## On the asymptotic distribution of large prime factors (1993)

Venue: | J. London Math. Soc |

Citations: | 16 - 0 self |

### BibTeX

@ARTICLE{Donnelly93onthe,

author = {Peter Donnelly and Geoffrey Grimmett},

title = {On the asymptotic distribution of large prime factors},

journal = {J. London Math. Soc},

year = {1993},

volume = {47},

pages = {395--404}

}

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### OpenURL

### Abstract

A random integer N, drawn uniformly from the set {1,2,..., n), has a prime factorization of the form N = a1a2...aM where ax ^ a2>... ^ aM. We establish the asymptotic distribution, as «-» • oo, of the vector A(«) = (loga,/logiV: i:> 1) in a transparent manner. By randomly re-ordering the components of A(«), in a size-biased manner, we obtain a new vector B(n) whose asymptotic distribution is the GEM distribution with parameter 1; this is a distribution on the infinite-dimensional simplex of vectors (xv x2,...) having non-negative components with unit sum. Using a standard continuity argument, this entails the weak convergence of A(/i) to the corresponding Poisson-Dirichlet distribution on this simplex; this result was obtained by Billingsley [3]. 1.