclid: 1 July 2020

zbMATH: 07235506
MathSciNet: MR4132594
Digital Object Identifier: 10.1215/00192082-8591576

Subjects:
Primary: 11N37
Secondary: 11N36, 11N64

Rights: Copyright © 2020 University of Illinois at Urbana-Champaign

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Vol.64 • No. 3 • September 2020
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