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November, 1984 Ordered Prime Divisors of a Random Integer
Stuart P. Lloyd
Ann. Probab. 12(4): 1205-1212 (November, 1984). DOI: 10.1214/aop/1176993149

Abstract

Without using the prime number theorem, we obtain the asymptotics of the $r$th largest prime divisor of a harmonically distributed random positive integer $N$; harmonic asymptotics are obtained from asymptotics of the zeta distribution via Tauberian methods. (Knuth and Trabb-Pardo need a strong form of the prime number theorem to obtain the distributions when $N$ is uniformly distributed.) A trick brings in Poisson variates, and then we can use the methods developed for the fractional length of the $r$th longest cycle in a random permutation.

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Stuart P. Lloyd. "Ordered Prime Divisors of a Random Integer." Ann. Probab. 12 (4) 1205 - 1212, November, 1984. https://doi.org/10.1214/aop/1176993149

Information

Published: November, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0551.10042
MathSciNet: MR757777
Digital Object Identifier: 10.1214/aop/1176993149

Subjects:
Primary: 60B99
Secondary: 10K20

Keywords: $r$th largest prime divisor

Rights: Copyright © 1984 Institute of Mathematical Statistics

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Vol.12 • No. 4 • November, 1984
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